460 Mr, H. Wedgwood on a System of Geometry 



the conception of sameness of direction at different points of a 

 system ; the species defined in the two first being traced out by 

 a point actually moving in the same direction under certain con- 

 ditions, and in the third by a point moving under the sole con- 

 dition of a negation of motion in a certain constant direction. 



When a point is made to describe a line of any shape, it must 

 begin to move in some one direction ; and as long as the stroke 

 is continued in the same unaltered direction, the result will be a 

 straight line under the proposed definition. So if a straight 

 line AB be given, and it be required to draw a straight line 

 parallel to it through a point C, the definition instructs us to 

 draw the line CD in the same direction from C with that marked 

 by the straight line AB at A. If the problem be to draw a 

 plane through A under the guidance of the definition, a direction 

 AN must be given in which the generating point is to be wholly 

 void of motion on setting out from A ; and if B be any subse- 

 quent point in the plane, the generating point in passing through 

 B must be wholly void of motion in a direction identical with 

 that marked by the line AN at A. 



As the motion of a point along a straight line is constantly 

 directed to the point at the extremity of the line, little difficulty 

 is commonly felt with respect to the first definition, but an am- 

 biguity is suspected when the term is applied to parallel lines. 

 It is asserted that the only meaning of the sameness of direction 

 of two straight lines passing through the points A and B, and 

 having no points in common, is the equality of the angles made 

 with the straight line joining AB ; and it is asserted, that there 

 is a covert assumption that if the straight lines make equal 

 angles with AB, they will make equal angles with a straight line 

 intersecting the parallels at any other points. 



To others, perhaps, the objection does not present itself in so 

 definite a form ; but they feel uncertain whether the expression 

 of ' sameness of direction ' be not used in a somewhat different 

 sense in respect of parallel lines than when applied to successive 

 portions of the same straight line, and in the former case they 

 are inclined to doubt whether the standard of identity of direc- 

 tion be not derived from the notion of parallelism rather than 

 vice versd. 



In answer to these objections, I propose to indicate the sense 

 in which identity of direction is uniformly to be understood in 

 these definitions, and the reasoning founded upon them, by 

 reference to a moveable scale which admits of application to 

 directions measured from different points in a system, and en- 

 ables us to fonn an estimate of their identity or divergence. 



One main cause of the difficulty which has been felt in deter- 

 mining the elementary composition of the simplest species of 



