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Ada Augusta, |
Countess of Lovelace |
NOTES BY THE TRANSLATOR |
Note A |
The particular function whose integral the Difference Engine was constructed to tabulate, is |
\Delta^7u_x=0 |
The purpose which that engine has been specially intended and adapted to fulfil, is the computation of nautical and astronomical tables. |
The integral of |
\Delta^7u_x=0 |
being uz = a+bx+cx2+dx3+ex4+fx5+gx6, |
the constants a, b, c, &c. are represented on the seven columns of discs, of which the engine consists. |
It can therefore tabulate accurately and to an unlimited extent, all series whose general term is comprised in the above formula; and it can also tabulate approximatively between intervals of greater or less extent, all other series which are capable of tabulation by the Method of Differences. |
The Analytical Engine, on the contrary, is not merely adapted for tabulating the results of one particular function and of no other, but for developing and tabulating any function whatever. |
In fact the engine may be described as being the material expression of any indefinite function of any degree of generality and complexity, such as for instance, |
F(x, y, z, log x, sin y, x p, &c.), |
which is, it will be observed, a function of all other possible functions of any number of quantities. |
In this, which we may call the neutral or zero state of the engine, it is ready to receive at any moment, by means of cards constituting a portion of its mechanism (and applied on the principle of those used in the Jacquard-loom), the impress of whatever special function we may desire to develope or to tabulate. |
These cards contain within themselves (in a manner explained in the Memoir itself) the law of development of the particular function that may be under consideration, and they compel the mechanism to act accordingly in a certain corresponding order. |
One of the simplest cases would be for example, to suppose that |
F(x, y, z, &c. &c.) |
is the particular function |
\Delta^nu_z=0 |
which the Difference Engine tabulates for values of n only up to 7. |
In this case the cards would order the mechanism to go through that succession of operations which would tabulate |
uz = a + bx + cx2 + ··· + mxn−1 |
where n might be any number whatever. |
These cards, however, have nothing to do with the regulation of the particular numerical data. |
They merely determine the operations to be effected, which operations may of course be performed on an infinite variety of particular numerical values, and do not bring out any definite numerical results unless the numerical data of the problem have been impressed on the requisite portions of the train of mechanism. |
In the above example, the first essential step towards an arithmetical result would be the substitution of specific numbers for n, and for the other primitive quantities which enter into the function. |
Again, let us suppose that for F we put two complete equations of the fourth degree between x and y. |
We must then express on the cards the law of elimination for such equations. |
The engine would follow out those laws, and would ultimately give the equation of one variable which results from such elimination. |
Various modes of elimination might be selected; and of course the cards must be made out accordingly. |
The following is one mode that might be adopted. |
The engine is able to multiply together any two functions of the form |
a + bx + cx2 + ··· + pxn. |
This granted, the two equations may be arranged according to the powers of y, and the coefficients of the powers of y may be arranged according to powers of x. |
The elimination of y will result from the successive multiplications and subtractions of several such functions. |
In this, and in all other instances, as was explained above, the particular numerical data and the numerical results are determined by means and by portions of the mechanism which act quite independently of those that regulate the operations. |
In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated. |
It is well to draw attention to this point, not only because its full appreciation is essential to the attainment of any very just and adequate general comprehension of the powers and mode of action of the Analytical Engine, but also because it is one which is perhaps too little kept in view in the study of mathematical science in general. |
It is, however, impossible to confound it with other considerations, either when we trace the manner in which that engine attains its results, or when we prepare the data for its attainment of those results. |
It were much to be desired, that when mathematical processes pass through the human brain instead of through the medium of inanimate mechanism, it were equally a necessity of things that the reasonings connected with operations should hold the same just place as a clear and well-defined branch of the subject of analysis, a fundamental but yet independent ingredient in the science, which they must do in studying the engine. |
The confusion, the difficulties, the contradictions which, in consequence 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 illustration 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 results 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 separate 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 signification. |
They may signify either relations that are the consequences of a series 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 complicated 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 themselves; and with respect to the second head, whenever numbers meaning operations and not quantities (such as the indices of powers) are inscribed on any column or set of columns, those columns immediately act in a wholly separate and independent manner, becoming connected with the operating mechanism exclusively, and re-acting upon this. |
They never come into combination with numbers upon any other columns meaning quantities; though, of course, if there are numbers meaning operations upon n columns, these may combine amongst each other, and will often be required to do so, just as numbers meaning quantities combine with each other in any variety. |
It might have been arranged that all numbers meaning operations should have appeared on some separate portion of the engine from that which presents numerical quantities; but the present mode is in some cases more simple, and offers in reality quite as much distinctness when understood. |
The operating mechanism can even be thrown into action independently of any object to operate upon (although of course no result could then be developed). |
Again, it might act upon other things besides 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 instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression 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 bearing 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 subsequently 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 practical result which it appears to us must be greatly facilitated by the independent manner in which the engine orders and combines its operations: 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 independent in their mode of acting, it must be easy, by means of a few simple provisions, and additions in arranging the mechanism, to bring out a double set of results, viz.—1st, the numerical magnitudes which are the results of operations performed on numerical data. |
(These results are the primary object of the engine.) |
2ndly, the symbolical results to be attached to those numerical results, which symbolical results are not less the necessary and logical consequences of operations performed upon symbolical data, than are numerical results when the data are numerical. |
If we compare together the powers and the principles of construction of the Difference and of the Analytical Engines, we shall perceive that the capabilities of the latter are immeasurably more extensive than those of the former, and that they in fact hold to each other the same relationship as that of analysis to arithmetic. |
The Difference Engine can effect but one particular series of operations, viz. |
that required for tabulating the integral of the special function |
\Delta^nu_z=0; |
and as it can only do this for values of n up to 7, it cannot be considered as being the most general expression even of one particular function, much less as being the expression of any and all possible functions of all degrees of generality. |
The Difference Engine can in reality (as has been already partly explained) do nothing but add; and any other processes, not excepting those of simple subtraction, multiplication and division, can be performed by it only just to that extent in which it is possible, by judicious mathematical arrangement and artifices, to reduce them to a series of additions. |
The method of differences is, in fact, a method of additions; and as it includes within its means a larger number of results attainable by addition simply, than any other mathematical principle, it was very appropriately selected as the basis on which to construct an Adding Machine, so as to give to the powers of such a machine the widest possible range. |
The Analytical Engine, on the contrary, can either add, subtract, multiply or divide with equal facility; and performs each of these four operations in a direct manner, without the aid of any of the other three. |
This one fact implies everything; and it is scarcely necessary to point out, for instance, that while the Difference Engine can merely tabulate, and is incapable of developing, the Analytical Engine can either tabulate or develope. |
The former engine is in its nature strictly arithmetical, and the results it can arrive at lie within a very clearly defined and restricted range, while there is no finite line of demarcation which limits the powers of the Analytical Engine. |
These powers are co-extensive with our knowledge of the laws of analysis itself, and need be bounded only by our acquaintance with the latter. |
Indeed we may consider the engine as the material and mechanical representative of analysis, and that our actual working powers in this department of human study will be enabled more effectually than heretofore to keep pace with our theoretical knowledge of its principles and laws, through the complete control which the engine gives us over the executive manipulation of algebraical and numerical symbols. |
Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connexion together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst: those who thus think on mathematical truth as the instrument through which the weak mind of man can most effectually read his Creator's works, will regard with especial interest all that can tend to facilitate the translation of its principles into explicit practical forms. |
The distinctive characteristic of the Analytical Engine, and that which has rendered it possible to endow mechanism with such extensive faculties as bid fair to make this engine the executive right-hand of abstract algebra, is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs. |
It is in this that the distinction between the two engines lies. |
Nothing of the sort exists in the Difference Engine. |
We may say most aptly, that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves. |
Here, it seems to us, resides much more of originality than the Difference Engine can be fairly entitled to claim. |
We do not wish to deny to this latter all such claims. |
We believe that it is the only proposal or attempt ever made to construct a calculating machine founded on the principle of successive orders of differences, and capable of printing off its own results; and that this engine surpasses its predecessors, both in the extent of the calculations which it can perform, in the facility, certainty and accuracy with which it can effect them, and in the absence of all necessity for the intervention of human intelligence during the performance of its calculations. |
Its nature is, however, limited to the strictly arithmetical, and it is far from being the first or only scheme for constructing arithmetical calculating machines with more or less of success. |
The bounds of arithmetic were however outstepped the moment the idea of applying the cards had occurred; and the Analytical Engine does not occupy common ground with mere “calculating machines.” |
It holds a position wholly its own; and the considerations it suggests are most interesting in their nature. |
In enabling mechanism to combine together general symbols in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science. |
A new, a vast, and a powerful language is developed for the future use of analysis, in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind than the means hitherto in our possession have rendered possible. |