552 
Proceedings of the Royal Society 
number of unknowns, he makes those terms different in sign which 
have only one factor in common. 
The contributions, therefore, which Leibnitz here makes to 
algebra may be looked upon as three in number : — 
(1) A new notation , numerical in character and appearance, for 
individual members of an arranged group of magnitudes ; the two 
numbers which constitute the notation being like the Cartesian 
co-ordinates of a point in that they denote any one of the said 
magnitudes by indicating its position in the group, . . . (i.) 
(2) A rule for forming the terms of the expression which equated 
to zero is the result of eliminating the unknowns from a set of 
simple equations, (ii.) 
(3) A rule for determining the signs of the terms in the said 
result (hi.) 
The last of these is manifestly the least satisfactory. In the first 
place, part of it is awkwardly stated. Making those terms different 
in sign which have only as many factors alike as is indicated by the 
number ivhich is less by one than the number of unknown quantities 
is exactly the same as making those terms different in sign ivhich 
have only two factors different. Secondly, in form it is very unprac- 
tical. The only methodical way of putting it in use is to select a 
term and make it positive ; then seek out a second term, having all 
its factors except two the same as those of the first term, and make 
this second term negative ; then seek out a third term, having all 
its factors except two the same as those of the second term, and 
make this third term positive ; and so on. 
Although there is evidence that Leibnitz continued, in his 
analytical work, to use his new notation for the coefficients of an 
equation (see Letters xi., xii., xiii. of the said correspondence), and 
that he thought highly of it (see Letter viii. “ chez moi c’est une 
des meilleures ouvertures en Analyse ”), it does not appear that by 
using it in connection with sets of linear equations, or by any other 
means, he went further on the way towards the subject with which 
we are concerned. Moreover, it must be remembered that the little 
he did effect had no influence on succeeding workers. So far as is 
known, the passage above quoted from his correspondence with De 
L’Hospital was not published until 1850. Even for some little 
