PRESENTMENT AND PROOF IN GEOMETRY • A 

 STUDY OF THE ASSOCIATED CIRCLES OF A 

 TRIANGLE. 



By Rev. Frederick Charles Kolbe, B.x\., D.D. 



There has been for a long time an earnest endeavour to 

 l)etter the teaching of Geometry. The movement has not, I 

 thiiik, gained (juite as much success as many of its enthusiasts 

 had hoped for. If we may judge by the most-used text-books 

 v.'hich the movement has protkiced, there seems to be some 

 mental confusion as to the i)hilosophy and psychology of the 

 sul»ject. or at least there has Ijeen some inconsistency in the 

 a])plicati()n of the psychological principles. 



1. We nuist hrst note a marked distinction in Mathematics 

 itself corresponding to our root ideas of Space and Time. We 

 always remain in physical touch with Space : Time, or Number, 

 tends to become purely abstract. Working with sense-realities 

 and working with symbols are two dilierent processes, and 

 often a mind that loves the one will not love the other. I 

 would almost say that Algebra is purr .Mathematics, while ( ieo- 

 metrv is applied — (|uite as much api)lied as the Tlieory of the 

 Laws of Motion. Of course ( leometry, hand in hand with 

 Algebra, l)eci)mes a powerful instrument of symbolic calculation: 

 even the Greeks, having no algebraic notation, made skilful use 

 of it for this purpose, as we see in Euclid's Second and Fifth 

 Books. But this was not the aspect of Geometry that Plato 

 valued for education. It was because it is the first and most 

 fundamental exercise of the idealising power, selecting from 

 Nature and iliinking away the complexities and irregularities 

 and imperfections, and so taking the first stei:)S towards that 

 reaching after the One in the Many, the Absolute amid the 

 contingent, the Infinite and Eternal within the limited and 

 transient, whicii is tlie work of Philosophy. By its keeping 

 hold of the tangible it goes beyond mere logic: it is, therefore, 

 something more than proofs and puzzles. It is a Knowledge 

 as well as a Process, and my conn)laint is that the process side 

 of it is exaggerated. 



Symbolic calculation, like formal Logic, lakes no account 

 of any relation between the contents of its terms. You have 

 to inter])ret the realities into symbols, and with these you work 

 away with the indifiference of a machine. If no error is made, 

 your result will be consistent with the data, and you have then 

 to retranslate it into realities. But the process is often wider 



