February 26, 1904.] 



SCIENCE. 



329 



how it represents uniform velocity. Show 

 that the results reached at one point on the 

 curve are general and apply equally well 

 to every point and that everywhere on your 

 curve the geometrical tangent and your 

 analytic limit interpret each other and give 

 the rate or velocity of the falling body. 



Note that the tangents are changing, that 

 the corresponding numbers are changing 

 and that these constitute a rate of change 

 of velocities. Show graphically the oblique 

 straight line representing the changing 

 velocities. Give its graphical, its numerical 

 and its nature interpretation. In the same 

 way study the line parallel to the axis of 

 abscissas representing gravity. Study the 

 graphs and their relation to each other. 

 Study the series of numbers resulting from 

 the selection of equal increments along the 

 X-axis, the relation, therefore, of these 

 operations to the theory of number series. 

 Connect the first differential coefScient 

 with the tangents and with rates, the sec- 

 ond with the changes of tangents or of 

 rates of tangents, and thus with the thing 

 in this problem that produces the changes 

 of velocities, that is, with the force of grav- 

 ity. Note the deformation of the original 

 curve if the resistance of the air had been 

 considered and its influence accounted for 

 by some simple laAV. Construct the curve 

 of the body projected upwards. Let up and 

 down destroy each other, so that the ordi- 

 nates at each point will be the algebraic 

 sum of opposite motions. Note the point 

 in the curve when the projected body is for 

 an instant stationary in the air. Observe 

 its connection with the first diiferential 

 coefficient. Note the deformation of the 

 curve due to the resistance of the air acting 

 according to some assumed law. 



Similarly, construct approximately the 

 smooth integral curve which represents 

 the movement of a steam railroad train 

 from station to station fifty miles apart. 

 Connect the contour of the curve with ve- 



locities and with forces, including in the 

 latter the steam in the cylinder, gravity 

 assisting or retarding, friction and air re- 

 sistance always retarding. Note how the 

 second differential coefficient carries us 

 back to steam in the cylinders, the third to 

 the causes leading to a variation of the 

 artificial forces, such as fuel, skill in stok- 

 ing, etc. Pursue maxima and minima prob- 

 lems in the same way. But now, instead 

 of a rate of change directly dependent upon 

 a conventional unit of time, we have rela- 

 tive rates of change and we quickly enlarge 

 our ideas of the meaning and application 

 of the first and second differential coeffi- 

 cient. We can safely begin the formal 

 element of the subject. Even then we 

 should continue the diagram and its in- 

 terpretation, though we may be utterly un- 

 able to set the highly artificial equation 

 over against any definite problem known 

 to exist in nature. 



Just as differentiation always has a sym- 

 bolic interpretation in tangents and rates, 

 so the integration of any expression may 

 be interpreted as the finding of an area. 



From engineering we have a remarkable 

 series of connected quantities and these may 

 be selected, as given by Professor W. K. 

 Hatt in the Railroad Gazette of December 

 23, 1898, for illtistrating the cumulative 

 effect of successive integrations. Five siic- 

 cessive diagrams used in engineering prac- 

 tice are connected by integrations. These 

 are in their order the load diagram, the 

 shear diagram, the moment diagram, the 

 slope diagram and the deflection diagram. 



But it is not necessary to enter further 

 upon specific illustration. The higher 

 analysis is replete with problems which 

 the skilled teacher may use as stepping 

 stones by which he may help the student 

 to pass with safety to higher and higher 

 mathematical attainment. Step by step he 

 masters his method while he is gaining a 



