2l6 



THE GRAMMAR OF SCIENCE 



to the speed of the train. We thus reach two new con- 

 ceptions which need definition and measurement, namely, 

 those of steepness and speed. In Fig. 1 1 we have a 

 horizontal straight line AB, and a sloping line AC. 



c Clearly the greater the angle 



J BAC the steeper AC will be, 

 c |. and the greater will be the 

 f height we shall ascend for the 

 horizontal distance AB. IfAB 

 be loo feet and CB the vertical 

 through B be 20 feet, we shall have ascended 20 feet for 

 a horizontal 100, or since the steepness of AC is the same 

 at all points, we shall ascend 2 feet in 10 feet, or 200 feet 

 in 2000 feet, or \ of a foot in i foot.^ Now, by 

 elementary arithmetic the ratios of 20 to 100, 2 to 10, 

 200 to 2000, and \ to i are all equal and may be 

 expressed by the fraction 1 This is termed the slope of 

 the straight line AC, and is a fitting measure of its steep- 

 ness. The slope is clearly the number of units or the 

 fraction of a unit we have risen vertically for a unit of 

 horizontal distance. If slope be a fit measure of steep- 

 ness for a straight line, we have next to inquire how we 

 can measure the steepness of a curved line. Let A and C 

 in Fig. 1 2 be two points on a 

 curved line, the curve showing 

 no abrupt change of direction 

 at the point A.^ Now draw 

 the line, or so-called chord, 

 AC; then, whether we go 

 up the curve from A to 

 C or along the chord - 

 from A to C, we shall 

 have ascended the same vertical piece CB for the same 

 horizontal distance AB. The slope of the chord AC 



^ This statement depends on the proportionality of the corresponding sides 

 of similar triangles (see Euclid vi. 4). 



2 A must be in the " middle of continuous curvature," as Newton expresses 

 it. This condition is important, but for a full discussion of the steepness of 

 curves we must refer the reader to pp. 44-7 of Clifford's Elenie)its 0/ Dynamic, 

 part i. 



Fig. 12. 



