REMAINING LAWS OF NATURE. 157 



small a number of original premises : why it is that we can set 

 out from only one characteristic property of each kind of 

 phenomenon, and with that and two or three general truths 

 relating to equality, can travel from mark to mark until we 

 obtain a vast body of derivative truths, to all appearance 

 extremely unlike those elementary ones. 



The explanation of this remarkable fact seems to lie in 

 the following circumstances. In the first place, all questions 

 of position and figure may be resolved into questions of 

 magnitude. The position and figure of any object are deter- 

 mined, by determining the position of a sufficient number of 

 points in it; and the position of any point may be deter- 

 mined by the magnitude of three rectangular co-ordinates, 

 that is, of the perpendiculars drawn from the point to three 

 planes at right angles to one another, arbitrarily selected. By 

 this transformation of all questions of quality into questions 

 only of quantity, geometry is reduced to the single problem 

 of the measurement of magnitudes, that is, the ascertainment 

 of the equalities which exist between them. Now when we 

 consider that by one of the general axioms, any equality, 

 when ascertained, is proof of as many other equalities as 

 there are other things equal to either of the two equals ; and 

 that by another of those axioms, any ascertained equality is 

 proof of the equality of as many pairs of magnitudes as 

 can be formed by the numerous operations which resolve 

 themselves into the addition of the equals to themselves or 

 to other equals; we cease to wonder that in proportion as 

 a science is conversant about equality, it should afford a 

 more copious supply of marks of marks ; and that the sciences 

 of number and extension, which are conversant with little 

 else than equality, should be the most deductive of all the 

 sciences. 



There are also two or three of the principal laws of space 

 or extension which are unusually fitted for rendering one 

 position or magnitude a mark of another, and thereby con- 

 tributing to render the science largely deductive. First ; the 

 magnitudes of inclosed spaces, whether superficial or solid, 

 are completely determined by the magnitudes of the lines 



