SUFFICIENT REASON. 



SUFFICIENT REASON. 



88 J 



called the mxt of lufficieut reason ; and even thin term may appear 

 inatcuratc, for it ihould be the want of any potntU amount of reason. 

 Since, however, all that takes place must hare a sufficient reason 

 (whether we know it or not) for iu happening, and everything which 

 u aaMrted must be capable, if true, of being shown to hare a sufficient 

 rauon, there is no objection to our using the words " want of sufficient 

 uason " in the nose of absolute want of reason, in all matters connected 

 with the exact sciences. If A be equal to B, there must not only be 

 reason, but reason enough for it : anything short of reason enough is no 

 reason at all, and anything short of proof enough in no proof at all. 



The use of the word reason in the statement of thin principle may 

 itself be fairly objected to. We are in the habit of speaking of mathe- 

 matical consequences in the same manner as of those to which the 

 not i. in of cause and effect applies. When one proposition is made to 

 subserve the proof of another, we call the first, one of the reasons of 

 the second, just as we should say that the reason of a flood was the 

 preceding heavy rain. But this mode of speaking must be objection- 

 able if the word reason be used in the same sense in both places. For, 

 first, we are at liberty to deny the effect on denying that cause ; if the 

 rain had not fallen, the flood would not have taken place. But when 

 we say that one mathematical proposition is the reason of another, in 

 which position do we stand if we make an hypothetical denial of the 

 first ! Simply in that of persons who assert a contradiction of terms, 

 and try tu make rational consequences. Thus, the equality of the 

 angles at the base of an isosceles triangle is one of the reasons (so 

 called) why the tangent of a circle is at right angles to the radius ; 

 rationally, the first is one of the simpler propositions, the necessity of 

 which, when seen, helps us to see the necessity of the second and more 

 complicated one. But the necessity of the first is not previous to that 

 of the second, except in the order of our perceptions, when we follow 

 Euclid. Suppose we were to ask, if the angles at the base of the 

 isosceles triangle had not been equal, what effect would that circum- 

 stance have had upon the position of the tangent of a circle ? We 

 might as well inquire what would our geometry have been if two 

 straight lines had been capable of inclosing a space ? We remember a 

 book of arithmetic in which it was gravely asked, by way of exercise 

 for the student, " If 6 had been the third part of 12, what would the 

 quarter of 18 have been ? " a question which can only be paralleled by 

 " If a thing were both to exist and not to exist at one and the same 

 moment, how many other non-existences would therefore become 

 existences t " 



Secondly, the term reason, in the sense of previous'cause, is wrong 

 as applied to mathematical propositions, because when any one is made 

 to prove the second, it generally happens that the second, when 

 granted, may be made to prove the first. Thus [RIGHT ANGLE] of the 

 two propositions, " all right angles are equal," and " two lines which 

 coincide between two points, coincide beyond them," one must be 

 assumed, and the other will then follow : but either may be the one 

 assumed ; the other will follow. Now it is absurd to say that of two 

 things each is the previous cause of the other. The whole of this con- 

 fusion may be remedied by any one who will remember that one pro- 

 position is not the cause of another, but it is our perception of the one 

 which is made the instrument of bringing about our perception of the 

 other. The constitution of our faculties is the previous cause of the 

 necessity of mathematical propositions, but not of one before another, 

 though in arriving at the perception of this necessity our cognisance of 

 the necessity of one is made the previous cause of that of the necessity 

 of another.' To say that B is the consequence of A, is only to say that 

 our knowledge of the truth of B is the consequence of our knowledge 

 of that of A. 



Taking care to use the word reason in the sense just alluded to, we 

 assume that whatever is necessary has a possibility of being shown to 

 be necessary, and that whatever is true has a possibility of being shown 

 to be true. If this be a legitimate assumption, it then follows that 

 whatever it is impossible to show to be true, must be false. But can 

 there be such a thing as a proposition of which there shall be seen, not 

 its falsehood, but the impossibility of demonstrating its truth ? Can 

 there arise a case in which we shall be so completely cognisant of all 

 that may possibly be said for or against an assertion, as to affirm a 

 necessary incapability of demonstration of one side or the other .' 

 Such cases are universally admitted by mathematicians to exist ; and 

 the final assertion which is made on the known impossibility of proving 

 a contradiction, is said to be made on the principle of the want of 

 tvfficitnt reason. But this very dangerous weapon is never put into 

 the hands of a beginner, in mathematics at least. And when we call 

 it a dangerous weapon, we do not deny its utility, but we only state 

 what is well known to every mathematical teacher, that a student who 

 is allowed to proceed one step by this principle will soon ask pe.r- 

 mission to make it the universal solvent of difficulties, and will be 

 quite ready to urge that a proposition cannot be shown to lie fa/ne, in 

 preference to seeking for or following the demonstration that it it true. 

 A beginner can easily admit a sound use of this principle, but can 

 hardly distinguish it from the thousand inaccurate applications which 

 his ignorance will make, if it be left in his own hands. 



But we can imagine we hear it said that this principle, though some- 

 times employed in pure physics, is never introduced into mathematical 

 reasoning except afttr direct demomtralion, in order to confirm the 

 mind of the learner by making him see how difficult it would have 



been to imagine the possibility of any contradiction being successfully 

 maintained against the proposition jiut prurtd. We believe, indeed, 



that this principle is seldom employed, aixl always without necessity, 

 so that we could wish its use were entirely abandoned. But we can 

 show that a tacit appeal to it is sometimes made ; and this is the v, , >i ~t 

 possible mode of employing it. If the principle be dangerous, and 

 liable to be unsoundly used, it should be most carefully stated when 

 it is used. Whenever we see a proposition assumed, not as an r 

 postulate, but in a definition for instance, or as a self-evident truth, wo 

 may trace the operation of this principle on our minds. For instance, 

 take the proposition which is, if there be such a thing in any one pro- 

 position, a digest of all the methods of mathematics, namely, that if the 

 same operations be performed on equal magnitudes, the resulting mag- 

 nitudes are equal. Try to imagine this not true, and want of sufficient 

 reason interferes to prevent success. What can make a difference '! In 

 this question the principle chums to be applied. 



Now, first, in examining the definitions of Euclid, we find an asser- 

 tion of theorems which we can hardly suppose that Euclid overlooked, 

 though it is very possible that the impossibility of imagining other- 

 wise may have been his gtu'de. For instance, the assertion of the 

 equality of the two parts into which a diameter divides a circle, follow- 

 ing immediately upon the definition of a circle ; and the definition of 

 equal solids as those which are contained by the same number of 

 plane figures equal each to each. These and such little matters have 

 been, or may be, corrected ; but we will now point out a use of the 

 principle which exists in our elementary works of the present day in an 

 unacknowledged form. 



In proving the celebrated proposition of Albert Girard relative to 

 the dependence of the area of a spherical triangle upon the sum of its 

 angles, it is assumed that two spherical triangles which have their sides 

 and angles equal, each to each, are equal in area. Now it is easily 

 shown [SYMMETRY] that there may be two such triangles of which it 

 is impossible to make one coincide with the other, nor is any process 

 ever given for dividing each into parts, so that the parts of one may be 

 capable of coinciding with those of the other. Let the angular points 

 of one be placed upon the angular points of the other (which is always 

 possible), and the triangles will not coincide ; in common language, 

 they will bulge in different directions. When the triangles ore so 

 placed, and the common chords drawn, there is no difficulty in seeing 

 that if ever a want of sufficient reason can be granted upon perception, 

 it is for there being any inequality of the areas of the two triangles. 

 And the equality of these areas is accordingly assumed : for instance, 

 in the proposition above alluded to, a pair of unsymmetrically equal 

 triangles always occurs, except when the given triangle is isosceles. 

 And thus the appeal to this principle may be avoided ; for it is easy to 

 make the given triangles into the sum or difference of isosceles triangles, 

 in which each of one set is capable of being actually applied to one of 

 the other. 



Leaving the subject of pure mathematics, let us now consider the 

 application of this principle in physics. We have observed [STATICS] 

 that the line of separation between pure mathematics and the more 

 exact parts of mathematical physics is very slight indeed : this means 

 as to the clearness and fewness of the first principles, and the rigour of 

 the demonstrations. If we cut the link which ties the sciences of 

 statics and dynamics to the properties of the matter which actually 

 exists around us, we may go farther, and say that we have not only 

 pure sciences, but pure sciences in which the principle of the want of 

 sufficient reason is strictly applicable, because it is our own selves who 

 have, by express hypothesis, excluded sufficient reason. In propo- 

 sitions of pure mathematics, we have seen that we cannot invent or 

 deny for any hypothetical purpose ; it and must be, is not and cannot 

 be, are synonymes, in all the truths which these sciences teach. But 

 the properties of matter which are not also those of space, are not, in 

 our conceptions, necessary : we can imagine them other than they are, 

 without any contradiction of ideas. 



We shall now proceed to consider the point mentioned in STATUS, 

 namely, the character of the axioms of that science. Are they " self- 

 evidently true," and " not to be learnt from without, but from within ?" 

 We will not here inquire whether the first must be the second, not 

 being sufficiently clear as to what is meant by knowledge " from 

 without " and knowledge " from within," to enter upon any such inves- 

 tigation. It will do for our purpose to take knowledge " from within " 

 to be a phrase descriptive of such truths as that two straight lines can- 

 not inclose a space, and knowledge " from without " another phrase 

 indicating such truths as are found, say in the facts of political history 

 or geography. Let us separate from the rest one axiom of statics, say 

 " equal weights at the ends of equal arms of a horizontal straight lever 

 balance one another." First, " equal weights " is a synonyme for equal 

 and parallel pressures. We have no objection to placing the idea of 

 pressure on the same footing as that of a straight line, for be the name 

 we give the former conception what it may, it is probable that those 

 powers of communication with the external world which are certainly 

 necessary to the development, at least, of pressure, are not less 

 necessary to that of straightness. Nor arc equal pressures difficult of 

 definition ; let them be those which are interchangeable, so that 

 cither may be put in the place of the other. The rest of the terms of 

 the axiom are geometrical, and to balance each other is to produce no 

 motion, motion, independently of producing causes, being, we think, 



