223 SECTION MECHANICS. 



mind, a solid boundary, and from that comes one of the great 

 material advantages which you have heard described by Sir Joseph 

 Whitworth in accurate end-measurement. The next to that in facility 

 of perception probably is the boundary of a plane by straight lines, 

 or of a line by cross lines, for there is no such thing observable by us 

 as any reasonable approach to a point. The accuracy of lines is not 

 so easily felt as a surface. A surface, if highly polished, probably 

 presents a greater approach to an accurate boundary than anything 

 we can have in the way of lines. On lines there are more or less, 

 roughly especially in regard to material lines, of which I am now 

 speaking two sides to the line. They are not generally well defined, 

 but we can average the line by finding its middle, and in that consists 

 the accuracy of line-measurement. When we come, however, to 

 lines carelessly drawn perhaps, or ill-defined, to which we cannot give 

 great precision, we find we are dealing with a very indefinite thing 

 indeed. For instance, if you take a hard pencil, and draw as fine a 

 line as you can with it on a piece of paper, and put it under a micro- 

 scope, you will find a series of dots scattered about an irregular 

 breadth, more or less as if you had been dropping seed out of a hopper 

 in going across a field. If you do the same with a mathematical pen 

 you will then find under the microscope that you have got something 

 equivalent to a cart track on a road, with a rut on each side. Still, if 

 it is done with great care and precision, you are able to get a veiy 

 definite middle to that line. Now, with surface measurement, as ordi- 

 narily presented, we have a very different thing. The surfaces we have 

 to measure, are generally defined in a very different way, either by an 

 edge, which we have not ourselves had the means of making accurate, 

 or by a line which has been drawn for us with more or less accuracy, 

 generally with very small accuracy, because there is no very good 

 means of drawing with any degree of accuracy any line but a straight 

 line or a circle. Therefore, in this measurement of areas we have not 

 as yet felt the necessity of any instruments at all approaching in pre- 

 cision either the solid measures used and so greatly perfected by the 

 Warden of the Standards, or the still more accurate linear-measures, 

 whether those of the French or English Commission of Standards, or 

 Sir Joseph Whitworth's end-measures. 



To return to the subject of ordinary plane measurement. The first 



