176 INDUCTION. 



8. It is a matter of something more than curi- 

 osity to consider to what peculiarity of the physical 

 truths which are the subject of geometry, it is owing 

 that they can all be deduced from so small a number 

 of original premisses : 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 is determined, by determining the posi- 

 tion of a sufficient number of points in it ; and the 

 position of any point may be determined by the mag- 

 nitude of three rectangular co-ordinates, that is, of 

 the perpendiculars drawn from the point to three axes 

 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 ascer- 

 tained, 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 opera- 

 tions 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 



