6 LECTURES AND ESSAYS [1869- 



find it to be that two triangles, having two sides of each 

 equal, are equal in all respects ; and from this theorem 

 he may at once conclude, by his own fourth formula 

 [&quot; angles having their sides coincident, coincide &quot;], that 

 angles contained by equal straight lines are equal ! It is 

 clear that Mr. Mill did not see that the point is to show 

 that, the triangles ABE, ACD, remaining rigid, AB may 

 be applied to AC, and AE to AD at the same time. But 

 this can only be brought out by figuring to oneself AB 

 moved round to coincide with AC, and then the triangle 

 ABE rotated about AB through two right angles ; and 

 this process was not competent to Mr. Mill, whose theory 

 bound him to prove the equality of the triangles by pure 

 syllogism from the two formulas, &quot; equal straight lines, 

 being applied to one another, coincide,&quot; and &quot; straight 

 lines, having their extremities coincident, coincide.&quot; 



But Mr. Mill may say, &quot; I have only to add, that equal 

 angles applied to one another coincide.&quot; 



Very well, you have then three syllogisms : 

 Equal straight lines coincide I Equal straight lines coincide 



if applied ; if applied ; 



AC, AB, are equal. AD, AE, are equal. 



Equal angles coincide if applied ; 



CAD, BAE, are equal. 



Logically these three syllogisms can give only three 

 independent conclusions : 



AC, AB coincide if applied. | AD, AE coincide if applied. 



The angles CAD, BAE coincide if applied : 

 but by no means the ONE conclusion that the rigid figure 

 ABE, ACD coincide if applied. If Mr. Mill still contends 

 that there is no need for intuition here, let him sub 

 stitute for the words &quot; equal straight lines,&quot; &quot; equal arcs 

 of great circles.&quot; The premises of his syllogisms are still 

 all right ; but, owing to circumstances that must be seen 

 to be understood, the spherical triangles cannot be made 

 to coincide. 



