370 INDUCTION. 



tain 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 being found to be anything 

 but the impossibility it was once supposed to be ; and the notion of 

 the superior certainty of geometry being an illusion arising from the 

 ancient prejudice which in that science mistakes the ideal data from 

 which we reason, for a peculiar class of realities while the correspond- 

 in,f ideal data of any deductive physical science are recognized as 

 what they really are, mere hypotheses. 



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

 have been ascertained by generalizing from observation and experi- 

 ment, 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 was obvious to the most careless observer, and which compose 

 the first principles and ultimate premisses 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 following terms: Whatever is equal to any one of a number of 

 equal magnitudes, is equal to any other of them. To these two must 

 be added, in geometry, 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 cannot agree. The equality of 

 two geometrical magnitudes cannot differ fundamentally in its nature 

 from the equality of two weights, two degrees of heat, or two portions 

 of duration, to none of which would this pretended definition of equal- 

 ity be suitable. None of these things can be so apphed to one another 

 as to coincide, yet we pertectly 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 ob- 

 jects to each other in the one case, like t\ie 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 principles or axioms, the remainder 

 of the premisses of geometry consist of the so-called definitions, that is 

 to say, propositions asserting the real existence of tho various objects 

 therein designated, together with some one property of each. In some 

 cases more than one property is commonly assumed, but in no case is 

 more than one necessary. It is assumed that there are such things in 

 natm'e as sti-aight lines, and that any two of them setting out from the 

 same point, diverge more and more without limit. This assumption, 

 (which includes and goes beyond Euclid's axiom that two straight 



