694 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR 
numerical quantities; but the present mode is in some cases more sim- 
ple, and offers in reality quite as much distinctness when understood. 
The operating mechanism can even be thrown into action indepen- 
dently of any object to operate upon (although of course no result 
could then be developed). Again, it might act upon other things be- 
sides number, were objects found whose mutual fundamental relations 
could be expressed by those of the abstract science of operations, and 
which should be also susceptible of adaptations to the action of the 
operating notation and mechanism of the engine. Supposing, for in- 
stance, that the fundamental relations of pitched sounds in the science 
of harmony and of musical composition were susceptible of such ex- 
pression and adaptations, the engine might compose elaborate and 
scientific pieces of music of any degree of complexity or extent. 
The Analytical Engine is an embodying of the science of operations, 
constructed with peculiar reference to abstract number as the subject 
of those operations. The Difference Engine is the embodying of one 
particular and very limited set of operations, which (see the notation 
used in note B) may be expressed thus,(+,+, +, +, +, +), or thus, 
6(+). Six repetitions of the one operation, +, is, in fact, the whole 
sum and object of that engine. It has seven columns, and a number 
on any column can add itself to a number on the next column to its 
right-hand. So that, beginning with the column furthest to the left, 
six additions can be effected, and the result appears on the seventh 
column, which is the last on the right-hand. The operating mechanism 
of this engine acts in as separate and independent a manner as that of 
the Analytical Engine ; but being susceptible of only one unvarying and 
restricted combination, it has little force or interest in illustration of 
the distinct nature of the science of operations. The importance of 
regarding the Analytical Engine under this point of view will, we think, 
become more and more obvious, as the reader proceeds with M. 
Menabrea’s clear and masterly article. The calculus of operations is 
likewise in itself a topic of so much interest, and has of late years been 
so much more written on and thought on than formerly, that any bear- 
_ing which that engine, from its mode of constitution, may possess upon 
the illustration of this branch of mathematical science, should not be 
overlooked. Whether the inventor of this engine had any such views 
in his mind while working out the invention, or whether he may sub- 
sequently ever have regarded it under this phase, we do not know; 
but it is one that forcibly occurred to ourselves on becoming acquainted 
with the means through which analytical combinations are actually 
attained by the mechanism. We cannot forbear suggesting one prac- 
tical result which it appears to us must be greatly facilitated by the 
independent manner in which the engine orders and combines its ope- 
rations: we allude to the attainment of those combinations into which 
imaginary quantities enter. This is a branch of its processes into 
which we have not had the opportunity of inquiring, and our conjecture 
therefore as to the principle on which we conceive the accomplishment 
of such results may have been made to depend, is very probably not in 
accordance with the fact, and less subservient for the purpose than 
some other principles, or at least requiring the cooperation of others. 
It seems to us obvious, however, that where operations are so indepen- 
dent in their mode of acting, it must be easy by means of a few simple 
