Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.
There must be some other possibility than death or lifelong penance... some meeting, some intersection of lines; and some cowardly, hopeful geometer in my brain tells me it is the angle at which two lines prop each other up, the leaning-together from the vertical which produces the false arch. For lack of a keystone, the false arch may be as much as one can expect in this life. Only the very lucky discover the keystone.
There must be some other possibility than death or lifelong penance ... some meeting, some intersection of lines; and some cowardly, hopeful geometer in my brain tells me it is the angle at which two lines prop each other up, the leaning-together from the vertical which produces the false arch. For lack of a keystone, the false arch may be as much as one can expect in this life. Only the very lucky discover the keystone.
The object of geometry in all its measuring and computing, is to ascertain with exactness the plan of the great Geometer, to penetrate the veil of material forms, and disclose the thoughts which lie beneath them? When our researches are successful, and when a generous and heaven-eyed inspiration has elevated us above humanity, and raised us triumphantly into the very presence, as it were, of the divine intellect, how instantly and entirely are human pride and vanity repressed, and, by a single glance at the glories of the infinite mind, are we humbled to the dust.
To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.