668 PHILOSOPHICAL TRANSACTIONS. [anNO 1742-3. 



a negative number, the product is positive. When we inquire into the pro- 

 portion of lines in geonnetry, we have no regard to their position or form; and 

 there is no ground for imagining any other proportion between a positive and 

 negative quantity in algebra, or between an increment and a decrement, than 

 that of the absolute quantities or numbers themselves. The algebraic expres- 

 sions, however, are chiefly useful, as they serve to represent the effects of the 

 operations ; and such expressions are not to be supposed equal that involve equal 

 quantities, unless the operations denoted by the signs are the same, or have 

 the same effect. Nor is every expression to be supposed to represent a certain 

 quantity ; for if the -/ — 1 should be said to represent a certain quantity, it 

 must be allowed to be imaginary, and yet to have a real square; a way of speak- 

 ing which it is better to avoid. It denotes only, that an operation is supposed 

 to be performed on the quantity that is under the radical sign. The operation 

 is indeed in this case imaginary, or cannot succeed ; but the quantity that is 

 under the radical sign, is not less real on that account. The author mentions 

 those things briefly, because they belong rather to a treatise of algebra than of 

 fluxions, wherein the common algebra is admitted. 



In order to avoid the frequent repetition of figurative expressions in the alge- 

 braic part, the fluxions of quantities are here defined to be any measures of 

 their respective rates of increase or decrease, while they are supposed to vary 

 (or flow) together. These may be determined by comparing the velocities of 

 points that always describe lines proportional to the quantities, as in the first 

 book; but they may be likewise determined, without having recourse to such 

 suppositions, by a just reasoning from the simultaneous increments or decre- 

 ments themselves. While the quantity a increases by differences equal to a, 

 2a increases by differences equal to la, and, supposing m and n to be invariable, 

 — increases by differences equal to — , and therefore at a greater or less rate 

 than a, in proportion as m is greater or less than n. Thus a quantity may be 

 always assigned that shall increase at a greater or less rate than a, i. e. shall have 

 its fluxion greater or less than the fluxion of a, in any proportion; and a scale 

 of fluxions may be easily conceived, by which the fluxions of any other quan- 

 tities of the same kind may be measured. 



Let B be any other quantity whose relation to a can be expressed by any alge- 

 braic form ; and while a increases by equal successive differences, suppose b to 

 increase by differences that are always varying. In this case, b cannot be sup- 

 posed to increase at any one constant rate; but it is evident, that if b increase 

 by differences that are always greater than the equal successive differences by 

 which — increases at the same time, then b cannot be said to increase at a less 



