14 OBJECTS, ADVANTAGES, AND 



tion, and one closely connected with this, is, that the truths which 

 Mathematics teach us are necessarily such, they are truths of them- 

 selves, and wholly independent of facts and experiments, they depend 

 only upon reasoning 1 ; and it is utterly impossible they should be other- 

 wise than true. This is the case with all the properties which we find 

 belong- to numbers and to figures 2 and 2 must of necessity, and 

 through all time, and in every place, be equal to 4 ; those numbers must 

 necessarily be always divisible by 3 without leaving any remainder over, 

 which have the sums of the figures they consist of divisible by 3 ; and 

 circles must necessarily, and for ever and ever, be to one another in the 

 exact proportion of the squares of their diameters. It cannot be other- 

 wise ; we cannot conceive it in our minds to be otherwise. No man 

 can in his own mind suppose to himself that 2 and 2 should ever be 

 more or less than 4 ; it would be an utter impossibility a contradiction 

 in the very ideas. The other properties of number, though not so plain at 

 first sight as this, are proved to be true by reasoning, every one step of 

 which follows from the step immediately before, as a matter of course, 

 and so clearly and unavoidably, that it cannot be supposed or even 

 imagined to be otherwise ; the mind has no means of fancying how it 

 could be otherwise : the final conclusion from all the steps of the rea- 

 soning or demonstration, as it is called, follows in the same way from 

 the last of the steps, and is therefore just as evidently and necessarily 

 true as the first step, which is always something self-evident, as that 2 

 and 2 make 4, or that the whole is greater than any of its parts, but 

 equal to all its parts put together. It is by this kind of reasoning, step 

 by step, from the most plain and evident things, that we arrive at the 

 knowledge of other things which seem at first not true, or at least not 

 generally true ; but when we do arrive at them, we perceive that they 

 are just as true, and for the same reasons, as the first and most obvious 

 matters ; that their truth is absolute and necessary, and that it would 

 be as absurd and self-contradictory to suppose they ever could, under 

 any circumstances, be not true, as to suppose that 2 added to 2 could 

 ever make 3, or 5, or 100, or any thing but 4 ; or, which is the same 

 thing, that 4 should ever be equal to 3, or 5, or 100, or any thing but 4. 

 To find out these reasonings, to pursue them to their consequences, 

 and thereby to discover the truths which are not immediately evident, 

 is what science teaches us ; but when the truth is once discovered, it is 

 as certain and plain by the reasoning, as the first truths themselves 

 from which all the reasoning takes its rise, on which it all depends, 

 and which require no proof because they are self-evident at once, the 

 instant they are understood. 



But it is quite different with the truths which Natural Philosophy 

 teaches. All these depend upon matter of fact ; and that is learnt by 

 observation and experiment, and never could be discovered by reason- 

 ing at all. If a man were shut up in a room with pen, ink, and paper, 

 he might by thought discover any of the truths in arithmetic, algebra, 

 or geometry ; it is possible, at least ; there would be nothing absolutely 

 impossible in his discovering all that is now known of these sciences ; 

 and if his memory were as good as we are supposing his judgment and 

 conception to be, he might discover it all without pen, ink, and paper, 

 and in a dark room. But we cannot discover a single one of the flmda- 



