404 



INDUCTION. 



one time supposed to constitute a 

 radical distinction between them and 

 physical truths, the latter, as resting 

 on merely probable evidence, being 

 deemed essentially uncertain and un- 

 precise. The advance of knowledge 

 has, however, made it manifest that 

 physical science, in its better under- 

 stood branches, is quite as demon- 

 strative as geometry. The task of 

 deducing its details from a few com- 

 paratively simple principles is found 

 to be anything but the impossibility 

 it was once supposed to be ; and the 

 notion of the superior certainty of 

 geometry is 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 corresponding 

 ideal data of any deductive physical 

 science are recognised as what they 

 really are, hypotheses. 



Every theorem in geometry is a law 

 of external nature, and might have 

 been ascertained by generalising from 

 observation and experiment, which in 

 this case resolve themselves into com- 

 parison 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 na- 

 ture, the certainty and universality 

 of which are obvious to the most 

 careless observer, and which compose 

 the first principles and ultimate pre- 

 mises 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 appli- 

 cable 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 con- 

 sequences 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 coin- 

 cide, are equal. Some writers have 

 asserted that this law of nature is a 

 mere verbal definition ; that the ex- 

 pression "equal magnitudes" mcan^ 

 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 geo- 

 metrical magnitudes cannot differ fun- 

 damentally in its nature from the 

 equality of two weights, two degrees 

 of heat, or two portions of duration, 

 to none of which would this defini- 

 tion of equality be suitable. None of 

 these things can be so applied to one 

 another as to coincide, yet we per- 

 fectly 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 bring- 

 ing them into a position in which our 

 senses can recognise deficiencies of 

 exact resemblance that would other- 

 wise escape our notice. 



Along with these three general 

 principles or axioms, the remainder 

 of the premises of geometry consists 

 of the so-called definitions : that is 

 to say, propositions asserting the real 

 existence of the various objects there- 

 in 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 nature as 

 straight 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 lines cannot enclose a space,) 

 is as indispensable in geometry, and 

 as evident, resting on as simple, 

 familiar, and universal observation, 



