BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 297 



Students in this Science, to draw up the following Treatise ; wherein 

 I have endeavored to enlarge the Boundaries of Analyticks, and to 

 make some Improvements in the Doctrine of Curved Lines. 



surely a sufficiently modest introduction of perhaps the most 

 important step in the progress of mathematical science. 



Something further as to the evolution of his theory of Fluxions 

 may be indicated, without too much technical detail, by the fol- 

 lowing passages from Brewster : 



Having met with an example of the method of Fermat, in Schoo- 

 ten's Commentary on the Second Book of Descartes, Newton suc- 

 ceeded in applying it to affected equations, and determining the pro- 

 portion of the increments of indeterminate quantities. These incre- 

 ments he called moments, and to the velocities with which the quan- 

 tities increase he gave the names of motions, velocities of increase, and 

 fluxions. He considered quantities not as composed of indivisibles, 

 but as generated by motion ; and as the ancients considered rectangles 

 as generated by drawing one side into the other, that is, by moving 

 one side upon the other to describe the area of the rectangle, so Newton 

 regarded the areas of curves as generated by drawing the ordinate into 

 the abscissa, and all indeterminate quantities as generated by con- 

 tinual increase. Hence, from the flowing of time and the moments 

 thereof, he gave the name of flowing quantities to all quantities which 

 increase in time, that of fluxions to the velocities of their increase, and 

 that of moments to their parts generated in moments of time. 



Newton then proceeds to show the application of the propositions 

 to the solution of the twelve following problems, many of which were 

 at that time entirely new : 



1. To draw tangents to curve lines. 



2. To find the quantity of the crookedness of lines. 



3. To find the points distinguishing between the concave and 

 convex portions of curved lines. 



4. To find the point at which lines are most or least curved. 



5. To find the nature of the curve line whose area is expressed 

 by any given equation. 



6. The nature of any curve line being given, to find other lines 

 whose areas may be compared to the area of that given line. 



7. The nature of any curve line being given, to find its area when 



