36

HISTORICAL VIEW OF THE

methods and principles were suggested, which have beenof the most important use.

In order, however, to proceed satisfactorily in our ac-count of these and several further investigations, wemust premise some more general considerations.

The proper object of contemplation in geometry isextension, in its several species of length, area, andsolidity. The course which the geometer has to pursue,is to consider, in the first instance, the different com-oinations of the simplest conceptions which producethe different species of extended figures : these willbe lines, straight or curved, and the spaces enclosed be-tween such lines. Extended surface, again, may besuch as to lie all in one plane, or it may be variouslycurved or rounded : under such surfaces, again, will becontained variously shaped solids. The geometer, then,distinguishes all such cases of these as he can reduce toany sort of regularity in their structure or mode offormation. Whenever any such principle of formationhas been assigned, he can proceed to the consequenceswhich will result fror the supposed method of construc-tion, and thus establish the properties belonging to suchfigures. All this, it must be borne in mind, applies tofigures, lines, and magnitudes, such as are the pure crea-tures of intellectual conception, and entirely abstractedfrom all gross physical ideas, derived from our notionsof material substances. The line of the geometer isthe abstract conception of the length of a thing, entirelydiscarding all reference to its breadth, thickness, or anyther property whatsoever ; and so, in like manner, theideas of surface, of solidity, &c. And in a corre-sponding way, the deductions which he makes, and themethod he pursues, for tracing their dimensions, andestablishing their properties, are not like those of actualmechanical measurement or numerical computation, butthe abstract and rigorous logical conclusions from thefirst definitions and assumptions. It is in this way thatthe high superiority of geometry as a school of closereasoning is displayed ; and, if regarded solely as a re-