TOli. XLII.] PHILOSOPHICAL TRANSACTIONS. 635 



was to be continued uniformly from that term without any variation. In order 

 to determine that space, and consequently the velocity which is measured by it, 

 four axioms are proposed concerning variable motions, two concerning motions 

 that are accelerated, and two concerning such as are retarded. The first is, 

 that the space described by an accelerated motion is greater than the space 

 which would have been described in the same time, if it had not been accelerated, 

 but had continued uniform from the beginning of the time. The second is, 

 that the space which is described by an accelerated motion, is less than the 

 space which is described in an equal time by the motion which is acquired by 

 that acceleration continued afterwards uniformly. By these, and two similar 

 axioms concerning retarded motions, the theory of motion is rendered applica- 

 ble to this doctrine with the greatest evidence, without supposing quantities in- 

 finitely little, or having recourse to prime or ultimate ratios. The author first 

 demonstrates from them all the general theorems concerning motion, that are 

 of use in this doctrine ; as, that when the spaces described by two variable mo- 

 tions are always equal, or in a given ratio, the velocities are always equal, or in 

 the same given ratio ; and conversely, when the velocities of two motions are 

 always equal to each other, or in a given ratio, the spaces described by those 

 motions in the same time are always equal, or in that given ratio ; that when a 

 space is always equal to the sum or difference of the spaces described by two 

 other motions, the velocity of the first motion is always equal to the sun) or 

 difference of the velocities of the other motions ; and conversely, that when a 

 velocity is always equal to the sum or difference of two other velocities, the 

 space described by the first motion is always equal to the sum or difference of 

 the spaces described by these two other motions. In comparing motions in this 

 doctrine, it is convenient and usual to suppose one of them uniform ; and it is 

 here demonstrated, that if the relation of the quantities be always determined 

 by the same rule or equation, the ratio of the motions is determined in the 

 same manner, when both are supposed variable. These propositions are de- 

 monstrated strictly by the same method which is carried on in the ensuing 

 chapters for determining the fluxions of the figures. 



In chap. 1, a triangle that has two of its sides given in position, is supposed 

 to be generated by an ordinate moving parallel to itself along the base. When 

 the base increases uniformly, the triangle increases with an accelerated motion, 

 because its successive increments are trapezia, that continually increase. There- 

 fore, if the motion with which the triangle flows, was continued uniformly from 

 any term for a given time, a less space would be described by it than the incre- 

 ment of the triangle which is actually generated in that time by axiom 1, but u 

 greater space than the incremeiit which was actually generated in an equal time 



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