769 



AXIOM. 



AXIS. 



770 



themselves are much clearer than the axioms of psychology on which 

 the opposition to them is grounded. For it is not to be supposed 

 that the opponents of axioms take first principles which are more evi- 

 dent than that " the whole is greater than its part," or that " two 

 straight lines cannot inclose a space." 



The necessity that there should be some axioms is evident from the 

 process of reasoning. The deduction of propositions from the com- 

 parison of other propositions must have a beginning somewhere, so 

 that there must be at least two propositions to begin with, the evidence 

 of which is derived from other sources than reasoning. Every attempt 

 which has been made to dispense with axioms altogether, has, as might 

 be expected, proved unsuccessful ; somewhere or other in the process 

 assumed theorems have been found. 



The more modern discussions which have arisen about axioms appear 

 to us to proceed from some fallacy of this sort, that the idea conveyed 

 by the whole of a sentence must be more complicated than that con- 

 veyed by any one of its parts ; or at least, that it must always be 

 necessary to enter separately upon the consideration of the auxiliary 

 forms of speech in which a simple idea is conveyed, before that idea 

 can be said to be explained. As an instance, in that most simple of 

 all propositions, " two and two are the same as four," which by itself is 

 comprehended as soon as spoken, we have the (by itself) difficult phrase 

 " are the same," implying identity, and leading, if pursued far enough, 

 to many very abstruse metaphysical considerations. These, in their 

 proper science, and considered with reference to other objects, are not 

 misplaced ; but, as applied to geometry, are not only unnecessary, but 

 subversive of the natural order of reasoning ; for however much may 

 be said upon maxims, axioms, first principles, or by whatever name 

 they may be called, there remains the simple proposition, " two and 

 two are the same as four," clearer, as a whole, than any one of the 

 explanations, illustrations, or comments, which have been brought to 

 ite aid. There is however this to be said for many writers who have 

 endeavoured to make such points better known than they are already ; 

 namely, that the older writers, in their love of what is called the 

 A priori method, had filled their books with notions against which it 

 was necessary to contend ; whence sprung a confirmed habit of reason- 

 ing xipon the nature of self-evident propositions. Locke (book iv. 

 chap. 7), ' On Maxima,' can hardly be intelligible to a reader who has 

 not some knowledge of what the school writers have said upon our 

 simplest perceptions, which rendered it necessary to contend both 

 against words without meaning, as when they said some such thing as 

 that " knowledge is the likeness of the thing known, formed in the 

 knowing faculty ; " and also against assumptions of a very dubious 

 character, such as " general propositions are known, at least sometimes, 

 before particular ones." 



All the oldest manuscripts of Euclid, the summary of Boethius, the 

 commentary of Proclus, the Arabic translations, and the earlier 

 European editions, agree in what is no doubt Euclid's plan, of distin- 

 guishing assumptions distinctly relating to space, under the name of 

 piaiulates (cuT^^aTo), from assumptions which equally relate to other 

 kinds of magnitude, under the name of common nations (xoivai (vvouu). 

 We cannot find out who first made the alteration which Robert 

 Simeon has adopted : it appears in Gregory's Greek text. This modern 

 alteration converts the postulate into an assumed problem, and the 

 axiom into an assumed theorem ; but the distinction of propositions 

 into problems and theorems does not exist in Euclid's work ; it is an 

 addition of editors. The more recent Greek texts have returned to 

 Euclid's distinction, and we hope translations will in time follow them. 

 We give Euclid's collection of postulates and common notions at 

 length. 



Puttulatet. 1. Let it be granted, from any point to any point, to 

 draw a straight line. 2. Also, to lengthen a finished straight line, and 

 continue it straight. 3. Also, with any centre and radius (JiArnjjia, 

 meaning interval measured from that centre) to describe a circle. 4. Also, 

 all right angles are equal to one another. 5. Also, if a straight line, 

 falling upon two straight lines, make the angles which are within and 

 upon the same side less than two right angles, the two straight lines, 

 being lengthened without end, shall meet one another upon that side 

 on which the angles are less than two right angles. 6. Also, two 

 straight lines cannot inclose a space. 



Common Notions. 1. Tilings equal to the same are equal to one 

 another. 2. Also, if equals be added to equals, the wholes are equals. 



3. Also, if from equals equals be taken, the remainders are equals. 



4. Also, if to unequals equals be added, the wholes are unequals. 



5. Also, if from unequals equals be taken, the remainders are un- 

 equals. 6. Also, things which are double of the same are equal 

 to one another. 7- Also, things which are halves of the same are 

 equal to one another. 8. Also, things which fit one another (have 

 the same boundary) are equals. Also, the whole is greater than 

 the part. 



Euclid has not stated all. the properties of space which he takes 

 fo granted. It is our belief that his work was not written for ele- 

 mentary students, but was a controversial treatise on the question, 

 Can geometry be formed into a demonstrated system, resting upon 

 definite postulates ? We imagine that when he collected his postu- 

 lates, six in number, and put them forward at the head of the first 

 book, he did not thereby intend to collect everything which he 

 assumed, but only his own selection from the theorems the postulation 



ARTS ASD BCI. DIV. VOL. I. 



of which had been matter of discussion. The theorems which all 

 parties had admitted without questisj among postulates, we suspect 

 him to have left unnoticed. In the following list will be found 

 Euclid's postulates, with the substitutions we have recommended 

 above, and all the postulates which are tacitly assumed ; giving, we 

 believe, a full account of all the theorems relating to space and figure 

 only, which the student of the first six books will be required to 

 assume : 



1. Any two points may be joined by a straight line. 2. Any termi- 

 nated straight line may be indefinitely lengthened. 3. A circle may be 

 drawn with any centre, and any distance terminated at that centre as a 

 radius. 4. Any point is within or without a circle, according as its 

 distance from the centre is less or greater than the radius. 5. A line 

 drawn from a point within a figure to a point without, cuts the 

 boundary of the figure. 6. A straight line which passes through a 

 point within a figure, will, if sufficiently produced, cut the boundary of 

 the figure in two points, one on each side of the point. 7. A figure 

 may be removed without any alteration of figure from one part of the 

 plane to another, and may be turned round before removal. 8. If two 

 straight lines coincide in two points, they coincide altogether, both 

 between the points and beyond them. 9. A straight line being indefi- 

 nitely produced both ways, any line drawn from a point on one side of 

 it to a point on the other, must cut the straight line. 10. Two lines 

 which cut one another cannot both be parallel to any third line. 11. If 

 a smaller area be cut out from a larger, the area left is the same from 

 whatever part of the larger the smaller may be taken. 



It would hardly be possible to make a list of all the " common 

 notions " which Euclid employs. The postulates, or notions concerning 

 space and figure, are the things on which it is most important to dwell 

 with precision. 



What is required to be conceded in the first three postulates, is not 

 that a straight line or circle can be imagined to be^lrawn, in the sense 

 usually attached to these words, but that the geometrical line can be 

 drawn, which is lenyth without breadth. This is impossible, mechanically 

 speaking, the line being a conception of the mind which cannot be 

 executed. [LINE.] 



The last of the " postulates " is a self-evident property of the straight 

 line, a term incapable of other definition than that which is contained 

 in its properties ; that is, we can make no use of the obvious notion 

 conveyed in the words " straight line," unless we admit some one or 

 other of its distinguishing characteristics, which is more definite than 

 saying that it lies evenly between its extreme points. We might 

 appear to avoid an axiom by saying, let the name " straight " line be 

 given to that species, no two of which can, under any circumstances, 

 inclose a space ; but in that case we should need another axiom 

 namely, we should require it to be granted that there is such a thing 

 as the straight line so defined, and that we have not assumed any 

 contradiction in supposing the above species of lines to exist. It must 

 be remembered, that though the definitions are placed at the beginning 

 in Euclid, it is not thereby implied that the terms defined are really 

 possible. " Let lines which, being in the same plane, do not meet, 

 though ever so far produced, be called parallels," does not mean us to 

 assume that such lines do exist, but only, that when they shall have 

 been proved to exist, the name by which it is agreed to call them 

 has been given. But some of the definitions, which ought therefore to 

 be distinguished from the rest, are tacitly accompanied by the assump- 

 tion of existence of the things defined. 



The 4th postulate is a theorem of more difficulty than the subject 

 requires ; since, with one additional assumption respecting the straight 

 line, it admits of proof. The assumption previously discussed, namely, 

 that two straight lines cannot enclose a space, amounts to assuming 

 that if two straight lines coincide in two points, or if two different 

 points of the one can be made to lie upon two different points of the 

 other, the portions of the straight lines lying between these points will 

 also coincide entirely. Let it be granted, in addition, that the parts 

 which are not between these points will coincide (an equally evident 

 proposition), and the 4th postulate of Euclid admits of proof. Euclid's 

 editors, in taking this postulate for granted, make use of it to prove our 

 additional assumption, which, as they phrase it, is " no two lines can 

 have a common segment ; " that is, two lines cannot coincide between 

 two points and not coincide elsewhere. But, of two propositions, one 

 of which it is found necessary to assume, that one should be the more 

 simple of the two. 



The 5th postulate, which is a theorem of some difficulty, neither 

 self-evident, nor even easily made evident, is not at all required in the 

 form given, even in Euclid. For he proves, without its assistance, 

 that if the two lines there mentioned meet, it must be on the side on 

 which the angles are less than two right angles. But it may be reduced 

 to a very evident form as follows : If a straight line be taken, and a 

 point exterior to it, of all the straight lines which can be drawn through 

 the point, one only will be parallel to the first-mentioned straight line. 

 The whole assumption lies in the word only ; for Euclid shows, without 

 the help of this axiom, that a parallel can be drawn, and how to 

 draw it. 



This axiom is the cause of the celebrated discussion on the theory of 

 PARALLELS, under which head it will be more fully treated. 



AXIS, AXE. This word is used in so many different senses, that it 

 may be defined as follows : Any line whatsoever which it is convenient 



3D 



