710 TRANSLATOR’S NOTES TO M. MENABREA’S MEMOIR. 
nation of quantities, is brought into use during a calculation. We easily 
ascertain this, from the inspection of any vertical column or columns of 
the diagram in which that quantity may appear. Thus, in the present 
case, we see that all the data, and all the intermediate results likewise, 
are used twice, excepting (mx! — m!'n), which is used three times. 
The order in which it is possible to perform the operations for the 
present example, enables us to effect all the eleven operations of which 
it consists, with only three Operation-cards; because the problem is 
of such a nature that it admits of each class of operations being 
performed in a group together; all the multiplications one after an- 
other, all the subtractions one after another, &c. The operations are 
{6(x), 3(—), 2(+)}- 
Since the very definition of an operation implies that there must be 
two numbers to act upon, there are of course two Supplying Variable- 
cards necessarily brought into action for every operation, in order to 
furnish the two proper numbers. (See Note B.) Also, since every ope- 
ration must produce a result, which must be placed somewhere, each ope- 
ration entails the action of a Receiving Variable-card, to indicate the 
proper locality for the result. Therefore, at least three times as many 
Variable-cards as there are operations (not Operation-cards, for these, as 
we have just seen, are by no means always as numerous as the operations) 
are brought into use in every calculation. Indeed, under certain con- 
tingencies, a still larger proportion is requisite ; such, for example, would 
probably be the case when the same result has to appear on more than 
one Variable simultaneously (which is not unfrequently a provision ne- 
cessary for subsequent purposes in a calculation), and in some other 
cases which we shall not here specify. We see therefore that a great dis- 
proportion exists between the amount of Variable and of Operation- 
cards requisite for the working of even the simplest calculation. 
All calculations do not admit, like this one, of the operations of the 
same nature being performed in groups together. Probably very few 
do so without exceptions occurring in one or other stage of the pro- 
gress; and some would not admit it at all.. The order in which the 
operations shall be performed in every particular case, is a very inter- 
esting and curious question, on which our space does not permit us 
fully to enter. In almost every computation a great variety of arrange- 
ments for the succession of the processes is possible, and various con- 
siderations must influence the selection amongst them for the pur- 
poses of a Calculating Engine. One essential object is to choose 
that arrangement which shall tend to reduce to a minimum the time 
necessary for completing the calculation. 
It must be evident how multifarious and how mutually complicated 
are the considerations which the workings of such an engine involve. 
There are frequently several distinct sets of effects going on simultane- 
ously ; all in a manner independent of each other, and yet to a greater — 
or less degree exercising a mutual influence. To adjust each to every — 
other, and indeed even to perceive and trace them out with perfect — 
correctness and success, entails difficulties whose nature partakes to — 
a certain extent of those involved in every question where conditions — 
are very numerous and inter-complicated; such as for instance the — 
estimation of the mutual relations amongst statistical pheenomena, and — 
of those involved in many other classes of facts. A. A. L. 
