CHAP. XI.] OF SECONDARY PROPOSITIONS. 175 



way as that of secondary propositions has been established upon 

 the notion of time. Perhaps, had this been done, the analogy 

 which we are contemplating would have been in somewhat closer 

 accordance with the view of those who regard space and time 

 as merely " forms of the human understanding," conditions of 

 knowledge imposed by the very constitution of the mind upon 

 all that is submitted to its apprehension. But this view, while 

 on the one hand it is incapable of demonstration, on the other 

 hand ties us down to the recognition of " place," TO TTOU, as an 

 essential category of existence. The question, indeed, whether 

 it is so or not, lies, I apprehend, beyond the reach of our faculties; 

 but it may be, and I conceive has been, established, that the 

 formal processes of reasoning in primary propositions do not re- 

 quire, as an essential condition, the manifestation in space of the 

 things about which we reason; that they would remain appli- 

 cable, with equal strictness of demonstration, to forms of exis- 

 tence, if such there be, which lie beyond the realm of sensible 

 extension. It is a fact, perhaps, in some degree analogous to this, 

 that we are able in many known examples in geometry and dy- 

 namics, to exhibit the formal analysis of problems founded upon 

 some intellectual conception of space different from that which is 

 presented to us by the senses, or which can be realized by the 

 imagination.* I conceive, therefore, that the idea of space is not 



* Space is presented to us in perception, as possessing the three dimensions 

 of length, breadth, and depth. But in a large class of problems relating to the 

 properties of curved surfaces, the rotations of solid bodies around axes, the vi- 

 brations of elastic media, &c., this limitation appears in the analytical investi- 

 gation to be of an arbitrary character, and if attention were paid to the processes 

 of solution alone, no reason could be discovered why space should not exist in 

 four or in any greater number of dimensions. The intellectual procedure in 

 the imaginary world thus suggested can be apprehended by the clearest light of 

 analogy. 



The existence of space in three dimensions, and the views thereupon of the 

 religious and philosophical mind of antiquity, are thus set forth by Aristotle: 

 MtyeOoQ Se. TO ptv <j> 'iv, ypa/jjurf, TO $' iiri dvo tiriirtdov, TO 5' ITTI Tpia <ra>fj.a' 

 Kat TTaoct Tavra OVK tffnv aXXo jttsysOof, $ia TO Tptd TrdvTa ilvai KOI TO TDIQ 

 rrdvTy. KdOairep yap Qacri /cat ol HvOa-yoptioi, TO TTCLV icai TO, irdvTa TOIQ Tpiaiv 

 tipiaTcti. TeXevrj) yap Kai fiiaov ical px 7 ) TOV &pt6pbv t\f.i T'OV TOV TraVTOQ' 

 TaiJTa dt TOV TYIQ TpidSog. Aio Trapd TTJQ <f>vaui)g tiXrjtyoTte uHnrep vofiovg tKeivrjQ, 

 Kat Trpogrde dyiaTtiaQ xP^/^a T &v Qttiv r< dpiOfjuf TOVT(. De Ccelo, 1. 



