ON BABBAGE’S ANALYTICAL ENGINE. 693 
ingredient in the science, which they must do in studying the engine. 
The confusion, the difficulties, the contradictions which, in conse- 
quence of a want of accurate distinctions in this particular, have up to 
even a recent period encumbered mathematics in all those branches 
_ involving the consideration of negative and impossible quantities, will 
at once occur to the reader who is at all versed in this science, and 
would alone suffice to justify dwelling somewhat on the point, in 
connexion with any subject so peculiarly fitted to give forcible illus- 
tration of it, as the Analytical Engine. It may be desirable to explain, 
that by the word operation, we mean any process which alters the mutual 
relation of two or more things, be this relation of what kind it may. 
_ This is the most general definition, and would include all subjects in 
_ the universe. In abstract mathematics, of course operations alter those 
particular relations which are involved in the considerations of number 
and space, and the reswlts of operations are those peculiar results which 
correspond to the nature of the subjects of operation. But the science 
_ of operations, as derived from mathematics more especially, is a science 
of itself, and has its own abstract truth and value ; just as logic has its 
_ own peculiar truth and value, independently of the subjects to which 
__ we may apply its reasonings and processes. Those who are accustomed 
_ to some of the more modern views of the above subject, will know 
_ that a few fundamental relations being true, certain other combinations 
of relations must of necessity follow ; combinations unlimited in variety 
_ and extent if the deductions from the primary relations be carried on far 
enough. They will also be aware that one main reason why the sepa- 
rate nature of the science of operations has been little felt,and in general 
little dwelt on, is the shifting meaning of many of the symbols used in 
_ mathematical notation. First, the symbols of operation are frequently 
_ also the symbols of the results of operations. We may say that these 
symbols are apt to have both a retrospective and a prospective signifi- 
cation. They may signify either relations that are the consequence of 
_ aseries of processes already performed, or relations that are yet to be 
_ effected through certain processes. Secondly, figures, the symbols of 
numerical magnitude, are frequently also the symbols of operations, as 
_ when they are the indices of powers. Wherever terms have a shifting 
meaning, independent sets of considerations are liable to become com- 
plicated together, and reasonings and results are frequently falsified. 
Now in the Analytical Engine the operations which come under the 
first of the above heads, are ordered and combined by means of a 
notation and of a train of mechanism which belong exclusively to them- 
_ selves ; and with respect to the second head, whenever numbers mean- 
_ ing operations and not quantities (such as the indices of powers), are 
inseribed on any column or set of columns, those columns immediately 
act in a wholly separate and independent manner, becoming con- 
nected with the operating mechanism exclusively, and re-acting upon 
this. They never come into combination with numbers upon any 
other colunins meaning quantities; though, of course, if there are num- 
Meaning operations upon 7 columns, these may combine amongst 
each other, and will often be required to do so, just as numbers mean- 
ing quantities combine with each other in any variety. It might have 
been arranged that all numbers meaning operations should have ap- 
peared on some separate portion of the engine from that which presents 
