m 
 
THE 
 
 FOUNDATIONS OF GEOMETRY. 
 
 BY 
 
 EDWARD T. DIXON. 
 
 CAMBRIDGE: 
 
 DEIGHTON, BELL AND CO. 
 LONDON : GEOEGE BELL AND SONS. 
 
 1891 
 

 \Y\oociv 
 
 PRINTED BY C. J. CLAY, M.A. AND SONS, 
 AT THE UNIVERSITY PRESS. 
 
CONTENTS. 
 
 PART I. 
 
 PAGE 
 
 ON THE LOGICAL STATUS OF THE SCIENCE OF GEOMETRY 1 
 
 PART II. 
 
 A SUBJECTIVE THEORY OF GEOMETRY DEDUCED FROM THE TWO 
 FUNDAMENTAL CONCEPTS, POSITION AND DIRECTION . 32 
 
 PART III. 
 
 ON THE APPLICABILITY OF THE FOREGOING SUBJECTIVE GEOMETRY 
 TO THE GEOMETRY OF MATERIAL SPACE . 87 
 
EKRATA. 
 
 p. 12, 1. 7, for CEEC read CEFC. 
 
 p. 12, 1. 10, for PE read FE. 
 
 p. 15, 1. 23, for ECP read EC* 1 . 
 
 p. 18, 1. 11, (twice) for Helmholz read Helmholtz. 
 
 p. 26, last line but two, erase not. 
 
PKEFACE. 
 
 T BELIEVE that the system of geometry I have set forth in 
 " this book is logically sound, and that consequently the more 
 it is discussed and criticised, the more firmly will it become 
 established. I shall therefore be very glad to see any criticisms 
 of my views, whether friendly or hostile, either in the public 
 press, or addressed to me privately, at the address given below. 
 But I have already found that, the subject being such a wide 
 one, criticism is apt to become discursive ; and with a view to 
 keeping it to the point I would suggest to my critics and 
 opponents in argument that they should consider categorically 
 the following questions : 
 
 (i) Do you accept the requirements I have laid down for 
 a logical definition ? (see p. 21). 
 
 (If not, please state which of them you object to, why 
 you object to it, and what you would propose to substitute 
 for it.) 
 
 (ii) Do you entertain a mental concept (which I shall call 
 by the name direction ) such that the assertion " A Vector is a 
 given amount of transference in a given direction, irrespective 
 of the point of departure," is intelligible, to you ? 
 
 (iii) If so, does not this concept fulfil all the four require 
 ments of my definition of direction ? 
 
 812184 
 
iv PREFACE. 
 
 (Whether you think these properties are established by 
 Euclid s geometry, or not, is immaterial. If you grant this 
 you have granted my Axiom II. ; for this does not assert 
 any objective fact at all.) 
 
 (iv) Do you accept the logical accuracy and permissibility 
 of my remaining definitions and axioms ? 
 
 (Objections on the score of convenience and simplicity had 
 better be considered elsewhere.) 
 
 (v) Do you admit the formal accuracy of the proofs of 
 propositions in my Books I. and II. ? 
 
 (N.B. If you admit this there can no longer be any doubt 
 as to the sufficiency of my premises.) 
 
 (vi) Do you admit the objective applications of my three 
 Axioms, and therefore of my system of geometry, as discussed 
 in Chap. I. of Part. III. ? 
 
 (vii) If you admit that there is a theoretical doubt as to 
 the objective counterpart of my second Axiom, please give 
 any criticisms which may occur to you on the remainder of 
 Part III. 
 
 Now, if there is no flaw in the line of argument I have 
 adopted, it follows that my conclusions are true, and con 
 sequently that any objection taken to them outside this line of 
 argument, however specious it may sound, must contain a fallacy. 
 I might therefore refuse to discuss such an objection. But the 
 objector might truly urge that, conversely, if his objection was 
 irrefutable, there must be some hidden fallacy in my argument. 
 And therefore, though I prefer arguing in my own way, having 
 devoted a good deal of thought to the subject, and having come 
 to the conclusion that my line of argument is the most direct, 
 and the easiest to discuss; I shall nevertheless feel bound 
 to give the best answer I can to any reasonable objection 
 whatever. 
 
PREFACE. v 
 
 I must here point out that, this book being intended for the 
 study of geometricians, I have not entered upon the question 
 whether beginners could readily be brought to understand it or 
 not. If it is not logically sound, to discuss such a question 
 would be useless. But if it is acknowledged to be logical, I 
 have no doubt that it could be drummed into the head of the 
 average schoolboy as easily as Euclid. But I prefer to postpone 
 this question till the more important one is at least on a fair 
 way towards settlement; when I shall, I hope, bring out a 
 text-book for beginners founded on my method. 
 
 EDWARD T. DIXON. 
 
 12, BARKSTON MANSIONS, 
 SOUTH KENSINGTON, 
 January, 1891. 
 
PART I. 
 
 ON THE LOGICAL STATUS OF THE SCIENCE 
 OF GEOMETRY. 
 
 CHAPTER I. 
 
 CAN we be absolutely certain of anything in this world ? 
 Or is all our knowledge only empirical and approximate ? Is 
 there such a thing as necessary truth, and if so how are we 
 to know when we have attained it ? 
 
 These questions open up perhaps the most disputed branches 
 of Logic and Metaphysics. Under one form or another the 
 contest has been raging round them ever since the time of 
 Aristotle. The line of battle has sometimes shifted forward, 
 sometimes back, sometimes it has changed front, so that quite 
 new issues seemed to be at stake. But the status of the Science 
 of Geometry has always been the key of the position ; though 
 the combatants on both sides have often confined their energies 
 to flank attacks, in despair of making any impression on the 
 citadel. 
 
 It was a prevailing idea among the ancients, though 
 perhaps it was never distinctly formulated, that if a man only 
 had a perfectly clear brain, if, that is, he could only always 
 think logically, he could know everything. Hence the import 
 ance which was ascribed to formal logic, and hence the 
 systems of philosophy founded upon data supposed to be known 
 a priori These fallacious methods would probably have been 
 abandoned much earlier than they were, had it not been for 
 the apparent success obtained in one instance, namely, in 
 
 D. 1 
 
2 PART I. 
 
 Geometry. Here at least it seemed that a real knowledge 
 of the external world had been deduced from a priori con 
 siderations alone. When, barely 300 years ago the great 
 Francis Bacon vigorously denounced such a priori reasoning, 
 and founded the school of experimental science, neither he nor 
 his disciples appear to have applied their reasonings to upset 
 this accepted view. And that he did not at once revolutionise 
 scientific thought is evident from the celebrated dictum of 
 Descartes, put forward half a century later, that everything 
 which we can clearly and distinctly conceive, is true. Bacon 
 indeed advocated experiment as a means to help the imperfect 
 deductive power of man, rather than as an end. It was left 
 to later philosophers, of whom Mill may be taken as a type, to 
 make a fetish of this means, under the name of Induction, to 
 exalt it above deduction and even to assert that all knowledge 
 whatever is gained by it alone. Such a theory of course involved 
 Mill in an attack on the deductive citadel the science of Geo 
 metry. It is not necessary here to examine the details of this 
 attack (which has moreover been most ably done in a recent 
 work by Prof. Jevons) as the general considerations to be 
 advanced presently will I think be sufficient to repel it. Other 
 philosophers have been driven to take up rather different ground. 
 Kant for example divided all judgements into analytic judge 
 ments, in which what is asserted in the predicate is already 
 contained in the connotation of the subject, and synthetic 
 judgements, in which something is added to this connotation. 
 And he maintained that there are certain judgements of the 
 latter class whose truth could be known a priori. Among 
 these he classed the Axioms of Geometry, and so cut the 
 Gordian knot. But to further exemplify his views he was 
 ill-advised enough to put the axioms of Arithmetic into the 
 same category, and in this latter case the fallacy is easily 
 exposed, as I hope to show presently. 
 
 But let us first consider how we come to know anything at 
 all. Philosophers of the inductive school say By induction 
 from experience. But what is experience ? It is easy to put 
 a case on paper "in so many instances antecedent A was 
 followed by consequent a " and so on. But how do we know 
 anything about either A or a ? A little thought must convince 
 anybody that the only things of which we have any direct 
 
CHAPTER I. 3 
 
 cognisance whatever are our own subjective impressions, our 
 sensations and thoughts, and that no one can be absolutely 
 certain that such a thing as an objective world exists at all. If 
 I say I had a tooth -ache this morning, I cannot be absolutely 
 certain that the ache was due to a bad tooth, I cannot even be 
 absolutely certain that there is any objective entity corre 
 sponding to what I mean by my tooth at all. Besides I may 
 have only dreamed the tooth-ache, or my memory may have 
 played me a trick, perhaps it was not this morning that I felt 
 the ache. But at any rate while the ache is going on I can be 
 absolutely certain of its subjective existence ; which subjective 
 existence is no less real even if I am dreaming the while. And 
 though in the case of a tooth-ache this truth is painfully 
 evident, it is not less true in the case of the feeblest sensation 
 or thought. We have direct cognisance, each of us, of what 
 goes on in his or her mind, and though we may subsequently 
 forget it, though we may misrepresent it in trying to convey it 
 to the mind of another person, while it is passing in our mind 
 it is to us absolute truth, that is, the thoughts or sensations are 
 absolutely there ; though they may, or may not correspond 
 to objective realities in the external world. If therefore we 
 confine Descartes dictum to purely subjective applications it is 
 strictly true. 
 
 Here then we have absolute certainty. Each of us can be 
 absolutely certain of what he is feeling or conceiving at any 
 given moment, and if there are any of his conceptions which he 
 can call up at will he can be absolutely certain that those 
 conceptions have to him a real existence ; subjectively, in his 
 own mind, be it understood ; though not necessarily objectively, 
 outside it. 
 
 In view of these considerations the old classification of 
 branches of knowledge, that is, sciences, as deductive and 
 inductive will need modification. The old designations are 
 indeed very misleading, for there hardly exists a branch of 
 knowledge in which deduction does not play an important part. 
 Perhaps there was little or no true deduction in Astrology ; or 
 at least it was an excessive and unwarranted use of induction 
 that led its votaries into error. Probably to the old salt 
 " weather wisdom " is purely inductive. But it is only since 
 deductive methods have been applied to them that Astrology 
 
4 PART I. 
 
 and weather wisdom have entered the ranks of the sciences, as 
 Astronomy and Meteorology. In the same way it was deduction 
 not induction that raised Alchemy to Chemistry, and the labour 
 of cataloguing animals and plants to the science of Biology. 
 The real distinction between what used to be classed as 
 inductive and deductive sciences lies, not in their methods, 
 but in the premises upon which they are founded. In the one 
 case the premises are drawn from observation and experiment, 
 and are not only inductive but objective, in the other they are 
 directly apprehended by the mind and can therefore only be 
 subjective. 
 
 It was maintained by some philosophers of the inductive 
 school that inasmuch as the conclusion of a syllogism is 
 contained, or implied, in the premises, no new knowledge can 
 be attained by deduction. Such an argument is however the 
 merest quibble. It is of course true that if you grant the truth 
 of the premises you grant the conclusion. But to say that 
 therefore you know the conclusion because you know the 
 premises is a flagrant non sequitur. Why otherwise do we 
 ever learn anything beyond the premises of a science ? Can 
 we, because we know the premises of Arithmetic, be said to 
 know the product of 3*1416 x 2*7183 before we have multiplied 
 out ? It most certainly is possible, very materially, to increase 
 our knowledge by deduction alone ; and if there are in our 
 minds any suitable subjective certainties which we can use as 
 premises, it may be possible to establish extensive (subjective) 
 sciences, whose (subjective) conclusions will be necessary truths, 
 as their premises are. 
 
 Let us then examine what kinds of premises a deductive 
 science may have. They may be classed under four heads 
 (1) Objective facts, (2) Postulates, (3) Definitions, (4) Axioms. 
 
 Under the head of objective facts I mean to include all facts 
 known by observation and experiment, that is therefore, all 
 objective propositions which we may regard as established 
 truths. I have however already shown that they cannot be 
 necessary truths and with them therefore we are not at present 
 concerned. 
 
 Postulates are objective propositions about whose truth 
 there yet remains some doubt, or whose truth is only provision 
 ally asserted until all the consequences that would flow deduc- 
 
CHAPTER I. 5 
 
 lively from it have been examined. Or else they are subjective 
 propositions, dealing with an imaginary state of things ; and in 
 this case the conclusions drawn from them are their necessary 
 consequences, if the deduction has been strict, though they 
 are not necessarily true. Thus the wave theory of light is 
 founded on the postulate that a certain elastic medium, the 
 asther, pervades all space. This postulate now-a-days almost 
 ranks as an objective fact, for the consequences which flow 
 deductively from it have been found to tally so exactly with 
 what is actually observed. But if it had not been so, the 
 postulate might still have been assumed subjectively, and, as a 
 mental exercise, an imaginary wave-theory of light might have 
 been worked out. 
 
 The logical status of a definition has long been a matter of 
 debate. The common view is that, it being a purely verbal 
 proposition, it can convey nothing but philological knowledge. 
 But just as I have shown that knowledge can be truly increased 
 by deduction, so it may be by definition. Thus if I am told 
 that an even number is one which can be divided into two equal 
 (integral) parts, this definition may suggest to me a completely 
 new conception, from which I may be able to deduce a whole 
 new science, which without it I could never have attained. 
 But apart from suggesting something new, if a definition 
 enables us to analyse a concept which we formerly entertained 
 only vaguely, it may enable us to deduce results from it which 
 perhaps we already knew were true, but which we could not 
 before connect with that concept formally. But, lastly, it is a 
 characteristic of that marvellous instrument, Language, that by 
 its aid we are actually able to reason accurately about things 
 we do not clearly conceive ; and this we are enabled to do by 
 defining a word or symbol by its attributes, without necessarily 
 conceiving its meaning as a whole at all. For example, we may 
 define the symbol *J 1 in symbolic language thus 
 
 and from this definition most important results can be deduced, 
 such as the series from which the values of the trigonometrical 
 ratios are calculated, and this without ever attaching any 
 definite denotation to the symbol J 1. 
 
 Thus all that is logically required for a definition is one or 
 
6 PART I. 
 
 more assertions with regard to the word to be denned or its attri 
 butes. Having made the assertions it may be that there is no 
 objective reality corresponding to the word there may not be 
 anything in the world possessing the named attribute or combina 
 tion of attributes. It may further happen that we know of 110 
 subjective concept corresponding to it; that our minds are not 
 able to form a concept combining the attributes ; but unless the 
 assertions are inconsistent, that is unless the falsehood of one 
 can be deduced from the truth of another, the definition remains 
 logically sound, and we may deduce theorems from it which 
 may, or may not, turn out useful or interesting, but which at 
 least are logically true. 
 
 In such definitions by assertion it is not necessary that the 
 word to be defined should be the subject of each proposition, as 
 is usually the case ; but it is necessary that it should be clearly 
 understood which word is being defined. It might indeed be 
 possible by two assertions each about two unknown words to 
 define them both, just as by two simultaneous equations we 
 may define two unknown quantities ; but there would seldom 
 be any advantage in such a course. And in any case the 
 meanings of all the other words in the definition must be 
 known. The propounder of a scientific theory is not of course 
 expected to teach his readers to speak, it is only necessary for 
 him to define the terms peculiar to his science, or those to 
 which he wishes to attach peculiar meanings. He may therefore 
 assume that the meanings of all other words are known to his 
 readers. 
 
 It having been taken for granted that a definition could not 
 impart any new knowledge, many propositions which are really 
 definitions have been commonly classed as axioms, especially 
 where the word to be defined does not appear as the subject of 
 the sentence, or where it is not quite clear which word requires 
 definition. Thus Euclid s first eight axioms do not assert anything 
 which is not contained in the connotation of the words equal 
 whole part &c. So, though they would be very bad definitions 
 of these words, they are logically nothing more. So also in his 
 ninth axiom the only question is which of the words is supposed 
 to be unknown ; for this proposition also only expresses what is 
 part of the connotation of its terms. 
 
 A definition, then, lays down the connotation which a word 
 
CHAPTER I. 7 
 
 is to bear, but does not assign to it its denotation ; it does not 
 assert that there is any objective thing, or even subjective idea, 
 corresponding to it. It is indeed possible to have a science 
 founded partly, or even entirely, on definitions without asserting 
 that there exist any things corresponding to the terms or 
 symbols denned, and such a science will be ready to hand if 
 any things are found possessing the attributes connoted by the 
 definitions. Algebra is strictly such a science. The symbols 
 used, a, 6, x, y &c. are defined, that is, their connotation is 
 laid down, when the distributive commutative laws, and law 
 of indices are asserted about them. But no denotation is laid 
 down for them it is a mistake to allow beginners to suppose 
 that they are merely numbers in disguise. It is indeed true 
 that, within certain limits, numbers do possess the attributes 
 ascribed to algebraical symbols; and, within those limits, 
 algebraical results are arithmetically true. Within other limits 
 vectors obey the same laws, and hence some algebraical formulae 
 may be given geometrical interpretations. But pure algebra is 
 independent of such applications, and consists in results deduced 
 from definitions alone, and its logical basis would be just the 
 same even if it had no practical applications. 
 
 The last class of premise we have to consider is the Axiom, 
 or necessary truth. Kant defined an apodictic truth as one 
 the negation of which was inconceivable. This involved him 
 in a difficulty, for though you or I may be unable to conceive 
 the negation of a given proposition, perhaps some one else, in a 
 future generation if not now, may be able to do so. Thus any 
 one might assert that " J 1 is an impossibility " and might 
 maintain that the assertion was apodictically true until some 
 one found a conceivable interpretation for the symbol. We 
 have however found a better criterion of necessary truth above. 
 We have seen that a necessary truth can only refer to the 
 subjective concepts of an individual, which are actually present 
 to his mind, and for an abiding conviction of its truth it is 
 therefore necessary that the concepts be such as his mind can 
 call up at will ; and then the realisation of its truth depends 
 upon the direct apprehension of the relations of subject and 
 predicate. For example if I say cogito, ergo, sum I have a 
 perfect subjective consciousness that I do think, and I conclude 
 with absolute certainty that I do (subjectively) exist. But if I 
 
8 PART I. 
 
 were to conclude that my body existed, objectively, I should be 
 guilty of a fallacy. And obviously it would be merely begging 
 the question to say cogitat, ergo est/ 
 
 The simplest form of axiom, then, is that which merely 
 asserts that a thing, defined in a certain way, is conceivable ; 
 or may conceivably undergo certain operations. Every man 
 can pronounce with certainty upon such propositions whether 
 they are true to him or not, if he understands the terms in 
 which they are stated. In the case of those axioms I shall 
 bring forward I cannot doubt that everybody will assent to 
 their truth ; but if any exceptional individuals should be found 
 to deny them, that is to deny that they are able to conceive the 
 things named, their denial will only reflect on their individual 
 mental capacities, and need not raise any doubts in the minds 
 of others. 
 
 Let us then apply these considerations to the simplest 
 deductive science, that of Arithmetic. Kant maintained that 
 some of the dicta of Arithmetic were synthetic judgements a 
 priori! He instanced the equation 7 -f 5 = 12. He was 
 indeed correct in saying that the conception of 12 is not the 
 same thing as that of (7 + 5). But the truth of the above 
 equation can be directly deduced from the conceptions of 7, 5, 
 and 12. This is therefore only another proof that new know 
 ledge may be gained by deduction. For we may define 7 as 
 1+1 + 1 + 1 + 1 + 1 + 1 and 5 as 1+1 + 1 + 1 + 1, and if these 
 definitions be substituted in the left side of the above equation, 
 it becomes the definition of 12. It is thus evident that as far 
 as simple addition goes, and so also with simple subtraction, 
 Arithmetic is purely a matter of definition. When however we 
 come to multiplication and division we require two axioms 
 which may be stated thus : 
 
 (1) Units may be conceived to be added together to form 
 numbers, which may themselves be treated as units. 
 
 (2) Any unit may be conceived to have been formed by 
 the addition of any number of sub-units, which may themselves 
 be treated as units. 
 
 Thus, to prove that "2x3 = 6" the symbols mean "consider 
 3 s as units, and add two of them together. The result will be 
 the same as six of the original units" which follows from the 
 definitions of 2, 3, and 6. Thus also 3x2 = 6 and therefore 
 
CHAPTER I. 9 
 
 2x3 = 3x2. In this way the commutative law of Arithmetic 
 is established, and the distributive law may be proved equally 
 easily, and so any rule of Arithmetic. 
 
 It might indeed be urged that the above two axioms were 
 in reality only two more implicit definitions, that they only 
 helped to define unit and number. That they are, however, 
 something more may be shown by taking an instance in which, 
 though they might still conceivably be true, they might also be 
 false. Thus, Are two twopenny apples worth fourpence ? 
 May we add each of the two pennies for each apple together to 
 form twopences, and then add the twopences as if they were 
 units ? The answer is No ; there may be a reduction on taking 
 a quantity, or if the vendor wants to eat one of the apples 
 himself he may ask more than fourpence for the two. When 
 he said one apple for twopence he did not mean that each 
 twopence might be treated as an unit to make fourpence for the 
 two apples. 
 
 We have then established a science whose conclusions, 
 interpreted subjectively are necessary truths. As a matter of 
 fact the premises of Arithmetic are so commonly true among 
 the objective units, as we believe them to exist, that its 
 conclusions are often thought to be as true objectively as they 
 are subjectively. If you grant that there are two apples each 
 worth twopence, and that the twopences may be added together 
 just as the individual pennies were, then the two apples are 
 worth fourpence, but the above objective assumptions are not 
 necessary truths. Besides the perfect subjective Arithmetic we 
 have therefore an objective Arithmetic, whose objective premises 
 are established by induction, and which is therefore imperfect, 
 in the sense of not dealing only with necessary truths. 
 
 Well then, in Geometry ought we not to find precisely the 
 same thing? Ought there not to be a perfect subjective 
 Geometry, as well as an applied objective one, the applicability 
 of the former to the latter being a matter to be determined by 
 induction from observation and experiment ? It is the object 
 of this book to show that such is the case, to establish the 
 perfect Geometry, and to examine the grounds on which we 
 may believe that it applies to the objective space in which we 
 live. 
 
CHAPTER II. 
 
 IT would be folly to attempt to substitute a new thing for an 
 old one, unless one is first convinced that there was something 
 bad in the old one, which will be remedied in the new. Before 
 therefore attempting to introduce a new system of Geometry I 
 must point out what I consider the defects of the old ones. 
 
 That there are some defects has long been recognised. It 
 has for a long while past been pretty generally admitted that 
 Euclid s last Axiom is not a necessary truth. Great is the 
 ingenuity which has been expended by Geometricians in 
 attempting to find an apodictic substitute for this Axiom. 
 
 Legendre thought he had proved its truth from the other 
 data of Geometry, but he himself had afterwards to admit that 
 his argument was fallacious. After so conspicuous a failure 
 most Geometricians seem to have reconciled themselves to the 
 idea that the Axiom could never be made apodictic, and to 
 have regarded attempts to do so in the same light as attempts 
 to square the circle. This is, I suppose, the reason that they 
 have overlooked a proof given by M. Vincent of Paris (quoted 
 by Mr F. W. Newman in his " mathematical tracts ") which I 
 give, with certain trifling modifications, at the end of this 
 chapter. 
 
 But, on the other hand, a great many Geometricians still 
 imagine that the proposition " Two straight lines cannot enclose 
 a space " is an a priori necessary truth. If it is only taken to 
 refer to subjective concepts it may indeed be so regarded, if it 
 is taken as the definition of c straight ; and though it would be 
 by no means a complete definition, as far as it goes it would be 
 irrefutable. But it is perfectly evident that Geometricians do 
 not regard it only subjectively. They deduce objective con 
 clusions from it, and it can only be held by them to refer to 
 
CHAPTER II. 11 
 
 objective straight lines. Now it is a sufficient definition of an 
 objective straight line to say that two straight lines of a given 
 small length cannot enclose a space. From this definition it is 
 possible to mechanically construct straight edges of the given 
 length, on the principle on which Whitworth constructs his 
 surface plates ; and two of these straight edges being joined 
 together with only a small part of their respective ends over 
 lapping, we get a longer straight line. By continuing this 
 process it is theoretically possible to produce a straight line to 
 any length whatever, (unless it should at some point rejoin 
 itself.) And how can we be certain that two such straight 
 lines diverging from a given point will never meet again? 
 There is no a priori reason why this should be so. All we can 
 say off-hand is that, as a matter of fact straight lines have 
 never been known to behave in such a manner ; but we are 
 bound to confess that they have been traced only through an 
 infinitesimal portion of the infinite space we dwell in, if so be 
 that they do not intersect again. 
 
 As however this axiom has generally been assumed to be a 
 necessary truth, it has been deduced from it that parallel 
 straight lines, as Euclid defines them, are necessarily possible. 
 For it may be shown that two straight lines which have a 
 common perpendicular must either intersect on both sides of 
 that perpendicular or on neither, since the figure is symmetrical 
 with respect to it. That the reality of parallel lines depends so 
 directly upon this Axiom has not however been generally recog 
 nised, simply because Euclid does not prove it in the above way, 
 but by means of his 16th proposition, the proof of which, as 
 it stands, merely begs the question at 
 issue. To complete the proof it is A 
 
 necessary to appeal not only to his 
 10th Axiom but to another which he 
 does not lay down ; to the effect that, 
 
 in a plane, it is impossible to pass . 
 
 from one side of an unterminated line B 
 to another, without intersecting it an omission which I am 
 glad to see a recent editor 1 has noticed, though he has not 
 noticed the want of the Axiom in this proposition. For in Euc. 
 i. 16, having shown that the triangles CEF, AEB are con- 
 
 1 The Pitt Press Euclid by H. M. Taylor. 
 
12 PART I. 
 
 gruent, and therefore the angles EOF, EAB equal, Euclid goes 
 on " But the angle ECD is greater than the angle ECF(Ax. 9)." 
 Now Ax. 9 only says " the whole is greater than its part," 
 and to assume that the angle EOF is part of the angle ECD 
 is to beg the question at issue. The proof should be completed 
 thus < Since the line CEFC is unterminated, if any part of the 
 straight line CD is within CEEC it must intersect its boundary 
 twice ; for if it is produced far enough there must be points in 
 it in either direction outside this limited space (new Axiom). 
 But CD already intersects PE in B, and EC and FC in 0, and 
 so cannot intersect either of them again (Ax. 10). Therefore 
 CD is wholly without the triangle EOF, and the angle EOF is 
 therefore part of the angle ECD, and is less than it (Ax. 9). 
 
 Thus, granting neither of the two Axioms, but defining a 
 straight line as one which is in general determined if two points 
 on it are given, we have two alternatives to Euclidian Geo 
 metry we might either have more than one straight line 
 through a given point fulfilling Euclid s definition of parallel, with 
 respect to a given straight line, or we might have none at all, 
 in which case any two straight lines which intersected at all 
 would intersect twice. These alternatives have been already 
 examined by Mr G. Chrystal 1 though his treatment of the sub 
 ject led him to rather erroneous conclusions. For he seems to 
 consider that the former alternative, which he calls hyperbolic 
 Geometry, is the only one worth practical consideration, whereas, 
 as we shall see, it is in reality inconsistent with the self-congru 
 ence of space, which he himself assumes, and may be shown 
 to be untenable by the proof given at the end of this chapter of 
 Euclid s Axiom 12. 
 
 Let us examine Euclid s premises more closely, in which, as 
 we have seen above, are included his definitions as much, if not 
 more, than his postulates and axioms. 
 
 A term generally denotes a thing or idea, and connotes its 
 attributes. Hence also a definition may indicate the denotation, 
 or detail the connotation of a term. 
 
 We have already seen that for deductive purposes the connota 
 tion is the only important thing to lay down. Thus Euclid s first 
 definition " A point is that which has no parts or no magnitude" 
 
 1 Paper on Non Euclidian Geometry in Proceedings of lioyal Society of 
 Edinburgh, Vol. x., 187980. 
 
CHAPTER IT. 13 
 
 though it does not tell us what a point is, is nevertheless a 
 useful definition, as it tells us at any rate that it does not 
 connote magnitude or the possibility of subdivision. The defi 
 nition ought not however to have been expressed as if it ex 
 plained the denotation of the term. An instant of time cannot 
 be said to have parts or magnitude, but a point is not an in 
 stant. The definition ought merely to assert, as part of the 
 connotation of the term "A point has no parts and no magnitude." 
 Similarly it is not true that a line is length. Length is only one 
 of the attributes of a line. The same remark applies to his defi 
 nition of a surface. Now how comes it that neither of these 
 two fundamental definitions is ever referred to throughout 
 Euclid s elements? There are I think three reasons. First, 
 Euclid has expressed the most essential part of the connotation 
 of the term straight line elsewhere, namely in his 10th Axiom. 
 Secondly, because he assumes the items of connotation of the 
 term surface, when necessary, without referring to any definition 
 or Axiom. But chiefly it is because, if Euclid ever tried to 
 make any use of these definitions, he would be confronted with 
 the question What are length, breadth, and thickness? A 
 question to which he provides no answer. 
 
 Euclid s definition of a straight line is generally acknow 
 ledged to be useless. It is merely a paraphrase of the word 
 straight, and by no means a good one. Euclid himself 
 defines a plane surface in a similar manner the definition 
 substituted for it in modern editions being the only one of the 
 first nine from which any deductions are made, and this one is 
 used illogically, for he never postulates or proves that the thing 
 defined as a plane can exist. 
 
 Next we read "A plane angle is the inclination of two 
 lines to one another in a plane, which meet together, but are 
 not in the same direction." What does this mean ? If an 
 angle is only another word for an inclination how does this 
 definition assist matters? Or does Euclid recognise the in 
 clination of two lines which do not meet ? What has the plane 
 got to do with it ? Is the angle between lines which meet but 
 are not in a plane, as between the meridians of longitude at 
 the pole, essentially different from the angle between two 
 intersecting arcs of circles in the same plane ? Why does 
 Euclid here admit the word direction which he has so 
 
14 PART I. 
 
 carefully banished from the rest of his work? Why does he 
 trouble to give a special definition of a plane rectilineal angle, 
 unless it is that in this case he can dispense with the awkward 
 word, and pretend that straightness has nothing to do with 
 direction, for which it has been substituted ? 
 
 Again, why does he speak of angles as contained by straight 
 lines? Why did he in I. 16 quoted above, call the angle ECF 
 part of the angle ECD ? 
 
 The answers to all these questions are really simple enough, 
 if we only make up our minds to use Euclid s own word, 
 direction/ a little more freely. By inclination Euclid means 
 difference of direction. This is why two curved lines must 
 extend in different directions from their common point, to have 
 an inclination at that point, and it is because he tacitly assumes 
 that a straight line always extends in the same direction from 
 any point in it, that in the case of the definition of a plane 
 rectilineal angle the word direction may be left out. An angle 
 is something different from an inclination, which is used to 
 measure it, just as a two-foot rule is not a distance, but is used 
 to measure distances To Euclid there can be little doubt that 
 an angle meant a portion of a plane, a sector cut off between 
 two intersecting straight lines. In the same way a figure, 
 such as a triangle, or circle, was to him not an arrangement of 
 lines, but a piece of a plane ; as it were a figure cut out of 
 paper, of which the lines were merely the boundaries. Thus 
 when in i. 4 he says " The whole triangle ABC coincides with 
 the whole triangle DEF, and is equal to it " he clearly refers to 
 them as triangular pieces of planes whose areas are thus 
 proved to be equal. Euclid s angles are really corners of such 
 figures cut out of a plane. This is confirmed by the easy way 
 in which he extends the use of the word angle to the corner of 
 a solid figure where three or more planes meet, calling it a 
 solid angle as opposed to a plane angle, a solid figure being 
 merely bounded by the planes, as a plane figure is merely 
 bounded by lines. That this view of an angle is confessedly 
 taken by some Geometricians will appear from M. Vincent s 
 proof quoted below, and from the fact that Mr Newman accepts 
 it without comment. Though perhaps Euclid never acknow 
 ledged it to himself, these angles, these sectors of planes, were 
 of course areas, of infinite extent, but such that they bore 
 
CHAPTER IT. 15 
 
 finite ratios to each other, which ratios measured the inclina 
 tions of the bounding lines ; some given sector, or the whole 
 plane, being taken as an unit. Thus if he had ever considered 
 curvilinear angles after defining them, he would either have 
 had to explain that a curvilinear angle was equal to the 
 rectilinear angle between the tangents at the angular point, 
 or he would have become involved in the doctrine of limits. 
 
 According to this view then, in the figure of I. 16 the angle 
 EOF actually is a part of the angle ECD. But another com 
 mon interpretation of the word angle is that it is a quantity 
 of revolution, from one direction to another. This is the way 
 it is regarded in Trigonometry, and it is a way which has many 
 advantages. But when Geometricians introduce it into ele 
 mentary text books they almost invariably forget that a straight 
 line may be revolved from one direction to another in various 
 ways, describing various conical surfaces the while, and perform 
 ing various quantities of turning, and that there is therefore 
 no reason why the amount of turning which measures the 
 inclination of CE to OF should be part of that which measures 
 that of CE to CD, unless the method of rotation is further 
 defined. This is sometimes done by saying the rotation must 
 take place in the plane containing the lines, and that therefore 
 in the above instance the angle ECP is only part of the angle 
 ECD because CF is in the plane of CE and CD. This would 
 be right enough if the word plane were so defined that we 
 could deduce from the definition that, in revolving in it from 
 CE to CD a straight line must pass once, and only once 
 through CF. But if this result can be obtained otherwise, the 
 conception of a plane may be altogether omitted, and with 
 advantage, as it is by no means a simple concept, and an angle 
 is a sufficiently difficult thing for a beginner to grasp without 
 any unnecessary complications being involved in it. 
 
 It is obvious from the above considerations, that an inclina 
 tion can never be greater than that measured by two right- 
 angles, or as it is in some modern books conveniently called, 
 a straight angle. This inclination is that between two opposite 
 directions. An Euclidian angle, that is, a sector of a plane, could 
 strictly speaking never be greater than a whole plane, that is, 
 four right angles, unless we are permitted to count some parts 
 of the plane twice. But a Trigonometrical angle, that is, a 
 
16 PART I. 
 
 quantity of revolution (under certain conditions), may be as 
 great as we please, one or more whole revolutions. Thus one 
 inclination may be measured by at least two Euclidian angles, 
 or by any number of Trigonometrical angles. A great deal of 
 the difficulty of Elementary Geometry is really caused by the 
 confusion of these three distinct concepts. My definition of an 
 angle is so framed as to restrict the meaning of the term to the 
 smallest positive trigonometrical angle. 
 
 There is little to be said about Euclid s remaining defini 
 tions. In that of Parallel straight lines we again come across 
 that curious condition " In the same plane " ; a condition 
 which in the subsequent axiom about straight lines which are 
 not parallel, is, curiously enough, omitted from all text books, 
 as far as I know, which gives the axiom at all ! The definition 
 of parallel lines is logically good, but of course it remains to 
 be proved that such things really exist. It has already been 
 pointed out how this may be done by the aid of Euclid s 10th 
 Axiom, though Euclid s own method is unsatisfactory. 
 
 As to Euclid s postulates, they are logically needed, since 
 definitions cannot be taken to assert the reality of the things 
 they define. But I think it is improbable that it was for this 
 purpose that Euclid wrote them down. To him they merely 
 represented permission to perform certain operations in Geo 
 metrical drawing. This is why he has no postulate about 
 drawing planes he knew of no practical method of drawing 
 them, as straight lines are drawn on the black board, and 
 accordingly he quite overlooks the logical necessity of a 
 postulate or theorem to prove that they are conceivable. 
 
 Most of the Axioms have already been discussed. The 
 tenth I have shown not to be a necessary truth, if it is taken 
 objectively, as clearly it is intended to be. The eleventh is 
 quite unnecessary, as it may be proved as a theorem, by the 
 method of superposition. The twelfth however may be proved, 
 if the self-congruence of space is admitted, that is, if we assume 
 that geometrical figures may be carried about in space without 
 any change of size or shape. This proof is virtually that of 
 M. Vincent referred to above. 
 
 We have seen already that two straight, that is self-con 
 gruent, lines in a plane which have a common perpendicular, 
 must either intersect on both sides of the perpendicular, or on 
 
CHAPTER II. 
 
 17 
 
 neither. In the latter case, the straight lines are of infinite 
 length, and enclose a part of the infinite plane in which they 
 lie, which we may call a band. If OA, PB be two straight 
 lines enclosing half such a band, on one side of the common 
 perpendicular OP, and OP be indefinitely produced, to X, and 
 Q taken in it so that QP = PO, and QC drawn likewise perpen 
 dicular to OP ; QC, PB enclose another half-band which may 
 be shown by -superposition to be equal to that enclosed by PB 
 and OA. And so other bands may be cut off from the plane 
 by straight lines RD...&C. But it may be shown by superposi 
 tion that the remaining quadrant of the infinite plane XRD is 
 still equal to XOA. Hence the area of a half band is infini 
 tesimal compared with that of a quadrant of the infinite plane, 
 that is with an Euclidian light angle. 
 
 But the area of any finite Euclidian angle is comparable 
 with that of a right angle. 
 
 Therefore the area of any finite Euclidian angle is infinitely 
 greater than that of any band of finite width. 
 
 Hence if any straight line OK on the same side of OA as 
 PB make any finite angle with OA, the whole area of the angle 
 KOA is greater than that of the band BPOA and therefore 
 PB must intersect OK. 
 
 This proves Euclid s 12th Axiom. The proof does not 
 seem quite satisfactory, chiefly because the accepted definitions 
 of the terms involved do not admit of its being stated in strictly 
 formal language. But as I prove the same thing by a more 
 comprehensive and formal method later on, there need be no 
 doubt as to its correctness. 
 
 D. 
 
CHAPTER HI. 
 
 BUT though I have criticised Euclid s premises, and in them 
 those of most systems of geometry, in some detail, I have not 
 yet mentioned what I take to be his fundamental misconception, 
 and that of nearly all subsequent writers on the subject. Even 
 though it is not explicitly stated in Euclid, it seems that he, in 
 common with most modern philosophers, regarded the concep 
 tion of space as a fundamental attribute of the human mind, 
 which requires no definition or discussion, being necessarily 
 one, and unalterably the same, to every one. This is the view 
 that was taken by Kant, and so energetically combated by 
 Helmholz, and lately again by Mr Herbert Spencer. Helmholz 
 indeed attempted to analyse the conception of space, and got 
 so far as to call it a " Dreifachige Manigfaltigkeit," a three-fold 
 multiplicity, which at least expresses the fact that its constitu 
 tion is somehow intrinsically connected with the number 3. 
 But Euclid s expression, applied by him to a solid/ that is, 
 a limited portion of space, that it has "length, breadth and 
 thickness," really means as much, and might be made to mean 
 a great deal more if length, breadth and thickness were 
 properly defined. As I have already kicked over the traces 
 in making use of the word direction/ I may as well say at 
 once that they are measurements made in three different, 
 that is, completely different, or as I call them, independent 
 directions. If therefore we say Space extends from every 
 position in it in three independent directions/ and if further 
 we add and the directions in which it extends from any two 
 positions in it are the same/ we have a definition of the concept 
 space in terms of the concept s position and direction, 
 from which it will be seen all the known properties of space 
 can be deduced. And it follows that the conception of space 
 
CHAPTER III. 19 
 
 is not a fundamental one at all, but that those of position and 
 direction are more fundamental than it. 
 
 It is not necessary, even if it were possible, to explain what 
 the terms position and direction denote. Every one who 
 speaks English must have some idea of their denotation, and 
 for logical purposes it is only necessary to lay down their 
 connotation strictly. Every one knows more or less what is 
 meant by the position of a thing, or the direction of one thing 
 from another, in contradistinction to the thing or things them 
 selves. We may talk of the position of the Solar system, when 
 comparing it with the positions of the fixed stars. But, if we 
 wish to compare the positions of things within a more limited 
 space, it will be evident that there are, within the solar system, 
 many different bodies whose positions differ from each other. 
 Again, within one of these bodies, the earth for example, there 
 are many positions, and if we are dealing with positions within 
 a yet smaller compass, we shall have to use yet more minute 
 bodies to indicate them. Though physicists used to tell us 
 that the atoms of matter were indivisible, the theory has 
 lately been advanced that each atom is a vortex ring of 
 seething aether; and the theory of such vortex rings involves 
 the consideration of differences of position within an atom. 
 When therefore I say that a position may be conceived to be 
 indicated by a minute particle of matter I do not mean to 
 assert any objective truth, but merely to suggest a convenient 
 way of picturing a position to oneself. The second part of the 
 definition of position (for which see Part II.) being added to 
 guard against any confusion of a position with any material 
 thing or point, a position not being a material entity at all. 
 
 In the same way a direction does not consist of two 
 points the points are only used as a convenient way of indi 
 cating it. A direction should be conceived as something purely 
 abstract, so that we may speak of two straight lines as extending 
 in identically the same direction, not in equal, parallel, or 
 similar directions. This conception of direction has proved a 
 stumbling-block to many but chiefly, I am convinced, because 
 from their youth up they have been taught to look at geometry 
 from the Euclidian point of view, which attempts only to recog 
 nise material dimensions, lengths, areas and Euclidian angles, 
 that is, pieces of planes, and to discard abstract distances and 
 
20 PART I. 
 
 inclinations, that is, differences of position and direction. So 
 much is this idea ingrained in our language and ways of 
 thought, that it is difficult to persuade many, even of those 
 who have never formally learnt Euclid, and much more those 
 who have spent the greater part of their lives in learning and 
 teaching it, that it is possible so to grasp Direction, as an 
 abstract concept. Even when confronted by the enormous 
 development of modern vector theories, even while themselves 
 advocating these theories as one of the greatest developments 
 in modern scientific thought, they will still argue that sameness 
 of direction is, as a matter of fact, an idea deduced from Euclid s 
 definitions of a straight line, and of parallel straight lines ; 
 although Euclid himself had spoken of sameness of direction 
 in his eighth definition, prior to giving any workable definition 
 of straightness, or of parallelism. To such objectors it may be 
 replied that it is of very little consequence how they obtained 
 the conception of sameness of direction, so long as they have 
 got it ; but seeing that, except in the one instance referred to 
 above, Euclid carefully evades the word direction, and that in 
 no text-book or course of oral instruction, so far as I know, is 
 any effort made to explain that sameness of direction means 
 Euclidian parallelism, it is at least very doubtful that they 
 acquired the concept through Euclidian teaching. Nor can it 
 fairly be maintained, until it has been proved by experiment, 
 that those who have not learned Euclid, or become imbued 
 with Euclidian ideas, are incapable of forming this conception. 
 On the contrary, I have tried the following experiment on a 
 few persons who had not studied Euclid standing a little way 
 from them I extend my arm in any direction, and ask them to 
 extend theirs " in the same direction." Excepting those who 
 were too shy to make the attempt, all have made a more or 
 less intelligent effort to do as I requested them, showing that 
 though their conception of sameness of direction was vague, 
 and probably inaccurate, the conception was there, and only 
 wanted training to develop it. That is to say, they had some 
 idea of its denotation, though the connotation of the term 
 required to be exactly defined to them. 
 
 To do this I have laid down four assertions about direction. 
 I have already pointed out that I am logically free to define a 
 word by laying down any assertions I please with respect to it, 
 
CHAPTER III. 21 
 
 as long as they are (i) not demonstrably incompatible with each 
 other. But if the definition is to form the basis of a deductive 
 science it is further advisable (ii) that the assertions should be 
 independent, as, if one or more could be deduced from the 
 remainder they might with advantage be omitted, and after 
 wards proved as theorems. And, where it is required to define 
 logically a term whose denotation is already known, it is further 
 necessary not only that (iii) the assertions should be commonly 
 accepted as true with respect to it, but that (iv) they should 
 restrict the meaning of the term exactly to its accepted denota 
 tion, neither more nor less, and should do so in the simplest 
 manner that can be devised. Now my four assertions form a 
 good definition, for (i) they are certainly not incompatible, for 
 all geometricians will admit that (iii) they are as a matter of 
 fact true, about what is commonly understood by the terms 
 direction and sameness of direction, whether they prove this 
 by Euclid or otherwise. The only questions to be examined 
 then are (ii) whether they are independent and (iv) whether 
 they restrict the denotation correctly. I think I may fairly 
 claim to shift the burden of proof of the former proposition on 
 to the shoulders of anyone who denies their independence, 
 merely observing that it has been the attempt to deduce the 
 fourth assertion from the other three which has been the true 
 cause of failure in all previous attempts to make use of 
 1 direction in elementary geometry. But if anyone should 
 succeed in deducing one of my assertions in the definition of 
 direction from the others, he will in no wise have damaged my 
 theory of geometry, but on the contrary, he will have improved 
 it, by enabling me to substitute a somewhat simpler definition 
 of direction for the one I here present. And as to the fourth 
 question, the sufficiency of the definition, the proof of the 
 pudding will be found in the eating, for no one will say that it 
 restricts the denotation too much, since they have granted 
 clause (iii), and, if the results I deduce are deduced fairly, 
 there can be no doubt as to the sufficiency of the premises. 
 
 But though I think these considerations and arguments 
 might be enough to enable my theory of directions to compete 
 with Euclid, if I had a fair start ; he unfortunately has the 
 advantage of me by some two thousand years ; and I must 
 therefore adduce a few more considerations to show the 
 
22 PART I. 
 
 extreme importance, and fundamental character of the concept 
 Direction. 
 
 The idea of a straight line is in common life associated 
 either with the idea of self-congruence, of a line which can 
 twist about itself without apparent motion, of a stretched 
 string, or of a ray of light. From all these conceptions, 
 definitions may be framed, which, as I show in Part III., all 
 lead to the same result. But there remains one conception of 
 a straight line ; that namely which Newton had in his mind 
 when he said that if a material particle were unacted on by 
 any forces, it would remain at rest, or continue to move with 
 uniform velocity in a straight line. There can, I think, be 
 little doubt that the words a constant direction might be 
 substituted for the last three words in this law of motion, 
 without altering the sense. He did not mean a straight line 
 as defined by any of the four methods indicated above, for 
 results can be deduced from the law of motion which are not 
 necessarily possible if straight lines are only so defined. From 
 the law of motion it follows that it would be theoretically 
 possible (by gyroscopes, Faucoult s pendula, or other methods) 
 to determine the velocity, and axis, of rotation of the earth 
 absolutely, and hence to fix absolutely a direction in space, 
 although we cannot fix a position. And if we had two particles 
 the resultant force on each of which vanished, but which 
 remained at the same distance apart, each must be at rest, or 
 moving with uniform velocity in a constant direction. Thus 
 the direction from one to the other is a fixed direction, and 
 since we cannot say they are absolutely at rest, both may be 
 moving in the same direction, though not in the same straight 
 line. 
 
 In view of these facts it must be difficult for anyone to 
 maintain that direction is a less fundamental concept than 
 even position. But even if any one should deny the bearing 
 of such a dynamical argument on the elements of geometry, it 
 surely must settle at once the question whether or not a 
 definite meaning can be, or commonly is, attached to the 
 epithet same when applied to directions of straight lines 
 which do not intersect ! 
 
 The propriety of including the fourth of my assertions in 
 the definition of direction, given in Part II., has been shown 
 
CHAPTER III. 23 
 
 logically to follow from the fact that it is acknowledged to be 
 true of what is generally denoted by that word, and that if it 
 were omitted the definition would include more than the 
 word direction is truly held to denote. But it will probably 
 none the less be maintained, by some supporters of Euclidian 
 methods, that the attribute ascribed to direction in this fourth 
 assertion is, to use old fashioned language, an accident/ not 
 a property of the term. I have indeed already pointed out 
 that, if so, its truth ought to be deducible from the remainder 
 of the definition ; and that if this can be done, so far from 
 upsetting my theory, it would only strengthen it. But I think 
 I can show direct reason for believing that the attribute in 
 question is a property ; and that as a matter of fact it is always 
 assumed by anybody who uses the expression extending in the 
 same directions in the same sense as parallel. 
 
 For surely, if it is conceivable that a point may move in a 
 given direction from A to B, or else in two other directions, 
 from A to C and C to B respectively, then must it not follow 
 that if a point may also move from A to B in the same 
 direction as A to B, it may also move from A to some point C 
 in the direction A C and from C to B in the direction CB ? In 
 books on the theory of vectors, not only is this assumed, but it 
 is further assumed that the transferences in the respective 
 directions must be proportional, or at least, (which comes to 
 the same thing), it is assumed that, in vector language, 
 
 And so far from attempting to justify this assumption by an 
 appeal to Euclid, it is usual to commence by proving Euclid s 
 proposition that the straight lines which join equal and 
 parallel straight lines towards the same parts are themselves 
 equal and parallel/ by it ; the above equation being in fact 
 merely Euclid s proposition in vector language. It will be seen 
 therefore that my fourth assertion about direction does not assert 
 as much as the assumption upon which vector theories are based, 
 for it asserts nothing as to the magnitude of the transferences. 
 Again, if it were required to determine whether two straight 
 lines in space, say two poles stuck in the ground, were in the 
 same direction or not, the first thing that would occur to 
 anyone to do would be to place himself in a line with them, 
 
24 PART I. 
 
 and see whether they covered correctly or not. That is, he 
 would see whether, when one of two intersecting rays of light 
 to his eye (say one along the ground) intersected both poles, 
 and another (say to the top of the far pole, to prevent the 
 chance of its going over the top of either of them) intersected 
 one of the poles, it would also intersect the other. In fact, he 
 would at once apply the test supplied by my fourth assertion 
 about direction. But this assertion has also another aspect. 
 In the former one it corresponds to Euclid s assertion about 
 parallel lines, that they are in one plane. In its other aspect 
 it corresponds to .what has been proposed as a substitute for 
 Euclid s 12th Axiom, being in fact only another way of 
 wording Playfair s Axiom, namely : If a straight line in the 
 same plane with two parallel straight lines intersects one of 
 them, it shall also intersect the other, and this of course is the 
 only aspect of the assertion which is likely to be combated. 
 
 But it is here that the difference of the definitions of the 
 terms straight line, parallel, and so on, by direction, or other 
 wise, comes in. Taking the former system of definitions, but 
 omitting my fourth assertion about direction, if AB, CD be the 
 two poles stuck in the ground in A 
 the same direction, and OBD be 
 the straight line of sight along 
 the ground intersecting them in 
 B and D, the point B being 
 between 0, the observer s posi 
 tion, and D, then as we have 
 
 seen 00 will intersect AB, but OB D 
 
 the question remains, will OA necessarily intersect DO pro 
 duced ? We can only conceive its not doing so by imagining 
 either OA or BO to be bent outwards that is, no longer to 
 extend constantly in the same directions. And, as we have not 
 predicated anything concerning the size of the figure, but only 
 concerning its shape (see Part II. def. 15) the difficulty will not 
 be in any way altered on however large a scale the figure be 
 conceived. But if a straight line is defined by any of the 
 other methods, and CD is only asserted to be parallel to AB in 
 the sense that however far they may be produced they do not 
 meet, nothing has been predicated concerning the shape of the 
 figure, and though when drawn of a moderate size its shape 
 
CHAPTER III. 25 
 
 may be, as near as we can tell, the same as if we had used the 
 definitions by direction, this is no guarantee that the shape 
 would remain the same if its size were indefinitely increased. 
 To prove this would require propositions not proved by Euclid 
 till his sixth book ! 
 
 (I hope it is unnecessary to remind the reader that the 
 above appeal to his imagination is not made to prove a 
 substitute for Euclid s famous axiom, but merely in the hope 
 of calling his attention to the fact that in reality my fourth 
 assertion about direction is already part of the connotation he 
 unconsciously ascribes to that term.) 
 
 I have been unable to find in any work on elementary 
 geometry any formal or satisfactory definitions of the terms 
 line, surface, and solid, or space. I have already pointed out 
 that Euclid s definition is of no use, because no definite con 
 notation is ascribed to the terms length, breadth and thickness. 
 It is however common in more modern text-books to try at 
 least to describe the denotation of the terms line, surface and 
 solid in one of two ways. The one way is to begin with a solid, 
 with which the reader is supposed to be familiar, or with space, 
 and to say that that which bounds a solid, or separates one 
 part of space from another, is a surface, and that that which 
 bounds a surface, or separates one part from another is a line, 
 and that which bounds a line, or separates one part from 
 another, is a point. The other way is to begin with a point 
 and to say that a moving point describes a line, a moving line 
 a surface, and a moving surface a solid, or space. Often both 
 methods are given without any effort to show that they lead to 
 the same results. The first method was formerly, at any rate, 
 the commonest, but it has one fatal drawback it assumes that 
 everybody s conception of a solid, or space, must necessarily be 
 one and unalterably the same, which I think I can show not to 
 be the case. The second method pretends to define the con 
 ception of space in terms of that of a point, or position ; which 
 latter conception certainly is of so elementary a character that 
 we may fairly assume it to be the same to everybody. But in 
 this method also there is a fatal fallacy it is, namely, impossible 
 to define space in terms of a position alone ; direction must also 
 be taken into account. The subterfuges that some geometricians 
 make use of to disguise this fact are positively amusing. One 
 
26 PART I. 
 
 of them for example thinks he has done so by making use of 
 the word way instead of direction ! Thus, he defines space 
 as " a three-way spread, with points as elements." It is indeed 
 true that a point moving any way, that is, in any direction, 
 from each position in it, describes a line of some sort, and that 
 such a line moving in any way (unless it everywhere moves 
 along itself) describes a surface of some sort ; which surface in 
 moving describes a solid of some sort. And if the line is a 
 plane curve and moves in its own plane, it may indeed be said 
 to describe that plane, but parts of the plane may be described 
 twice or three times over, and other parts not at all. And so a 
 curved surface in space, moving in space, may describe part of 
 that space two or three times over and other parts not at all. 
 In order that such a definition of space may be satisfactory and 
 complete, it is necessary to stipulate that the moving point 
 shall move always in the same way (that is, direction) or the 
 opposite ; that the moving line shall always continue to point 
 the same way (that is, that it shall remain parallel to itself), 
 and that a given point in it shall move always in one way or 
 the opposite, different from the ways in which the moving line 
 points. And similar restrictions must be placed upon the 
 movement of the surface so described, in order that it may 
 describe space. These conditions of course involve the concep 
 tion of direction as I define it, and to evade the use of the 
 word by speaking of so many way spreads is not only begging 
 the question but is positively inaccurate. For, the description 
 of a plane as a two-way spread and a solid as a three-way one 
 is not enough, since a line, even a straight one, has two ways, 
 namely forwards, and backwards, and a plane has not only two, 
 but an infinite number of ways. It is therefore clear that 
 something analogous to my definition of independent direc 
 tions is necessary to distinguish the three ways which a 
 spread must have to enable it to rank as a solid, besides some 
 workable definition of what a way is. And this definition 
 must clearly contain a clause corresponding to my fourth 
 assertion, for if not the surface, described as above by the 
 motion of a straight line, would not possess the property that 
 any straight line having two points in it would not lie wholly 
 in it ; that is, it would not be a plane according to the accepted 
 definition. 
 
CHAPTER IV. 
 
 THERE are a few more points among my premises to which 
 I must briefly call attention. The conception of opposite 
 directions follows as an obvious consequence of the fact that 
 two points are required to indicate a direction, and that 
 therefore when we name one, we at the same time indicate 
 a second direction, which stands in a peculiarly intimate 
 relation to the first. From the definition of dependence of 
 directions we see that a direction is dependent upon its 
 opposite direction, but is independent of any other single di 
 rection ; for the movements contemplated in that definition 
 are merely movements in certain straight lines, and may be 
 either one way or the other along them. (N.B. there is no 
 assumption, or assertion about this it is merely a matter of 
 verbal definition.) This definition of independence of direction 
 may at first seem wordy, and arbitrary ; but the reader will 
 soon see that it merely expresses in precise language what we 
 meant (nine) pages back when we agreed that space might be 
 said to extend in three ( independent directions from any given 
 position in it. 
 
 I draw a distinction between distance and a linear dimension, 
 such as length, which I think will be found convenient. A 
 distance is the difference 1 between two positions, and is an 
 
 1 I feel bound to protest here against a criticism of the late Prof. De Morgan 
 in a review of a book of Mr Wilson s, which he published in the Athenaum, and 
 to express my surprise that so accurate a thinker could have been guilty of it. 
 He says apropos of a definition of Mr Wilson s, "Is a direction a magnitude? 
 Is one direction greater than another ? We should suppose so, for an angle, a 
 magnitude to be halved and quartered, is the difference of direction of two 
 straight lines which meet one another. " Clearly neither Mr Wilson nor I use 
 the word difference in the sense it is used in Arithmetic, when we say that 5 is 
 the difference of 12 and 7 ! If two positions or directions are not the same, 
 they must be different, and that difference is called a distance, or inclination, as 
 the case may be. These differences are not magnitudes, but the magnitudes 
 used to compare them are lengths, and angles, respectively. If Mr Wilson did 
 not make this clear, I hope at least that I have done so. 
 
28 PART L 
 
 abstract notion it does not necessarily imply that there is, or 
 even might be, a straight line between them. Thus it may be 
 measured by an amount of transference, under fixed conditions. 
 But length is a concrete measure of an actual figure ; it is a 
 piece of a straight line of given dimension, which may be carried 
 about and compared with other straight lines. So other 
 dimensions may be regarded as material things, pieces of 
 planes or solids. On this view also an inclination is not a 
 dimension but an abstract notion, analogous to a distance ; but 
 an Euclidian angle, a piece of a plane, even though an infinite 
 piece, is precisely analogous to an area. These conceptions are 
 brought out clearly enough in the definitions. 
 
 The assigned methods of measuring distances and inclinations 
 are purely arbitrary, and that they are feasible has to be deduced 
 from the axioms, later on. It is also shown in the text that the 
 methods are convenient, as they measure the quantities with 
 the minimum amount of transference and twisting respectively. 
 The method of measuring a straight angle is an obvious ex 
 tension of that used for any other angle ; but, in as far as it is 
 arbitrary, it too is shown to be feasible and convenient. 
 
 One of the features of my method is that I do not separate 
 the text in the usual manner into Plane and Solid Geometry, 
 but rather into the geometry of lines, and the geometry of 
 surfaces and space. In ordinary Plane Geometry all the figures 
 are supposed to be in a plane. This is rather hard on a 
 beginner, who can hardly be expected to have any clear idea of 
 what a plane is, and the result is that he is apt to miss the 
 force of this important restriction. Even his teachers do the 
 same thing, as witness the fact already referred to that in, I 
 believe, no text-book is the necessary condition noted that the 
 straight lines in Euclid s 12th Axiom must be in one plane. 
 Again in Euc. I. 4, we read, " Therefore the whole triangle ABC 
 coincides with the whole triangle DEF". I am sure that nine 
 people out of ten miss the meaning of this sentence, which is, 
 that their planes coincide. For why should they do so ? Did 
 Euclid ever lay down an axiom that two planes cannot enclose 
 a space ? And if this fact is deducible from his other axioms, 
 where is the theorem in which he deduces it ? Yet this little 
 sentence, which I am convinced most people overlook as a piece 
 of Euclidian circumlocution, is the basis of a very large portion 
 
CHAPTER IV. 29 
 
 of Euclid s reasoning, including his celebrated 47th proposition, 
 and the greater part of his second arid sixth books ! 
 
 In my method I postpone all mention of planes, even their 
 definition, till the second book. In the first book, however, I 
 prove everything Euclid proves in his first book, which does 
 not refer to plane areas; and two propositions which Euclid 
 gives (though in slightly different forms) in his eleventh book. 
 I am thus able to postpone the definitions of the terms surface, 
 plane, &c., to the second book ; where also I give definitions of 
 terms which are either not defined at all in ordinary text-books, 
 or whose definitions would require modification to be brought 
 into accordance with my system. In my second book I prove 
 everything in Euclid s eleventh (except one proposition referring 
 to proportion) and several other highly important propositions 
 which do not appear in any of the ordinary text-books. 
 
 My three axioms, being the foundations of a purely sub 
 jective science, merely assert the power of the human mind to 
 conceive certain things, the nature of the things in question 
 being explained in the definitions. They do not assert any 
 objective fact at all. Of course it would be possible for any 
 given man to deny that he could conceive those things. I do not 
 anticipate coming across many such people, but as they would 
 be no more capable of understanding Euclid than of under 
 standing my geometry, I do not at present care to argue with 
 them. 
 
 My second axiom might be paraphrased thus Space may 
 be conceived to extend from every position in it in the same 
 directions. For if the positions can be conceived, of course 
 points may be conceived to occupy them, forming straight lines 
 &c. The full import of this axiom can only be explained later 
 on, but in case anyone should object that Euclid proves that 
 a straight line can be drawn through any point parallel to 
 any given straight line, whereas I have to assume it, I must 
 refer him to my remarks above, where I show that Euclid s 
 pretended proof really rests on the debatable assumption con 
 tained in his 10th Axiom. 
 
 I have deferred stating my third axiom till the beginning of 
 the second book. I do this, not only because the beginner 
 would not easily understand it until he had become more 
 familiar with the idea of independent directions, but also to 
 
30 PART I. 
 
 emphasise the fact that the propositions in my first book in no 
 way depend upon it. 
 
 As the system of geometry I present in Part II. is a purely 
 subjective one, I do not of course require any powers to draw 
 straight lines, or perform any constructions as Euclid does. I 
 only ask the reader to conceive the lines and figures, not to 
 draw them, and only add the diagrams in the book to aid his 
 imagination he is on no account to suppose that the words in 
 the text refer to the actual lines in the diagrams. Euclid 
 himself virtually abandons his geometrical drawing in his 
 eleventh book, and falls back on hypothetical constructions. 
 But on the other hand my imaginary constructions are hardly 
 hypothetical. Each of them is strictly justified by appeals to 
 the axioms of the science. Thus, I prove by those axioms that 
 an angle may be conceived to be measured by the method I lay 
 down in the definition, that a plane may be conceived through 
 any given position extending in any two given independent 
 directions, that a plane may be conceived to revolve about a 
 fixed straight line in it, and so on. These, and similar points, 
 are overlooked in every other geometry with which I am 
 acquainted, even where actual constructions are made, as in 
 Euclid. 
 
 I have presented Part II. of this book almost in the form I 
 should propose to give to an elementary text-book of geometry, 
 though perhaps a few of these explanations would have to be 
 incorporated with it. A beginner might very well assume its 
 objective application at first, it would not be necessary or 
 advisable to trouble him with such metaphysical subtleties. 
 It would evidently be easy upon this foundation to build up 
 the remainder of a complete work of Elementary Geometry. 
 But in the meanwhile, all the fundamental propositions having 
 been demonstrated (including the fact that two planes cannot 
 enclose a space, required in Euc. I. 4), the superstructure may 
 be bodily taken from the ordinary text-books. 
 
 NOTE. 
 
 I feel it necessary here to refer briefly to some previous attempts to 
 make use of direction in elementary treatises on geometry, and to the 
 strictures which have been passed upon them. For I have already found 
 a disposition on the part of my opponents in argument to save themselves 
 
CHAPTER IV. 31 
 
 trouble by shifting the above-mentioned criticisms bodily on to my 
 shoulders. Now I am of opinion that the reviews to which I specially 
 refer, where they resist the temptation to be funny, are in the main just 
 and logical. Neither Mr Wilson, nor Mr Willock, the chief exponents of 
 the direction theory make any attempt to give a workable definition of 
 direction, or sameness of direction ; and their definitions of angle are 
 hopeless jumbles of the three concepts I have taken pains to distinguish 
 above. The consequence is that they lay themselves open to the accusation 
 of having defined parallel straight lines as such as make equal angles with 
 any transversal. Such a charge could not lie against me, for I on the 
 contrary, define sameness of direction, that is parallelism, first, and angle 
 by it. Besides this, not having defined sameness of direction properly 
 they either assume Euclid s 12th Axiom after all, or are led into positive 
 fallacies in the attempt to do without it. 
 
 But in any case I think I may claim from my critics the courtesy of 
 first-hand criticism, and that they shall not dish up against me the stuff 
 that somebody else wrote, against some other theory, at some other time. 
 
PABT II. 
 
 A SUBJECTIVE THEORY OF GEOMETRY, DEDUCED 
 
 FROM THE TWO FUNDAMENTAL CONCEPTS, 
 
 POSITION, AND DIRECTION. 
 
 BOOK I. 
 ON STRAIGHT LINES, AND ANGLES. 
 
 DEFINITIONS. 
 
 1. IMPLICIT definition of Position : 
 
 (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 contradic 
 tion in terms. 
 
 2. Implicit definition of Direction : 
 
 (a) A direction may be conceived to be indicated by 
 naming two points, as the direction from one to the other. 
 
 (b) If a point move from a given position constantly in 
 a given direction, there is only one path, or series of positions 
 along which it can pass. (Such a path may be called a direct 
 path/ and a continuous series of points occupying such positions, 
 a straight line ; see definition 4.) 
 
 (c) If the direction from A to B is the same as the 
 direction from B to (7, that from A to C is also that same 
 direction. 
 
BOOK I. 33 
 
 (d) If two unterminated straight lines which intersect, 
 are each intersected by a third straight line in two separate 
 points, any unterminated straight line extending in the same 
 direction as this last one, which intersects one of the two 
 former, shall also intersect the other. 
 
 3. The direction from B to A is said to be opposite to that 
 from A to B. 
 
 4. A straight line is a continuous series of points extending 
 from each of them in the same two opposite directions. 
 
 (NOTE. Since a straight line which extends from A to B 
 also extends from B to A, it must always extend in two opposite 
 directions.) 
 
 5. If a point may be conceived to move from one position 
 to another in a given direction, or else along a series of straight 
 lines extending in a succession of directions, the single direction 
 is said to be dependent upon those other directions. But if it 
 is impossible to move the point from the one position to the 
 other by any series of straight lines whatever, in those 
 directions, the single direction is said to be independent of 
 them. 
 
 6. The difference between two positions is called the 
 distance between them. 
 
 It is conventionally measured by the amount of transference 
 required to move a point from the one position to the other in 
 a constant direction, that is, along a straight line. (X.B. The 
 propriety of this convention will appear when it is shown, in 
 Prop. 16, that the amount of transference so required is a 
 minimum.) 
 
 7. The difference between two directions is called their 
 inclination to one another. 
 
 The measure of an inclination is called an angle and is the 
 amount of twisting required to turn a straight line from the 
 one direction to the other, with the condition that in turning it 
 shall pass through a continuous series of directions all inter 
 mediate between and dependent upon the directions whose 
 inclination is to be measured. (N.B. The possibility of so 
 measuring an inclination is shown in Prop. 5, and its propriety 
 appears when it is shown in Prop. 17 that the amount of 
 twisting so required is a minimum.) 
 
 D. 3 
 
34 PART II. 
 
 8. The measure of the inclination of two opposite direc 
 tions is called a straight angle. 
 
 A straight angle is measured by the twisting of a straight 
 line from the one direction to the other keeping it always in 
 directions dependent upon the opposite directions and some 
 third direction chosen arbitrarily. (N.B. Since no third 
 direction can be dependent upon two opposite directions, the 
 ordinary convention fails in measuring a straight angle. The 
 propriety of this supplementary convention appears however 
 when it is shown in Prop. 6, that whatever third arbitrary 
 direction is chosen the result will be the same.) 
 
 9. An angle equal to half a straight angle is called a right 
 angle. 
 
 10. The phrase parallel to may be taken to be equivalent 
 to extending in all the same directions as... 
 
 Hence two straight lines are said to be parallel, if they both 
 extend in the same two opposite directions. 
 
 11. The phrase perpendicular to... may be taken to be 
 equivalent to equally inclined to all the directions in which... 
 extends. 
 
 12. A triangle is the figure formed by the three straight 
 lines joining three points, which are not in one straight line* 
 two and two. The points are called the corners of the triangle. 
 The angle between the directions from one corner to the two 
 others is called an interior angle (or merely an angle) of the 
 triangle, and that between the direction from one corner to a 
 second, and that from the second to the third is called an 
 exterior angle. 
 
 13. A quadrilateral consists of four straight lines joining 
 four points, no three of which are in a straight line, two and 
 two, in order. If the sides form two pairs of parallel straight 
 lines, it is called a parallelogram (for names applied to other 
 quadrilaterals, see end of Prop. 23 of the first book.) 
 
 14. If two figures can be conceived, (either at once or at 
 different times,) to occupy exactly all the same positions, they 
 are said to be congruent to one another. 
 
 15. The size of a geometrical figure is determined by the 
 distances between the various points in it; its shape by the 
 inclinations of the directions from one to another of the various 
 pairs of points in it. 
 
BOOK I. 35 
 
 Axioms. 
 
 I. Any geometrical figure may be conceived to be moved 
 from any one part of space to any other, except in so far as it is 
 restricted by the other axioms (see Book n. Prop. 14, corollary 
 (iii)), without its size or shape being in any way altered. 
 
 II. A straight line may be conceived to extend from any 
 given position to any distance, in any given direction. 
 
 (N.B. A third axiom is added at the beginning of Book n.) 
 
 NOTE. 
 
 A direction may be given by naming two points, as the direction 
 from one to the other, (D. 2. a.) or if it is not important to distinguish it 
 from the opposite direction, it is enough to name a straight line extending 
 in it. Hence by the second axiom we may conceive a straight line through 
 one given point extending to another, or, as it is called, joining the two 
 points ; we may conceive a terminated straight line to be produced either 
 way as far as we like ; we may conceive a straight line through one given 
 point, extending in the direction from a second given point to a third ; or 
 we may conceive a straight line through a given point parallel (see D. 10.) 
 to a given straight line. 
 
 The expression Join AB } is used as an abbreviation for Conceive a 
 straight line joining the points A and B. 3 
 
 A position, or point, is denoted by a single letter of the alphabet, as 
 the point A. 
 
 The distance between two positions, or the terminated straight line 
 between them when considered in reference to its length, is denoted by the 
 two letters denoting the positions or extremities of the line, with a bar 
 over them thus, the distance AB, 3 or the straight line CZ). } 
 
 A direction is denoted by the letters denoting the points which indicate 
 it, taken in order, as the direction AB. 3 Thus the opposite direction to AB 
 is the direction BA. 
 
 An angle may be denoted as the angle between the directions AB, and 
 CD. But if it is between the directions from one point to two others it is 
 merely denoted by the three letters denoting the points, that denoting 
 the point from which the directions lie being placed in the middle. Thus 
 the angle between the directions AB and AC is called simply the angle 
 BAC or the angle CAB. A straight line may be denoted by the letters 
 denoting two or more points in it. If there are only two letters the order 
 is indifferent, but if there are more than two they are (if possible) placed 
 in the order they actually lie, either forwards or backwards, but this rule 
 need not be followed where the order of the points themselves is not 
 determinate. 
 
 32 
 
36 PART II. 
 
 N.B. It must be observed that a direction (and consequently an 
 inclination and an angle also) has no position in space. Thus we may 
 freely talk of the angle between the directions AB and CD even if the 
 straight lines AB and CD do not intersect, and to speak of the angle 
 between AB and AC as the angle at A (as Euclid sometimes does) is 
 incorrect, as also it is to talk of the angle between two straight lines/ 
 Moreover as each straight line has two opposite directions, there are four 
 angles between them, not one, if we consider the phrase merely an 
 abbreviation for between their directions. 
 
 PKOPOSITION I. 
 
 Two parallel straight lines cannot intersect, nor can any two 
 straight lines intersect in more than one point. 
 
 For, if two parallel straight lines had a common position, and 
 from this position a point were to move in either of the two 
 opposite directions in which they both extend from it, it could 
 only move along one path [D 2 (&)] and therefore there could 
 not be more than one straight line through this position in the 
 given directions. 
 
 And if any straight line extend from a point A in a given 
 direction to the next point B, and from B to the next point G } 
 the direction BC is the same as AB [D 4], and therefore the 
 direction AC is the same direction [D 2 (c)]. So it may be 
 proved that the direction AD to the next point, and so to any 
 point on the same side of A is the direction AB. In the same 
 
 A C 
 
 MM 
 
 B O 
 
 way the direction from A to any point on the other side of A 
 isBA. 
 
 Therefore no two straight lines can have two common 
 points, for if they had they would extend in the same directions, 
 and also intersect, which is impossible (see above). 
 
 Therefore Two parallel straight lines &c. Q. E. D. 
 
 Corollary. Two straight lines cannot enclose a space. 
 
 NOTE. The references in brackets refer to the premises or previous 
 propositions: D = definition, A, axiom, P, postulate, c, corollary. 
 
BOOK 1. 37 
 
 PROPOSITION II. 
 
 If each of two unterminated straight lines intersect each of 
 two intersecting straight lines in two separate points, they shall 
 either be parallel to one another, or shall intersect 
 
 For if two straight lines AB, A B each intersect each of the 
 two intersecting straight lines A A , BB , which intersect in 0, 
 in the separate points A, B and A , B , 
 
 Conceive a straight line through A parallel to AB. [A 2. 
 
 Then the straight lines OAA , OB which intersect in 0, are 
 
 met by the straight line AB in separate points, and one of them, 
 OAA is met by the parallel straight line A C in A . 
 
 Therefore OB is also met by this straight line in some point 
 C [D 2 (d)]. Now, if C is the same point as B , the straight 
 line A C is the same as A B , [I. 1], which latter is therefore 
 parallel to AB. 
 
 But if C is not the same point as B , then the two straight 
 lines CB B, A B , which intersect in B , are met by the straight 
 line A C in separate points, and one of them, OBB is met by 
 the parallel straight line AB in B. 
 
 Therefore the unterminated straight lines AB, A B must 
 intersect. [D 2 (d). 
 
 Therefore If each of two &c. Q. E. D. 
 
 PROPOSITION III. 
 
 If each of two unterminated straight lines intersect each of 
 two parallel straight lines, they shall either be parallel to one 
 another, or shall intersect. 
 
38 PART II. 
 
 For if two unterminated straight lines AB, A B each 
 
 intersect each of two parallel straight lines A A , BB , in A, B, 
 and A , B , 
 
 Let C be any third point in AB. Join CB . [A 2. 
 
 Then the straight lines CB A, CB which intersect in C, are 
 met by the straight line BB in separate points, and one of 
 them, CBA is met by the parallel straight line A A . 
 
 Therefore the other, CB , also intersects A A in some point 
 D. [D 2 (d). 
 
 Now, if A is the same point as D, the straight line A B is 
 the same as the straight line CB D, and therefore intersects 
 AB in C, [I. 1. 
 
 But if A is not the same point as D, the unterminated 
 straight lines AB, A B each intersect each of the two inter 
 secting straight lines AA D and CB D in separate points. 
 
 Therefore they are either parallel, or intersect. [I. 2. 
 
 Therefore If each of two &c. Q. E. D. 
 
 PROPOSITION IV. 
 
 If each of two straight lines which intersect are intersected by 
 a transverse straight line in separate points, any straight line 
 through their point of intersection, ivhich also intersects the 
 transverse straight line, is in a direction dependent upon their 
 directions. 
 
 And conversely, any straight line through their point of 
 intersection in a direction dependent on their directions either 
 intersects, or is parallel to, any such transverse straight line. 
 
 For if OA, OB be two straight lines intersecting in 0, and 
 
BOOK I. 39 
 
 AB be a transverse straight line intersecting OA in A, and OB 
 
 n 
 
 And if OX be any other straight line through 0, intersecting 
 AB in X, 
 
 Through X conceive an unterminated straight line XZ, 
 parallel to OA. [A 2. 
 
 Then, since the straight lines BXA, BO, which intersect in 
 
 B, are met by the straight line AO in separate points, and one 
 of them BXA is met by the parallel straight line XZ, in X } 
 
 Therefore BO is also met by XZ in some point Z [D 2 (d)]. 
 Hence a point may be conceived to move from to X in the 
 direction OX along the straight line OX, or along the straight 
 lines OZ, ZX, in the directions OB, OA. 
 
 Therefore the direction OX is dependent upon the directions 
 OA, OB. [D 5. 
 
 And conversely, 
 
 If the direction OX is dependent upon the directions OA, 
 OB, 
 
 Then it must be possible to move a point from to some 
 point X in. OX in the direction OX, or else to move it to X by 
 a series of straight lines OZ, ZX in the directions OB, OA. 
 
 [D5. 
 
 Let B be any other point in OZB, join BX. 
 
 Then since the straight lines BZO, BX, which intersect in 
 B are met by the straight line ZX in separate points ; and one 
 of them BZO is met by the parallel straight line OA in 0, 
 
40 
 
 PART II. 
 
 Therefore BX is also met by OA in some point A [D 2 (d)] 
 and therefore OX intersects the transverse straight line AB. 
 
 And if A B be any other transverse straight line, intersecting 
 OA in A , and OB in B , 
 
 Then the two straight lines AB, AB each intersect each of 
 the two intersecting straight lines OAA , OBB in separate 
 points. 
 
 They are therefore either parallel, or they intersect. [I. 2. 
 
 But if they are parallel, 
 
 Then the straight lines OBB , OX which intersect in 0, are 
 met by the straight line BX in separate points, and one of 
 them, OBB , is met by the parallel straight line B A in B . 
 
 Therefore OX also intersects B A . [D 2 (d). 
 
 And if AB, A B are not parallel, let them intersect in P. 
 Therefore the straight lines PA , OX each intersect each of the 
 intersecting straight lines PAX, A AO in separate points. 
 
 Therefore PA B and OX are either parallel, or they 
 intersect. [I. 2. 
 
 Therefore If each of two &c. Q. E. D. 
 
 PROPOSITION V. 
 
 A straight line may be conceived to be twisted from any one 
 given direction to any other which is independent of it, so as to 
 measure the angle between them, in one, and only one way. 
 
 For if be any fixed point, straight lines OA, OB may be 
 conceived in the given directions from to points A, B [A 2], 
 which will not be one and the same straight line, since the 
 directions are independent. 
 
 And a straight line may be conceived joining AB, and also 
 one from to any point X in AB. [A 2. 
 
BOOK I. 41 
 
 And the point X may be conceived to move along AB from 
 A to B, that is in the constant direction AB, in one and only 
 one way. p 2 (6). 
 
 As it does so OX is twisted from the direction OA to OB in 
 one, and only one way. 
 
 And since the directions in which OX extends are always 
 intermediate between, and dependent upon the directions OA and 
 OB [I. 4], the twisting of OX measures the angle AOB. [D 7. 
 
 Therefore A straight line &c. Q. E. D. 
 
 Corollary. If any direction be intermediate between and 
 dependent upon two others, the sum of the angles it makes with 
 those two directions is equal to the angle they make with each 
 other. For the twistings of a straight line to measure the 
 former angles are parts of the twisting which measures the 
 latter. 
 
 PROPOSITION VI. 
 
 The inclination of two opposite directions is a constant 
 inclination; and the conventional method of measuring it ahuays 
 gives the same angle. 
 
 For if OA, OB be straight lines from any point to points 
 A, B, in opposite directions, 
 
 And if O A , O B be two other straight lines from any point 
 to points A , B , in any other two opposite directions, 
 
 Then since the direction A is opposite to OA [D 3], it is 
 the same as OB. Therefore AOB is one straight line. 
 
 So also A O B is one straight line. 
 
 Therefore if the straight line AOB be conceived to be moved 
 so that falls on and A in A B [A 1], the straight lines 
 will become one and the same straight line [I. 1], that is OB, 
 O B will also coincide. 
 
 Therefore the inclination of OA to OB is the same as that 
 if O A to O B in the new position, and the inclination was not 
 altered by the movement of AOB [A 1]. Therefore the incli 
 nations of the opposite directions were the same. 
 
 And if the inclination of OA to OB be measured by twisting 
 a straight line from OA to OB, keeping it in directions dependent 
 on the directions OA, OB, OC, [D 8, 
 
42 PART II. 
 
 And if the inclination of O A to O B be measured by a 
 
 A/ 
 
 straight line keeping it in directions dependent on O A , O B, 
 G \ 00, O G being any arbitrarily chosen directions, [D 8. 
 
 Then either the angles AOC, A O G must be equal, or one 
 must be greater than the other, say AOC than A O G . 
 
 In the latter case as a straight line OX revolves to measure 
 the angle A OB, before reaching OG it will reach some situation 
 OX such that the angle AOX is equal to A O G . 
 
 Hence if the whole figure AGB be conceived to be moved 
 and placed so that coincides with [A 1], AOB with 
 A O B , either OG, or in the latter case OX, may be made to 
 coincide with O G . 
 
 Therefore in the former case the figures are congruent, and 
 therefore the twisting of the same straight line measures both 
 straight angles, which are therefore equal. 
 
 In the latter case the straight angle AOB measured by 
 a straight line twisted through directions dependent on OA t 
 OB, and OX is equal to the straight angle A O B , and since 
 this straight line in revolving passes through OG [I. 4], it gives 
 the same angle as one twisted through directions dependent on 
 OA, OB and OG. 
 
 Therefore The inclination &c. Q. E. D. 
 
 Corollary, (i) Hence the sum of the angles between any 
 direction and two opposite directions is a straight angle. 
 
 (ii) Hence also a right angle, which is half a 
 straight angle, is a constant angle. 
 
BOOK L 
 
 43 
 
 PROPOSITION VII. 
 
 The angle between two directions is equal to that between the 
 directions opposite to them. 
 
 For if AOA , BOB , be any two straight lines through any 
 point ; OA, OB being two given directions, and OA OR the 
 directions opposite to them, 
 
 If we conceive one end, OX of a straight line XOX to 
 revolve from OA to OB, measuring the angle A OB, the other 
 end OX will at the same time be revolving from OA to OB 
 
 and measuring the angle A OB . 
 
 Therefore the angle between OA and OB is equal to that 
 between OA and OB . 
 
 Therefore The angle &c. Q. E. D. 
 
 PROPOSITION VIII. 
 
 The sum of the interior angles of any triangle is a straight 
 angle. 
 
 For if ABC be any triangle, 
 
 Conceive the side BC to be produced to D, [A 2. 
 
 And through C conceive a straight line CE in the direction 
 BA, to any point E. 
 
44 PART II. 
 
 Then because the directions CE, GD are respectively the 
 same directions as BA, BC, 
 
 The angle DGE is the same as the angle CB A. 
 
 And because the directions CE, CA are respectively opposite 
 to the directions AB, AC, 
 
 The angle EGA is equal to the angle BA C. [I. 7. 
 
 But since a point can be conceived to move from B to A in 
 the direction CE, or else by straight lines BC, GA in the 
 directions CD, CA, 
 
 The direction CE [D 5] is dependent on the directions CD, 
 CA ; and it is intermediate between them. 
 
 Therefore the sum of the angles DGE, EGA is equal to the 
 angle DC A. [I. 5 c. 
 
 Therefore the sum of the angles CBA, BAG, AGB is equal 
 to the sum of the angles DC A, AGB. 
 
 That is ; to a straight angle. [I. 6 c. (i). 
 
 Therefore The sum &c. Q. E. D. 
 
 Corollaries, (i) Hence the exterior angle of a triangle, as 
 ACD is equal to the sum of the two interior opposite angles CBA, 
 BAG. 
 
 (ii) Hence also the sum of any two angles of a triangle is 
 less than a straight angle, and any exterior angle is greater than 
 either of the interior opposite angles. 
 
 PROPOSITION IX. 
 
 If each of two straight lines intersect each of two intersecting 
 straight lines, or of two parallel straight lines, in separate points, 
 and 
 
 (i) if their directions on the same side of one of the trans 
 verse straight lines are equally inclined to one of its directions ; 
 or 
 
 (ii) if they make with its opposite directions angles whose 
 sum is a straight angle ; or 
 
 (iii) if their directions on opposite sides of the transverse 
 line make equal angles with its opposite directions ; 
 
 the straight lines shall be parallel. 
 
 For if each of two straight lines AB, CD intersect each of 
 the two straight lines EF, KL, which either intersect or are 
 parallel, 
 
BOOK I. 45 
 
 Then AB, CD either intersect or are parallel. [I. 1 and 2. 
 ,E 
 
 But they cannot intersect, for if they did, in say : 
 (Then if EF intersect AB in G and CD in H, OGH would 
 be a triangle, and therefore) 
 
 (i) The exterior angle EGO would be greater than the 
 interior opposite angle GHO [I. 8 c. (ii)], whereas it is not, if 
 the directions GB, HD of the straight lines AB, CD on the 
 same side of EF are equally inclined to the direction HE, or 
 GE,ofEF: 
 
 (ii) The sum of the interior angles BGH, DHG of the 
 triangle OHG would be less than a straight angle. [I. 8 c. (ii). 
 
 And since the sum of the four angles AGE, BGH, CHG, 
 DHG is two straight angles [I. 6 c. (i). 
 
 The sum of the angles AGH, CHG would be greater than 
 a straight angle, 
 
 Whereas in both cases they are equal to a straight angle, if 
 the directions of AB and CD on the same side of EF make 
 with its opposite directions angles whose sum is a straight 
 angle, 
 
 (iii) And the angle AGH would be greater than the 
 interior opposite angle DHG of the triangle OHG [I. 8 c. (ii)], 
 whereas it is not, if the directions GA, HD of the straight lines 
 AB, CD on opposite sides of EF make equal angles with its 
 opposite directions. 
 
 So it may be proved that none of the conditions in the 
 enunciation can hold unless AB is parallel to CD. 
 
 Therefore If each of two &c. Q. E. D. 
 
 Corollary. And, conversely, it is evident that if AB is 
 parallel to CD all the conditions in the enunciation will hold, 
 from the definitions of the terms, or as special cases of Proposi 
 tions 6 cor. (i), and 7. 
 
46 PART II. 
 
 PROPOSITION X. 
 
 If two triangles have two sides of the one respectively equal 
 to two sides of the other, and the angles between the directions of 
 those sides equal, they shall be congruent to each other. 
 
 For if the sides AB, AC of a triangle ABC are respectively 
 equal to the sides DE, DF of a triangle DEF, 
 
 And the angles BAG 1 , EDF between the directions of these 
 sides are equal, 
 
 Then the triangle ABC may be conceived to be moved, 
 
 [A 1], and placed upon DEF, so that the corner A falls upon 
 D, the side AB upon DE and, since the angles BAO, EDF are 
 equal, the side A C upon the side DF. 
 
 And since AB is equal to DE and AC to DF, the corner B 
 will fall upon E, and C on F. 
 
 Therefore BC will coincide with EF [I. 1] and consequently 
 the triangles are congruent. [D 14. 
 
 Therefore, If two triangles &c. Q. E. D. 
 
 Corollary. Hence if two sides of a triangle are equal, two of 
 its angles, namely, those between the directions of the equal sides 
 and the third side, are equal. 
 
 For if AB in the above proof had been equal to AC, the 
 triangle ABC might also have been moved so that AB fell on 
 DF, and AC on DE, and the triangles would also have been 
 congruent so. Hence both the angles ABC and ACB would be 
 shown to be equal to DEF, and therefore to each other. 
 
BOOK L 47 
 
 PROPOSITION XI. 
 
 If two triangles have two angles of the one respectively equal 
 to two angles of the other, and have two corresponding sides 
 equal, iliey shall be congruent to each other. 
 
 For if ABC, DEF be t\vo triangles which have two angles 
 of the one respectively equal to two angles of the other, 
 
 Then since the sum of the three angles of every triangle is 
 a straight angle, rj g 
 
 The third jmgles_ of the two triangles are also equal. 
 
 Hence if BC, EF are corresponding equal sides, 
 
 The triangle ABC may be conceived to be moved [A 1] and 
 placed so that the corner B falls on E, the side BC on EF, and 
 since the angle ABC is equal to the angle DEF, the side BA 
 upon the side ED. 
 
 Then since BC is equal to EF, C will fall upon F: and 
 since A is in BA, it will fall upon ED, or ED produced. 
 
 Hence if a straight line through F be twisted from the 
 direction FE towards the direction FD [I. 4], measuring the 
 angle EFD [I. 5], and always intersecting ED, or BA, it will 
 also measure the angle EGA. And since the angles EFD, 
 BOA are equal, it must reach FD and CA both at once. 
 
 Therefore A must coincide with D, and the triangles be 
 congruent. 
 
 Therefore If two triangles &c. Q. E D 
 
 Corollary. Hence if two angles of a triangle are equal, the 
 two opposite sides are equal. 
 
 For if the angle ABC in the above proof had been equal to 
 the angle ACB, the triangle ABC might also have been moved 
 so that C fell upon E, and B on F, and the triangles would also 
 have been congruent so. ^lence both the sides AB, AC would 
 be shown to be equal to DE, and therefore to each other. 
 
48 PART II. 
 
 PROPOSITION XII. 
 
 The greater side of every triangle is opposite to the greater 
 angle, 
 
 And conversely, the side which is opposite to the greater 
 angle is the greater side. 
 
 For if one side AC of a triangle ABC be greater than 
 another AB, 
 
 There must be some point D between A and C such that 
 AD is equal to AB. Join BD. [A 2. 
 
 Then because D is in AC between A and C, the direction 
 BD is intermediate between and dependent upon the directions 
 BA,BC. [1-4. 
 
 Therefore the angle ABC is greater than the angle ABD. 
 
 [I. 5 c. 
 
 But since the sides AB, AD of the triangle ABD are equal, 
 
 The angle ABD is equal to the angle ADB. [I. 10 c. 
 
 And as the angle ADB is an exterior angle of the triangle 
 BCD, it is greater than the interior opposite angle BCD. 
 
 [I. 8 c. (ii). 
 
 Much more therefore is the angle ABC greater than the 
 angle BCD, or AGE. 
 
 Therefore the greater side of every triangle is opposite the 
 greater angle. 
 
 And conversely, the side which is opposite to the greater 
 angle is the greater side. 
 
 For, if not, it must be either equal to or less than the other. 
 
 But if it were equal, the opposite angles would be equal 
 [I. 10 c], whereas they are not. 
 
 And if it were less, the angle opposite to it would be the 
 less (see above), whereas it is the greater. 
 
 Therefore The greater side &c. Q- E. D. 
 
BOOK I. 49 
 
 PROPOSITION XIII. 
 
 If two triangles have two sides of the one respectively equal 
 to two sides of the other, but the angle between the directions of 
 those sides greater in the one triangle than in the other, the third 
 side of the former triangle shall be greater than the third side of 
 the other. 
 
 And conversely, if the third side of the one triangle be 
 greater than the third side of the oilier, the angle opposite to 
 it in the former triangle shall be greater than the corresponding 
 angle of the other. 
 
 For if the sides AE,^AC of a triangle ABC are respectively 
 equal to the sides DE, DF of a triangle DEF, 
 
 But the angle BAG be greater than the angle EDF, 
 
 Then, of the sides AB, AC, one must be not less than the 
 other. Suppose A C to be that one. 
 
 Through A conceive a straight line to be twisted from A B 
 towards AC, always intersecting BC in X [I. 5], until it has 
 described an angle equal to EDF. 
 
 A D 
 
 Then since the angle EDF is less than the angle BAG, X 
 will be between B and C. [I. 5 c. 
 
 Then since AC is not less than AB, the angle ABC is not 
 less than the angle ACE. [I. 12. 
 
 And since AXC is the exterior angle of the triangle ABX, 
 
 The angle AXC is greater than the interior opposite angle 
 ABC. [I. 8 c. (ii). 
 
 Therefore it is greater than the angle ACB. 
 
 Therefore the side A C of the triangle AXC is greater than 
 the side AX. [I. 12. 
 
 Hence if G be the point in AX, in the same direction from 
 A as X is, and at the same distance from it as C, the point X 
 will be between A and G. 
 
 Therefore the direction CX is intermediate between and 
 dependent upon the directions CA and CG. [I. 4. 
 
 D. 4 
 
50 PART II. 
 
 Therefore the angle A CG is greater than the angle X CG. 
 
 [I. 5 c. 
 
 And similarly, because X is between B and C, the angle 
 BGC is greater_than the angle_Z0. [I. 5 c. 
 
 But since AG is equal to AC, 
 
 The angle AGG is equal to the angle AGG. [I. 10 c. 
 
 Therefore the angle BGG is greater than the angle AGG 
 and much more therefore is it greater than BCG. (See above.) 
 
 Therefore the opposite side BG of the triangle BGC is 
 greater than the side BG. _ [I. 12. 
 
 But since the sides AB, AG of the triangle ABG are 
 respectively equal to the sides DE, DF of the triangle DEF, 
 and the angle BAG is equal to the angle EDF, 
 
 ^Therefore the triangles are congruent, and BG is equal to 
 EF. [I. 10. 
 
 Therefore BG is greater than EF. 
 
 And conversely, if BG is greater than EF, the angle BAG 
 shall be greater than the angle EDF. 
 
 For, if not, it must be equal or less. 
 
 But if it were equal BC would be equal to EF [I. 10], 
 which it is not, 
 
 And if it were less, BC would be less than EF (see above), 
 whereas it is greater. 
 
 Therefore the angle BAG must be greater than the angle 
 EDF. 
 
 Therefore If two triangles &c. Q. E. D. 
 
 PROPOSITION XIV. 
 
 If two triangles have the three sides of the one respectively 
 equal to the three sides of the other, they shall be congruent. 
 
 For if one angle of the one were either greater or less than 
 the corresponding angle of the other, the side opposite to this 
 angle would be greater or less than the corresponding side of 
 the other triangle, whereas it is equal to it. [I. 13. 
 
 Therefore the angles between the directions of any two 
 sides of the one triangle must be equal to the angle between 
 those of the two sides equal to them of the other triangle. 
 
 Hence the triangles are congruent. [I. 10. 
 
 Therefore // two triangles &c. Q. E. D. 
 
BOOK /. 51 
 
 PKOPOSITION XV. 
 
 If two triangles have two sides of the one respectively equal to 
 two sides of the other, and an angle of the one triangle opposite 
 one of those sides equal to that opposite the equal side of the 
 other triangle, then the angles of the two triangles opposite their 
 other equal sides shall either be equal, or their sum shall be a 
 straight angle. 
 
 For if two sides AB, AC of a triangle ABC are respectively 
 equal to two sides DE, DF of a triangle _DEF,_ and if the 
 angles ABC, DEF opposite the equal sides AC, DF be equal, 
 
 The triangle ABC may be conceived to be moved [A 1] 
 and placed so that the corner B falls on E, BA on ED. and 
 BC on EF, since the angle ABC is equal to DEF. 
 
 And since AB is equal to DE, A will fall upon D. 
 
 And since C falls on EF, either it falls upon F, and the 
 triangles are congruent, 
 
 Or else it falls at some other point in EF, and DGF is a 
 triangle. 
 
 Of the two points G and F, one of them, say G, must be 
 between the other and E. 
 
 Then since the sides DG, DF of the triangle DGF are 
 equal, the angle DGF is equal to the angle DFG. [I. 10 c. 
 
 But the sum of the angles DGE, DGF is a straight angle. 
 
 [I. 6 c. (i). 
 
 Therefore the sum of the angles DGE, that is ACB, and 
 DFE is a straight angle. 
 
 Therefore If two triangles &c. Q. E. D. 
 
 Corollaries. Hence (i) if besides the conditions in the enun 
 ciation it is known that the doubtful angles are either both 
 greater, or both less, than a right angle, the triangles are 
 congruent ; for then the sum of the doubtful angles cannot be 
 a straight angle. 
 
 42 
 
52 PART II. 
 
 (ii) If the known equal angles are either right angles, or 
 greater than right angles, the triangles are congruent ; for since 
 the sum of the known and the doubtful angle in either triangle 
 is less than a straight angle [I. 8 c. (ii)], both the doubtful 
 angles must be less than right angles. 
 
 (iii) Or, if it is known that one of the doubtful angles is a 
 right angle, the triangles are congruent ; for then in either case 
 the other doubtful angle is a right angle and therefore the 
 triangles are congruent. [I. 11. 
 
 PROPOSITION XVI. 
 
 Any two sides of a triangle are together greater than the 
 third. 
 
 For if ABC be any triangle, 
 
 Then if its sides are all equal, obviously any two of them 
 together are greater than the third. 
 
 But if not, one of them, say AB, must be not less than 
 either of the others, and greater than one of them, say A G. 
 
 Then there must be some point D between A and B, at 
 the same distance from A as G is. Join CD. [A 2. 
 
 Then since the sides AC, AD of the triangle ACD are equal, 
 
 The angles ACD, ADC are equal. [I. 10 c. 
 
 And therefore since their sum is less than a straight angle, 
 each of them is less than a right angle. [I. 8 c. (ii). 
 
 But the sum of the angles CDB, CD A is a straight angle. 
 
 [I. 6 c. 
 
 Therefore the angle BDC is greater than a right angle. 
 
 And the sum of the angles BDC, BCD is less than a straight 
 angle. [I. 8 c. (ii). 
 
 Therefore the angle BCD is less than a right angle. 
 
 Therefore the angle BDC is greater than the angle BCD. 
 
BOOK I. 53 
 
 Therefore the side BC of the triangle BCD is greater than 
 BD. [I. 12. 
 
 To these unequals add the equals AC and AD respectively. 
 
 Therefore BC, CA are together greater than BD, DA, that 
 is, than BA. 
 
 And since no side of the triangle is greater than BA, any 
 two sides of the triangle are together greater than the third. 
 
 Therefore A ny two &c. Q. E. D. 
 
 Corollary. Hence a straight line is the shortest path between 
 two positions. For any other path may, within any assignable 
 limits of accuracy, be supposed to be made up of a succession 
 of short straight lines. And instead of traversing such a 
 cornery path, it would, by this proposition, always be shorter 
 to miss out the next coming corner, and to go straight to the 
 next but one. A fortiori would it be shorter to miss out all 
 the corners, and go straight from the one position to the other. 
 
 (NOTE. This is why the distance between two positions 
 was denned as the amount of transference along a straight line 
 between them.) 
 
 PROPOSITION XVII. 
 
 Any two of the angles between three independent directions 
 are together greater than the third. 
 
 For if be any position, and if straight lines OA, OB, OC 
 be conceived extending from it in any three independent 
 directions, 
 
 Then, if the three angles between them are all equal, 
 obviously any two of them are together greater than the third. 
 
 But if not, one of them, say AOB, must be not less than 
 either of the others, and greater than one of them, say AOC. 
 
 Conceive a straight line [A 2] joining any two points A 
 and B in the given directions from 0, and conceive a straight 
 line to revolve from OA towards OB, always intersecting AB 
 in D, until the angle AOD is equal to AOC. [I. 5. 
 
 Then D will be between A and B, since the angle AOB is 
 greater than AOC. [I. 5 c. 
 
 Let C be the point in the third given direction from 0, at 
 the same distance from it as D is. Join CA, CB. [A 2. 
 
Then since the direction OC is independent of the directions 
 OA, OB ; OC cannot intersect AS, and G is not in AB. [I. 4. 
 
 Therefore ACB\^ a triangle. [D 12. 
 
 Therefore AC, CB are together greater than AB. [I. 16. 
 
 And since the sides OA, OC of the triangle OAC are 
 respectively equal to the sides OA, OD of the triangle OAD, 
 and the angle A OC is equal to the angle AOD, 
 
 _The triangles are congruent, and therefore AC is equal to 
 AD. [I. 10. 
 
 But AC, CB are together greater than AD, DB. 
 
 (See above.) 
 
 Therefore CB is greater than DB. 
 
 But the sides OC^OB^of the triangle OBC, are respectively 
 equal to the sides OD, OB of the triangle OBD. 
 
 Therefore the angle COB is greater than the angle DOB. 
 
 [I. 13. 
 
 To each of these unequals add the equals AOC, AOD re 
 spectively. 
 
 Therefore the angles BOG, CO A are together greater than 
 BOD, DO A, that is, the angle BOA. [I. 5 c. 
 
 And since none of the three angles are greater than BOA, 
 any two of them are together greater than the third. 
 
 Therefore Any two &c. Q. E. D. 
 
 Corollary. Hence the shortest way of twisting a straight line 
 from one direction to another is as in measuring the angle 
 between them. 
 
 For any other way of twisting it may, within any assignable 
 limits of accuracy, be supposed to be made up of a succession 
 of short twistings as in measuring the angles between a series 
 of directions. And instead of twisting it through such a 
 succession of directions it would, by this proposition, always be 
 shorter to miss out the next coming direction, and to twist it 
 
BOOK I. 55 
 
 as if measuring the angle to the next but one. A fortiori 
 would it be shorter to twist it at once as if measuring the 
 angle from the first direction to the last. 
 
 (NOTE. This is why an angle was defined as the amount 
 of twisting through a continuous series of directions inter 
 mediate between and dependent upon the extreme directions.) 
 
 PROPOSITION XVIII. 
 
 The sum of the angles between any three independent direc 
 tions is less than tivo straight angles. 
 
 For if be any position and if straight lines be conceived 
 through extending in any three independent directions 
 OA, OB, 00 to points A, B, C } 
 
 Join AB, BO, OA ; 
 
 Then since the direction OA is independent of the direc 
 tions OB, 00 ; OA cannot intersect BO. [I. 4. 
 
 And since AO does not intersect BO, the direction AO is 
 independent of the directions AB and AO. [I. 4. 
 
 Therefore the angles OAB, OAO are together greater than 
 BAG. [I. 17. 
 
 Similarly the angles OB A, OBC are together greater than 
 ABC and the angles OCB, 00 A are together greater than BOA. 
 
 Therefore the six angles OAB, OAO, OB A, OBC, OCB, 
 OCA together, are greater than the three angles of the triangle 
 ABC, 
 
 That is, than a straight angle. [I. 8. 
 
 To each of these unequals add the three angles AOB, BOO, 
 COA, 
 
 Then the nine angles of the three triangles OAB, OBC, 
 OCA, that is 3 three straight angles, are together greater than 
 one straight angle together with the three angles AOB, BOC, 
 COA. 
 
56 PART IL 
 
 Therefore these three angles are together less than two 
 straight angles. 
 
 Therefore The sum &c. Q. E. D. 
 
 PROPOSITION XIX. 
 
 One, and only one, straight line can be conceived to a given 
 straight line from a given point without it, in a direction per 
 pendicular to it. 
 
 For if be any point outside a straight line AB, 
 
 Let G, D be any two points in AB. Join 00. 
 
 Then if the angle OCA is equal to the angle OCB, 00 is 
 perpendicular to AB. [D 11. 
 
 But if not, then since the sum of the angles OCA, OCB is 
 a straight angle [I. 6 c. (i)], one of them must be greater and 
 the other less than a right angle. [D 9. 
 
 Let OCB be the one that is less than a right angle. 
 
 Since every straight line through in a direction dependent 
 on the directions OC and OD either intersects AB, or is parallel 
 to it, [I. 4. 
 
 If a straight line OP through revolve from OC towards 
 B, remaining in directions dependent upon OC and OD, it will 
 continue to intersect AB on that side of C on which B is, until 
 it is parallel to AB, 
 
 That is, until the sum of the angles POC, OCB is a straight 
 an g!e. [I. 9 c. 
 
 But since the angle OCB is less than a right angle, before 
 this OP will reach a direction such that the sum of the angles 
 POC, OCB is a right angle. In this direction let it intersect 
 AB in E. 
 
 Then OEB is the exterior angle of the triangle OCE and 
 
BOOK L 57 
 
 is therefore equal to the sum of the two interior opposite 
 angles EOC, OCE [I. 8 c. (i)], that is, to a right angle. 
 
 But the sum of the angles OEB, OEA is a straight angle 
 [I. 6 c.], that is, two right angles. [D 9. 
 
 Therefore the angles OEB, OEA are equal, and OE is 
 perpendicular to AB. [D 11. 
 
 But no other straight line from to AB can be so, for if 
 one as OD were, the angles ODE, OED would be both right 
 angles, and therefore their sum a straight angle, whereas being 
 interior angles of a triangle their sum must be less than a 
 straight angle. [I. 8 c. (ii). 
 
 Therefore One, and only one, &c Q. E. D. 
 
 PROPOSITION XX. 
 
 The shortest path to a straight line from a point without it is 
 the straight line perpendicular to it. 
 
 And of all other straight lines to it from the given point, 
 those which meet it at equal distances from the foot of the 
 perpendicular are equal, and those which meet it nearer the foot 
 of the perpendicular are shorter than those which meet it farther 
 
 off. 
 
 For if be any point outside a straight line AB, 
 
 A D F C E B 
 
 Conceive a straight line 00 from to C in AB, in a 
 direction perpendicular to AB. [I. 19. 
 
 Conceive a straight line from to any other point D in AB. 
 
 Then since ODC is a triangle, the sum of the angles ODC, 
 OCD is less than a straight angle. [I. 8 c. (ii). 
 
 But the angle OCD is half a straight angle. [I. 6 & D 11. 
 
 Therefore the angle ODC is less than half a straight angle, 
 and therefore less than OCD. 
 
 Therefore the side OC of the triangle OCD is less than the 
 side OD. [I- 12. 
 
58 PART II. 
 
 Therefore OG is the shortest straight line, and a fortiori 
 the shortest path from to AB. [I. 16 c. 
 
 And if E be any other point in AB at the same distance 
 from C as D is,_ 
 
 The sides OG, GD of the triangle OGD are respectively 
 equal to the sides OG, GE of the triangle OCE, and the angles 
 OGD, OGE are equal. 
 
 Therefore the triangles are congruent, and OD is equal to 
 OE. [I. 10. 
 
 And if F be any point nearer to G than D or E is, say on 
 the same side as D, 
 
 Then it may be shown as above that the angle OGF being 
 a right angle, the angle OFG is less than a right angle. 
 
 [I. 8 c. (ii). 
 
 Therefore the angle OFD is greater than a right angle. 
 
 But the angle ODF is less than a right angle, and therefore 
 less than OFD. 
 
 Therefore the side OF of the triangle OFD [I. 12] is less 
 than the side OD. 
 
 Therefore The shortest path &c. Q. E. D. 
 
 PROPOSITION XXI. 
 
 The straight lines which join those extremities of two equal 
 and parallel straight lines which lie in opposite directions from 
 the others, bisect one another ; 
 
 And those which join those extremities which lie in the same 
 directions from the others, are themselves equal and parallel. 
 
 For if AB, GD be two equal and parallel straight lines, the 
 direction from A to B being the same as that from G to D, 
 
 Then if a straight line through A be conceived to revolve 
 from the direction AC towards AD and so on, remaining 
 always in directions dependent on AC and AD, it will continue 
 to intersect CD on the same side of C as D is, until it is 
 parallel to CD, that is, coincident with AB. [I. 4. 
 
 Therefore the direction AD is intermediate between, as 
 well as dependent upon, the directions AC, AB. 
 
 Therefore AD must intersect CB between C and B, in 0, 
 say. [I. 4. 
 
BOOK I. 59 
 
 Similarly CB intersects AD between A and D, in 0. 
 
 Then, since the directions AB, AD are respectively opposite 
 to the directions DC, DA, 
 
 The angle BAO is equal to the angle CDO. [I. 7. 
 
 Similarly the angle ABO is equal to the angle DCO, 
 
 And the sides AB, CD of the triangles OAB, ODC are equal. 
 
 Therefore the triangles are congruent. [I. 11. 
 
 Therefore AO is equal to OD and BO to OC, that is, AD, 
 CB bisect one another in 0. 
 
 And since OA is equal to OD and OB to OC, and the 
 angles AOC, DOB of the triangles AOC, DOB are equal, 
 
 Therefore the triangles are congruent. [1. 10. 
 
 Therefore A C is equal to BD, 
 
 And the angle A CB is equal to the angle CBD. 
 
 Hence the two straight lines AC, BD each intersect each of 
 the intersecting straight lines AD, BC in separate points, and 
 their directions CA, BD on opposite sides of one of these 
 straight lines, make equal angles with its opposite directions 
 CB, BC. 
 
 Therefore they are parallel. [I. 9. 
 
 And they have been shown to be equal. 
 
 Therefore Tfie straight lines &c. Q. E. D. 
 
 PROPOSITION XXII. 
 
 The perpendicular distance of any point in a straight line 
 from a straight line parallel to it, is constant. 
 
 For if A, B be any two points in a straight line to which a 
 straight line CD is parallel, 
 
 Conceive the straight line A C from A to CD perpendicular 
 to CD, [I. 19. 
 
 And let D in CD be in the same direction from C as B is 
 from A, and at the same distance from it. 
 
60 PART II. 
 
 Then AB, CD are equal and parallel straight lines. 
 A B 
 
 Therefore also AC and ED are equal and parallel. [I. 21. 
 Therefore BD is the perpendicular to CD, and is equal to 
 
 AB. 
 
 Therefore The perpendic alar &c. 
 
 Q. E. D. 
 
 PROPOSITION XXIII. 
 
 The opposite sides and angles of a parallelogram are equal 
 to one another. 
 
 For if ABDC be a parallelogram, AB is parallel to CD. 
 
 [D13. 
 
 Let X in CD be the point in the same direction from C as 
 B is from A, and at the same distance from it, 
 
 x D 
 
 Then AB, CX are equal and parallel straight lines. 
 
 Therefore AC and BX are equal and parallel straight lines. 
 
 [I. 21. 
 
 But BD is parallel to AC. [D 13. 
 
 Therefore BX and BD extend in the same directions from 
 B, and they are therefore one and the same straight line. [I. 1. 
 
 And as both X and D are in CD also, they must be one 
 and the same joint. [I. 1. 
 
 Therefore AB and CD are equal, as also are AC and BD. 
 
 And since AB is parallel to CD, the sum of the adjacent 
 angles of the parallelogram BAG, ACD is a straight angle. 
 
 [I. 9 c. 
 
BOOK I. 61 
 
 And for the same reason the sum of the angles ACD, CDB 
 is also a straight angle. 
 
 Therefore the angle BAC is equal to the angle CDB. 
 
 And similarly the angle ACD is equal to the angle DBA. 
 
 Therefore The opposite sides &c. Q. E. D. 
 
 Corollaries, (i) Hence if it is possible for a point to move 
 from one position to another by moving certain distances in two 
 given directions successively, these movements may be taken in 
 either order. 
 
 (ii) Therefore also, if it is possible for a point to move from 
 one position to another by moving certain distances in any 
 number of given directions, these movements may be taken in 
 any order luhatever. For any desired change of order may be 
 effected by repeatedly interchanging the order of two conse 
 cutive movements. We see therefore that the dependence or 
 independence of a given direction on certain other directions, 
 is an intrinsic relation between the directions, and has nothing 
 to do with the magnitude, or order of the transferences selected 
 to test it by. 
 
 (iii) If two adjacent sides of a parallelogram are equal, all 
 its sides are equal. Such a figure is called a rhombus. 
 
 (iv) If two adjacent angles of a parallelogram are equal, 
 all its angles are equal, and they are all right angles. For the 
 sum of two adjacent angles is a straight angle. Such a figure 
 is called a rectangle. A figure which combines both the 
 properties of a rhombus and a rectangle is called a square. 
 
BOOK II. 
 
 ON PLANES AND SPACE. 
 
 LEMMA. 
 
 If each one of any group of directions is dependent upon a 
 certain number of them, each of ivhich is independent of the 
 remainder in that number ; then each one of the whole group of 
 directions is dependent upon any similar number of independent 
 directions chosen from among the group. 
 
 Let the letters of the alphabet ab z denote the direc 
 tions in the group, and let a certain number of them, ab k 
 
 say, denote the certain number of directions, each of which is 
 independent of the remainder, upon which all the directions 
 are dependent. 
 
 The proposition will be proved by showing that for any 
 one, a, of the independent directions a k, we may substitute 
 any one, z, of the remainder, and each one of the whole group 
 will be dependent upon b k and z\ provided only that z is 
 independent of 6 k. 
 
 For by hypothesis, z is dependent upon a k. 
 
 Therefore it is possible to move a point from some position 
 
 B 
 
 A 
 
 A to another Z, by a straight line AZ ex tending in the direction 
 
BOOK II. 63 
 
 z, or by a succession of straight lines AB, BC, CD KZ in 
 
 the successive directions a, 6, c /,-. [D 5. 
 
 The distances to be moved along some of these lines may 
 perhaps be zero, but since the direction z is independent of the 
 directions b k, the distance along AB cannot vanish, for else 
 it would be possible to move a point from A to Z by straight 
 lines in directions b k only. 
 
 Therefore it is possible to move a point from A to B by a 
 straight line in the direction a, or else by straight lines AZ, 
 ZK DC, CB in directions z and k c, 6 ; respectively. 
 
 Therefore the direction a is dependent upon the directions 
 z and b k. And if x be any other direction whatever in the 
 group, then by hypothesis a movement in direction x may be 
 replaced by movements along straight lines in directions a k. 
 But we have already seen that a movement along a straight 
 line in direction a may be replaced by movements along 
 straight lines in the directions z and 6 k. 
 
 Therefore a movement in direction x may be replaced by 
 movements along straight lines in directions z and b k, for 
 the order of the movements is indifferent. [I. 23 c. (ii). 
 
 Therefore If each one &c. Q. E. D. 
 
 NOTE. We see from this proposition that directions may be 
 classified in groups, as follows : 
 
 (a) Groups dependent upon one independent direction. 
 Such a group contains only two directions, the given direc 
 tion, and the one opposite to it. But there are any number of 
 such groups (see A 3). 
 
 (b) Groups dependent upon two independent directions. 
 Such a group contains all the directions in which a straight 
 
 line extends while revolving to measure a straight angle. 
 There are also any number of such groups (see A 3). 
 
 (c) Groups dependent upon three independent directions. 
 Material space, as we ordinarily conceive it, consists of only 
 
 one such group, and it is with such a space that we are here 
 concerned (see A 3). 
 
 Such groups may, for brevity, be spoken of as groups of one, 
 two, or three independent directions. 
 
64 PART II. 
 
 Additional Axiom. 
 
 III. Space may be conceived as extending from every posi 
 tion in it in three and only three independent directions. 
 
 [Since (Ax. II.) a straight line may be conceived to extend 
 in any given direction from any given position, all directions 
 conceived by virtue of Axiom III. belong to one group of three 
 independent directions, and the directions in which space extends 
 are the same from every position in it.] 
 
 Further definitions. 
 
 16. A continuous series of positions extending from each 
 one of them in a complete group of one, two, or three indepen 
 dent directions, is called a spread! If the directions in which 
 it extends from every position in it are the same, it is called a 
 regular spread. 
 
 17. A spread of one independent direction is called a line, 
 and a regular spread of one independent direction a straight 
 line (see also D 4). If however the directions in which it 
 extends vary gradually from point to point in the line, it is 
 called a curved line. 
 
 18. A spread of two independent directions is called a 
 surface, a regular spread of two independent directions a plane 
 surface, or merely a plane. If however the directions, or some 
 of them, in which it extends vary gradually from point to point 
 in the surface, it is called a curved surface. 
 
 19. The spread of three independent directions called space 
 is a regular spread (see A 2, 3). Any limited portion of it is 
 called a geometrical solid. 
 
 20. A dimension is a measurement made upon a figure 
 with a view to determining its size. 
 
 21. Linear dimensions are measures of the extension of 
 portions of spreads of one independent direction, that is, of 
 lines. 
 
 22. Among linear dimensions, length, breadth, and thick 
 ness, are measured along three straight lines in independent 
 directions. Circumference is measured round a closed line 
 surrounding the figure, &c. 
 
BOOK II. 65 
 
 23. Superficial dimensions are measures of the extension 
 of portions of spreads of two independent directions, that is, 
 of surfaces. 
 
 24. Among superficial dimensions plane areas are measured 
 over planes, and curved areas over curved surfaces, &c 
 
 25. Volumetric dimensions or volumes are measures of the 
 extension of portions of spreads of three independent directions, 
 that is, of solids. Since by Axiom III. we have only conceived 
 one such spread, we need only conceive one kind of volume. 
 
 26. If two planes intersect in a straight line, the angle 
 between directions in each perpendicular to that straight line 
 is called the angle bettueen the planes. 
 
 27. A tetrahedron is a figure consisting of six straight lines 
 joining four points, which are not in one plane, two and two. 
 
 28. A parallelepiped is a figure consisting of twelve straight 
 lines forming the intersections of three pairs of parallel planes. 
 
 (The terms tetrahedron and parallelepiped are often also 
 used to denote the envelopes formed by the planes in which 
 these straight lines are, or the geometrical solids which they 
 enclose.) 
 
 29. Two positions "are said to be on opposite sides of a 
 point when they lie in opposite directions from it; in other 
 words, when the point is in the straight line joining the posi 
 tions, and between them ; 
 
 So two positions are said to be on opposite sides of a line or 
 surface if the straight line joining them intersects the line or 
 surface once between them ; 
 
 But if the line or surface intersect the straight line joining 
 the positions twice between them, each of the positions is on 
 the opposite side of the line or surface to a position in the 
 straight line between the two intersections, and they are there 
 fore on the same side of the line or surface as each other. 
 
 So generally, two positions are said to be on the same, or 
 opposite sides of a line or surface, according as the straight 
 line joining them intersects that line or surface an even or an 
 odd number of times, between them. 
 
 NOTE. 
 
 With reference to the definitions of the words parallel and perpen 
 dicular, the reader is requested to observe that they apply equally to 
 D. 5 
 
66 
 
 PART II. 
 
 planes, or to a plane and straight line. Thus one plane is parallel to 
 another if it extends in all the same directions as it. So a plane is parallel 
 to a straight line if it extends in its two directions ; but it would not be 
 accurate on this definition to speak of the straight line as parallel to a 
 plane. From the definition of perpendicular, it is not strictly accurate 
 to talk of anything but a direction as perpendicular to anything. But 
 since if two straight lines are at right angles, each of the directions of 
 either is perpendicular to the other, the straight lines may be called 
 perpendicular. So with a straight line and a plane. But it would be 
 wrong to speak of two planes which are at right angles, as perpendicular to 
 each other. 
 
 A plane is denoted by three letters denoting three points in it which 
 are not in a straight line (see n. 3). 
 
 PROPOSITION I. 
 
 A plane may be conceived through any given position extend 
 ing in any two given independent directions. 
 
 For if be a given position, 
 
 Then straight lines may be conceived through to points 
 A, B, in the given directions from 0. [A 2. 
 
 And a transverse straight line may be conceived joining 
 AB. [A .2. 
 
 And straight lines may be conceived from to every point 
 in AB, and also one through parallel to AB. [A 2. 
 
 These straight lines are all in directions dependent upon 
 the directions OA, OB, and their directions include all which 
 are so dependent. [I. 4. 
 
 And the directions form a continuous series. For clearly 
 those of the straight lines which intersect AB form a continuous 
 
BOOK II. 67 
 
 series, since the positions in AB form a continuous series. [D 4. 
 And the remaining directions, namely, the two in which AB 
 extends, form part of the continuous series, the straight lines 
 in which intersect any other transverse straight line AB not 
 parallel to AB. 
 
 Therefore a continuous series of positions has been conceived 
 extending from in a complete group of two independent 
 directions, dependent on the directions OA, OB. 
 
 Moreover, it extends from any other point P in it in the 
 same, and no other directions. 
 
 For, since the straight line OP is wholly in the series of 
 positions, it extends from P in the directions PO and OP. 
 
 And if it extends from in any other direction to any 
 point X, 
 
 Through P conceive a straight line PY parallel to OX, and 
 let Y be any point in it. Join Y. [A 2. 
 
 Then since it is possible to move a point from to Y by 
 the straight line OF, or else by the straight lines OP, PY, 
 
 The direction OF is dependent on the directions OP, PF, 
 
 [D5. 
 
 That is, on the directions OP, OX. 
 
 But each of these is dependent on the directions OA, OB. 
 
 Therefore the direction OF is dependent on the directions 
 OA and OB. [Lemma. 
 
 Therefore the point F is in the series of positions. 
 
 Similarly any point in PF is in the series of positions. 
 Therefore the series extends from P in the direction PF, that 
 is, the direction OX, and so in any direction in which the series 
 extends from 0. 
 
 And it extends in no other direction. 
 
 For if it extends in any direction PZ, and Z be any point in 
 this direction from P, in the series of positions, 
 
 Then the direction OZ is dependent on the directions OA, 
 OB, as also is the direction OP. 
 
 Therefore the direction PZ, which is dependent on the 
 directions OP, OZ, is dependent on the directions OA, OB. 
 
 [D 5, Lemma. 
 
 Therefore A plane may be conceived &c. Q. E. D. 
 
 Corollaries, (i) Hence an unterminated straight line may 
 be conceived to revolve in a plane about a faced point in itself, 
 
 52 
 
68 PART II. 
 
 describing an angle, and before it has completed a straight angle 
 it will have passed once, and only once, through every point in the 
 plane. 
 
 (ii) Hence also any number of straight lines through a 
 fixed point, in a plane, may be conceived to revolve similarly, and 
 if they revolve at the same rate their situations relative to each 
 other will remain unaltered. 
 
 (iii) And therefore any portion of a plane may be con 
 ceived to revolve in that plane, round any fixed point in it, and 
 may therefore be moved to any part of the plane without any 
 alteration of shape or size, so far as this can be accomplished by 
 translation and such rotation alone. 
 
 PROPOSITION II. 
 
 If a straight line intersect a plane which is parallel to it, or 
 if a straight line intersect any plane in two separate points, it is 
 wholly in the plane. 
 
 For if a straight line intersect a plane which is parallel to 
 it, a point moving from the intersection in either of the 
 directions in which the straight line extends would move along 
 the straight line, and also along the plane, for both extend in 
 these two directions. [D 10. 
 
 Therefore the straight line is wholly in the plane. 
 
 And if a straight line intersect a plane in two separate 
 points, both straight line and plane extend in the directions 
 from one of the points to the other, and the straight line 
 extends in no other directions. 
 
 Therefore the plane is parallel to the straight line and 
 therefore, as above, it wholly contains it. 
 
 Therefore // a straight line &c. Q. E. D. 
 
 PROPOSITION III. 
 
 One plane, and only one, can be conceived through 
 (i) any three points which are not in a straight line ; 
 
 (ii) any straight line, and a point without it ; 
 (iii) any tiuo straight lines which intersect one another ; 
 
 (iv) any two parallel straight lines. 
 For (i) since the three points are not in a straight line, 
 
BOOK II. 69 
 
 The directions from one of them to the other two are 
 independent directions [D 5], and therefore a plane may be 
 conceived through this point extending in those independent 
 directions, and it will contain the other two points. [II. 1. 
 
 And any plane containing the three points must also extend 
 from the first point in the directions of the other two, and 
 therefore be one and the same plane. 
 
 (ii) An}i two points being taken in the straight line, one 
 and only one plane can pass through these and the given point, 
 and as the straight line intersects this plane in two separate 
 points it is wholly in it. [II. 2. 
 
 (iii) Besides their point of intersection a point may be 
 selected in each of the intersecting straight lines, and one and 
 only one plane conceived through these three points (see above) 
 which will contain both the straight lines. [II. 2. 
 
 (iv) Any point being selected in one of the parallel straight 
 lines, one and only one plane may be conceived through this 
 point and the other straight line (see above), and since this 
 plane is parallel to the first straight line, and the straight line 
 intersects it, it is wholly in it. [II. 2. 
 
 Therefore One plane, and only one &c. Q. E. D. 
 
 PROPOSITION IV. 
 
 Two unterminated straight lines in a plane either intersect or 
 are parallel. 
 
 For if AB, CD be any two unterminated straight lines in a 
 plane, which are not parallel, 
 
 Conceive a straight line [A 2] joining any point A in one 
 and any point C in the other. 
 
 Then the plane extends from A only in the independent 
 directions AB, CD, and in directions dependent upon them. 
 
 [D18. 
 
70 PART II. 
 
 Therefore since it extends in the direction AC, this direction 
 is dependent upon the directions AB, CD. 
 
 Therefore since it is possible to conceive a point to move 
 from A to C along AG it is possible to conceive it to do so 
 along straight lines in the directions AB, CD. [D 5. 
 
 Therefore the straight lines AB and CD, which are in those 
 directions, must intersect each other. 
 
 Therefore Two unterminated &c. . Q. E. D. 
 
 PROPOSITION V. 
 
 Any straight line, to ivhich a plane is not parallel, intersects 
 that plane in a single point. 
 
 For if A B be any straight line, arid CDE a plane which is 
 not parallel to it, 
 
 Conceive a straight line joining any point A in AB to any 
 point C in CDE. 
 
 Then since the plane is not parallel to AB, the direction 
 
 AB is independent of the directions CD, CE in which the plane 
 extends. 
 
 Therefore the direction AC is not independent of the three 
 independent directions AB, CD, CE. [A 3. 
 
 Therefore, since it is possible to conceive a point to move 
 from A to C along AC, it is possible to conceive it to do so 
 along straight lines in the directions AB, CD, CE. [D 5. 
 
 The first of these motions is along the straight line AB, and 
 the others along straight lines in the plane CDE. 
 
 Therefore AB and CDE intersect, 
 
 And since the plane CDE is not parallel to AB, AB is not 
 wholly in it. [D 18. 
 
 Therefore it cannot intersect it in more than one point. [II. 2. 
 
 Therefore Any straight line &c. Q. E. D. 
 
BOOK II. 71 
 
 PROPOSITION VI. 
 
 Two parallel planes cannot intersect one another, 
 
 And any two planes which are not parallel intersect in a 
 straight line. 
 
 For if two planes which are parallel had a common point, 
 each would extend from that point in all the same directions, 
 and they would therefore be one and the same plane. [D 10. 
 
 But if they are not parallel, let A be a point in one of them, 
 AEG, which is not in the other, DEF. 
 
 Then the plane DEF cannot extend in both the independent 
 
 directions AB, AG, for it is not parallel to the plane ABG. 
 
 [D10. 
 
 Let it therefore not extend in the direction AB. 
 
 Therefore a straight line AB will intersect it in a single 
 point P. [II- 5. 
 
 And if the plane DEF extends in the direction AG, a 
 straight line through P in this direction is wholly in both 
 planes. [II- 2. 
 
 But if not, a straight line A C also intersects the plane GDE 
 in a single point Q [II. 5], which is not the same as P, since A 
 is the only point common to AB and AC [I. l],and A is not in 
 the plane GDE. 
 
 Therefore a straight line PQ intersects both planes in two 
 separate points, and is therefore in both of them. [II. 2. 
 
 And no point outside this straight line can be in both 
 planes, or they would be one and the same plane. [II. 3. 
 
 Therefore Two parallel planes &c. Q. E. D. 
 
 Corollary. Two planes cannot enclose a space. 
 
72 
 
 PART II. 
 
 PROPOSITION VII. 
 
 If a plane intersect two parallel planes it does so in two 
 parallel straight lines. 
 
 For if it intersects them at all, it does so in two straight 
 lines. [II 6. 
 
 And these straight lines being in one plane must either 
 intersect or be parallel. [II. 4. 
 
 But they cannot intersect, since they are in parallel planes, 
 which do not intersect. [II. 6. 
 
 Therefore they are parallel. 
 
 Therefore If a plane &c. Q. E. D. 
 
 PROPOSITION VIII. 
 
 If a direction be perpendicular to each of two intersecting 
 straight lines, it is perpendicular to a plane determined by them. 
 
 For if a direction be perpendicular to each of two straight 
 lines AO, BO which intersect in 0, 
 
 Through conceive a straight line COG in this direction 
 and let C, C , be any two points in it equidistant from 0. 
 
 And if OP be any other direction in which the plane extends 
 which is determined by OA, OB, 
 
 Through P a straight line may be conceived in the plane 
 
 which is not parallel to OP, OA t or OB [II. 1 c. (i)], and which 
 
BOOK II. 73 
 
 therefore does not pass through 0, but which does intersect OA , 
 and OB, in two separate points A and B. [II. 4. 
 
 Join the points C, G to the points J., B, and P. 
 
 Then since the direction OC is perpendicular to AO, the 
 direction .40 is also perpendicular to COG . 
 
 Therefore A is equidistant from the points C, (7 , which are 
 equidistant from 0, the foot of the perpendicular. . [I. 20. 
 
 Similarly it may be shown that B is equidistant from C, C . 
 
 Therefore in the triangles AGE , AG B the sides AC, (7Z[are 
 respectively equal to the sides AC , G B and the side AB is 
 common. 
 
 Therefore the triangles are congruent. [I. 14. 
 
 But if the triangle ACB were placed so as to coincide with 
 AG B, PC would_ coincide with_PC . 
 
 Therefore PC is equal to PC . 
 
 And in the triangles PCO, PC O the sides CO, C 7 ^ are also 
 equal, and the side PO is common. 
 
 Therefore the triangles are congruent [I. 14] and the angle 
 POC is equal to the angle POC , and therefore each of them is 
 a right angle. [I. 6. c. (i). 
 
 Therefore the inclination of OC to OP is the same as to OA 
 or OB. [I. 6. c. (ii). 
 
 Thus the direction OC is equally inclined to every direction 
 in which the plane determined by OA, OB extends. 
 
 Therefore the direction CO is perpendicular to the plane. 
 
 [D 11. 
 
 Therefore If a direction &c. Q. E. D. 
 
 PROPOSITION IX. 
 
 In a plane, through any given point in it, one and only one 
 straight line can be conceived in a direction perpendicular to 
 a given straight line in the plane. 
 
 If the given point is without the given straight line the 
 proposition has already been proved [1. 19]; for the perpendicular 
 from the point to the straight line is in the plane containing 
 them. 
 
 But if not, a straight line through the given point may be con 
 ceived to revolve in the given plane, starting from the directions 
 
74 PART II. 
 
 of the given straight line, and describing an angle. And before 
 completing a straight angle it will have passed through every 
 point in the plane, and therefore have extended in every direction 
 in which it extends. [II. 1. c. (i). 
 
 And during this revolution, when it has twisted through 
 a right angle, its directions will be equally inclined to the 
 opposite directions of the given straight line [I. 6. c. (ii)] ; that 
 is, they will be perpendicular to it once [D 11], and once only, 
 during the revolution. 
 
 Therefore In a plane &c. Q. E. D. 
 
 PROPOSITION X. 
 
 Through any given point one, and only one straight line may 
 be conceived in a direction perpendicular to a given plane. 
 
 For if be the given point and OA a straight line through 
 in any direction in which the given plane extends, 
 
 Through conceive a plane [II. 1] OAB parallel to the 
 given plane, and in it conceive the straight line OB in a 
 direction perpendicular to OA. [II. 9. 
 
 Conceive any point G in a direction from independent 
 of the directions OA, OB. [A 3. 
 
 Through 00, OB conceive a plane, and in this plane con 
 ceive the straight line OD perpendicular to OB. [II. 3. 
 
 Then since the planes OB A, OBG, are not parallel, they 
 intersect only in the straight line OB. [II. 6. 
 
 Therefore the straight lines OD, OA are in independent 
 directions. 
 
 Through OA, OD therefore conceive a plane [II. 3], and in 
 this plane conceive the straight line OE, in a direction perpen 
 dicular to OA. [II. 9. 
 
BOOK II. 
 
 75 
 
 Then since the direction OB is perpendicular to the straight 
 lines OA and OD, it is perpendicular to the plane AOD, and 
 therefore to the straight line OE in this plane. [II. 8. 
 
 Therefore OE is perpendicular to OB, and it is so also to 
 OA, and therefore to the plane AOB, and to the given plane to 
 which AOB is parallel. [II. 8. 
 
 And if any other straight line OF through could be in a 
 direction perpendicular to the given plane, or to AOB, 
 
 A plane could be conceived through OE and OF [II. 3], 
 which would not be parallel to the plane AOB, and would 
 therefore intersect it in a straight line [II. 6], to which the 
 directions OE, OP would be perpendicular. [II. 8. 
 
 But the straight lines OE, OF are in one plane with this 
 intersection, and therefore cannot both be in directions perpen 
 dicular to it. [II. 9. 
 
 Therefore Through a given point &c. Q. E. D. 
 
 PROPOSITION XI. 
 
 Through any given point one, and only one plane may be con 
 ceived to which a given direction is perpendicular. 
 
 For if be the given point and OA a straight line through 
 in the given direction, 
 
 Through conceive two more straight lines, [A 2], such 
 
 that the directions OA, OB } 00 are independent, [A 3. 
 
 Through OA, OB, and OA, 00 conceive two planes [II. 1], 
 
 and in them conceive the straight lines OD, OE in directions 
 
 perpendicular to OA. [II. 9. 
 
76 PART II. 
 
 Since the planes AOB, A 00 are not parallel, they only 
 intersect in the straight line OA. [II. 6. 
 
 Therefore the directions OD, OE are independent. Conceive 
 a plane through OD, OE. [II. 3. 
 
 Then OA is perpendicular to both OD and OE, and there 
 fore to the plane DOE. [II. 8. 
 
 And if it were perpendicular to any other plane through 0, 
 and F were any point in this plane and not in DOE, 
 
 Conceive a plane through OA, OF [II. 3]. Then this plane 
 would not be parallel to the plane DOE, and therefore it would 
 intersect it in a straight line OG. [II. 6. 
 
 And OA would be perpendicular to OF and OG. 
 
 But OF and OG are in the same plane as OA, and therefore 
 could not both be perpendicular to it. [II. 9. 
 
 Therefore Through any given point &c. Q. E. D. 
 
 PROPOSITION XII. 
 
 The shortest path to a plane from a point without it is a 
 straight line in a direction perpendicular to the plane. 
 
 And of all other straight lines to it from the given point, 
 those which meet it at equal distances from the foot of the per 
 pendicular are equal, and those which meet it nearer the foot of 
 the perpendicular are shorter than those which meet it farther off. 
 
 For if be any point outside a plane ABC, 
 
 Conceive a straight line OP [II. 11] in a direction perpen 
 dicular to ABC. Then since ABC is not parallel to OP it 
 must intersect it in a single point P. [II. 5. 
 
 If OA be any other straight line to A in the plane, from 0, 
 
 O 
 
 then AP is in the plane [II. 2], and therefore OP is perpen 
 dicular to PA. [D 11 
 
BOOK II. 
 
 77 
 
 Therefore OP is less than OA. [I. 20. 
 
 Thus OP is the shortest straight line, and a fortiori the 
 shortest path from to the plane. 
 
 And if B be any other point in the plane, at the same 
 distance from P as A is, 
 
 Then the_ sides AP, PO of the triangle APO are equal to 
 the sides BP, PO of the triangle BPO respectively, and the 
 angle APO is equal to the angle BPO, since OP is perpen 
 dicular to the plane APB. 
 
 Therefore the triangles are congruent, and OA is equal to 
 OB. [I. 10. 
 
 And if C be any point in the plane nearer to P than A, 
 
 Some point D in AP is at the same distance from P as C is, 
 and is therefore nearer than A is. 
 
 Therefore the straight line OD is equal to OC (see above), 
 and less than OA. [I. 20. 
 
 Therefore also OC is less than OA. 
 
 Therefore The shortest path &c. Q. E. D. 
 
 PROPOSITION XIII. 
 
 The perpendicular distance of any point in a straight line 
 or plane from a plane which is parallel to it, is constant. 
 
 For if A t B be any two points in a straight line or plane to 
 which a plane CDE is parallel, 
 
 Conceive a straight line AC in a direction perpendicular to 
 the plane CDE. [II. 11. 
 
78 
 
 PART II. 
 
 Then as CDE is not parallel to AC, it will intersect it in a 
 single point C. [II. 5. 
 
 Through C conceive a straight line CD equal and parallel 
 to AB. [A 2. 
 
 Then since the plane CDE is parallel to AB, D will be in 
 this plane. 
 
 Join BD. Then BD is also equal and parallel to AC. 
 
 [I. 21. 
 
 Therefore it is the perpendicular from B to the plane CDE. 
 
 [II. 10. 
 
 And it is equal to AC, the perpendicular from A. 
 
 Therefore The perpendicular distance &c. Q. E. D. 
 
 PROPOSITION XIV. 
 
 An unterminated plane may be conceived to revolve round any 
 fixed straight line in it, describing an angle; and before com 
 pleting a straight angle it will have passed once and only once 
 through every point in space. 
 
 For if AB be any fixed straight line in a plane and any 
 point in it, 
 
 Conceive a plane COD through 0, to which AB is perpen 
 dicular. [II. 11. 
 
 Then a straight line XO Y through may be conceived to 
 revolve in this plane, describing an angle. [II. 1 c. 
 
 And in every position of it a plane may be conceived 
 through it and AB. [II. 3. 
 
 And since as XOY revolves in the plane COD, XOY is 
 
 always in a direction perpendicular to AB, the intersection of 
 any two situations of the plane. 
 
 Therefore the plane XOA in revolving describes an angle, 
 
BOOK II. 79 
 
 [D 26], namely the same as that described by XOY in the 
 plane COD. 
 
 And if P be any point in space, 
 
 Through P conceive a plane PO Q parallel to the plane 
 COD. [II. i. 
 
 As this is not parallel to AB, it will intersect it in a single 
 point . [II. 5. 
 
 And the plane XOA will intersect the parallel planes COD, 
 PO Q in parallel straight lines XOY, X O Y. [II. 7. 
 
 Hence as the plane XOA revolves describing an angle, 
 Z ^F^will revolve in the plane PO Q describing the same 
 angle. 
 
 But as X O Y revolves, before completing a straight angle 
 it will pass once and only once through every point in the 
 plane, and therefore through P. [II. 1 c . (i). 
 
 Hence as the plane XOA revolves, before completing a 
 straight angle it will pass once through P. 
 
 And it cannot pass through P except when X O Y does, for 
 if it did, it would coincide with the plane PO Q [II. 3] which it 
 does not, since it contains AB, which is not wholly in that 
 plane, but perpendicular to it. 
 
 Therefore An unterminated plane &c. Q. E. D . 
 
 Corollaries, (i) Hence also any number of planes through 
 a fixed straight line may be conceived to revolve similarly, and if 
 they revolve at the same rate their situations relative to one 
 another will remain unaltered. 
 
 (ii) And consequently any geometrical solid may be conceived 
 to revolve in space round any fixed straight line in space, and 
 may therefore be moved from any one part of space to any other, 
 without alteration of shape or size, so far as this can be accom 
 plished by translation and such rotation alone. 
 
 (iii) Hence any plane figure may be conceived to be moved 
 into any given plane, so that any point in it shall fall upon a 
 given point in that plane, any straight line through that point, on 
 a given straight line through the given point, and any other point 
 in it on either side of the given straight line in the given plane. 
 For any point in the figure may be moved to the given point 
 by a translation, that is, without altering the direction from 
 any one point to any other in the figure. And then the plane 
 of the figure and the given plane will intersect in a straight line, 
 
80 PART IT. 
 
 [II. 6], by a rotation round which the plane figure may be brought 
 into the given plane. The figure may now be rotated in the 
 given plane [II. 1 c. (iii)] round the given point till any straight 
 line through it coincides with the given straight line. And 
 lastly, if any other point in the figure is on the wrong side of 
 the given straight line, it may be brought on to the right side 
 by revolving the plane figure through a straight angle round 
 the given straight line. 
 
 This corollary shows that none of the movements of figures 
 we have conceived in the past propositions have been incon 
 sistent with axioms II. and III. 
 
 PKOPOSITION XV. 
 
 It is impossible to pass from one side of an unterminated 
 surface in space to the other, without passing through the surface. 
 
 For if 0, P, be any two points on opposite sides of an 
 unterminated surface in space, 
 
 Then a straight line OP intersects the surface an odd 
 number of times between and P. [D 29. 
 
 The proposition will be proved by showing that however P 
 may move in space, if it does not pass through the surface, the 
 number of intersections between and P will always remain 
 odd, and can therefore never vanish. 
 
 For in whatever direction P may move, unless in the 
 direction of or the opposite direction, OP will commence to 
 generate a plane. [II. 1. 
 
 Therefore however P moves, except in the direction of or 
 the opposite, OP will describe a spread which from every point 
 in it extends in the same directions as a plane, that is, a surface. 
 
 And if P only moves in the direction of and the opposite, 
 the number of intersections between and P will remain 
 unaltered, unless P passes through one of them, that is, through 
 the given surface. 
 
 Therefore as P moves OP will generate a surface bounded 
 by the path of P, the point 0, and two situations of the straight 
 line OP. 
 
 And from any of the intersections of OP with the given 
 surface, this surface and that described by OP extend in all the 
 directions in which two planes would ; 
 
BOOK IL 81 
 
 Their intersection therefore extends in all the directions in 
 which the intersection of two planes would ; 
 
 That is, in two opposite directions. [II. 6. 
 
 Therefore the intersection of the two surfaces consists of one 
 or more lines which are not terminated, unless by the boundaries 
 of the surface generated by OP. 
 
 Hence if A be one of the intersections between and P, as 
 OP revolves A will describe a continuous line on the surface 
 generated by OP, and the intersection A can only vanish if at 
 some point B this line bends back as OP moves forward, 
 
 And in this case, before reaching B, OP must have inter 
 
 sected the continuous line traced by A in a second point A , 
 which therefore vanishes together with A as OP passes the 
 point B. 
 
 And the line described by A cannot be terminated in the 
 path described by P, unless P passes through the given surface, 
 nor can it be terminated at 0, since is not in the given 
 surface. 
 
 Thus as OP revolves the intersections can only appear or 
 disappear two at a time. 
 
 Therefore It is impossible &c. Q. E. D. 
 
 Corollary. In the same way it may be shown that in a plane 
 it is impossible to pass from one side to the other of an untermi- 
 nated line without passing through the line. 
 
 NOTE. In this proposition contacts of the second order must be 
 counted double intersections, of the third order triple, &c. Also if the 
 given surface intersects or touches itself it must be counted double at such 
 
82 PART II. 
 
 points, &c. It is not necessary to discuss these special cases here, but 
 they might perhaps be a reason for omitting this proposition from an 
 elementary text-book. 
 
 PROPOSITION XVI. 
 
 // two tetrahedra have their six corresponding sides respec 
 tively equal, they shall be equal in every respect, that is to say, 
 the angles between the directions of corresponding sides shall be 
 equal, and the angles between corresponding plane faces shall 
 also be equal. 
 
 And if their corners are similarly situated with respect to 
 each other in space, they shall be congruent to each other. 
 
 For since the corresponding sides of the tetrahedra are 
 respectively equal, they consist of four pairs of corresponding 
 triangles, whose three sides are respectively equal. 
 
 Therefore these corresponding triangles are congruent to 
 each other. [I- 14. 
 
 Therefore also the angles between the directions of corre 
 sponding sides which meet at a corner are equal, for they are 
 angles of corresponding triangles. 
 
 Conceive one of the tetrahedra to be moved and placed so 
 that one of its triangles coincides with the corresponding 
 triangle of the other, which is congruent with it. [A. 1. 
 
 Let ABC be this triangle, and D, E the remaining corners 
 of the two tetrahedra. 
 
 Now since the triangles ADB, AEB are congruent (see 
 above), 
 
 If they were applied to each other so as to coincide, one 
 
BOOK II. 83 
 
 perpendicular [I. 19] could be conceived from their common 
 corner D, or E, to the side AB, meeting it in one point P. 
 
 Therefore even though D, E do not coincide, the perpen 
 diculars from them on AB meet it in the same point P. 
 
 Similarly, conceive perpendiculars from D and E on AC 
 meeting it in one point Q. 
 
 And in the plane ABC conceive straight lines through P 
 and Q in directions perpendicular to AB, AC respectively. 
 
 [II. 9. 
 
 These straight lines are not parallel, for if they were, in the 
 plane ABC two straight lines AB, AC through A would be in 
 directions perpendicular to the same straight line, which is 
 impossible. 
 
 Therefore they intersect in some point 0. [II. 5. 
 
 Join DO, EO, 
 
 Then as the direction AB is perpendicular to both PD and 
 PO it is perpendicular to a plane containing them [II. 8], and 
 therefore to DO in this plane. 
 
 Similarly the direction AC is perpendicular to DO. 
 
 Therefore the direction DO is perpendicular to both AB and 
 AC, and therefore to the plane ABC. [II. 8. 
 
 Similarly EO is perpendicular to the plane ABC. 
 
 But only one straight line may be conceived through a given 
 point, in a direction perpendicular to a given plane. [II. 10. 
 
 Therefore DOE is one straight line. 
 
 And since the triangles DAB, EAB are congruent, 
 
 (see above) 
 
 The perpendiculars DP and EP are equal. 
 
 And the side PO is common to the .two triangles DPO, 
 EPO. 
 
 Also the angles DOP, EOP opposite the equal sides DP, 
 EP are equal, and moreover they are right angles. 
 
 Therefore the triangles are congruent [I. 15. c (ii)], and 
 therefore the angles DPO, EPO are equal. 
 
 But these angles are the angles between the planes DBA, 
 EBA, since PD, PO, PE are perpendicular to AB. [D 26. 
 
 So it may be shown that the angles between any two 
 corresponding planes are equal. 
 
 Through A conceive a straight line AR parallel to BC, to 
 any point R Join DR, ER, OR. 
 
 6-2 
 
84 PART II. 
 
 Then since DOE is perpendicular to the plane ABC (see 
 above), and OR is in this plane, it is perpendicular to OR. 
 
 Therefore RO is the perpendicular from R on DOE. 
 
 But it was shown above that the triangles DPO, EPO are 
 congruent. Hence D and E are equidistant from 0. 
 
 Therefore RD is equal to RE. [I. 20. 
 
 And also AD is equal to AE, and AR is common to the 
 triangles DAR, EAR. 
 
 Hence the triangles are congruent [I. 14], and therefore the 
 angle DAR is equal to the angle EAR. 
 
 Therefore the angles between the directions of corresponding 
 sides of the tetrahedra which do not meet at a corner, as the 
 sides BC, AD and BO, AE, are also equal. 
 
 And since it has been shown above that DO is equal to 
 EO, and is in the same straight line with it, 
 
 If the corners of the tetrahedra are similarly situated with 
 respect to each other, D and E will be on the same side of 0, 
 and will therefore be one and the same point. 
 
 Therefore the tetrahedra are congruent. 
 
 Therefore If two tetrahedra &c. Q. E. D. 
 
 Corollaries, (i) Hence if two tetrahedra have three sides 
 meeting at one corner of the one respectively equal to the corre 
 sponding three of the other, and have the three corresponding 
 angles between their directions respectively equal, they shall be 
 equal in every respect. For the third sides of each of the three 
 triangles meeting at the corner, will be respectively equal 
 also. [I. 10. 
 
 (ii) Or if they have the six angles of each, between the 
 corresponding sides which meet at two corners, respectively equal, 
 and the sides between those corners equal, they shall be equal in 
 every respect For the other four sides meeting at those two 
 corners are respectively equal [I. 11], and therefore the tetra 
 hedra are equal in every respect. [c. (i) above, 
 (iii) Hence a direction may be determined with reference to 
 two given independent directions by giving its respective inclina 
 tions to them, if it is also known which side it is of a plane 
 determined by those given directions. 
 
BOOK II. 85 
 
 PROPOSITION XVII. 
 
 The four diagonals of a parallelepiped, that is, the four 
 straight lines joining its opposite corners, intersect in a single 
 point, which bisects each of them. 
 
 For a parallelepiped consists of the intersections of three 
 pairs of parallel planes. [D 28. 
 
 And each pair intersects each of the other planes in a pair 
 of parallel straight lines. [II. 7. 
 
 Hence the parallelepiped consists of twelve straight lines, 
 four of which extend in each of three independent directions, 
 
 Or of twelve straight lines joining eight points, two and two, 
 three meeting at each point and extending from it in three 
 independent directions. 
 
 Hence if AB, CD, EF, GH be four parallel sides of a 
 
 parallelepiped, AE, EF, CG, DH a second four, and AC, BD, 
 EG, FH the remainder, 
 
 ABDC is a parallelogram. [D 13. 
 
 Therefore AB is equal to CD. [I. 23. 
 
 Similarly CD is equal to GH. 
 
 Therefore AB, GH are two equal and parallel straight lines. 
 
 Therefore AH and BG bisect one another, in say. [I. 21. 
 
 Similarly AC, FH are equal and parallel straight lines. 
 
 Therefore AH and CF bisect one another, and they there 
 fore do so in also. 
 
 Similarly ED passes through and is bisected by it. 
 
 Therefore The four diagonals &c. Q. E. D. 
 
86 PART II. 
 
 TABLE OF THEOREMS WITH THEIR EQUIVALENTS IN EUCLID. 
 
 Proposition 
 
 Euclid s equivalent 
 
 Proposition 
 
 Euclid s 
 equivalent 
 
 I. 1 
 
 I. D 35 and A 10 
 
 1.21 
 
 I. 33 
 
 2 
 
 none 
 
 22 
 
 none 
 
 3 
 
 none 
 
 23 
 
 I. 34 
 
 4 
 
 none 
 
 II. 1 
 
 none 
 
 5 
 
 none 
 
 2 
 
 I. D7 
 
 6 
 
 I. 14 and c. A 11 
 
 3 
 
 XL 2 (?) 
 
 7 
 
 I. 15 
 
 4 
 
 I. D35 
 
 8 
 
 I. 32 and c. I. 16, 17 
 
 5 
 
 none 
 
 9 
 
 I. 29 and c. I. 27, 28 
 
 6 
 
 XI. D 8, P. 3 
 
 10 
 
 I. 4 and c. I. 5 
 
 7 
 
 XI. 16 
 
 11 
 
 I. 26 and c. I. 6 
 
 8 
 
 XL 4 
 
 12 
 
 I. 18, 19 
 
 9 
 
 I. 11 
 
 13 
 
 I. 24, 25 
 
 10 
 
 XL 6, 11, 12 
 
 14 
 
 1.8 
 
 11 
 
 XL 5 
 
 15 
 
 (analogous to VI. 7) 
 
 12 
 
 none 
 
 16 
 
 I. 20 
 
 13 
 
 none 
 
 17 
 
 XI. 20 
 
 14 
 
 none 
 
 18 
 
 XI. 21 
 
 15 
 
 none 
 
 19 
 
 I. 19 
 
 16 
 
 none 
 
 20 
 
 none 
 
 17 
 
 none 
 
 Proofs are given of all Euclid s theorems (and such of his 
 problems as are of theoretical importance,) in his first and 
 eleventh books, with the exception of those in his first book 
 which deal with plane areas, and of one in his eleventh book 
 which treats of proportion, and of one other theorem ; this one 
 is his I. 21, a theorem of no importance. It may however be 
 easily deduced, after my I. 16, if an equivalent condition is 
 substituted for the implied one that the point inside the 
 triangle is in the same plane as the triangle, or as it stands 
 after my II. 3. 
 
PART III. 
 
 ON THE APPLICABILITY OF THE FOREGOING 
 
 SUBJECTIVE GEOMETRY TO THE GEOMETRY 
 
 OF MATERIAL SPACE. 
 
 CHAPTER I. 
 
 HAVING thus established the foundations of subjective 
 geometry of three independent directions, we may now confront 
 the question : Is there a corresponding objective geometry ; 
 and if so, does it follow the same laws ? 
 
 The question whether there is or is not an objective world 
 at all, is a metaphysical one of the abstrusest character, and 
 involves the discussion of the logical basis of the inductive 
 method in its most abstract form. We need hardly, however, 
 attack this question directly here, as no practical man will 
 doubt that the outer world has a real objective existence. But 
 it may be useful to consider briefly how we attained to our 
 knowledge of its objectivity. 
 
 I think it will be generally agreed that the lower forms 
 of animal, and possibly the higher forms of vegetable life, 
 though they can hardly have formed the conception of 
 an objective external world, do feel certain subjective im 
 pressions of pleasure and pain, of heat and cold, and so on. 
 At what point in the scale of existence automatic and reflex 
 actions are divided from voluntary ones, and whether even 
 voluntary actions necessarily imply a consciousness of objective 
 existence, is open to dispute. But those, at any rate, who 
 believe in Evolution can scarcely doubt that, after a certain 
 period of development with only a subjective consciousness, 
 after an epoch of many generations of life with experiences 
 only recognised as subjective, the perception that these were 
 
88 PART III. 
 
 due to an objective environment must have sprung into being. 
 And as such a perception must have been of great advantage 
 to the percipient, it would only require natural selection to 
 confirm and extend it. Thus we might have, and probably 
 have, an instinctive consciousness of the objectivity of our 
 environment, without this consciousness constituting a direct 
 apprehension of an objective fact; and this consciousness there 
 fore does not add anything to our reasons for believing that 
 our environment is objective. Such a consciousness might 
 indeed seem to Kant to constitute an apodictic truth, but it 
 does not constitute a necessary one. 
 
 Granted then that there is an objective universe, is its 
 geometry identical with the subjective geometry we have been 
 considering ? The question we have to answer is merely this : 
 Are the axioms of that subjective geometry true in the material 
 universe ? 
 
 The first axiom, that any geometrical figure may be con 
 ceived to be moved from any one part of space to any other, 
 except in so far as it is restricted by the other axioms, without 
 its shape or size being altered, is subjectively a truism ; but 
 considered objectively it raises several difficult points. We 
 may indeed take it to be sufficiently established by induction 
 that a material body may be moved to any part of space ; and 
 further we have doubtless sufficient ground for believing that 
 any forces required to produce such motions are accounted for 
 otherwise than by any change of size or shape which the body 
 is required to undergo by the nature of space, on being moved. 
 But we have no a priori reason for assuming that such a change 
 of size or shape would require force to produce it, and to discuss 
 whether it is produced or not we are thrown back upon first 
 principles. 
 
 Size and shape depend on distances and inclinations respec 
 tively, which in our subjective geometry were measured by 
 amounts of transference and twisting. In virtue of the first 
 axiom we were enabled to substitute direct comparison by the 
 method of superposition for these methods of measurement. 
 But if the axiom is not objectively true, presumably the size 
 and shape of the figure applied would alter, on being moved, 
 in the same way as that to which it was applied, and so the 
 method of superposition would not prove that two congruent 
 
CHAPTER I. 89 
 
 figures were of the same size or shape when in different parts 
 of space. If we attempt to define amount of transference or 
 of twisting strictly we can only do so by the aid of Newton s 
 first law of motion, which would introduce the conception of 
 time. For just as I have already shown that the words in a 
 straight line in his law can only be construed as meaning what 
 I mean by the phrase in a constant direction, so the words 
 uniform motion mean nothing but motion with a constant 
 amount of transference in any given portion of time. And the 
 only measure we have of time is the motion of a body not 
 acted on by any forces (the rotation of the earth is the ordinary 
 standard, and very closely approximates to an ideal one). 
 Thus we should seem to be defining in a vicious circle; and 
 though this is not quite the case, for the law of motion implies 
 a consistency between the measures of distance or inclination, 
 and time, deduced from it, yet the criterion it affords is difficult 
 to apply as a practical test. However, since from this law of 
 motion the most remote results have been deduced, and have 
 been found to correspond with objective facts, its truth, at 
 least in so far as it asserts that a body not acted on by any 
 effective forces moves a constant distance in a constant time, 
 may fairly be assumed. And so the first axiom of subjective 
 geometry may be translated into the first objective fact of the 
 geometry of material space. 
 
 It will be convenient to consider the third axiom next, and 
 to discuss whether material space is a spread of three inde 
 pendent directions, that is, whether it does extend from every 
 position in it in a complete group of three independent direc 
 tions. That it does do so is established by a pure induction. 
 The objectivity of my arm being granted, I know I could 
 extend it in three independent directions, and in any direction 
 dependent upon them, from my present position in space, or 
 from any position I have ever occupied (where I was not 
 obstructed by material objects). And by induction I infer that 
 the rest of space is like those parts of it with which I am 
 acquainted. Other instances and facts might be adduced 
 proving the same thing ; but the logical proof must be of the 
 same nature, namely, a generalisation from a limited number 
 of known instances, to an unlimited number of unknown 
 ones, and though of course it is amply good enough for 
 
90 PART III. 
 
 all practical purposes, it establishes only an objective fact, 
 not a necessary truth. It applies with equal force to prove 
 that material space extends from no position in it in less than 
 three independent directions, or from no position in more. 
 Thus we have established the second objective fact on which 
 to base the geometry of material space. 
 
 I have already pointed out in Part I that the remaining 
 axiom may be paraphrased into a form of which the objective 
 counterpart would be : Material space extends from every 
 position in it in the same directions. Now we have already 
 granted that from every position in it it extends in three 
 independent directions, and in all directions dependent upon 
 them. Therefore if from any position B in it it extended in 
 any direction in which it does not extend from a position A, 
 this direction could not be dependent upon the three indepen 
 dent directions in which material space extends from A. And 
 we should therefore have four independent directions to deal 
 with. 
 
 Now I am aware that there exists, not only among the 
 ordinary public, but even more among geometricians, a rooted 
 prejudice against what they would call Geometry of Four 
 Dimensions, but which I prefer to call geometry of four inde 
 pendent directions. If therefore any of my readers is of opinion 
 that A Fourth Dimension or a Fourth independent direction 
 is a priori inconceivable, and an impossibility ; then I have 
 no more to say. He has ipso facto granted the objective truth 
 of my remaining axiom, and consequently admitted that my 
 subjective geometry may also be applied objectively to the 
 geometry of material space which is all I desire to prove. 
 The remainder of this book is not for him : if he reads it, 
 and thinks it transcendental folly, let him at least not presume 
 to criticise what, by his own confession, he does not understand. 
 
CHAPTER II. 
 
 I HAVE already had occasion to point out that, if \ve are 
 careful to reason with formal accuracy, it is not essential that 
 we should be able to picture to ourselves clearly every link in 
 the deductive chain, and that intelligible results may be truly 
 deduced by the aid of symbols whose denotation is unintelli 
 gible to us. It is therefore no argument against the theory of 
 geometry of four independent directions which I am about 
 to advance, to say that a fourth independent direction is 
 inconceivable. Personally, I believe it to be by no means 
 inconceivable, and I shall subsequently give my reasons for 
 this belief. But lest the reader should imagine that the 
 validity of my investigation into the foundations of the science 
 of geometry in any way depends upon this conceivability, I put 
 the investigation, or the essential part of it, first, and the ex 
 planation afterwards. 
 
 If such an investigation has never been made before, it is 
 simply because the foundations of geometry, especially those of 
 what is generally spoken of as geometry of three dimensions 
 have never before been laid down with formal accuracy. We 
 have already seen that even in his first book Euclid assumes 
 several things which are not stated in his postulates or axioms. 
 But in his eleventh book he seems to abandon all attempts at 
 laying down his premises with formal accuracy, and simply 
 appeals to the supposed intuitive knowledge of his reader. 
 The first three propositions of his eleventh book are most 
 extraordinary examples of such slipshod reasoning, and what is 
 more extraordinary still, is that scarcely any effort has been 
 made by more modern geometricians to improve them ! The 
 existence, even, of the first proposition is only due to a strange 
 confusion of ideas. Up to this Euclid has been drawing straight 
 
92 PART IIL 
 
 lines freely without ever considering whether they might or 
 might not have loose ends hanging out of the plane in which 
 he tacitly assumes all his former figures to have been drawn. 
 It is only now dawning on him that the existence of this plane, 
 which is really nothing but the pedagogue s black-board, is a 
 matter which he ought not so airily to have taken for granted. 
 Again, the second proposition contains simply no formal reason 
 ing whatever, but is merely an appeal to the experience or 
 intuitive knowledge of the reader. Take the second paragraph l 
 verbatim, thus " Let any plane pass through the straight line 
 
 EB " Where is the postulate by which it is allowed to 
 
 conceive, or draw, any plane through a given straight line ? 
 
 " and let the plane be turned round EB " How do we 
 
 know that a plane can be turned round a fixed straight line ? 
 
 " produced if necessary " (i.e. the plane, not the straight 
 
 line). But there is no postulate about producing planes " 
 
 until it pass through the point C." Now in this last respect 
 alone, as far as I know, has it occurred to any one materially to 
 improve this proposition. In a recent work 2 an axiom is 
 specially interpolated to cover this difficulty. The axiom in 
 question I prove in a theorem (II. 14), namely, that before a 
 plane revolves through a straight angle round a fixed straight 
 line in it, it will pass through every point in space. Again, in 
 the next proposition, what right has Euclid to assume, as he 
 does, that if two planes cut one another their common section 
 must be a line, at all ? Why should it not be a single point ? in 
 which case his proof would break down. It is the adroit manner 
 in which Euclid begs this question which saves him from having 
 to lay down anything corresponding to my third axiom. 
 
 The geometry of three dimensions, as usually put forward, 
 is not then a formal deductive science, though apparently it 
 satisfies the wants of most men. But what I maintain about 
 my geometry of three independent directions is that it is 
 strictly formal, the truth of every proposition in it depends 
 solely on the truth of the stated premises. And if one of those 
 premises were altered we could see at once which of the propo 
 sitions would have to be altered, and we should probably be 
 able to determine the alteration required. If the third axiom 
 
 1 Todhunter s Euclid, Macmillan, 1875. 
 
 2 Geometry in Space, by Mr K. C. J. Nixon. 
 
CHAPTER II. 93 
 
 is to be altered, it is at once evident that no change is produced 
 in the first book, as the axiom was not even stated till the 
 beginning of the second. And it is easy to trace which of the 
 propositions in the second book depend upon it. 
 
 They are namely these : V., that part of VI. which asserts that any 
 two planes which are not parallel intersect in a straight line, VII., for 
 without the axiom the intersections might only be single points, X. and 
 XL, when they assert that only one plane or straight line may be conceived, 
 XIV., when it asserts that the revolving plane passes through every point 
 in space, XV. ; and from XVI. we may remove the restriction that the 
 corners of the tetrahedra must be similarly situated in space, in order 
 that they may be congruent. 
 
 Now let us substitute for the third axiom the following 
 postulate, 
 
 Let it be granted that the positions of points may vary 
 in four, but not in more than four independent directions. 
 
 And let us see what happens to the above seven proposi 
 tions. Thus it appears that we are no longer able to assert in 
 proposition V. that the direction AC cannot be independent of 
 the three independent directions AB, CD, CE. But if AB 
 intersects the plane CDE, it will be so dependent. Therefore 
 AB does not necessarily intersect the plane. It may pass from 
 one side to the other without intersecting it at all. That is 
 to say, the positions of points which appear to us the same, 
 even though picturing to ourselves a three dimension model, 
 instead of a diagram on paper, may in reality be different ; just 
 as when looking at an ordinary diagram representing a three 
 dimension figure, positions appear the same in it which may 
 yet be different in reality. Thus within the bounding surface 
 of a solid of three dimensions we must allow for an infinite 
 number of positions which do not form part of that solid, but 
 lie away from it in a fourth independent direction. Therefore 
 the whole of the three dimension space we are in the habit of 
 considering is only an infinitesimal portion of the space of four 
 independent directions assumed by the postulate, just as a plane 
 is only an infinitesimal portion of a space of three independent 
 directions. 
 
 Let us therefore invent a special name for such limited 
 portions of the new space, and lay down the following new 
 definition. 
 
94 PART III. 
 
 A spread of three independent directions is called a form, a 
 regular spread of three independent directions a regular form. 
 
 The spread of four independent directions, the new space, is 
 by the second axiom a regular spread, for this axiom is, of 
 course, still to hold good. 
 
 Thus the space we have hitherto been considering is a 
 regular form, and to make the seven propositions enumerated 
 above still hold true we have only to prefix the words, " In a 
 regular form " to their enunciations, just as the enunciations of 
 several other propositions in the same book commence with the 
 words, " In a plane." 
 
 It will not be necessary to give all the elementary proposi 
 tions of geometry of four independent directions in extenso as I 
 have done those of three. For the logical processes by which 
 they are established are in most cases obvious extensions of 
 those used in Book II., and the reader can if necessary readily 
 enough supply them himself. But in the subjoined syllabus of 
 enunciations one or two of the most important proofs have been 
 sketched out. 
 
 SYLLABUS OF ENUNCIATIONS FOR BOOK III. 
 
 [In this syllabus the letters r. f. mean regular form, p. plane, s. 1. 
 straight line, i. d. independent direction. P. postulate]. 
 
 I. One, and only one, r. f. can be conceived through any given 
 point, extending in any three given i. d s. 
 
 This is only another way of enunciating II. 14. 
 
 II. (i) If a p. intersect a r.f. in three points not in a s. I.; or 
 (ii) if a s. I. intersect it in two separate points ; or 
 
 (iii) if either a p. or a s. L to which it is parallel inter 
 sect it at all, then they shall be wholly in it. 
 
 III. One and only one r.f. can be conceived through 
 (i) four points not in one p., 
 
 (ii) a p. and a point without it, 
 (iii) two p s. which intersect in as. L, and so on. 
 
 IV. Any s. I. to which a r. f. is not parallel intersects it in 
 a single point. 
 
CHAPTER II. 
 
 95 
 
 For if A be any point in a s. 1. A B to which a r. f. CDEF is not parallel, 
 the four directions AB, CD, CE, CF are independent. Therefore the 
 direction AC is not independent of them (P). That is, it is possible to 
 move from AC by a s. L AC, or by s. 1 s. in directions AB, CD, CE, and 
 
 CF (D 5). The first of these motions is in the s. 1. AB, the others hi the 
 r. f. CDEF. Therefore they intersect. And they cannot intersect in more 
 than one point. [III. 2. 
 
 V. Any p. to which a r. f. is not parallel intersects it in a 
 single s. I. 
 
 For if A be a point in the p. ABC not in the r. f. DEFG ; since the 
 r. f. is not parallel to the p. it is not parallel to both AB, AC. Therefore 
 one of them, AB, intersects it in a single point P (III. 4). If the r. f. is 
 parallel to AC, a s. 1. through P parallel to AC is in both p. and r. f. 
 
 (III. 4). If not AC intersects the r. f. in a single point Q, and the s. 1. 
 PQ is in both p. and r. f. And they cannot intersect in any point outside 
 PQ (HI. 2). 
 
 VI. Two parallel r. f s. cannot intersect; and any two 
 which are not parallel intersect in a p. 
 
 For if ABCD, EFGH be two r. f s. which are not parallel, and A be 
 any point in the first which is not in the second, then EFGH is not 
 
 parallel to all of AB, AC, AD. Therefore it intersects at least one of them 
 
96 PART III. 
 
 in a single point, P (III. 4). And EFGH is not parallel to either of the 
 p s. ABC, ABD. Therefore it intersects each of them in a s. 1. PQ, PR 
 (III. 5). Therefore also the p. PQR is in both r. f s. And they cannot 
 intersect in any point outside this p. (III. 2). 
 
 VII. Any two p s. extending in four i. d s. intersect in a 
 single point. 
 
 For if A BC, DEF be two p s. which extend in four i. d s., and if A be 
 any point in the first which is not in the second, then the direction AD 
 is not independent of the four i. d s. AB, AC, DE, DF (P). Therefore 
 it is possible to move from A to D by a s. 1. AD or by s. 1 s. in these four 
 directions (D 5). But the first two of these motions are in ABC, and the 
 other two in DEF. Therefore the planes intersect, and they cannot 
 intersect in more than one point, else a r. f. might be conceived through 
 them [III. 3], in which case they would not extend in four i. d s. 
 
 Corollary. If two p s. extend in only three i. d s. they do not 
 intersect at all, unless both are in one r.f. 
 
 For if they only extend in three i. d s. either they are both in one r. f. or 
 two parallel r. f s. may be conceived through them. And as these parallel 
 r. f s. do not intersect, neither do the p s. they contain. 
 
 VIII. If each of two i. d s. in one plane be at right-angles to 
 each of two in another, any direction in the one p. is perpendicular 
 to the other p. 
 
 This follows at once from the definition of perpendicular. Two such 
 planes may be called perpendicular to each other. 
 
 IX. One, and only one, p. may be conceived through a given 
 point perpendicular to a given p. ; and it intersects in a single 
 point. 
 
 For if D be the given point and ABC the given p. conceive a s. 1. DP 
 to a point P in a s. 1. AB in the p. ABC, perpendicular to AB (I. 19) ; and 
 in the p. ABC conceive PC in a direction perpendicular to AB (II. 9), 
 
 and conceive DQ to a point Q in PC, in a direction perpendicular to PC 
 (I. 19). Then DQ is perpendicular to the p. ABC (II. 8). And if R be 
 any point in a direction from D independent of QD, AB and AC in the 
 
CHAPTER IT. 97 
 
 same way RS may be conceived to S in ABC, in a direction perpendicular 
 to it. Then since R is not in a r. f. A BCD, SR is in a direction independent 
 of QD, AB, AC. Therefore it is not parallel to QD. Therefore a p. DEF 
 may be conceived through D in the directions QD, SR which is perpen 
 dicular to the p. ABC (III. 8 and 7), and which intersects it in a single 
 point Q. And if any other p. through D were perpendicular to ABC, it must 
 also contain DQ, and therefore be in one r. f. with the p. DEF (III. 5). 
 But the p. ABC would intersect this r. f. in a single s. 1. which therefore 
 would be perpendicular to both the p s. through DQ, and in one r. f. with 
 them, which is impossible (II. 11). 
 
 X. If a direction be perpendicular to each of three i. d s. in 
 a r.f. it is perpendicular to the r. f. 
 
 A mere extension of my II. 8, or Euclid s XI. 4. 
 
 XI. One, and only one, s. I. may be conceived through a given 
 point in a direction perpendicular to a given r. f. And conversely 
 one and only one r.f. may be conceived through a given point to 
 which a given direction is perpendicular. 
 
 XII. The perpendicular s. I. is the shortest path from a 
 given point to a given r.f. And, of all other s. Z s....and so on. 
 
 XIII. An unterminated r. f. may be conceived to revolve 
 about any fixed p. in it describing an angle. And before it has 
 completed a straight angle, it will have passed once, and only 
 once, through every position in the new space of four i. d s. we are 
 considering. 
 
 For if ABC be the fixed p. and any point in it, through one, and only 
 one, p. DEF can be conceived perpendicular to ABC (III. 9) and a s. 1. 
 XOT in this p. may revolve describing an angle, and a r. f. may always be 
 conceived through this and ABC (III. 3), which will revolve with it, describing 
 the same angle. And a p. may be conceived through any point P in the 
 new space, parallel to the p. in which XOT revolves (III. 9), which will 
 intersect the p. ABC in a single point (7 and the r. f. XOAB intersects 
 this p. in a s. 1. parallel to XOT. And so on, as in II. 14. 
 
 Corollaries, (i) Hence any number of r. fs. may be conceived 
 to revolve similarly about the same p. 
 
 (ii) And consequently any figure of four dimensims may 
 be conceived to revolve about a fixed p. in it. 
 
 Hence two tetrahedra which have their six sides respectively equal 
 must be congruent. For their bases being made to coincide, the vertex of 
 one may be revolved about its base till it coincides with the vertex of the 
 D. 7 
 
98 PART III. 
 
 other. In doing so the moving vertex will however have to leave the r. f. 
 containing the other tetrahedron, only returning to it every time it has 
 revolved through a straight angle. 
 
 XIV. If two pentangular bodies have their ten sides re 
 spectively equal, they shall be equal in every respect, and if their 
 corners are similarly situated with respect to each other, they 
 shall be congruent. 
 
 By a pentangular body I mean the figure formed by the straight lines 
 joining five points not in a r. f. two and two. Thus there are j o that is 
 10 sides. The proof is precisely analogous to II. 16. 
 
CHAPTER III. 
 
 IN the last chapter the elements of geometry of four 
 independent directions were carried as far as I had previously 
 carried those of geometry of three : and I think it must now be 
 evident to the reader that they might without difficulty be 
 carried further without our having any clear idea of what the 
 words a fourth independent direction, mean. I shall therefore 
 now attempt to show how an idea can be formed of the 
 meaning of those words, and if even I fail, the reader will 
 recognise that my main argument is in no way damaged by the 
 failure. 
 
 Perhaps the most striking change introduced into geometry 
 by conceiving variations of position in a fourth independent 
 direction, is the possibility of getting past an unterminated 
 surface without passing through it. To take the simplest case, 
 conceive a material plane, and a material point moving towards 
 it, and getting past it. What does one mean by asserting that 
 at some moment the point must have intersected the surface ? 
 That at that moment the position of the point must have been 
 identical with some position now occupied by a point belonging 
 to the plane. The plane and point being material bodies this 
 would seem self-evident, but so far from being a necessary 
 truth, it is only one of the experimental facts upon which we 
 found our belief in the second Objective Fact of material 
 geometry ; and if the plane and point are not supposed to be 
 material, but merely mental concepts, the assertion no longer 
 has a leg to stand on. We are undoubtedly able to conceive a 
 space of three independent directions, and if we imagine points 
 not to be material, we may conceive any number of them 
 apparently occupying the same position in such a space, but 
 their positions in reality differing in a direction in which we 
 have not power to move material points. In just the same 
 
100 PART III. 
 
 way if we were to look at a motionless view through a hole in 
 a screen, it would have just the appearance of a perfect picture, 
 painted on a plane surface. We should see its length and 
 breadth but be unable to judge its depth. We might see two 
 lines cross each other, without being able to assert positively 
 that they intersected, for at their apparent point of intersection 
 one might be at a greater depth than the other. To get a true 
 idea of what was before us we should need to regard the view, 
 or picture, from another point of view under ordinary circum 
 stances indeed, we do regard every picture from two points of 
 view, namely from our two eyes. So also if we actually had 
 before us a figure of four dimensions, we could not find it out, 
 because we could not move so as to get another view of it. 
 But as the four dimension figures we have to deal with are not 
 actually before us, but are merely mental concepts, we may 
 conceive ourselves, mentally, to walk round them, or to twist 
 them round so as to expose various views to us, each of which 
 views we can mentally picture to ourselves as a model in 
 three dimensions. As these models are merely geometrical 
 projections, their shape can be determined by the methods of 
 formal geometry we have indicated above. For we have seen 
 that the material space we commonly conceive is, geometrically, 
 a regular form. The differences of position in a figure of four 
 dimensions, which we do not perceive in a model of only three, 
 are dimensions of the figure measured in a fourth independent 
 direction, which is perpendicular to this regular form. Hence 
 the model is found by projecting every point in the real figure 
 by a straight line perpendicular to material space. Thus the 
 apparent length of any straight line in the three dimension 
 model is its true length, multiplied by the cosine of the angle 
 it makes with its projection in material space ; and so on, just 
 as in ordinary orthogonal projection. 
 
 I am rather shy of here introducing a device, which however 
 I have found most useful myself, to aid me in conceiving figures 
 of four dimensions. For I have found it very difficult to 
 prevent people from taking the shadow for the substance, from 
 thinking that what I mean to put forward merely as an illus 
 tration, is in some way intended as physical fact. But having 
 warned the reader, I must run the risk, and trust to him not to 
 misunderstand me. 
 
CHAPTER III. 101 
 
 Let us then picture to ourselves figures of four dimensions 
 by conceiving models, or diagrams in material space (which for 
 this purpose we conceive as a regular form), which diagrams 
 are orthogonal projections of the true figures.- arid kt us 
 conceive the distance of every point in the - figure , from .a 
 regular form parallel to material space (but ail- on ona sidfi Or 
 the figure), to be represented in the diagram by the degree of 
 blackness of the corresponding point in the diagram. The 
 fourth dimension of the figure, over and above the length 
 breadth and thickness of the diagram, we may call its solidity ; 
 for just as its thickness is the difference in height of its top 
 and bottom, so its new dimension is the differences of the new 
 measurements of its opposite boundaries in the new direction, 
 that is the difference between two degrees of blackness, which 
 we may accurately represent by a certain degree of density 
 given to each point. Care must be taken however to dis 
 tinguish between solidity which is a dimension of a body, and 
 the blackness which we have used to represent a measurement 
 from a fixed regular form which is in fact a coordinate and 
 which we may call concentration. Thus if we consider a three 
 dimension solid, such as a sphere, it is clear that corresponding 
 to any vertical line through it, it has only one thickness, but 
 its surface has two heights. So with a body of four dimensions 
 it will, at any point in its three dimension diagram, have only 
 one solidity, but its boundary will have two concentrations. 
 A representation of the body by its solidity cannot therefore 
 give a perfect idea of its shape, to do which a double diagram, 
 of concentrations of its two boundaries, is required. But any 
 such a diagram of concentrations may be regarded as a diagram 
 of solidities of a body bounded by one irregular boundary, 
 and by the regular form from which the concentrations are 
 measured. 
 
 Thus in a diagram of concentrations a line will be a line 
 of varying blackness, unless material space is parallel to it. 
 A straight line will not only necessarily appear straight in the 
 diagram, but its degree of blackness will vary uniformly from 
 one end to the other. Two parallel straight lines will appear 
 as parallel straight lines, of shade varying at the same rate and 
 in the same direction. A plane in the same way will appear as 
 a plane shaded in parallel lines of shade whose blackness varies 
 
102 PART III. 
 
 uniformly in one direction. Similarly a regular form will 
 appear as a cloud in space, stratified in parallel plane strata 
 whose blackness varies uniformly in one direction. Hence, 
 even if -two lines appear to intersect, they do not really do so, 
 unless .at their apparent intersection they are equally shaded. 
 Thus a -straight line appearing to intersect a plane need not 
 necessarily do so. Two planes may appear to intersect, but, 
 unless there are points of equal shade in their apparent inter 
 section, they do not do so in reality. As however their shadings 
 vary uniformly, they will vary uniformly along the straight line 
 in material space along which they appear to intersect. Hence, 
 unless the lines of shade are both parallel to this line, in which 
 case there is no variation of shade along it, or unless the 
 shadings in both planes vary at the same rate in the same 
 direction along it, one of the shadings will somewhere along the 
 apparent intersection overtake the other. And at this single 
 point we shall have a true intersection. In the other cases there 
 will in general be no intersection at all, for the planes are in 
 parallel regular forms, but if there is one point of intersection 
 there will be a whole straight line and both planes are in one 
 regular form. Again with a little effort of imagination two 
 regular forms may be conceived in space at once. If their 
 planes of uniform blackness are not parallel, each plane in one 
 regular form will intersect all the planes of the other, in 
 straight lines. Of all these however only one will be a true 
 intersection, namely where the plane intersects the other plane 
 of the same blackness. It may readily be seen that all those 
 real intersections of each plane in one form with the corre 
 sponding plane in the other form, together constitute a properly 
 shaded plane, the intersection of the two regular forms. If the 
 planes of uniform blackness in the two forms are parallel, any 
 one of them will not in general intersect the corresponding one 
 in the other form. But unless also the shading of the two forms 
 varies at the same rate in the same direction, that is unless 
 the regular forms are parallel, at some point the shading 
 of one form will overtake that of the other, and here a whole 
 plane of uniform blackness from each form will coincide, and 
 we get a plane of intersection to which material space 
 happens to be parallel. The intersections of straight lines and 
 planes with a regular form may be similarly imagined. 
 
CHAPTER III. 103 
 
 If we conceive two intersecting lines, the true angle between 
 their directions would not be the same as the apparent one, 
 just as in a plane diagram the angles between straight lines 
 are altered. The shading would however indicate the alter 
 ation. For if we took two points of equal shading in the two 
 straight lines, and joined them, material space would be 
 parallel to the straight line so found, and therefore by a 
 revolution round it, the pair of intersecting straight lines might 
 be brought into a regular form parallel to material space, when 
 the angle would appear at its true magnitude. Hence the 
 perpendicular from the intersection of the two straight lines on 
 the axis round which they were twisted, would increase in 
 length until they were in a regular form parallel to material 
 space, and therefore the angle within which this perpendicular 
 fell would be diminished, and the other angle increased by the 
 twisting, in a manner which may easily be understood from the 
 analogous case of plane projections with which we are all 
 familiar. It is to be observed that in the revolution here 
 contemplated no apparent revolution would take place. The 
 figure would appear to remain in the same plane, the angles 
 only expanding or contracting by the straight lines sliding in 
 the plane, and the parts remote from the axis becoming more, 
 or less, black until they assumed the shade appropriate to the 
 regular form in which the axis lay, parallel to material space. 
 
 In the same way we may conceive the revolution of a 
 tetrahedron about its base. If we suppose the base to be 
 fixed in material space, the three sides ending in the vertex 
 will all be shaded, growing either more or less black towards 
 the vertex, if this is not in material space. If therefore it 
 start from material space, say towards the black direction, it 
 will grow darker and darker. Since the perpendicular from 
 the vertex always reaches the base in the same point, and 
 since the angles it makes with directions in the base are all 
 equal, and being right angles are not foreshortened by the 
 projection, the perpendicular will not appear to revolve, but 
 only to become shorter, until the vertex apparently sinks into 
 the plane of the base, here attaining its maximum blackness, 
 namely so much blacker than the base as corresponds to a 
 difference of concentration equal to the length of the perpen 
 dicular. After this the vertex will emerge at the opposite side 
 
104 PART III. 
 
 of the base again, still apparently moving in the same straight 
 line, but getting less and less black, until as it returns to its 
 original shade, the perpendicular regains its original length. 
 By a similar succession of phases, the vertex becoming lighter 
 in shade however, instead of darker, the revolution may be 
 completed, and the vertex return to its original position. 
 
 In Book II. we saw that two tetrahedra might be equal in 
 every respect, but yet not congruent. This phenomenon is one 
 of which we have every-day experience as in the case of 
 right- and left-hand boot. And it may have struck the 
 reader that the expression used in the enunciation of my 
 proposition II. 16 If the corners are similarly situated in 
 space is rather an indefinite test of right- or left-handedness to 
 give in what professes to be a formal text-book. But the fact 
 is there is no formal geometrical test to apply, the only other 
 appeal would be to the right and left hand of the reader, or to 
 some other objective thing, which could have no place in a 
 subjective geometry. In that geometry we were conceiving 
 figures in a regular form, and we now see that it is a mere 
 chance which way we might happen to conceive them to be 
 placed in that form whether in the same or in the opposite 
 way to another figure equal in every respect. The same thing 
 in the case of plane figures is evident. No one can define a 
 scalene triangle as right- or left-handed, simply because it 
 might be put into a given plane either way up. If however 
 we had a set of triangles which never moved out of a plane, we 
 might label one right-handed, and every other triangle equal in 
 every respect would be either right-handed like it, or left- 
 handed. So it is with us in material space. I am in material 
 space and cannot get out of it. I therefore mentally label one 
 of my hands right and the other left and no confusion can 
 arise. But if I ever, by magic or other arts, were allowed to 
 move in a fourth independent direction, I might on my return 
 find my right hand where my left used to be I should have to 
 wear my right-hand boots on the foot I used to call my left 
 with the exception of the pair I wore during my mystical 
 journey, which would of course have turned with me. But 
 to return to serious reasoning, I commend the question of 
 right- and left-handedness to the consideration of Euclidian 
 geometricians, and ask them to discuss it from their point of 
 
= 10 faces, and a regular form connects every four, that 
 
 CHAPTER III. 105 
 
 view, and either to frame a geometrical definition of it, or to 
 explain why they cannot do so. 
 
 In geometry of two independent directions a point is fixed 
 if its distances from two fixed points remain constant. This 
 may be expressed by saying that a triangle is the simplest 
 rigid frame. Similarly in geometry of three independent 
 directions a tetrahedron is the simplest rigid frame, and in 
 geometry of four, a pentangular body. In a pentangular body 
 there are five comers, and a side connects every two, that is in 
 
 5 4 
 all ^ = 10 sides, a plane face connects every three, that is 
 
 5.4.3 
 1.2.3 
 
 is Y~~ T = 5 regular forms. This method is general, and 
 1. . Z 3 4 
 
 so we can calculate the corresponding figures for a rigid frame 
 of any number of dimensions. 
 
 A point moving in a constant direction generates a straight 
 line, this straight line moving parallel to itself in a second 
 direction, a plane. In the same way a spread of any number of 
 independent directions may be generated. But if each of these 
 motions is limited in amount, we generate a terminated straight 
 line, a parallelogram, a parallelepiped, and what we may call a 
 parallel body in geometry of four independent directions. A 
 parallel body will have twice as many corners as a parallele 
 piped, that is 16. At each comer four sides will meet, extending 
 in four independent directions, that is the number of sides is 
 16 x 4 x ^ = 32. Every two sides meeting at a corner determine 
 a plane face, each face however unites four corners. Hence the 
 
 number of plane faces is 16 . ^-^ - = 24. Every three sides 
 
 meeting at a corner determine a regular form, but each regular 
 form, or parallelepiped, unites eight corners. Hence the number 
 
 4321 
 
 of regular forms is 16 . ^ = 8. 
 1 . L . o o 
 
 The above results are summarised in the following table, 
 which it is not quite superfluous to give, as similar but incorrect 
 tables have been given before. 
 
106 
 
 PART III. 
 
 Number 
 
 
 
 
 
 
 
 ofi.d s. \ 
 
 
 
 
 
 
 
 of dimensions of most ex- 1 
 tended body 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 n 
 
 of Cartesian axes J 
 
 
 
 
 
 
 
 of axis planes 
 
 
 
 
 
 1 
 
 3 
 
 6 
 
 n(n- 1) 
 
 1.2 
 
 of axis forms 
 
 o 
 
 o 
 
 o 
 
 1 
 
 4 
 
 n(n- !)(,- 2) 
 
 
 
 
 
 
 
 1.2.3 
 
 of axis spaces of four i. d s. 
 
 
 
 
 
 
 
 
 
 1 
 
 n(n- !)( 2)(3) 
 
 1O O A 
 
 
 
 
 
 
 
 .2.3.4 
 
 and so on 
 
 
 
 
 
 
 
 of corners of rigid frame 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 + 1) 
 
 of sides 
 
 
 
 1 
 
 3 
 
 6 
 
 10 
 
 (n + 1 ) n 
 
 1.2 
 
 of plane faces 
 
 
 
 
 
 1 
 
 4 
 
 10 
 
 (n+l)n(nI) 
 
 1.2.3 
 
 of regular forms ,, 
 
 
 
 
 
 
 
 1 
 
 5 
 
 (n-t-l)n(ft !)(**- f) 
 
 1.2.3.4 
 
 and so on 
 
 
 
 
 
 
 
 of corners of parallel frame 
 
 1 
 
 2 
 
 4 
 
 8 
 
 16 
 
 y. 
 
 of sides 
 
 
 
 1 
 
 4 
 
 12 
 
 32 
 
 n.2-i 
 
 of plane faces 
 
 
 
 
 
 1 
 
 6 
 
 24 
 
 n(nV) On _ a 
 
 1.2 
 
 of regular forms ,, 
 
 
 
 
 
 
 
 1 
 
 8 
 
 *(-!)(- 2) 3 
 
 1.2.3 
 
 and so on 
 
 
 
 
 
 
 
 
 The reader will do well, if he really wishes to become 
 familiar with geometry of four independent directions, to 
 picture these figures, and the other more complex ones, to 
 himself from various points of view, and to picture them as 
 twisting from one view to another. If he commences to do 
 this by the aid of the shading convention he will find after a 
 little practice he will even be able to dispense with this aid, 
 and to truly conceive a fourth independent direction, in the 
 same way, if not with the same ease, as he conceives the third. 
 
CHAPTER IV. 
 
 WE may now discuss an alternative to the supposition that 
 material space is a regular form, which the two objective facts 
 we have accepted still leave open. When the gifted author of 
 Flatland conceived a race of beings living in a land of two 
 dimensions, he tacitly assumed that that land was a plane : 
 it does not seem to have occurred to him to conceive a race 
 living on the surface of a sphere, or to discuss the views of 
 geometry they would be likely to entertain. If the sphere 
 were very large compared with their powers of locomotion, they 
 might indeed never find out that it was not a plane for the two 
 objective facts of their material space corresponding to those we 
 have accepted as true about ours, would still hold good. In 
 line land there would indeed be two possible alternatives, the 
 line might either be a circle, or a helix. I propose then to show 
 at once that for us too there is an alternative, if we only grant 
 the two objective facts corresponding to my first and third 
 subjective axioms: and I shall subsequently show that this is 
 the only alternative consistent with the truth of those objective 
 facts. 
 
 The objective facts which we have accepted determine, the 
 first, that material space possesses the property that any portion 
 of it is congruent with any other equally extensive portion, for 
 any geometrical figure which exactly occupies the first portion 
 may be made to exactly occupy the second, without change of 
 size or shape. It is, in fact, " Ein in sich selbst congruenter 
 Raum," that is, a self-congruent space. The second objective 
 fact determines that that space is a form, whether a regular one 
 or not. 
 
 We are naturally led at once to seek for a self-congruent 
 form in the analogue of the sphere. We may define a circle as 
 
108 PART III. 
 
 the locus, in a spread of two independent directions, of all points 
 at a given distance from a given point in it ; and it may be 
 conceived to be generated by a point in a straight line revolving 
 about a fixed point in itsslf, in such a spread. A sphere is a 
 similar locus in a spread of three independent directions, and 
 may be conceived to be generated by the revolution of 
 a circle about a diameter, in such a spread. The locus of 
 all points in a spread of four independent directions at a 
 given distance from a given point in it may similarly be 
 conceived to be generated by the revolution of a sphere about 
 one of its diametral planes. For the regular form containing 
 the sphere will, before completing a straight angle, have passed 
 through every point in the spread of four independent directions, 
 and, as in each situation of it the sphere contains all points in it 
 at the given distance from the given point, or centre, the locus 
 generated, when the revolution through a straight angle is 
 complete, is the locus required. Moreover the locus is a form. 
 For if a plane be conceived through the radius vector to any 
 point in the locus, and extending in any direction perpendicular 
 to the radius vector, the extremity of the radius vector may 
 describe a circle in that plane, and in the locus. But in the 
 space of four independent directions there is a whole group of 
 three independent directions, each of which is perpendicular to 
 the radius vector; and therefore the extremity of the radius 
 vector may commence to move in any one of these directions. 
 Hence we may call the locus a circular form, and say it extends 
 from every point in it in a complete group of three independent 
 directions, each of which is perpendicular to the radius vector. 
 In more familiar language, the radius vector is perpendicular to 
 the tangent-regular-form at every point. 
 
 We can picture to ourselves the generation of a circular 
 form, just as we pictured the revolution of a tetrahedron about 
 its base. If the revolution of the sphere is about a diametral 
 plane to which material space is parallel, each point in the 
 sphere will merely appear to move in a straight line perpen 
 dicular to the axis plane, those on one side becoming lighter, 
 and the others darker in shade as they appear to approach 
 the axis plane. After reaching the axis plane, when the whole 
 sphere will merely appear as a circular disc, with a double 
 shading on it, becoming both lighter and darker towards the 
 
CHAPTER IV. 109 
 
 centre, the points of the sphere will appear respectively to 
 emerge from the axis plane on opposite sides, and their shading 
 to return to its original intensity as they approach material 
 space again. During the intermediate stages the shape of the 
 sphere will appear to be a more or less oblate spheroid. Thus 
 if the sphere was not originally in a regular form parallel to 
 material space, it would have begun by looking like an oblate 
 spheroid, one pole of which was more, and the other less, black 
 than the equator, which would be of a uniform shade. If then 
 the revolution did not take place about the equatorial plane, 
 each point in the sphere would not simply appear to move in a 
 straight line perpendicular to the axis plane, but would appear 
 to describe an ellipse of greater or less excentricity. If the 
 axis plane extended in the direction of concentration, the 
 sphere would always look like a doubly shaded circular disc, as 
 described above, and the revolution of the sphere would merely 
 appear like the ordinary revolution of such a disc about a 
 diameter though of course that diameter would be a doubly 
 shaded line, which in reality represented an equator of the 
 sphere. In any of these cases the same result would be produced ; 
 namely a spherical ball, doubly shaded from the circumference 
 towards the centre, the one shading growing blacker, and the 
 other less black, as it proceeded inwards. 
 
 It is not impossible to imagine two shaded clouds occupying 
 the same space, and forming thus a diagram of concentration of 
 a circular form. But we may also, at the cost of a certain 
 loss of information, simply conceive, as a diagram of solidity, a 
 spherical ball, of a density at the centre corresponding to the 
 thickness of the ball, that is the length of a diameter, and at 
 every other point in it to the length of a chord at right angles 
 to the straight line from the centre to that point. Thus the 
 density would vanish at the circumference of the ball, but it 
 would not fade away by imperceptible degrees, but would end 
 quite abruptly, since at the circumference the circular form 
 would extend in the direction of concentration. 
 
 But to proceed with the investigation of the geometrical 
 properties of the circular form, in continuation of Book III. 
 
 XV. If a regular form intersect a circular form, the inter 
 section is a sphere. For if a perpendicular be conceived from 
 the centre of the circular form to the regular form, all points in 
 
110 PART III. 
 
 the intersection must be equidistant from the foot of this 
 perpendicular, for if they were not they could not be equi 
 distant from the centre of the circular form (III. 12). If the 
 perpendicular is not less than the radius of the circular form 
 there will of course be no intersection, and if it is equal there 
 will be a contact. It will not be necessary to mention such 
 obvious special cases in the other propositions. 
 
 XVI. If two circular forms intersect, their intersection is a 
 sphere. For the centres of the two spheres and any point in 
 the intersection form a triangle, all of whose sides are constant 
 in length, and whose base is immoveable. And the perpendicular 
 from the vertex, the point in the intersection, must meet this 
 base always in the same point, and be of constant length, for 
 any two situations of the triangle are congruent to each other 
 (I. 14). Hence the intersection consists of all points at a fixed 
 distance from the fixed foot of the perpendicular, and in direc 
 tions from it perpendicular to the straight line joining the 
 centres of the spheres that is therefore, in a regular form 
 (III. 11). Therefore the intersection is a sphere. 
 
 XVII. If a plane intersect a circular form, the intersection 
 is a circle. This may be proved in the same way as proposition 
 XV. 
 
 XVIII. If a sphere intersect a circular form, the intersection 
 is a circle. For the intersection of the regular form in which 
 the sphere is and the circular form is a sphere (III. 15). And 
 that the intersection of this sphere and the given sphere, which 
 are both in one regular form, is a circle, may be proved as in 
 III. 16, unless the centres of the two spheres coincide. In this 
 case, since by hypothesis they intersect, they must be one and 
 the same sphere. 
 
 XIX. If three circular forms intersect, their intersection is 
 a circle. For the intersection of two of them is a sphere, and 
 the intersection of this with the third a circle (III. 18). 
 
 XX. If a straight line intersect a circular form, it does so 
 in two, and only two, points. For the intersection consists of 
 points equidistant from the foot of the perpendicular from the 
 centre on the straight line. If this perpendicular is equal in 
 length to the radius of the circular form there is of course only 
 
CHAPTER IV. Ill 
 
 one point common to the straight line and form but in this 
 case the straight line does not intersect, but only touches it. 
 
 If a regular form through the centre of a circular form 
 intersect it, the radius of the sphere of intersection is obviously 
 the same as that of the form. In all other cases it is less. 
 Hence such a sphere may be called a great sphere in the 
 circular form. In the same way the intersection of a plane 
 through the centre with a circular form may be called a great 
 circle, and the points of intersection of a straight line through 
 the centre, two poles. The intersection of two great spheres 
 is a great circle, and the intersection of two great circles, or 
 three great spheres (if they intersect at all), is a pair of poles. 
 Through two great circles which intersect, or any three points 
 which are not in the same great circle, a great sphere may be 
 conceived. Thus the great circles joining three points which 
 are not in one great circle form a spherical triangle, the radius 
 of whose sphere is that of the circular form. Hence it may be 
 shown as in spherical trigonometry that if A t B, and C be the 
 angles of the triangle, and A its spherical area, and p the radius 
 of the circular form, 
 
 A + B + C-7r=- t . 
 P 
 
 Now we have seen that a circular form may be conceived 
 to be generated by the revolution of a sphere round any 
 diametral plane. The whole circular form may therefore re 
 volve in any way about its centre without ever occupying any 
 new positions. Hence any figure in it may be carried from any 
 one part to any other part of a spread of this form, without 
 undergoing any change of size or shape. Moreover such a figure 
 may revolve about any diameter of the circular form in any way, 
 that is, remaining in the circular form it may revolve about 
 any point in it, that is a pole of such a diameter. Similarly 
 the figure may revolve about any diametral plane of the circular 
 form, that is, remaining in the circular form, it may revolve 
 about any great circle in it. It follows therefore that a space 
 of the shape of a circular form is a self congruent space of 
 three dimensions, and as far as the two objective facts we have 
 accepted as true go, there is nothing to distinguish material 
 space from such a space. If it were such a space it would follow 
 that the things we call straight lines were really great circles, 
 
112 PART III. 
 
 and the things we call planes great spheres of circular space. 
 Let us consider whether we really know anything to preclude 
 this possibility. 
 
 What do we actually know about the things we call straight 
 lines ? If I say that the edge of a certain ruler is straight I 
 may mean that it extends in the same direction from every 
 point in it, but if my assertion is challenged I have to fall back 
 upon some other test of straightness. There are four ways in 
 which the straightness of a line may be tested, and upon each 
 of them a geometrical definition of the term straight line has 
 been founded by one or other writer on the subject. The four 
 tests are these : 
 
 1. Take three approximate straight edges, and fit them 
 against each other two and two, noticing where they touch. 
 By scraping down the places where they touch each may 
 ultimately be made to coincide exactly with the other two all 
 along. In practice this method is used with plane surfaces, not 
 straight edges, and straight edges are got by the intersection 
 of two such surfaces. This may be called the Whitworth test 
 of straightness or flatness, for it was Sir Joseph Whitworth who 
 first made practical use of it. It corresponds to the definitions 
 Two straight lines, or planes, cannot (in general) enclose a 
 space. 
 
 2. Take a rigid rod, supposed to be straight, and twist it 
 about two fixed points in itself (the rod may be conceived as a 
 thin right circular cylinder and in the limit, as a line). If 
 straight, it will not change its position. Thus a turned wooden 
 ruler of uniform thickness, is straight. The corresponding 
 definition may be stated thus A straight line is such that 
 it may be twisted about two fixed points in itself without 
 change of position. 
 
 3. Take a fine light flexible thread and stretch it tightly 
 between two points. The straight line thus produced (the 
 sag of the string owing to its weight being neglected) is very 
 frequently used, by bricklayers, gardeners, and so on. It- 
 corresponds to Legendre s definition, which may be paraphrased, 
 in order not to use terms in a different sense from what I have 
 given to them in this book, thus A straight line is the shortest 
 path between two positions. 
 
 4. But perhaps the commonest test of all in every-day 
 
CHAPTER IV. 113 
 
 life is that a straight line is the path of a ray of light in an 
 uniform medium, that is in one in which it is propagated with 
 the same velocity in all directions. Hence it follows that, 
 whether on the wave theory of light or on the emission theory, 
 the locus of a given phase of a given wave, or of particles which 
 started at a given moment, as the case may be, is the locus of 
 a number of points whose distances from the origin of the 
 disturbance are equal. The path of a ray of light is an 
 orthogonal trajectory of a series of such loci, and a straight 
 line may therefore be denned as such a trajectory. 
 
 I believe one or other of these four tests corresponds to 
 every definition of a straight line which has been made use 
 of in geometry, except the definition by direction. Euclid s 
 definition may perhaps be excepted, as having no meaning 
 whatever, but he uses the first test of straightness above cited. 
 Plato said A straight line is that of which the extremity hides 
 all the rest, the eye being placed in the continuation of the 
 line. This is of course the fourth test. Legendre, and the 
 majority of foreign geometricians I believe, have adopted the 
 third test. One author only, as far as I have discovered 1 , 
 makes any theoretical use of the second test, deducing it from 
 the idea of the locus of the points of contact of two spheres 
 with given centres. Thus the test approximates very closely 
 to the fourth test. The same author defines a plane in an 
 analogous manner, which is equivalent to defining it as the 
 locus of all points at equal distances from two given points. 
 
 Now it would be easy to show that in a regular form all 
 these tests would be fair tests of the straightness of a line. 
 But it is not difficult to show that if space is a circular form all 
 the tests are equally applicable, but that all give, not a straight 
 line but, a great circle. 
 
 For (1) through three points which are not in a straight 
 line one plane only can be drawn. Hence through two points 
 not in a diameter of a circular form, and the centre, one plane 
 only can be drawn, and therefore through two such points in 
 a circular form only one great circle can pass. We have there 
 fore only to assume that material space is so large a circular 
 form that we have never attempted to draw more than one 
 straight line through two points at opposite ends of a dia- 
 
 1 Geometry without Axioms, T. P. Thomson. 
 D. 8 
 
114 PART III. 
 
 meter, and it would seem to us that two straight lines, 
 (though they were really great circles) could not enclose a 
 space. 
 
 (2) As a diametral plane of a circular form can be twisted 
 about itself without any change of position, so can a great circle 
 in the circular form. The way Mr Thomson deduces his defini 
 tion from the idea of a series of spheres touching each other 
 may also be followed out in a circular form. For a sphere in a 
 circular form is the intersection of that circular form with a 
 regular form. Its centre is the foot of the perpendicular from 
 the centre of the circular form upon this regular form, and is 
 therefore not in the circular form at all. The apparent centre 
 is a point in the circular form in the diameter through the 
 true centre, and the apparent radii are arcs of great circles 
 through this apparent centre. Let A, B therefore be the 
 apparent centres of two spheres in a circular form whose centre 
 is 0, and let P, Q be their true centres, and let the two spheres 
 touch each other at C, in the circular form. Then the 
 regular forms containing the spheres intersect in a plane 
 through C. And, since both spheres touch this plane, the 
 directions PC, QC are perpendicular to it. But since the 
 circular form extends from C in the same directions as this 
 plane does, the direction OC is also perpendicular to it. Now 
 since the directions OP, OQ are perpendicular to the straight 
 lines CP, CQ, these straight lines are not parallel. Therefore 
 they determine a plane, which must therefore be perpendicular 
 to the plane of contact of the two spheres, through (7, and 
 which must therefore contain also 00, which is perpendicular 
 to this plane. (III. 9.) Hence the point C is in the plane 
 OPQ, that is in the plane OAB. Therefore it is in the great 
 circle AB, and this great circle is therefore the locus of such 
 points of contact. 
 
 (3) That a great circle is the shortest path from one point 
 to another in a circular form may be proved in the same way 
 that it is proved in spherical trigonometry. Or, since all the 
 premises, from which the analogous proposition is proved for a 
 regular form, apply in a circular form, if the lines in the 
 figure are none of them as great as a semi-circumference of a 
 great circle, the proof that a straight line is the shortest path 
 in a regular form applies to prove that the great circle is the 
 
CHAPTER IV. 115 
 
 shortest path in a circular form, if it is not so great as a semi- 
 circumference. (The fact that the exterior angle of a triangle 
 is greater than the interior and opposite is proved in Euclid s 
 manner, supplemented in the way I have indicated in Part I.) 
 
 (4) I have already shown that the true centre of a sphere 
 in a circular form is not in that form, but in the radius vector 
 to the apparent centre, or pole, of the sphere. The centres of 
 a series of increasing spheres with the same apparent centre 
 will therefore be a series of points in this radius vector, ap 
 proaching the centre of the circular form. Any one of these 
 spheres, from any point in it, extends in directions which are 
 perpendicular both to its own radius vector from its true centre, 
 and to the radius vector from the centre of the circular form. 
 That is to say, it extends in directions perpendicular to a plane 
 containing its own true centre, the centre of the circular form, 
 and the point in question. This plane also contains its apparent 
 centre, and is the diametral plane, whose intersection with the 
 circular form constitutes the great circle, which is the apparent 
 radius vector to the given point. The sphere therefore extends 
 from the given point in directions to which the directions in which 
 this apparent radius vector extends, are perpendicular. Thus, 
 such a great circle, through the apparent centre of the series of 
 (apparently) concentric spheres, is an orthogonal trajectory of 
 the series ; and so fulfils the last test we have of straightness. 
 
 Thus the two objective facts we have accepted as true are 
 not enough to prove that material space is not a circular form. 
 Moreover if it was a circular form of large radius, we might 
 easily not find it out without having first logically investigated 
 the possibility of its being one, and its consequences. This will 
 be evident if we merely consider how many generations of men 
 lived and died without ever finding out that the surface of the 
 earth is a sphere, and not a plane. And yet they had far more 
 obvious means at hand wherewith to deal with the problem, 
 for their straight lines were at least straight enough to detect 
 the curvature of the surface of the earth. 
 
 The method to be employed in this case will however doubt 
 less have occurred to the reader at once. It is by measuring 
 the angles of triangles, for excess. The discussion of this 
 method is however better postponed until we see that its 
 results would really be conclusive. To do this we must in- 
 
 82 
 
116 PART III. 
 
 vestigate whether there are no other forms which are self- 
 congruent, besides the regular and circular forms; and this 
 when we conceive the possibility of the variation of positions 
 not only in four, but in any number of independent directions. 
 
 It would be difficult to conduct this investigation by the 
 geometrical methods I have hitherto employed, so I embrace 
 the opportunity of showing how readily the conception of direc 
 tion, as I have advanced it, adapts itself to the development of 
 the theory of analytical geometry; and I shall outline that 
 theory from the very beginning in order to show that its methods 
 are perfectly general, and that it is by a pure mathematical 
 induction, not by mere reasoning from particular to general 
 assertions, that its methods are applied to geometry of any 
 number of independent directions. 
 
CHAPTER V. 
 
 THE primary object of analytical geometry is the numerical 
 representation of differences of position and direction. For we 
 cannot even conceive such a thing as an absolutely fixed point, 
 we can only assume that a certain point is fixed, that is, that it 
 does not change its position, and determine the positions of all 
 other points by reference to it. So also, although the common 
 acceptance of Newton s first law of motion seems to make it 
 probable that we do conceive of absolutely fixed directions, we 
 have no geometrical test for fixity of direction, and so, geometri 
 cally, directions also can only be determined relatively. The 
 problem of analytical geometry is then to determine positions 
 and directions with reference to a given point, whose position is 
 assumed to be unalterable, and one or more given directions. 
 
 The most elementary conception of geometry is that of 
 a single position, of a single, fixed, point. Such a point, 
 the datum from which we start, is called the Origin. If 
 besides this we conceive any other points, they lie in one or 
 more directions from the origin. And having once conceived 
 a direction, we may conceive positions varying in this direction 
 from any position we have already conceived. 
 
 Let us then start with an origin A, and a second position B. 
 Then we have conceived the direction AB. Let C therefore be 
 another point in this direction from B. Since we have also 
 conceived the direction BA we may also conceive points in this 
 direction from A, that is we may conceive any point in the 
 unterminated straight line AB. 
 
 If now the distance AB be represented by unity and the 
 distance A C by 3, say, the positions of B and C with reference 
 to A as origin may be represented by the numbers 1 and 3 
 respectively. But now suppose we take B as origin, then 
 
118 PART III. 
 
 the position of C will be represented by the distance BC, 
 that is by (AC AB} or 2. If we apply the same rule to 
 the case of the position A, since its position was formerly 
 represented by 0, its position will now be represented by 
 (0 1), or 1. Hence if we represent positions on one side 
 of the origin by positive numbers, or coordinates as they are 
 called, and those on the opposite side by negative numbers, 
 a single number, with its proper sign, will represent any 
 position. And we obtain the following universal rule : 
 
 If the origin is moved to a position whose old coordinate 
 was a, a position whose coordinate was formerly x will now be 
 represented by (x a). 
 
 Thus we can analytically determine any position, in a given 
 direction or the opposite, from a given origin 0, by a single 
 (positive or negative) coordinate, x. If we now conceive 
 another position besides all these, it lies in a new direction 
 from 0, and we may conceive a similar series of positions 
 in this new direction, each determined by a single coordinate 
 y, from the origin 0, or from any of the positions formerly 
 determined by a coordinate x from 0. Thus we have a system 
 of positions determined by two measurements in independent 
 directions from ; x and y. By the first corollary to proposi 
 tion I. 23 the order in which these measurements are made is 
 indifferent. And as they are quite independent measurements 
 the rule about change in the coordinate consequent on changing 
 the origin, given above, will apply to each separately. Thus if 
 the origin is moved to a position whose old coordinates were 
 (a, b) the new coordinates of (x, y) are (x a), (y b). 
 
 If we now consider yet another position not included among 
 these, we conceive a third independent direction, and to deter 
 mine a position we require three coordinates. The rule for 
 change of origin applies as before to each separately. And 
 so we may go on ad libitum. Hence in a space of any number 
 of independent directions, if the origin is moved to a point 
 (abc...) and if (xyz...*) be the old and (x y z ...) the new coordi 
 nates of any point, we have 
 
 x = x a\ y y ~ b\ z = z c\...\ , . 
 
 , _ ,. _ ,/,. _ . f (I) 
 
 Straight lines through the origin in the given independent 
 
CHAPTER V. H9 
 
 directions in which the coordinates are measured are called the 
 coordinate axes. Every two of these determine a plane, called 
 an axis plane. Every three determine a regular form, called an 
 axis form. And so on. Thus in geometry of four independent 
 directions there are four axes, six axis planes, and four axis 
 forms. 
 
 The obvious way of indicating a direction analytically is by 
 giving the coordinates of a point at unit distance from the 
 origin. It follows from the principles of similar triangles 
 established by Euclid in his sixth book, that if (xyz...) be 
 the coordinates of any point at a distance s from the origin, 
 
 its direction from the origin is represented by ( - , - , -) 
 
 \o S S J 
 
 To find its direction from any other point (xyz...} whose 
 distance from it is s we have only to move the origin to the 
 point (xy z...), so that the coordinates of (xyz...) become 
 (x x) (y y ) (z -/)...; and therefore its direction from 
 
 (xyz...) is represented by f -, , - r~ t , ...J. Simi 
 larly, to find the direction of the point (x y z...) from (xyz...) 
 we move the origin to (xyz...) ; and as s, the distance between 
 the points, remains the same, the direction is represented by 
 
 f*__* y^y. __ .A that is, by the same numbers, or 
 V s s s J 
 
 direction coefficients as we may call them, as before, but with 
 their signs reversed. 
 
 To find an expression for the distance of any point from the 
 origin, let OL, LM, MN,... be the coordinates (xyz...) of any 
 position P at a distance s from the origin 0. Let (a./3y...) be 
 the angles which OP makes with the directions of the axes, and 
 conceive perpendiculars PA, PB, PC... from P to each of the 
 axes. 
 
 Then if each of the points L, M, N,... be projected upon OP 
 by straight lines perpendicular to it, it is obvious that OP is 
 equal to the sum of these projections, that is that 
 
 s = x cos a + y cos ft + z cos 7 + 
 
 OA OA 
 
 But cos a = T.-T, = . Hence 
 OP s 
 
120 PART III. 
 
 and OA is equal to the sum of OL and LA, that is to the sum 
 of x, and the sum of the projections of y, z... upon the axis of x, 
 by straight lines perpendicular to it. That is, if (xy) represent 
 the angle between the directions of x and y, and so on 
 
 s 2 = x . x + x . y cos (xy) + x . z cos (xz) + ...... 
 
 4- z . x cos (zx) + zy cos (zy) + z . z + ...... 
 
 .......................................... and so on 
 
 = x 2 + 2/ 2 + ^ + ...... + Zxy cos (xy) + %yz cos (ys) + ...... (2), 
 
 there being p terms of the first kind, and of the second. 
 If (Imn...) be the direction coefficients of the direction OP, 
 
 rp 77 
 
 dividing the above equation by s 2 we get, since I = - , m = - 
 
 s s 
 
 and so on, 
 P + m> + ri i + ... +ttmcQa(xy)+2mncoa(yz)+... = 1 ...... (3). 
 
 This equation of condition subsists then for all direction coeffi 
 cients. 
 
 The distance of (xyz...) from any other point (xy z r ...) can 
 be deduced at once from the above formulae by moving the 
 origin to (x y z ...), that is by writing (x of) for x, (y y } 
 for y and so on in the formula (2). If the axes are all at right 
 angles to each other the cosines of the angles between them are 
 all zero and so equations (2) and (3) become 
 
 s? = o?+y* + z z + ...... , I 2 -t- m 2 + n* + ...... = 1. 
 
 The condition that any number of directions 12...^ shall not 
 be independent of each other in a space of p independent direc 
 tions, may be found from the consideration that if they are not, 
 after moving a distance s lt from the origin in any one of the 
 given directions it must be possible to return to it by move 
 ments (positive or negative) s 2 s 3 ...s q in the remaining directions. 
 Hence given any value of ^ it must be possible to find values of 
 s 2 s 3 ...s q to satisfy the p equations 
 
 + S 3 1 3 + ...... S q lq = 
 
 q m q = 
 
 ........................ and so on 
 
 We have then to eliminate (q 1) quantities from p equations. 
 
CHAPTER V. 121 
 
 If we can do so, we may divide by s lt the remaining quantity, 
 and have left one or more equations of condition that the direc 
 tions shall not be independent. Thus if q is greater than p 
 these equations will contain some of the variables S 2 s 3 ... and 
 so a solution can always be found, i.e. the directions are not 
 independent. If q is equal to or less than p we obtain one 
 or more conditions, in the form of determinants equated to zero. 
 As such symmetrical determinants occur frequently in the next 
 
 few pages, I shall use the symbol < " for a determinant 
 
 formed of columns of Ts, ms and so on, the rows being distin 
 guished by the suffixes 123 ...... p. If the number of letters in 
 
 the numerator is less than the number of suffixes, the number 
 is to be made up with columns of units. 
 
 The condition then that p directions be not independent is 
 
 where all the p letters appear in the numerator. The conditions 
 that any less number, q, be not independent are of the same 
 form, but only q of the p letters appearing in each numerator 
 and of numbers in each denominator. Hence if p q = r, the 
 number of conditions may be written either 
 
 p.p-l.p-2 ..... .p-r+1 p.p-l.p-2 ...... p-q + 1 
 
 L/A , . 
 
 rl ql 
 
 but of this number only (r + 1) are independent, for they are all 
 deduced from p equations after eliminating (q 1) quantities. 
 
 The condition that two directions may be at right angles is 
 easily found. For if the directions 1 and 2 are at right angles 
 the distances of a point (l^n^. . .) from two points (Lmji*...) 
 (1 2 , m*, n 2 ,...) each at unit distance from the origin, must 
 be equal. Hence 
 
 + 2 (^ - 1) (7/ij - m 2 ) cos (xy) + 
 
 + 2 (^ 4- L) (m^ + w 2 ) cos (xy) + 
 From which we deduce the condition 
 IJ, Z + ??V7i 2 + n-jn* 4- . . . + (l-iin* + Lin^ cos (xy) + . . . = 0. . . - (5). 
 
122 PART III. 
 
 In the case of rectangular axes this reduces to 
 IJt + Wjraa + n^iz + ...... =0. 
 
 The condition that two directions may be identical or oppo 
 site may be put in a similar form, namely 
 
 I l l 2 + m 1 m 2 + n l n 2 + ...... = + 1 ............... (6). 
 
 For if to twice the above equation (6) with the lower sign 
 we add the equations 
 
 or if from these we subtract twice equation 6 with the upper 
 sign, we ge,t 
 
 (li Z 2 ) 2 + K m*y + (HI n^ + ...... &c. = 0, 
 
 which requires that each of the expressions in the brackets 
 should vanish. 
 
 It is easy in a similar manner to find the trigonometrical 
 ratios of the angle between any two directions. For the dis 
 tance between the points (l-jn-^n^) and (7 2 ra 2 tt 2 ), both being at 
 unit distance from the origin, is the chord of this angle. Hence 
 if the axes are rectangular we have 
 
 sin - = J s/(Zi - y + (X - w 2 ) 2 + (X - n^f + ...... &c. 
 
 A 
 
 cos- = 
 
 &c.), 
 
 &c.), 
 
 A ^ 
 
 cos A = cos 2 sin 2 = l^ + m^m^ + 7? a ?i 2 4- &c., 
 
 - 2 
 
 sin A = Jl (IJz 4- m-^n^ + n a ?i 2 -f &c.) 2 ; 
 
 and the latter expression may easily be shown to be equal to 
 
 
 &C > 
 
 
 the bracket containing determinants formed from all the com 
 binations of the coefficients two at a time. The conditions (5) 
 
CHAPTER V. 
 
 123 
 
 and (6) may therefore be replaced by equating this expression 
 to 1 and respectively. Similar expressions for oblique axes 
 may be found by exactly the same methods. 
 
 We have seen that the directions of the axes must be 
 independent directions, but with this restriction it is still 
 possible to twist them into various directions. Any possible 
 alterations in the directions of the axes may be performed by 
 successive twists of two axes at a time in their own axis plane, 
 and each of such twistings will only affect the coordinates 
 parallel to the axes twisted, though in the case of oblique axes 
 it will also affect the angles between these axes and the rest. 
 Suppose, for example, in a space of four independent directions 
 it is required to twist the axes into four new independent 
 directions 1, 2, 3, 4 that is, until they pass through the four 
 points (hmjnjci) (I 2 m 2 njc. 2 ) and so on, all at unit distance from 
 the origin. This can be done in six stages, thus 
 
 A rotation in brings the axis 
 
 that of y that of z 
 
 and that 
 
 the plane of 
 
 of x to 
 
 to 
 
 to 
 
 of w to 
 
 to) 
 
 ^ m 1 
 
 Z, 7W 2 
 
 0000 
 
 0000 
 
 (yz) Z^OO 
 
 1 2 m 2 n 2 
 
 m 3 >i 3 
 
 0000 
 
 (*v) 
 
 ^mjOO 
 
 1 2 m 2 n 2 
 
 m 3 n 3 k 3 
 
 m k 
 
 (*) 
 
 l m l Jc l 
 
 1 2 m 2 n 2 
 
 OwigW 3 3 
 
 l Z 4 & 4 
 
 (*f) 
 
 i ^i ^i i 
 
 1% m 2 w 2 
 
 ^3 m 3 n 3 ^3 
 
 l m ^ 4 
 
 (y) 
 
 ^wVh^i 
 
 1 2 m 2 n 2 k 2 
 
 JsWsWg&J 
 
 Z 4 m 4 rc 4 4 
 
 The coordinates in the table being all referred to the original 
 axes. This method is sometimes convenient. Or else general 
 formulae may be obtained thus, 
 
 We observe that if a point can move from the origin to a 
 
 given position by a movement s, in a direction (Imn ), it 
 
 can also move to it by movements Is, ms, ns, in the direc 
 tions of the axes of (#, y, z ) respectively, these being the 
 
 coordinates of the position. 
 
 Let it then be required to find the coefficients (\ fiv ) 
 
 of a direction referred to a set of axes (xy z ), whose 
 
 coefficients referred to a given set of axes (xyz ) were 
 
 (\fjiv ), the directions of the new axes referred to the old 
 
 being (l^i^ ) (I 2 m 2 n 2 ) and so on. 
 
 Then by hypothesis a point can move from the origin to 
 
124 
 
 PART III. 
 
 the point P, at unit distance from it, whose old coordinates are 
 (\IJLV ), either 
 
 (1) by movements (\JJLV ) in the directions of the axes 
 
 of (x, y, z ) respectively, or 
 
 (2) by movements (V//.V ) in the directions of the 
 
 axes of (oc y z ) respectively. 
 
 But these latter movements may be replaced as follows 
 
 \ by (li\ \ (wjV), (XV) in the directions (xyz ) re 
 spectively [for (A/00 ) is a point at distance V from the 
 
 origin in the direction (l^m^ )], 
 
 pf by (hp), (m^fjb), (n^i!) ... in the same directions, and so on. 
 Or, as the movements may, by (I. 23, cor. (ii)) be taken in 
 any order, a point may be moved from the origin to P by 
 movements 
 
 (ZjV + Izfi + l s v + ) in the direction of x ; 
 
 (wjA + w&2// + in$v f + ) in the direction of y ; 
 
 (r^V 4- n 2 fjf + n 3 v + ) in the direction of z ; 
 
 and so on. 
 
 These therefore are the coordinates of P referred to the 
 axes (xyz ). That is to say, 
 
 v = WjV + n 2 fjb + n s v + 
 
 and so on 
 
 Whence we get also 
 
 A IJL v ... 
 {. 
 
 Imn... . Imn... 
 
 A //, v ... 
 
 ^3 H-3 . 
 
 Imn.., 
 
CHAPTER V. 125 
 
 and so on ; where, in the determinants in the numerators of the 
 second expressions, one number, namely that corresponding to 
 the coefficient (X ) on the left of the equation, and one letter, 
 namely that corresponding to the coefficient (X) of each de 
 terminant, is omitted, leaving determinants of the order (p 1). 
 To deduce from these formulae the new coordinates 
 
 (x y z ) of a position whose old coordinates were (xyz ) 
 
 we observe that the distance of the position from the origin s, 
 is unaltered ; and its direction from the origin, referred to the 
 
 (CC II Z \ 
 
 - , - , - . Substituting: these in (8) 
 s s s J 
 
 we get the values of the coefficients of the same direction 
 
 /*/ nj 
 
 referred to the new axes, that is, of - , , and so on. As the 
 
 s s 
 
 equations are all homogeneous the s multiplies out, and the 
 result is the same as if we had written x for X, of for X and so 
 on, throughout equations (7) and (8). 
 
 If only two axes are rotated in their own plane the formulae 
 (7) and (8) reduce to 
 
 X = 
 / in, 
 
 12 
 
 and y.-.(9). 
 
 
 
 In the special case where both the old and new systems 
 of axes are rectangular, equations (8) may be much simplified. 
 This may be proved by the method of successive rotations of 
 two axes at a time, referred to above. In this case, since the 
 direction at right angles to (I, m) in the plane of (xy) is the 
 direction ( in, I), equations 9 become 
 
 X = l\ Nil* ] j X 7 = l\ + HILL } 
 
 , , ,V and , , > (10). 
 
 fj, = m\ + IfJL } fjL = ??lX + l/ji) 
 
 c . lm 
 for, d> = 
 
 Hence the effect on the coordinates (xyz ) of any posi 
 tion, of a twist of the axes of (xy) in their own plane till they 
 
 pass through the points (l^m^O ) (I 2 m. 2 00 ) that is, till 
 
 the axis of x extends in the direction 
 
 , 0, ) 
 
126 PART III. 
 
 is that they become respectively 
 
 l& + my . - m& + 1$ . . 
 
 ,/^T^ 2 V?T^ 
 
 And the new coefficients of the direction (l^n^ ...... ) are 
 
 l? + m^ ; 0; n,- ^ ; ...... &c. 
 
 Hence the next twist, in the axis plane (xz), has to bring the 
 axis of x into the direction 
 
 &, 
 
 A 1 + , + ," 
 
 Therefore the coordinates of P after this twist become 
 
 
 and the new coefficients of the direction (l{m^n^ ...... ) are 
 
 Jlf + ntf + n? , 0; 0; A;; ...... &c. 
 
 If we continue this process (p 1) times, that is, if we 
 twist the axis of x with reference to each of the other axes in 
 succession, the direction coefficients (l^m^n^...) become all zero- 
 except the first, which is unity since 
 
 Zi 2 + mf + r^ 2 + ...... (p terms) =1, 
 
 and the coordinate x of the position P becomes 
 
 x = ^x + 
 Similarly y = I 2 x + m$ + n^z + 
 
 and so 
 
 ...... i 
 
 o onj 
 
 and in these equations as before we may write X for # , and 
 X for x, and so on, and so get the new direction coefficients of a 
 direction. For example, the old coefficients of the old axis of x 
 were (1, 0, 0, 0, ...... ). Thus the new coefficients of the old 
 
 axis of x are (IfJ^ ...... ). Thus, considering the axes (x y z ...... ) 
 
 as the original ones, formula (11) gives for the coordinate x of a 
 position referred to the other axes 
 
 a = 
 which agrees with (7). 
 
CHAPTER VI. 
 
 IF (xyz ) is any point in a space of p independent 
 
 directions what is represented by a single equation 
 
 F(xyz )=0 (1).? 
 
 The equation obviously represents the locus of a number of 
 points obeying certain conditions. As (in general) the variables 
 may vary continuously in the equation, the locus will (in general) 
 be a spread of some kind. Let us then investigate the direc 
 tions in which it extends from any point in it. 
 
 If (x + A#) (y + Ay) (z + A,z) be a point in the locus 
 
 near (xyz ) at a distance As- from it, then the direction 
 
 from (xyz ) to it is represented by the direction coefficients 
 
 Aa? Ay Az 
 
 As As As " 
 
 If the distance As is indefinitely diminished, in the limit this 
 direction is a direction in which the spread extends from the 
 point (xyz ), and its coefficients are 
 
 dx dy dz 
 ds ds ds 
 
 Now s is some function of xyz Hence differentiating 
 
 (1) with respect to s we get 
 
 dF dx dF dy dF dz 
 dx ~ds + d^ ds + Tz ds + " (2) 
 
 dF 
 The quantities -T-, are constant at any given point in the 
 
 locus, and may be replaced by letters XYZ which are 
 
 understood to be dependent on xyz , the coordinates of the 
 
 dx 
 given point. The quantities -y- &c. are the direction coef- 
 
128 PART III. 
 
 ficients of any direction in which the spread extends from this 
 
 given point, and may be replaced by (\/JLV &>). Thus the 
 
 equation 
 
 X\+Yp + Zv + -0 (3), 
 
 determines in what directions the spread extends. 
 
 Now the p quantities (\fj,v o>) are not quite indepen 
 dent, for as we have seen, being direction coefficients, they are 
 connected by an equation of condition which we found in the 
 last chapter (formula 3). This condition also prevents their all 
 being zero. Hence we may consider one of them X, to be 
 determined by this condition in terms of the others, and not to 
 be zero, we may therefore divide equation (3) by X and have 
 left an equation of the first degree in (p 1 ) arbitrary variables, 
 
 fJL V CO 
 
 x x x 
 
 Now in equation (3) any one of the constant quantities 
 XT will be zero for every point in the locus, if the cor 
 responding coordinate does not appear in equation (1). But 
 even if it does appear, the quantity may be zero for a particular 
 point in the locus, for it is got by differentiating F with respect 
 to one of the variables in it, and then giving these variables 
 particular values, which may of course make the result vanish. 
 
 But suppose that at a given point (cc y z ) in the locus, q 
 
 out of the p quantities X, Y, do not vanish, including 
 
 X. Then (3) becomes an equation in (#1) arbitrary ratios 
 
 , -, . We may therefore assign any values we please 
 
 XX X 
 
 to all but one ( say ~ } of these ratios and also to all the (p q) 
 
 V X/ 
 
 coefficients which do not occur in the equation, and we may 
 
 then determine ^ . Suppose we ascribe the value zero to each 
 X 
 
 one of these arbitrary ratios. We have then found one direc 
 tion in which the spread extends, which we may denote by 
 
 (Xi/iiOO ) 
 
 Now vary each of the (p 2) arbitrary values we have 
 assumed in turn, making each in turn unity instead of zero for 
 instance. In the case of the (q 2) arbitrary ratios which appear 
 
 in the equation (3) a redetermination of ^ will be necessary ; 
 
CHAPTER VI. 129 
 
 but in the case of the (p q) which do not appear in the equa 
 tion this will not be necessary. We shall now have (p 1) 
 
 directions in which the spread extends from (x yz ). It 
 
 only remains to show that they are independent directions, 
 and that all other directions in which the spread extends are 
 dependent upon them. 
 
 Now the complete determinant formed from these (p 1) 
 
 direction coefficients (X^ o^) (\. 2 v. 2 &> 2 ) omitting 
 
 any one of the coefficients, /JL^ from each, may be de 
 veloped thus 
 
 . \v a) 
 
 9 
 
 12 (p-1) 
 
 
 12 (p-S) 
 
 Now none of the quantities \ , X, are zero, and of each 
 
 of the quantities (v o>) in each direction after the first, all 
 
 are zero except one, in each direction, whereas in the first 
 direction all are zero. Hence each determinant on the right 
 of the above equation has a complete row of zeros (that cor 
 responding to direction 1) except the first, the letters in which 
 are all zero except those on the leading diagonal. Therefore 
 this term on the right hand of the equation remains alone, and 
 is not zero. Therefore the (p 1) directions are independent 
 of each other. 
 
 Now let us select any other direction in which the spread 
 extends, and which therefore satisfies equation (3). We thus 
 have p directions in all. If the q coefficients of this new 
 direction which appear in the equation (3) are equal or pro 
 portional to the first q of any of the former (p 2) directions, 
 then the determinant 
 
 . \JJLV 7T 
 
 q 
 
 formed from these first q coefficients will have two rows equal 
 or proportional, and will therefore vanish. But if not we obtain 
 one more independent and consistent equation of form (3), that 
 is q equations in all, which suffice to eliminate the q quantities 
 D. 9 
 
130 PART III. 
 
 XT ...... none of which by hypothesis are zero, and so we are 
 
 left with the same condition as above, namely 
 
 Now if A/T be one of the remaining (p q) direction coef 
 ficients we have 
 
 
 123 
 
 And we gave the value zero to all the ratios ^- except one. As 
 
 A 
 
 xjr is not among the first q direction coefficients, it was one of 
 those after the q ih which was not zero, say the (q + l) th . Thus 
 all the terms on the right of the above equation after the first 
 vanish on account of the -v/r coefficient, and the first vanishes 
 because, as we have seen, its determinant <f> vanishes. In the 
 same way it can be proved that we may form a determinant 
 with yet another of the remaining (p q) coefficients, which 
 will vanish ; and by proceeding in this way to the end, we 
 find that 
 
 That is the p directions are not independent, and therefore the 
 spread represented by equation (1) extends from every point in 
 it in (p 1) and no more independent directions. We may call 
 such a spread one of the (p l) th order. 
 
 In the same way it may be shown that two simultaneous 
 independent and consistent equations 
 
 = 01 
 
 in general represent a spread of the (p 2) th order. 
 For as above we may deduce the equations 
 
 + = oi 
 
 + =01 
 
 And even if both these equations (5) contain all the p 
 
 coefficients (X//,i/ o>) we may eliminate one of them and 
 
 so obtain an equation in (p 1) of them, from which we may 
 
CHAPTER VI. 131 
 
 show as above that the locus extends in (p 2) independent 
 directions, for we may ascribe (p - 2) arbitrary values to the 
 ratios of (p 1) of the coefficients, determining the p th coef 
 ficient in each case from one of the equations (5). And if 
 equations (5) are independent we may show as above that 
 there cannot be more than (p 2) independent directions 
 which satisfy them. 
 
 Thus two simultaneous independent and consistent equa 
 tions in general represent a spread of the (p 2) th order. Or 
 we may say that two spreads of the (p l) th order in general 
 intersect in a spread of the order (p 2). 
 
 And so generally q equations represent in a space of p 
 independent directions a spread of the (p q) th order, or the 
 intersection of q spreads of the order (p 1) is in general a 
 spread of the order (p q). 
 
 Thus a line will in general be represented by (p 1) equa 
 tions, a surface by (p 2), a form by ( p 3) and so on. 
 
 The condition that a spread shall be a regular one is that 
 the equation or equations determining the directions in which 
 it extends should be the same for every point in the spread. 
 
 7 TTT 7 77T 
 
 That is, the differentials -7- , -j- and so on, must all be con- 
 
 dx dy 
 
 stants. Hence all the equations representing regular spreads 
 must be equations of the first degree. 
 
 The equations to a straight line may be got at in another 
 way. Let (abc ...... ) be any point in a straight line and 
 
 (Imn ...... ) the direction in which it extends, s the distance 
 
 of any other point in it from (abc ...... ). Then we have 
 
 y = b + ms ; z = c + ns ...... and so on. 
 
 ~ as a 11 b 2 c 
 
 Or j = -- = - -= ...... =s ........ .(6). 
 
 I m n 
 
 (The last equation is of course not independent of the 
 others.) 
 
 If this straight line passes through a second known point 
 (a b c ...... ) at a distance s from (abc ...... ) then 
 
 , a a b b 
 
 I = - 7 ; ??i = 
 s s 
 
 and so on. Hence the equations to the straight line joining two 
 points are, 
 
 92 
 
132 PART III. 
 
 X CL y }) _z c _ _s x,-x 
 
 , T7 1 ~~r I> \ /1 
 
 a a b be c s 
 
 Similarly if (abc ) be a known point in a plane which ex 
 tends in two known directions, 1 and 2, from it, then if it extend 
 in any third direction, we have 
 
 Imn _ mnk _ _ ~ 
 
 But 1 3 = X ~-~ , m 3 = ^ and so on. Hence substituting 
 
 s s 
 
 in the above equations and expanding, 
 
 v . win . nl , . . Im 
 
 (x a) $ -r-^ -f (y b) (j) ^ + (z c) 9 =-^ = 
 
 ^ ( \ rk^ m ( d\(h mn 
 
 and so on 
 
 are the equations to the plane. There will altogether be 
 
 nj ft ^ ft O 
 
 Q-*- q~T ~ f these equations, but only (p 2) of them 
 
 will be independent. 
 
 If we are given three points 1, 2, 3 in the plane which are 
 not in a straight line, then if s s" be the distances from 1 to 2 
 and to 3 respectively the conditions that the direction to any 
 point x from (abc...) shall be dependent on these directions may 
 be written 
 
 x a y b z c 
 s s s 
 
 l a 2 &! & 2 Cj C2 
 
 s s s f 
 
 and so on. 
 
 This may be reduced to the series of equations 
 
 123 
 
 123 
 123" 
 
 123 
 
 01 
 
 and so on. 
 
 .(9). 
 
CHAPTER VI. 133 
 
 And so we may find the equations to any regular spread. 
 These equations all apply to oblique as well as to rectangular 
 axes. 
 
 If the equation, referred to rectangular axes, 
 
 Ax + By+Cz+ ...... = K ............ (10), 
 
 represent a spread of the order (p 1) in a space of p indepen 
 dent directions, then the directions in which it extends from any 
 point are determined by the equation 
 
 +Cv+ ...... = 0. 
 
 Hence a direction whose coefficients are proportional to 
 (A, B, (7...) is at right angles to every direction in which 
 the spread extends, that is, it is perpendicular to the spread. 
 If therefore (linn...) be this direction, equation 10 may be 
 written 
 
 Now a straight line through the origin in this direction, 
 perpendicular to the spread is represented by 
 
 m n 
 
 Hence the intersection of this with (11) is at a distance s from 
 the origin, given by writing Is for x, ms for y and so on, in (11), 
 that is 
 
 I 2 s + m?s + n 2 5+ 
 
 But 2 + ?rc 2 +?i 2 + ...... =1. 
 
 Hence the equation to a spread the perpendicular from the 
 origin on which is in direction (Imn...) and of length s is 
 
 lx + my + nz + ...... = s ............... (12). 
 
 Similarly we may show that if we have q equations of the 
 first degree representing a regular spread of the order (p - q), 
 that the directions whose coefficients are respectively propor 
 tional to the coefficients of the variables in these equations, are 
 all perpendicular to the spread. And since we have q equations 
 which are independent and consistent, we shall get q directions 
 perpendicular to the spread, which may be shown to be inde- 
 
134 PART III. 
 
 pendent directions, since the equations are independent equa 
 tions. Thus we may have a spread of the q ih order perpendicular 
 to one of the order (p q). 
 
 Since the formulae we found in Chapter V. for the transfor 
 mation of coordinates on moving the origin or rotating the axes 
 in any way, are all formulae of the first degree in the variables, 
 it follows that no movement of the axes can alter the degree of 
 an equation representing a given spread. Hence the degree of 
 an equation is an intrinsic property of the spread it represents. 
 The obvious way therefore of investigating the properties of 
 spreads represented by equations of a given degree is to investi 
 gate whether the equation cannot be simplified by moving the 
 origin or axes. Thus in equations of the second degree we have 
 terms such as # 2 , and such as xy of the second degree. It will 
 be found that terms of the second class can always be got rid of 
 by rotating the axes, and that sometimes terms of the first can, 
 but that both classes can not be got rid of at once. Further, 
 by shifting the origin, it is always possible to get rid of terms 
 of the first degree if corresponding ones of the second degree 
 remain, and if this is not the case it is still possible to get rid 
 of all but one of the terms of the first degree, and of the constant 
 term. In this way spreads of the second degree may be divided 
 into those which can be reduced to terms of the second degree 
 and a constant term only, and those which can not. As in the 
 former class, if any point (abed...) is in the spread, the point 
 ( a b c d.,.) is also in it, this class may be called central 
 spreads, and the others non-central. 
 
 As this book does not profess to be a treatise on geometry 
 of more than three independent directions, I will merely 
 conclude by enumerating the chief characteristics of forms 
 of the second degree, that is spreads represented by equations 
 of the second degree in a space of four independent directions. 
 
 The equation to a central form may be reduced to the form 
 
 If all the upper signs are taken it may be called the elliptic 
 form. Ifa = b = c = d the equation may be written 
 
 a? 2 + 2/ 2 + ,z 2 -|-w 2 = a 2 .................. (14), 
 
 which, if the axes are rectangular, clearly represents a form 
 
CHAPTER VI. 135 
 
 every point in which is at a distance a from the origin, that is, 
 a circular form. 
 
 If one or more of the lower signs is taken in equation (13) 
 we get a hyperbolic form. There are three orders of these. 
 The first and third orders, that is the forms 
 
 are conjugate to each other. That is, they are asymptotic, both 
 to each other, and to the form 
 
 which I have called the ellipto-conical form. 
 
 The second order of hyperbolic forms are self conjugate, that 
 is each of the two forms 
 
 are hyperbolic forms of the second order, and are conjugate to 
 each other, and both asymptotic to the form 
 
 ,___ fl8) 
 
 a 3 + 6 3 c 3 d?~ 
 
 which I have called the hyperbole-conical form. 
 
 It may be shown that there are a number of straight lines, 
 or generating lines as they may be called, wholly in the hyper 
 bolic form of the first order, which do not however pass through 
 every point in the form. The form of the second order may be 
 completely generated by straight lines, as also may the ellipto- 
 conical form. The hyperbolo-conical form may be generated 
 similarly by planes, but there are neither generating planes nor 
 lines in the elliptic form, or in the hyperbolic form of the third 
 order. 
 
 The equation to a non-central form may be reduced to 
 
 If all the upper, or all the lower, signs are taken we have the 
 ellipto-parabolic form ; if the signs of the first three terms are 
 not all the same, the hyperbolo-parabolic form. As the sign of 
 
136 PART III. 
 
 the last term is indeterminate, these are the only material 
 variations possible. 
 
 The hyperbole-parabolic forms 
 
 are conjugate, and asymptotic, both to each other, and to the 
 cylindro-conical form 
 
 There are no generating lines in the ellipto-parabolic form, but 
 the hyperbolo-parabolic may be completely generated by straight 
 lines, and the cylindro-conical by planes. Besides these forms 
 the equation may represent other cylindrical forms, when one or 
 more coordinate vanishes from it, or it may reduce to one or two 
 regular forms, to a single point, or to an impossible locus. 
 
CHAPTER VII. 
 
 WE are now in a position to return to the main question 
 the objective truth or otherwise of my second axiom. 
 
 We have already granted that material space, the space in 
 which material bodies are free to move, does extend from every 
 position in it in a complete group of three independent direc 
 tions. Consequently if it does not extend from all positions in 
 it in the same directions, there must be some number, p, of 
 independent directions, greater than three, on which all the 
 directions in which it extends are dependent. Hence material 
 space is a form of some kind, in a space of p independent direc 
 tions, and its shape is therefore represented by (p 3) equations, 
 in p coordinates. 
 
 We may assume that the origin of the system of coordinates 
 is in material space; that the axes are rectangular; and that 
 the equations are all expressed in rational algebraical form. 
 
 Now there is one thing more that we know about material 
 space, namely that it is a self-congruent space. From this it 
 follows that from every position in material space its shape and 
 size must appear the same, and therefore that if we move the 
 origin to any point in material space it must always be possible, 
 by a suitable rotation of the axes, to reduce the equation to it 
 to the same form it had originally. 
 
 Now if we move the origin to any point (x y z ...), to 
 transform the equation we must write (x + x ) for x, (y + y) 
 for y, and so on, in all the equations. Hence if x n ; x m y n ~ be 
 terms of the highest degree, n, in the old equations, these will 
 become in the new equations, 
 
 and (x n y n ~ m + mx m - l x y n -* n + (n - in) x m yy n - m - 1 + . . .) 
 
138 PART III. 
 
 and the only terms of the highest degree, n, will still be x n , 
 x m y n ~ m , as before. But terms of any lower degree will in general 
 be altered. For each term of the degree n, produces terms of 
 all degrees below n, and these cannot be cancelled by others 
 arising from other terms for every possible movement of the 
 origin. These terms will therefore have to be got rid of by 
 rotating the axes, keeping them still rectangular, which may 
 be done by the formulae (11) or (13) of Chapter V. 
 
 For the sake of simplicity we will suppose the formulae (11) 
 for rotation of two axes at a time to be made use of repeatedly ; 
 that is, we write (Ix + my) for #, and (mx ly) for y (where 
 I 2 + m* = 1). 
 
 But, as the highest terms, of the degree n, were not changed 
 by the translation of the origin, neither must they be changed 
 by the rotation of the axes. Hence these terms must be of the 
 form A (x 2 + y 2 ) or some power of such an expression ; that is 
 
 A (# 2 +i/ 2 ) 2 , so that n must be an even number. But a similar 
 rotation in the axis plane (yz) shows that the terms of the 
 highest order must also be included in the form 
 
 and so on for all the coordinates. Hence each of the (p 3) 
 equations representing material space must be of the form 
 
 n 
 
 A (x 2 + ?/ 2 -M 2 ...) 2 + lower terms = ............ (1), 
 
 where A may in any one equation be zero, but if it is not, all 
 those coordinates appear within the bracket which are capable 
 of variation in material space. If more than one of the equa 
 tions contain terms of the n ih order, we may therefore eliminate 
 such terms from all but one of them, by dividing each of them 
 by their constant factor corresponding to A, and then subtract 
 ing them two and two. We have then one equation of the above 
 form, and (p 4) of lower degrees. 
 
 If we now move the origin to another point in material 
 space (x y z ...... ) the terms of the ?i th degree produce terms 
 
 in all the variables of every degree below the n th , and any 
 other terms in the old equation also produce new terms of 
 lower degrees. Since the old origin was in material space, the 
 old equations contained no constant term, and the condition 
 that the new origin is also in material space eliminates the 
 constant term from the new equation. But there remain terms 
 
CHAPTER VII. 139 
 
 of all degrees up to the (ti - l) th , to be eliminated by the ro 
 tations of the axes. Now these rotations of two axes at a time 
 
 afford us ? p equations, if q is the number of coordinates 
 J. . 
 
 which are capable of variation in material space. Hence the 
 rotation of the axes may be made to satisfy - ^ o conditions, 
 
 but not more. Now if n = 2 there are only terms of the first 
 degree to be eliminated from the equations. And since only q 
 coordinates vary in material space, it must be possible to find 
 constant values from the equations for the other (p q) which 
 values must be zero since the origin is in material space. 
 Hence there can not be more than q terms of the first degree, 
 
 and the ?~~^ conditions will be sufficient to bring them 
 
 l . 
 
 back to what they were in the original equation. But if n is 
 greater than 2, even if it is 4, besides the q terms of the first 
 
 degree, there will be ^ a of the second, and yet more of 
 
 the third degree. Hence it is impossible to bring them all 
 back to what they were originally. Therefore n cannot be 
 greater than 2, and equation (1) may be written 
 
 a? -f ?/ 2 + z z + w- + + terms of the first degree 
 
 = (2). 
 
 The remainder of the equations being of a degree below n 
 are of the first degree. As we have seen (p q) of them may 
 be reduced to the form 
 
 u = 0, (3). 
 
 And now let us revolve the axes so that three of them, x, y, 
 and z, are tangent to material space at the origin. Hence 
 
 -y , and j- must vanish when all the coordinates are equated 
 dx ay dz 
 
 to zero. Therefore there can be no terms of the first degree in 
 x, y, z in the equations. Thus the equations are reduced to 
 the form 
 
 a? + y~ + z~ + w 2 + v- -t- + aw+bv + = CTj (1 equation) 
 
 a^w -f b^v + = (q 4) equations V (4). 
 
 u = (p q) equations] 
 
 Now write (Iw + mv) for w and (mw Iv) for v, rotating the 
 
140 PART III. 
 
 axes of w, v, in their own plane, and determine -- by equating 
 
 lYv 
 
 the coefficient of w in the second of the above equations to zero, 
 that is by writing = - . Similarly rotate the next pair of 
 
 axes so as to eliminate w from the second equation. And as 
 there are (q 4) coordinates besides w in these (q 4) equations 
 we may repeat this process (q 4) times, and so eliminate w 
 from all the (q 4) equations, and leave only (q 4) variables 
 in them. They then suffice to determine constant values for 
 all these variables, which are obviously all zero, and so we have 
 finally left the equations to material space 
 
 x? + ?/ 2 + z- + w 2 + aw = (one equation)) 
 v = 0, u = ...... (p 4) equations}" 
 
 These (p 4) equations represent a regular spread of four 
 independent directions, through the origin. Hence we have 
 nothing more to do with the (p 4) independent directions in 
 which these coordinates are measured, but only with the spread 
 of four independent directions, and the first of the above equa 
 tions. 
 
 Hence, let us move the origin to the point (0, 0, 0, -^ ), in 
 
 \ ^/ 
 
 the spread of four independent directions, this equation becomes 
 
 and consequently : If Material space is not a regular form, the 
 only alternative consistent with the objective facts we have 
 granted is that it should be a Circular Form, and this in how 
 ever many independent directions points in it might be con 
 ceived to vary. 
 
 Mr Chrystal, starting on the definition of a straight line 
 merely as a self-congruent line, came to rather a different 
 conclusion. He deduced a theory of hyperbolic space, which I 
 have now shown to be inconsistent with the assumption that 
 space is self-congruent, which he nevertheless assumes in his 
 demonstration. This might indeed have been indirectly in 
 ferred from the proof given in Part I. of Euclid s 12th Axiom. 
 But I have now further shown that there cannot be more than 
 one kind of elliptic space, of which Mr Chrystal discussed 
 
CHAPTER VII. 141 
 
 several, and that kind is that which he calls double elliptic 
 space, but which is not elliptic, but circular. 
 
 But there are observed facts which prove that, if material 
 space is not a regular form, it is a circular one of such enormous 
 radius that the minute fraction of it of which we know anything 
 at all may without sensible error be considered a regular form, 
 for all practical purposes. 
 
 We have already seen that if material space is a circular 
 form of radius p } in any triangle 
 
 A 
 
 P 2 
 
 That is, the excess of a triangle varies as its area. Now the 
 largest triangle whose angles we could measure would be one 
 inscribed in the earth s orbit. And in such a triangle there 
 cannot be any great excess, or astronomers would ere now have 
 observed it. Therefore its area must be small compared with 
 that of a great sphere. Much more so therefore must the area 
 of those triangles be by which the distance of the sun from the 
 earth is determined. If therefore, as an approximation, we 
 treat these as plane triangles, we may assume that the distance 
 of the sun has been correctly calculated, at some 91 millions of 
 miles, say. To find the area of a triangle inscribed in the earth s 
 orbit, we may, as an approximation, consider it a plane triangle, 
 and the orbit as circular and of radius r. The area of an 
 
 equilateral triangle will then be ^ r 2 . 
 
 Q /Q //*\ 2 
 
 Hence the excess of the triangle is [*) 
 
 4 \pj 
 
 Hence if e be the circular measure of the excess 
 
 3V3 1)4 
 
 I) 
 
 e) 
 
 and so even if so considerable an excess as 10" could have been 
 overlooked, the radius of material space must be at least 160 
 times that of the earth s orbit. As the largest mundane tri 
 angle whose angles have been measured in a trigonometrical 
 survey has probably an area considerably less than that of an 
 equilateral triangle inscribed in a circle of 50 miles radius, the 
 
 excess of such a triangle would only be (srttnttt of tne 
 
142 PART III. 
 
 excess of a triangle inscribed in the earth s orbit, a quite 
 imperceptible quantity. 
 
 But there is another consideration which may assure us 
 that the radius of material space must in all probability be 
 very many times greater than this, if it is not infinite, that is 
 if material space is not a regular form. 
 
 For a long while the fixed stars were believed to have no 
 parallax, even with such a base as the diameter of the earth s 
 orbit. On the assumption that material space is a circular 
 form this might indeed be explained as being due to the excess 
 of the triangles used to measure their distances ; which would 
 mean that they must all lie upon or near the equatorial sphere 
 of which the solar system occupies one pole, and therefore be 
 separated from us by a distance of about a quadrant of a great 
 circle. Inexplicable as such a disposition of the stars in space 
 might seem, it is rendered still more so when we consider that 
 some stars have in recent years been found to have as much 
 parallax as 2", or even more. These stars must therefore be 
 separated by a distance less than a quadrant from us. But 
 in that case how is it that no stars have been found whose 
 distances are a little more than a quadrant ? For such stars 
 would have equal but negative parallaxes, a phenomenon which 
 has never been observed. 
 
 But if we reject this explanation of the smallness of the 
 parallax of the fixed stars, assuming, as seems so much more 
 probable, that they are more or less evenly distributed through 
 out space, then the most distant of them which is visible to us 
 must be many times as distant from us as the nearest, and its 
 distance is not greater than a quadrant of a great circle. Hence 
 the triangles used to determine the parallax of the nearest 
 fixed stars must be approximately plane, and their calculated 
 distances therefore approximately correct. Now the distance 
 of a star with 2" parallax would be about 100,000 times the 
 diameter of the earth s orbit, and consequently a quadrant 
 of circular space must be many times this that is, millions of 
 times the distance from us to the sun. Thus we cannot hope 
 to discover any excess, even in a triangle inscribed in the 
 earth s orbit ! 
 
 Now no objective knowledge can attain more decisive results 
 than this. The inductions on the strength of which we accepted 
 
CHAPTER VII. 143 
 
 my other two axioms as objectively true of material space, are 
 certainly no stronger, and before doubting the objective truth 
 of the remaining one it would therefore be necessary to re 
 consider them. We have then come to the following 
 
 CONCLUSIONS. 
 
 I. THERE EXISTS A SUBJECTIVE GEOMETRY, WHOSE SUBJECTIVE 
 
 CONCLUSIONS ARE NECESSARY TRUTHS. 
 
 II. THAT THE CONCLUSIONS OF THIS GEOMETRY ARE ALSO 
 
 APPLICABLE TO THE OBJECTIVE GEOMETRY OF MATERIAL 
 
 SPACE, is PROVED BY INDUCTIONS AS CONVINCING AS ANY 
 
 WE KNOW OF, EXCEPT PERHAPS THAT WHICH CONVINCES 
 US THAT THERE IS AN OBJECTIVE UNIVERSE AT ALL. 
 
 QUOD ERAT DEMONSTRANDUM. 
 
 THE END. 
 
 CAMBRIDGE: PRINTED BY c. j. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. 
 
14 DAY USE 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 ASTRONOMY. MATHEMATICS- 
 
 This book is due on lthelastme?eafiS^ff Below, or 
 
 on the date to which renewed. 
 Renewed books are subject to immediate recall. 
 
 
 
 LD 21-40m-10, 65 
 (F7763slO)476 
 
 General Library 
 
 University of California 
 
 Berkeley 
 
 
&A C 
 
 . U JLKHS, LEY LIBRARIES 
 
 III