VOL. XXI.] PHILOSOPHICAL TRANSACTIONS. 443 



solves those wherein it is necessary to consider quantities connected by the 

 signs + and — . And in the 4th, he considers indeterminate Problems. 



He prefixes to his first part some general rules, how to proceed in a geome- 

 tric investigation ; and because these rules contain what is most material in his 

 method, we think it not improper to relate them as he has laid them down 

 hirtself. 



1. An unknown line is always terminated in an unknown point; hence to 

 avoid confusion, the unknown points ought to be denoted with the last letters 

 of the alphabet v, z, y, x, &c. to distinguish them from the known points 

 a, b, c, d, &c. and if there is occasion, one and the same point may be de- 

 noted with two letters, when a known and unknown line concur in it. 



First definition. — Additive ratio is that whose terms are disposed to addition, 

 that is, to composition. Subtractive ratio is that whose terms are disposed to 

 subtraction, that is, to division. 



I i 1- ■ 



a b X c 



Let the line ac, be divided in the points b, and x, the ratio between ab, 

 and bx, is additive; because the terms ab, and bx, compose the whole ax ; 

 but the ratio between ax and bx is subtractive, because the terms ax, and bx, 

 differ by the line ab. 



2. The same order of the letters which is in the figure, ought to be kept 

 in the analysis, that so by mere inspection it may be known whether the ratio 

 is additive or subtractive ; and consequently whether you ought to compose 

 or divide. 



3. When you are to argue by proportions, and the proportion lies in a right 

 line, there is no other way to proceed on but by composition or division ; 

 therefore if both ratios are additive, you must argue by composition ; if both 

 subtractive, by division ; so as always to use that way of arguing which is the 

 fittest for the preservation of those terms that are known ; but when one ratio 

 is additive and the other subtractive, the additive must either be made subtrac- 

 tive, or the subtractive additive ; and this change is made by repeating either 



term. 



1 1 1— _, 



a b c d 



For if we design to change the additive ratio of ab to bd, into subtractive, 

 let be be made equal to ab, and thus the ratio of be to bd, tliat is, of ab to 

 bd, will be subtractive; and likewise, if the subtractive ratio of bd to be was 

 to be made additive, it is but making ab equal to be. 



4. This is always to be observed, when the terms of the ratio which is to be 

 reduced, are known ; but if they are unknown, and their sum or difference is 



3 L 2 



