434 INDUCTION. 



laws ave expressions, are of a kind peculiarly accessible lo the senses, and 

 suggesting eminently distinct images to the fancy. That geometry is a 

 strictly physical science would doubtless have been recognized in all ages, 

 had it not been for the illusions produced by two circumstances. One of 

 these is the characteristic property, already noticed, of the facts of geom- 

 etry, that they may be collected from our ideas or mental pictures of ob- 

 jects as effectually as from the objects themselves. The other is, the de- 

 monstrative character of geometrical truths ; Avhich was at one time sup- 

 posed to constitute a radical distinction between them and physical truths; 

 the latter, as resting on merely probable evidence, being deemed essentially 

 uncertain and unprecise. The advance of knowledge has, however, made 

 it manifest that physical science, in its better understood branches, is quite 

 as demonstrative as geometry. The task of deducing its details from a 

 few comparatively simple principles is found to be any thing but the im- 

 possibility it was once suppos.ed to be ; and the notion of the superior cer- 

 tainty of geometry is an illusion, arising from the ancient prejudice which, 

 in that science, mistakes the ideal data from which we reason, for a pecul- 

 iar class of realities, while the corresponding ideal data of any deductive 

 physical science are recognized as what they really arc, hyj^otheses. 



Every theorem in geometry is a law of external nature, and might have 

 been ascertained by generalizing from observation and experiment, which 

 in this case resolve themselves into comparison and measurement. But it 

 was found practicable, and, being practicable, was desirable, to deduce these 

 truths by ratiocination from a small number of general laws of nature, the 

 certainty and universality of which are obvious to the most careless ob- 

 server, and which compose the first principles and ultimate premises of the 

 science. Among these general laws must be included the same two which 

 we have noticed as ultimate principles of the Science of Number also, and 

 which are applicable to every description of quantity; viz.. The sums of 

 equals are equal, and Things which are equal to the same thing are equal 

 to one another; the latter of which may be expressed in a manner more 

 suggestive of the inexhaustible multitude of its consequences, by the fol- 

 lowing terms : Whatever is equal to any one of a number of equal magni- 

 tudes, is equal to any other of them. To these two must be added, in ge- 

 ometry, a third law of equality, namely, that lines, surfaces, or solid spaces, 

 which can be so applied to one another as to coincide, are equal. Some 

 writers have asserted that this law of nature is a mere verbal definition; 

 that the expression "equal magnitudes " means nothing but magnitudes 

 which can be so applied to one another as to coincide. But in this opinion 

 I can not agree. The equality of two geometrical magnitudes can not dif- 

 fer fundamentally in its nature from the equality of two weights, two de- 

 grees of heat, or two portions of duration, to none of which would this 

 definition of equality be suitable. None of these things can be so applied 

 to one another as to coincide, yet we perfectly understand what we mean 

 when we call them equal. Things are equal in magnitude, as things are 

 equal in weight, when they are felt to be exactly similar in respect of the 

 attribute in which we compare them: and the application of the objects to 

 each other in the one case, like the balancing them with a pair of scales in 

 the other, is but a mode of bringing them into a position in which our 

 senses can recognize deficiencies of exact resemblance that would otherwise 

 escape our notice. 



Along with these three general piinciples or axioms, the remainder of 

 the premises of geometry consists of the so-called definitions: that is to 



