480 
Proceedings of the Royal Society 
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 } AGD, 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, “ equal straight lines, being 
applied to one another, coincide,” and “ straight lines, having their 
extremities coincident, coincide.” 
But Mr Mill may say, “ I have only to add, that equal angles 
applied to one another coincide.” 
Very well, you have then three syllogisms : — 
Equal straight lines coincide if 
applied ; 
AC, AB, are equal. 
Equal straight lines coincide if 
applied ; 
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 figures ABE, 
ACD coincide if applied. If Mr Mill still contends that there is 
no need for intuition here, let him substitute for the words “ equal 
straight lines,” “equal arcs of great circles.” 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. 
There are only two courses open to Mr Mill — either to confess 
that the attempt to square geometry with a preconceived theory has 
forced him into a grossly erroneous demonstration, or to invent a 
new formula — viz., that if in two plane figures any number of con- 
secutive sides and angles taken one by one may be made to coincide, 
they may also be made to coincide as rigidly connected wholes. But, 
then, Mr Mill must maintain that the man who reads Euc. I 4 for 
