SOLID MEASUREMENTS. 229 



idea of measurement of plane surface is to reduce it to one or more 

 square units, that is, having obtained its linear dimensions, and 

 taken account of the irregularities of its area, to find out the number 

 of square units contained in it. That is generally done geometrically 

 by cutting it up into strips of some definite shape, either by parallel 

 lines, or else by lines radiating from a point, and then these strips are 

 each separately measured with great ease and with tolerable accuracy ; 

 and the characteristic of the measurement is such that the error on 

 each strip, although noticeable when altogether, is still very small and 

 diminishes very much more rapidly than the number of strips into 

 which the area is divided. Every person accustomed to the quadra- 

 ture of curves is well aware of that. That is to say, if you use any 

 tolerably accurate mode of measuring, and cut up an area, either 

 physically or mathematically, into ten strips, you have a certain definite 

 error of, say, perhaps one-tenth of a square inch. But if you cut it up 

 into 100 pieces, instead of reducing the error by ten, you generally, 

 if your arrangements are well made, reduce it by 100. Now 

 when we come to solid quantities, the ordinary method of dividing 

 them is to cut them up into slices. You add the slices together to 

 make the solid, in the same way as you add the solids together to 

 make the surface. But that is hardly the case in general application. 

 We do not cut these cylinders before us into strips, but we are obliged 

 to have resort to means of replacement, this means being, for various 

 reasons, far less accurate than any of the modes of linear-measure 

 we have just described. But yet in the hands of a gentleman like 

 Mr. Chisholm we can get a degree of accuracy which few persons 

 unaccustomed to consider the thing would dream pf probably far more 

 accuracy than we ordinarily meet with in surface measurement. 



Now, there are also certain other beautiful little instruments called 

 Planimeters, by which a task, which at first sight would hardly be 

 considered conceivable, has been successfully accomplished, namely 

 that by simply running a pen point round any irregular closed curve, 

 of which we wish to measure the area, a little wheel records the area 

 that the pencil -goes round. That seems conceivable enough if we 

 had to deal either with a circle or an ellipse, but it seems almost 

 inconceivable when we have to deal with curves with any amount of 

 irregularity. The most useful of these, as at present arranged, is 



