132 THE PRINCIPLES OF SCIENCE. [CHAP. 



We can always prove a proposition, if we find it more 

 convenient, by proving its equivalent. To assert that all 

 not-B's are not-A's, is exactly the same as to assert that all 

 A's are B's. Accordingly we may ascertain that A = AB by 

 first ascertaining that b = ab. If we observe, for instance, 

 that all substances which are not solids are also not capable 

 of double refraction, it follows necessarily that all double 

 refracting substances are solids. We may convince our- 

 selves that all electric substances are nonconductors of 

 electricity, by reflecting that all good conductors do not, 

 and in fact cannot, retain electric excitation. When we 

 come to questions of. probability it will be found desirable 

 to prove, as far as possible, both the original proposition 

 and its equivalent, as there is then an increased area of 

 observation. 



The number of alternatives which may arise in the 

 division of a class varies greatly, and may be any number 

 from two upwards. Thus it is probable that every s lib- 

 stance is either magnetic or diamagnetic, and no substance 

 can be both at the same time. The division then must be 

 made in the form 



A = ABc -I- A&C. 



If now we can prove that all magnetic substances are 

 capable of polarity, say B = BD, and also that all dia- 

 magnetic substances are capable of polarity, C = CD, it 

 follows by substitution that all substances are capable of 

 polarity, or A = AD. We commonly divide the class sub- 

 stance into the three subclasses, solid, liquid, and gas ; and 

 if we can show that in each of these forms it obeys Carnot's 

 thermodynamic law, it follows that all substances obey 

 that law. Similarly we may show that all vertebrate 

 animals possess red blood, if we can show separately that 

 fish, reptiles, birds, marsupials, and mammals possess red 

 blood, there being, as far as is known, only five principal 

 subclasses of vertebrata. 



Our inductions will often be embarrassed by exceptions, 

 real or apparent. We might affirm that all gems are in- 

 combustible were not diamonds undoubtedly combustible. 

 Nothing seems more evident than that all the metals are 

 opaque until we examine them in fine films, when gold and 

 silver are found to be transparent. All plants absorb 

 carbonic acid except certain fungi ; all the bodies of the 



