BEMARKS ON BOOK I.] 



MATHEMATICS. PLANE GEOMETRY. 



553 



assent must, in like manner, be given to certain funda- 

 mental and necessary truth-s; these are enumerated in the 

 A slums. You perceive, therefore, that the definitions fur- 

 nish the raw material worked upon, and that the postulates 

 and axioms furnish the implements worked with the 

 postulates supplying the elements of the constructions, 

 the axioms the elements of the reasonings. 



And here we must caution you against a very prevalent 

 mistake. Do not for a moment imagine that Euclid re- 

 quired his postulates to be granted, because the funda- 

 mental operations, under that head, are so easy of per- 

 formance ; nor that his axioms are to be assented to 

 because the truths so-called are so easy of proof. His 

 reasons for these preliminary stipulations were of a 

 directly opposite kind ; he bargains with you to grant 

 the possibility of his fundamental problems (the postu- 

 lates), solely because he is unable, practically, to perform 

 them ; and he calls upon you to admit, without proof, 

 his fundamental theorems (the axioms), solely because 

 he is unable to demonstrate them. A postulate and an 

 axiom should each have a twofold character : a postulate 

 should be a conceivable, but, at the same time, a really 

 impracticable operation ; an axiom should be a self- 

 evident, but, at the same time, an indemonstrable truth. 

 Euclid asks us to " grant that a straight line may be 

 drawn from any one point to any other," from sheer ne- 

 cessity ; the apparent simplicity of the operation is in 

 reality a cause of its difficulty. What operation, still 

 more simple, could be made subsidiary to the drawing of 

 a straight line ? And how could he direct the perform- 

 ;u of the latter, without some operation still more 

 ntary f liusides, an isolated straight line, accord- 

 ing to Euclid's strict definition, has no visible or external 

 existence. Euclid's line is merely the abstraction knyth; 

 a id length, unaccompanied by other dimensions, cannot, 

 of course, be actually exhibited. The finest line that 

 y<m could draw upon paper would be a solid bar of 

 ink ; and the finest line an artist could engrave upon 

 steel would be a sunken channel, with both breadth and 

 depth. You may possibly think that, as the physical or 

 material lines here adverted to are so very slender, it is 

 not worth while to make any objection to them on the 

 score of their width or thickness; but the " near enough," 

 or the "that'll do" system, has no place whatever in 

 "/'* system, which is one of rigid, uncompromising 

 accuracy. A line that you could draw and exhibit would 

 no more be regarded as a line that Euclid had defined, 

 than a beam of timber would be. Remember that with 

 him " a miss is as good as a mile." 



As with the postulates, so with the axioms ; they arc 

 inserted from necessity, and solely because Euclid was 

 unable to demonstrate them. That self-evidence alone 

 was not considered by him as sufficient to justify the 

 claim of a proposition to a place among the axioms, is 

 plain, from his uniform practice of demonstrating what- 

 ever can be demonstrated, and assuming only what can- 

 not ; taking care, however, in general, that the truth 

 assumed shall be, not only an ultimate truth, but also a 

 thing perceived to be true as soon as enunciated. We say, 

 in general, because there is one remarkable exception 

 the 12th axiom of the first book is indemonstrable, but 

 not self-evident ; it has one of the characteristics of an 

 axiom, but not the other. In the preceding book, we 

 have thought it prudent to keep this so-called axiom out 

 of sight, till tho2!)th proposition was reached; because up 

 to that point its aid is not required : and arguments, 

 abundantly sufficient to produce full conviction of its 

 truth, could be adduced then, though they could not 

 have been employed at the opening of the subject. 



Now you must not be surprised or disappointed that 

 geometry pre-eminently the science of demonstrated 

 truth should thus require to rest upon principles which 

 must be gratuitously admitted. No reasoning pr< 

 whatever can even be conceived to exist, unsupported by 

 a like foundation. A proposition may be affirmed on 

 the one side, and denied on the other ; but the matter 

 cannot be reasoned out it cannot be ai-yued, unloss 

 some common first principle or principles be at the 

 commencement agreed to by both parties. If every- 



thing be denied, there may be assertion and contradic- 

 tion, dispute and altercation, but certainly no argument. 

 The notic.-able thing in Euclid's first principles, or ax- 

 ioms, is, that, with the exception mentioned above, they 

 are such as nobody in his senses would think of contro- 

 verting, inasmuch as the truth of them is self-evident ; 

 that is, so immediately obvious, that nothing of the kind, 

 anterior to them in obviousness and simplicity, can pos- 

 sibly be adduced ; for if anything could, then that thing 

 being the more simple and elementary would itself 

 become the axiom, or first principle, by aid of which the 

 former might be demonstrated. You see, therefore, that 

 it is essential to the very nature of an axiom that it 

 should be too simple and elementary to admit of demon- 

 stration by help of anything more simple and elementary. 

 The axioms are, on this account, self-evident, indemon- 

 strable truths. Proposition II., of Book III. namely, 

 that " If any two points be taken in the circumference of 

 a circle, the straight line which joins them shall fall 

 H-'itliin the circle" is a proposition as self-evident, to 

 any one who has a clear conception of a circle, as that 

 which affirms that "two straight lines cannot inclose a 

 space ;" but as it is demonstrable, it ia very properly 

 placed in the body of the work. 



But it is time to give you a few words of advice as to 

 the disposition of mind with which you should sit down 

 to the study of geometry, and to notice some of the 

 intellectual advantages which you have a right to expect 

 from the time and attention devoted to the subject. 



You have already seen what the preliminary conditions 

 are, which Euclid makes with you : he considers these 

 to have been conceded, without the slightest qualification 

 or reserve. As far as he is committed, he will take care 

 that they are faithfully adhered to, and he stipulates 

 that they shall be equally binding upon you. They are 

 fully and fairly placed before you in the Definitions, the 

 Postulates, and the Axioms. In these matters he seems 

 to assume a sort of magisterial authority, from which he 

 allows no appeal. If you refuse to subscribe to the con- 

 ditions which he himself lays down, he, on his part, 

 refuses to be your guide : he can conduct you through the 

 rich domain you wish to explore upon no other terms : 

 he is inflexible as to his preliminary arrangements for 

 the journey ; and he as good as tells you, as he told King 

 Ptolemy of old, that ' ' there is no royal road to geo- 

 metry." 



If, therefore, you have any notion still lingering in 

 your mind, about a line, or a triangle, or a circle, <fec., 

 not strictly in accordance with what he authoritatively 

 declares to be the notion, you must dismiss either it or 

 Aim. If you have any scruples or misgivings about his 

 postulates or axioms, you must, in like manner, over- 

 come them : you must examine them, and re-examine 

 them, till you are fully convinced that the postulates are 

 really conceivable operations, an! that the axioms are 

 really unquestionable truths ; for, depend upon it, he 

 will tie you down most rigidly to the conditions, and 

 allow of no escape, evasion, or qualification. It mat ers 

 nothing to him by what process of mind you satisfy 

 yourself of the truth of his axioms, nor by what me- 

 chanical contrivances you seek to give an outward repre- 

 sentation of lines, triangles, and circles ; you may use 

 pen, ruler, compasses, or whatever you please, for your 

 own individual convenience or assistance ; but remember, 

 that he himself takes no cognizance of these things. If, 

 without any instrumental aid at all, and by mere freedom 

 of hand, you were to sketch the outline of an inclosed 

 figure, and make that the representation, to yourself, of 

 a circle, Euclid would find no fault with you, provided 

 only you still, in imagination, endowed it with the 

 characteristics of the true circle that he had defined. 

 You will find a carpenter, or a mason, much harder to 

 please, in a matter of this kind, than you would find 

 Euclid. If you have carefully read what has preceded, 

 you will see that we are quite justified in making this 

 statement ; for you will have learned that Euclid is not 

 concerned with the representations, but with the thiiigt 

 themselves, the purely intellectual conceptions. 



Taking it, then, as a settled matter, that you receivo 



4 B 



