284 MEMOIR OF AUGUSTUS DE MOL'GAN. 



1865. respecting curves of the second order, are applications of the 

 principles of pure Mathematics. They form, in fact, a particular 

 Jnaugiir.il branch of the application of first principles. . . . 



We nmst not mistake and misapprehend these internal appli- 

 cations ; we must not regard them as constituting entirely what 

 we are to turn our attention to. We have several things before 

 us besides these, which are very little attended to. One of these 

 is what may be called Logical Mathematics. We want a great 

 deal of study of the connection of Logic and Mathematics. 

 Where is any consideration of this question to be found ? If I 

 may be allowed to say more on a subject to which I have 

 devoted a good deal of time and thought, I would make a few 

 observations on this very important and yet very much neglected 

 one. 



There is exact Science in two branches : the Analysis of the 

 necessary Laws of Thought, and the Analysis of the necessary 

 Matter of Thought. The necessary Matter of Thought, that 

 without which we cannot think, consists of Space and Time. 

 These exist everywhere, and we can imagine no thought without 

 them. Space and Time are the only necessary Matter of Thought. 

 These form the subject-matter of the Mathematics. The con- 

 sideration of the necessary Laws of Thought, on the other hand, 

 constitutes Logic. These latter have been little studied hitherto, 

 even apart from the study of the necessary matter of thought. 



We mathematicians may very easily improve our reasoning 

 from the very beginning. For, though the Logic that Euclid 

 used is very accurate, there has been no inquiry made with 

 regard to it ; and the consequence is that for two thousand years 

 we have been proving, as we go through the Elements of Geometry, 

 that a thing is itself. That is to say, we have been proving, in 

 the Elements of Geometry, by help of a syllogism, a thing which 

 must be admitted before syllogism itself can be allowed to be 

 valid. Thus, does Euclid not prove that, when there is but one 

 A and but one B, if the A be the B, then the B is the A ? He 

 would not take such a thing as that without appearance of proof. 

 'A thing is itself;' that is the assertion, that is what Euclid 

 would not take without proof ! 



To take an example. Let us suppose that there is a village 

 which contains but one grocer and but one Post-Office. Then, 

 if the grocer's be the Post-Office, the Post-Omce is the grocer's. 

 For, if it be possible, let the Post-Office be somewhere else, say 

 at the chandler's. Then, because the Post-Office is the chandler's 



