of Edinburgh, Session 1885-86. 
555 
the fractions which express the values of the unknowns in a set of 
linear equations, . (iv.) 
(2) A rule for determining the sign of any individual term in the 
said common denominator (and, included in the rule, the notion of a 
“ derangement (in. 2) 
(3) A rule for obtaining the numerators from the expression for 
common denominator, . * » (v.) 
The problem which Cramer set himself at this point in his 
hook was exactly that which Leibnitz had solved, viz., the elimination 
of n quantities from a set of n + 1 linear equations. The solution 
which Cramer obtained, and which, he it remarked, was the solution 
best adapted for his purpose, was quite distinct in character from 
that of Leibnitz. Leibnitz gave a rule for writing out the final 
result of the elimination; what Cramer gives is a rule for writ- 
ing out the values of the n unknowns as determined from n of the 
n + 1 equations, after which we have got to substitute these values 
in the remaining (n+ l)th equation. The notable point in regard to 
the two solutions is, that Cramer’s rule for writing the common 
denominator of the values of the n unknowns (an expression of the 
ftth degree in the coefficients) is exactly Leibnitz’s rule for writing 
th q final result , which is an expression of the (n+ l)th degree. Had 
either discoverer been aware that the same rule sufficed for obtaining 
both of these expressions, he could not have failed, one would 
think, to note the recurrent law of formation of them. The result 
of eliminating w , x , y , z from the equations, 
a r w + + c r y + d^ = e r (r = 1, 2, 3, 4, 5) 
is, according to Leibnitz, if we embody his rule in a later 
symbolism, 
I a f i%dte h | = 0; 
whereas, according to Cramer, it is — 
a 1 e 2^3 C 4^5l _j_ £ \ a 2 e ?-, C jdc\ _j_ c jl^2^8 e 4^5l + g \ a f?, C A e ^ _ e * 
\ a 2 b s G A\ V2VAI l \ a A c A\ V2VAI 15 
and from the collocation of these the one natural step is to the 
identity 
- \af^d^ = aJeAcAl + + • • • • “ g iWA c A\ • 
The fate of Cramer’s rule was very different from that of Leibnitz. 
