lgrobol/openminuscule · Datasets at Hugging Face

text stringlengths 0 24k
This law may be of very various kinds.
We may propose to obtain our results in successive powers of x, in which case the general form would be
C + C1x + C2x2 + &c.;
or in successive powers of n itself, the index of the function we are ultimately to obtain, in which case the general form would be
C + C1n + C2n2 + &c.;
and x would only enter in the coefficients.
Again, other functions of x or of n instead of powers might be selected.
It might be in addition proposed, that the coefficients themselves should be arranged according to given functions of a certain quantity.
Another mode would be to make equations arbitrarily amongst the coefficients only, in which case the several functions, according to either of which it might be possible to develope the nth function of (5.), would have to be determined from the combined consideration of these equations and of (5.) itself.
The algebraical nature of the engine (so strongly insisted on in a previous part of this Note) would enable it to follow out any of these various modes indifferently; just as we recently showed that it can distribute and separate the numerical results of any one prescribed series of processes, in a perfectly arbitrary manner.
Were it otherwise, the engine could merely compute the arithmetical nth function, a result which, like any other purely arithmetical results, would be simply a collective number, bearing no traces of the data or the processes which had led to it.
Secondly, the law of development for the nth function being selected, the next step would obviously be to develope (5.)
itself, according to this law.
This result would be the first function, and would be obtained by a determinate series of processes.
These in most cases would include amongst them one or more cycles of operations.
The third step (which would consist of the various processes necessary for effecting the actual substitution of the series constituting the first function, for the variable itself) might proceed in either of two ways.
It might make the substitution either wherever x occurs in the original (5.), or it might similarly make it wherever x occurs in the first function itself which is the equivalent of (5.).
In some cases the former mode might be best, and in others the latter.
Whichever is adopted, it must be understood that the result is to appear arranged in a series following the law originally prescribed for the development of the nth function.
This result constitutes the second function; with which we are to proceed exactly as we did with the first function, in order to obtain the third function, and so on, n-1 times, to obtain the nth function.
We easily perceive that since every successive function is arranged in a series following the same law, there would (after the first function is obtained) be a cycle of a cycle of a cycle, &c. of operations, one, two, three, up to n-1 times, in order to get the nth function.
We say, after the first function is obtained, because (for reasons on which we cannot here enter) the first function might in many cases be developed through a set of processes peculiar to itself, and not recurring for the remaining functions.
We have given but a very slight sketch of the principal general steps which would be requisite for obtaining an nth function of such a formula as (5.).
The question is so exceedingly complicated, that perhaps few persons can be expected to follow, to their own satisfaction, so brief and general a statement as we are here restricted to on this subject.
Still it is a very important case as regards the engine, and suggests ideas peculiar to itself, which we should regret to pass wholly without allusion.
Nothing could be more interesting than to follow out, in every detail, the solution by the engine of such a case as the above; but the time, space and labour this would necessitate, could only suit a very extensive work.
To return to the subject of cycles of operations: some of the notation of the integral calculus lends itself very aptly to express them: (2.) might be thus written:—
(6.)
$(\div ),\sum (+1)^p(\times,-)$\hspace{1em}or\hspace{1em}$(1),\sum(+1)^p(2,3)$,
where p stands for the variable; (+ 1)p for the function of the variable, that is, for Φp; and the limits are from 1 to p, or from 0 to p-1, each increment being equal to unity.
Similarly, (4.) would be,—
(7.)
\sum (+1)^n\{(\div ),\sum (+1)^p(\times,-)\}
the limits of n being from 1 to n, or from 0 to n-1,
(8.)
or \sum (+1)^n\{(1),\sum (+1)^p(2,3)\}
Perhaps it may be thought that this notation is merely a circuitous way of expressing what was more simply and as effectually expressed before; and, in the above example, there may be some truth in this.
But there is another description of cycles which can only effectually be expressed, in a condensed form, by the preceding notation.
We shall call them varying cycles.
They are of frequent occurrence, and include successive cycles of operations of the following nature:—
(9.)
p(1,2\ldots m),\overline{p-1}(1,2\ldots m),\overline{p-2}(1,2\ldots m)\ldots\overline{p-n}(1,2\ldots m)
where each cycle contains the same group of operations, but in which the number of repetitions of the group varies according to a fixed rate, with every cycle.
(9.) can be well expressed as follows:—
(10.)
\sum p(1,2\ldots m), the limits of p being from p-n to p.
Independent of the intrinsic advantages which we thus perceive to result in certain cases from this use of the notation of the integral calculus, there are likewise considerations which make it interesting, from the connections and relations involved in this new application.
It has been observed in some of the former Notes, that the processes used in analysis form a logical system of much higher generality than the applications to number merely.
Thus, when we read over any algebraical formula, considering it exclusively with reference to the processes of the engine, and putting aside for the moment its abstract signification as to the relations of quantity, the symbols +, ×, &c. in reality represent (as their immediate and proximate effect, when the formula is applied to the engine) that a certain prism which is a part of the mechanism (see Note C.) turns a new face, and thus presents a new card to act on the bundles of levers of the engine; the new card being perforated with holes, which are arranged according to the peculiarities of the operation of addition, or of multiplication, &c.
Again, the numbers in the preceding formula (8.), each of them really represents one of these very pieces of card that are hung over the prism.
Now in the use made in the formulæ (7.), (8.)
and (10.), of the notation of the integral calculus, we have glimpses of a similar new application of the language of the higher mathematics.
Σ, in reality, here indicates that when a certain number of cards have acted in succession, the prism over which they revolve must rotate backwards, so as to bring those cards into their former position; and the limits 1 to n, 1 to p, &c., regulate how often this backward rotation is to be repeated.
A. A. L.
Note F
There is in existence a beautiful woven portrait of Jacquard, in the fabrication of which 24,000 cards were required.
The power of repeating the cards, alluded to by M. Menabrea, and more fully explained in Note C., reduces to an immense extent the number of cards required.
It is obvious that this mechanical improvement is especially applicable wherever cycles occur in the mathematical operations, and that, in preparing data for calculations by the engine, it is desirable to arrange the order and combination of the processes with a view to obtain them as much as possible symmetrically and in cycles, in order that the mechanical advantages of the backing system may be applied to the utmost.
It is here interesting to observe the manner in which the value of an analytical resource is met and enhanced by an ingenious mechanical contrivance.
We see in it an instance of one of those mutual adjustments between the purely mathematical and the mechanical departments, mentioned in Note A. as being a main and essential condition of success in the invention of a calculating engine.
The nature of the resources afforded by such adjustments would be of two principal kinds.
In some cases, a difficulty (perhaps in itself insurmountable) in the one department would be overcome by facilities in the other; and sometimes (as in the present case) a strong point in the one would be rendered still stronger and more available by combination with a corresponding strong point in the other.
As a mere example of the degree to which the combined systems of cycles and of backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes.
Suppose it be required to eliminate nine variables from ten simple equations of the form—
ax0 + bx1 + cx2 + dx3 + ··· = p (1.)
a1x0
+
b1x1 + c1x2 + d1x3 + ··· = p' (2.)
&c. &c. &c. &c.
We should explain, before proceeding, that it is not our object to consider this problem with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution.
The first step would be the elimination of the first unknown quantity x0 between the first two equations.
This would be obtained by the form—
(a1a-aa1)x0 + (a1b-ab1)x1 + (a1c-ac1)x2 +
+ (a1d-ad1)x3 + · · · · · · · · · · · · · · · · · · · · · · · · = a1p-ap1,
for which the operations 10 (×, ×, −) would be needed.
The second step would be the elimination of x0 between the second and third equations, for which the operations would be precisely the same.
We should then have had altogether the following operations:—
10(×, ×, −), 10(×, ×, −) =
20(×, ×, −)
Continuing in the same manner, the total number of operations for the complete elimination of x0 between all the successive pairs of equations would be—
9 · 10(×, ×, −) = 90(×, ×, −)
We should then be left with nine simple equations of nine variables from which to eliminate the next variable x1, for which the total of the processes would be
8 · 9(×, ×, −) =
72(×, ×, −)
We should then be left with eight simple equations of eight variables from which to eliminate x2, for which the processes would be—
7 · 8(×, ×, −) =
56(×, ×, −)
and so on.
The total operations for the elimination of all the variables would thus be—
9·10
+ 8·9 + 7·8 + 6·7
+
5·6 + 4·5 + 3·4 + 2·3 + 1·2 = 330.
So that three Operation-cards would perform the office of 330 such cards.
If we take n simple equations containing n−1 variables, n being a number unlimited in magnitude, the case becomes still more obvious, as the same three cards might then take the place of thousands or millions of cards.
We shall now draw further attention to the fact, already noticed, of its being by no means necessary that a formula proposed for solution should ever have been actually worked out, as a condition for enabling the engine to solve it.
Provided we know the series of operations to be gone through, that is sufficient.
In the foregoing instance this will be obvious enough on a slight consideration.
And it is a circumstance which deserves particular notice, since herein may reside a latent value of such an engine almost incalculable in its possible ultimate results.
We already know that there are functions whose numerical value it is of importance for the purposes both of abstract and of practical science to ascertain, but whose determination requires processes so lengthy and so complicated, that, although it is possible to arrive at them through great expenditure of time, labour and money, it is yet on these accounts practically almost unattainable; and we can conceive there being some results which it may be absolutely impossible in practice to attain with any accuracy, and whose precise determination it may prove highly important for some of the future wants of science, in its manifold, complicated and rapidly-developing fields of inquiry, to arrive at.

This law may be of very various kinds.

We may propose to obtain our results in successive powers of x, in which case the general form would be

C + C1x + C2x2 + &c.;

or in successive powers of n itself, the index of the function we are ultimately to obtain, in which case the general form would be

C + C1n + C2n2 + &c.;

and x would only enter in the coefficients.

Again, other functions of x or of n instead of powers might be selected.

It might be in addition proposed, that the coefficients themselves should be arranged according to given functions of a certain quantity.

Another mode would be to make equations arbitrarily amongst the coefficients only, in which case the several functions, according to either of which it might be possible to develope the nth function of (5.), would have to be determined from the combined consideration of these equations and of (5.) itself.

The algebraical nature of the engine (so strongly insisted on in a previous part of this Note) would enable it to follow out any of these various modes indifferently; just as we recently showed that it can distribute and separate the numerical results of any one prescribed series of processes, in a perfectly arbitrary manner.

Were it otherwise, the engine could merely compute the arithmetical nth function, a result which, like any other purely arithmetical results, would be simply a collective number, bearing no traces of the data or the processes which had led to it.

Secondly, the law of development for the nth function being selected, the next step would obviously be to develope (5.)

itself, according to this law.

This result would be the first function, and would be obtained by a determinate series of processes.

These in most cases would include amongst them one or more cycles of operations.

The third step (which would consist of the various processes necessary for effecting the actual substitution of the series constituting the first function, for the variable itself) might proceed in either of two ways.

It might make the substitution either wherever x occurs in the original (5.), or it might similarly make it wherever x occurs in the first function itself which is the equivalent of (5.).

In some cases the former mode might be best, and in others the latter.

Whichever is adopted, it must be understood that the result is to appear arranged in a series following the law originally prescribed for the development of the nth function.

This result constitutes the second function; with which we are to proceed exactly as we did with the first function, in order to obtain the third function, and so on, n-1 times, to obtain the nth function.

We easily perceive that since every successive function is arranged in a series following the same law, there would (after the first function is obtained) be a cycle of a cycle of a cycle, &c. of operations, one, two, three, up to n-1 times, in order to get the nth function.

We say, after the first function is obtained, because (for reasons on which we cannot here enter) the first function might in many cases be developed through a set of processes peculiar to itself, and not recurring for the remaining functions.

We have given but a very slight sketch of the principal general steps which would be requisite for obtaining an nth function of such a formula as (5.).

The question is so exceedingly complicated, that perhaps few persons can be expected to follow, to their own satisfaction, so brief and general a statement as we are here restricted to on this subject.

Still it is a very important case as regards the engine, and suggests ideas peculiar to itself, which we should regret to pass wholly without allusion.

Nothing could be more interesting than to follow out, in every detail, the solution by the engine of such a case as the above; but the time, space and labour this would necessitate, could only suit a very extensive work.

To return to the subject of cycles of operations: some of the notation of the integral calculus lends itself very aptly to express them: (2.) might be thus written:—

(6.)

$(\div ),\sum (+1)^p(\times,-)$\hspace{1em}or\hspace{1em}$(1),\sum(+1)^p(2,3)$,

where p stands for the variable; (+ 1)p for the function of the variable, that is, for Φp; and the limits are from 1 to p, or from 0 to p-1, each increment being equal to unity.

Similarly, (4.) would be,—

(7.)

\sum (+1)^n\{(\div ),\sum (+1)^p(\times,-)\}

the limits of n being from 1 to n, or from 0 to n-1,

(8.)

or \sum (+1)^n\{(1),\sum (+1)^p(2,3)\}

Perhaps it may be thought that this notation is merely a circuitous way of expressing what was more simply and as effectually expressed before; and, in the above example, there may be some truth in this.

But there is another description of cycles which can only effectually be expressed, in a condensed form, by the preceding notation.

We shall call them varying cycles.

They are of frequent occurrence, and include successive cycles of operations of the following nature:—

(9.)

p(1,2\ldots m),\overline{p-1}(1,2\ldots m),\overline{p-2}(1,2\ldots m)\ldots\overline{p-n}(1,2\ldots m)

where each cycle contains the same group of operations, but in which the number of repetitions of the group varies according to a fixed rate, with every cycle.

(9.) can be well expressed as follows:—

(10.)

\sum p(1,2\ldots m), the limits of p being from p-n to p.

Independent of the intrinsic advantages which we thus perceive to result in certain cases from this use of the notation of the integral calculus, there are likewise considerations which make it interesting, from the connections and relations involved in this new application.

It has been observed in some of the former Notes, that the processes used in analysis form a logical system of much higher generality than the applications to number merely.

Thus, when we read over any algebraical formula, considering it exclusively with reference to the processes of the engine, and putting aside for the moment its abstract signification as to the relations of quantity, the symbols +, ×, &c. in reality represent (as their immediate and proximate effect, when the formula is applied to the engine) that a certain prism which is a part of the mechanism (see Note C.) turns a new face, and thus presents a new card to act on the bundles of levers of the engine; the new card being perforated with holes, which are arranged according to the peculiarities of the operation of addition, or of multiplication, &c.

Again, the numbers in the preceding formula (8.), each of them really represents one of these very pieces of card that are hung over the prism.

Now in the use made in the formulæ (7.), (8.)

and (10.), of the notation of the integral calculus, we have glimpses of a similar new application of the language of the higher mathematics.

Σ, in reality, here indicates that when a certain number of cards have acted in succession, the prism over which they revolve must rotate backwards, so as to bring those cards into their former position; and the limits 1 to n, 1 to p, &c., regulate how often this backward rotation is to be repeated.

A. A. L.

Note F

There is in existence a beautiful woven portrait of Jacquard, in the fabrication of which 24,000 cards were required.

The power of repeating the cards, alluded to by M. Menabrea, and more fully explained in Note C., reduces to an immense extent the number of cards required.

It is obvious that this mechanical improvement is especially applicable wherever cycles occur in the mathematical operations, and that, in preparing data for calculations by the engine, it is desirable to arrange the order and combination of the processes with a view to obtain them as much as possible symmetrically and in cycles, in order that the mechanical advantages of the backing system may be applied to the utmost.

It is here interesting to observe the manner in which the value of an analytical resource is met and enhanced by an ingenious mechanical contrivance.

We see in it an instance of one of those mutual adjustments between the purely mathematical and the mechanical departments, mentioned in Note A. as being a main and essential condition of success in the invention of a calculating engine.

The nature of the resources afforded by such adjustments would be of two principal kinds.

In some cases, a difficulty (perhaps in itself insurmountable) in the one department would be overcome by facilities in the other; and sometimes (as in the present case) a strong point in the one would be rendered still stronger and more available by combination with a corresponding strong point in the other.

As a mere example of the degree to which the combined systems of cycles and of backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes.

Suppose it be required to eliminate nine variables from ten simple equations of the form—

ax0 + bx1 + cx2 + dx3 + ··· = p (1.)

a1x0

+

b1x1 + c1x2 + d1x3 + ··· = p' (2.)

&c. &c. &c. &c.

We should explain, before proceeding, that it is not our object to consider this problem with reference to the actual arrangement of the data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution.

The first step would be the elimination of the first unknown quantity x0 between the first two equations.

This would be obtained by the form—

(a1a-aa1)x0 + (a1b-ab1)x1 + (a1c-ac1)x2 +

+ (a1d-ad1)x3 + · · · · · · · · · · · · · · · · · · · · · · · · = a1p-ap1,

for which the operations 10 (×, ×, −) would be needed.

The second step would be the elimination of x0 between the second and third equations, for which the operations would be precisely the same.

We should then have had altogether the following operations:—

10(×, ×, −), 10(×, ×, −) =

20(×, ×, −)

Continuing in the same manner, the total number of operations for the complete elimination of x0 between all the successive pairs of equations would be—

9 · 10(×, ×, −) = 90(×, ×, −)

We should then be left with nine simple equations of nine variables from which to eliminate the next variable x1, for which the total of the processes would be

8 · 9(×, ×, −) =

72(×, ×, −)

We should then be left with eight simple equations of eight variables from which to eliminate x2, for which the processes would be—

7 · 8(×, ×, −) =

56(×, ×, −)

and so on.

The total operations for the elimination of all the variables would thus be—

9·10

+ 8·9 + 7·8 + 6·7

+

5·6 + 4·5 + 3·4 + 2·3 + 1·2 = 330.

So that three Operation-cards would perform the office of 330 such cards.

If we take n simple equations containing n−1 variables, n being a number unlimited in magnitude, the case becomes still more obvious, as the same three cards might then take the place of thousands or millions of cards.

We shall now draw further attention to the fact, already noticed, of its being by no means necessary that a formula proposed for solution should ever have been actually worked out, as a condition for enabling the engine to solve it.

Provided we know the series of operations to be gone through, that is sufficient.

In the foregoing instance this will be obvious enough on a slight consideration.

And it is a circumstance which deserves particular notice, since herein may reside a latent value of such an engine almost incalculable in its possible ultimate results.

We already know that there are functions whose numerical value it is of importance for the purposes both of abstract and of practical science to ascertain, but whose determination requires processes so lengthy and so complicated, that, although it is possible to arrive at them through great expenditure of time, labour and money, it is yet on these accounts practically almost unattainable; and we can conceive there being some results which it may be absolutely impossible in practice to attain with any accuracy, and whose precise determination it may prove highly important for some of the future wants of science, in its manifold, complicated and rapidly-developing fields of inquiry, to arrive at.