INTRODUCTORY DISCOURSE OF THE 



Nay, tiii* branch of mathematics may be said to apply still more extensively than even the other ; for it 

 baa no relation to space, which geometry has ; and, therefore, it is applicable to cases where figure and size 

 tre wholly out of the question. Thus you can speak of two dreams, or two ideas, or two minds, and can 

 calculate respecting them just as you would respecting so many bodies ; and the properties you find belong- 

 ing to numbers, will belong to those numbers when applied to things that have no outward or visible or 

 perceivable existence, and cannot even be said to be in any particular place, just as much us the same 

 numbers applied to actual bodies which may be seen and touched. 



It is quite otherwise with the science which we are now going to consider, Natural Philosophy. This 

 teaches the nature and properties of actually existing substances, their motions, their connections with each 

 other, and their influence on one another. It is sometimes also called Physics, from the Greek word 

 signifying Nature, though that word is more frequently, in common speech, confined to one particular 

 branch of the science, that which treats of the bodily health. 



We have mentioned one distinction between Mathematics and Natural Philosophy, that the former 

 does not depend on the nature and existence of bodies, which the latter entirely does. Another distinction, 

 and one closely connected with this, is, that the truths which Mathematics teach are necessarily such, they 

 are truths of themselves, and wholly independent of facts and experiments, they depend only upon reason- 

 ing; and it is utterly impossible they should be otherwise 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 

 necettarily, and for ever and ever, be to one another in the exact proportion of the squares of their 

 diameters. It cannot be otherwise ; 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 leas than 4 ; it would be an utter 

 impossibility a contradiction in the very ideas ; and if stated in words, those words would have no sense. 

 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 reasoning 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 ; for instance, 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 through 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 anything but 4 ; or, which is the same thing, that 4 should ever 

 be equal to 3, or 5, or 100, or anything 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, and must be assented to 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 reasoning 

 at all. If a man were shut up in a room with pen, ink, and paper, he might by thinking 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 fundamental properties of matter without 

 observing what goes on around us, and trying experiments upon the nature and motion of bodies. Thus, 

 the man whom we have supposed shut up, could not possibly find out beyond one or two of the very first 

 properties of matter, and those only in a very few cases ; so that he could not tell if these were general 

 properties of all matter or not. He could tell that the objects he touched in the dark were hard and 

 resisted his touch ; that they were extended and were solid : that is, that they had three dimensions, 

 length, breadth, and thickness. He might guess that other things existed besides those he felt, and that 

 those other things resembled what he felt in these properties ; but he could know nothing for certain, and 

 could not even conjecture much beyond this very limited number of qualities. He must remain utterly 

 ignorant of what really exists in nature, and of what properties matter in general has. These properties, 

 therefore, we learn by experience ; they are such as we know bodies to have j they happen to have them 

 they are so formed by Divine Providence as to have them but they might have been otherwise 

 formed ; tho great Author of Nature might have thought fit to make all bodies different in every respect. 



