296 mk de morgan, on the general principles of which the 



Multiply both sides by v{{l + t)v + l}, and tbe result is expressed by saying that 

 pendent, 



f'iV tV 



-—.ai{ai + l) is the same whether x be {\ + t)v or . That is, t and v being inde 



JtB « + 1 



f 01 = , k being constant : or m? = c . 



But fx +f ( - j = 1 for all values of .v ; whence k = 1, c = 1. 



whence all the usual forms can be obtained, and the theorem stated at the beginning may be 

 demonstrated. 



I now proceed to apply what has been done to the several cases. 



1. Successive translations of a point. The four postulates are in this case results of 

 thought. The diagonal law is so distinctly selfevident that its deducibility is worthy of note. 



2. Simultaneous translations, A point cannot undergo two translations at once : it 

 cannot be translated from A to B and also from ^ to C at one and the same time, B and C 

 being different points. The usual interpretation of what is called simultaneous translation 

 through AB and AC is translation from A to B while the line AB itself is translated, without 

 change of direction, from AB, to CD its equal and parallel. Thus each point of AB is 

 translated through an equal and parallel of AC, while each point in turn receives, as it were, 

 the moving point. The translation of a point, and that of a line, are tendencies which may 

 be aggregated, and the four postulates are intelligible and true of such tendencies considered 

 together. If by translation we merely understand removal from one to another of two 

 parallel lines, the postulates are still true, and simultaneous translation from each side of a 

 parallelogram to its opposite is translation from one to the other of any two parallels which 

 pass through the ends of the diagonal. It is not necessary, in aggregation of translations, 

 that the motion should be rectilinear : translation over AB may be understood as made by 

 motion over any curve line which has A and B for its two ends. 



The nearest notion to two linear translations of a point, simultaneously made, is as 

 follows. Let AB and AC both be divided into the same infinitely great number of in- 

 finitely small parts, and let translation over the first part of AB be followed by translation 

 over an equal and parallel to the first part oi AC; then the same of the second parts, and so 

 on. If equal subdivisions be made on each line, the diagonal will then be described in a 

 manner which gives a good notion of the pointed branch of a curve : and such continuity as 

 a pointed branch possesses may be affirmed of the mode of making the two translations. For 

 any finite part of time, however small, contains its due proportion of the translation parallel to 

 ABy and of that parallel to AC. 



