Mr. J. Walsh on the Twelfth Book of Euclid. 1 83 

 between two points; and it leads to the conclusion, that it can 

 only be determined by the immediate evidence of the senses 

 that science cannot afford any aid. ' 



The man who knows not A from B, perceives as clearly as 

 the most expert analyst, that when a straight line intersects 

 any one of two parallel straight lines, it will Intersect the other 

 also, produced, if necessary. The mind is mediately led to 

 th^ perception by the idea that a straight line does not bend 

 more to one side than to another. But some one more deeply 

 ofhl 1 ! SCien 'u'- ?V atisfied with the immediate evidence 

 or his senses, thinks he can prove the axiom mediately. He 

 talks of functions and homogeneity: he says angles are num- 

 bers, and that lines are not numbers. And then he thinks he 



thlFZ^t y n T? ei ' S an ? h T°S eneit > '.that two straight lines 

 that ai e not parallel to each other will meet when produced I I 

 cannot perceive the real tendency of such reasoning. It is se- 

 verely reprehended by the illustrious Newton, page 1 7, Motte's 

 Translation of the Principia: « Relative quantities," says he 

 are not the quantities themselves whose names they bear, but 

 tliose sensible measure, of them (either accurate or inaccurate) 

 whrch are commonly used instead of the measured quantities 



vhZTnT 1 T h T d °u dehle the P Urit ? of mathematical and 

 pinlosophical ruths, who confound real quantities themselves 

 with their relations and vulgar measures." Surely angles 

 are no more numbers than are sticks and stones. Physics 

 ion7 eS N t0 the . na , tUre ^ things; geometry into their dela- 

 tions. No magnitude can enter into calculation, but through 



avin^, IT f me ai ' bltr f y baSe of ^parison. The fi?st 



,ts ZJ T ' e n ? tlC f d is ,' that tbe whoIe is grater than 

 its part. The second, that the shortest distance between two 



C 1S I T ig ? lme ;- This axiom ' thou S h more c °mplex 

 than tl e first, as depending on it, yet is not susceptible of any 

 roof beyond the immediate evidence of the senses. The 

 third, that when a straight line intersects any one of two 

 parallel straight lines, it is not parallel to the other. This 

 axiom, though more complex still than the second, as de- 

 pending on the second, is notwithstanding incapable of any 



K't IS pr f sen l ed ^ our Perception of a straight 

 hne, that it does not bend any more to one side than it does 

 to another. Here I would be understood to avoid all dis- 

 cussion about the meaning of words. If I know the idea the 

 ™ U intended to impress, I care not whence it was derived, 

 or why different nations make use of different sounds to im- 

 I" ess the same idea. My object is to free the first principles of 

 geomrtry from sophistical cavilling; principles Sat arfS 

 -. en, to the illiterate, as to the inost profound mathtmatt 

 ' S " The 



