OpenLLM-France/wikimedia huggingface dataset:
# Economic Sophisms/244

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"surrounded, prevent our producing directly, and with reference to which we are placed in reality in the hypothetical position we have been examining with reference to iron. We produce at home neither tea, coffee, gold, nor silver. Is our industry
en masse diminished in consequence? No; only in order to create the counter-value of these imported commodities, in order to acquire them by means of exchange, we detach from our national labour a portion less great than would be required to produce these things ourselves. More labour thus remains to be devoted to the procuring of other enjoyments. We are so much the richer and so much the stronger. All that external competition can do, even in cases where it puts an end absolutely to a determinate branch of industry, is to economize labour, and increase our productive power. Is this, in the case of the foreigner, the road to domination?
If we should find in France a gold mine, it does not follow that it would be for our interest to work it. Nay, it is certain that the enterprise would be neglected if each ounce of gold absorbed more of our labour than an ounce of gold purchased abroad with cloth. In this case we should do better to find our mines in our workshops. And what is true of gold is true of iron.
The illusion proceeds from our failure to see one thing, which is, that foreign superiority never puts a stop to national industry, except under a determinate form, and under that form only renders it superfluous by placing at our disposal the result of the very labour thus superseded. If men lived in diving-bells under water, and had to provide themselves with air by means of a pump, this would be a great source of employment. To throw obstacles in the way of such employment, as long as men were left in this condition, would be to inflict upon them a frightful injury. But if the labour ceases because the necessity for its exertion no longer exists, because men are placed in a medium where air is introduced into their lungs without effort, then the loss of that labour is not to be regretted, except in the eyes of men who obstinately persist in seeing in labour nothing but labour in the abstract.
It is exactly this kind of labour which machinery, commercial freedom, progress of every kind, gradually supersedes;
# Economic Sophisms/247

<pagequality level="4" user="Zoeannl" /> style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"immediately, and as a necessary consequence of this very cheapness, there would rise up a thousand other branches of industry more profitable than the one which had been superseded.
We arrive, then, at the conclusion that domination by labour is impossible, and a contradiction in terms, seeing that all superiority which manifests itself among a people means cheapness, and tends only to impart force to all other nations. Let us banish, then, from political economy all terms borrowed from the military vocabulary: to fight with equal weapons, to conquer, to crush, to stifle, to be beaten, invasion, tribute, etc. What do such phrases mean? Squeeze them, and you obtain nothing. … Yes, you do obtain something; for from such words proceed absurd errors, and fatal and pestilent prejudices. Such phrases tend to arrest the fusion of nations, are inimical to their peaceful, universal, and indissoluble alliance, and retard the progress of the human race.
THE END.
EDINBURGH: PRINTED BY OLIVER AND BOYD.
# Economic Sophisms/248

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# Economic Sophisms/242

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"
"IN the same way that in time of war we attain the mastery by superiority in arms, do we not, in time of peace, arrive at domination by superiority in labour?"
This is a question of the highest interest at a time when no doubt seems to be entertained that in the field of industry, as in the field of battle, the stronger crushes the weaker.
To arrive at this conclusion, we must have discovered between the labour which is applied to commodities and the violence exercised upon men, a melancholy and discouraging analogy; for why should these two kinds of operations be thought identical in, their effects, if they are essentially different in their own nature?
And if it be true that in industry, as in war, predominance is the necessary result of superiority, what have we to do with progress or with social economy, seeing that we inhabit a world where everything has been so arranged by Providence that one and the same effect—namely, oppression—proceeds necessarily from two opposite principles?
With reference to England's new policy of commercial freedom, many persons make this objection, which has, I am convinced, taken possession of the most candid minds among us: "Is England doing anything else than pursuing the same end by different means. Does she not always aspire at universal supremacy? Assured of her superiority in capital and labour, does she not invite free competition in order to stifle
Continental industry, and so put herself in a situation to reign as a
# Economic Sophisms/245

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"not useful labour, but labour become superfluous, without object, and without result. On the contrary, protection sets that sort of useless labour to work; it places us again under water, to bring the air-pump into play; it forces us to apply for gold to the inaccessible national mine, rather than to the national workshops. All the effect is expressed by the words, deperdition of forces.
It will be understood that I am speaking here of general effects, not of the temporary inconvenience which is always caused by the transition from a bad system to a good one. A momentary derangement accompanies necessarily all progress. This may be a reason for making the transition gently and gradually. It is no reason for putting a stop systematically to all progress, still less for misunderstanding it.
Industry is often represented as a struggle. That is not a true representation of it, or only true when we confine ourselves to the consideration of each branch of industry in its effects upon similar branches, regarding them both in thought apart from the interests of the rest of mankind. But there is always something else to be considered, namely, the effects upon consumption, and upon general prosperity.
It is an error to apply to trade, as is but too often done, phrases which are applicable to war.
In war the stronger overcomes the weaker.
In industry the stronger imparts force to the weaker. This entirely does away with the analogy.
Let the English be as powerful and skilful as they are represented, let them be possessed of as large an amount of capital, and have as great a command of the two great agents of production, iron and fuel, as they are supposed to have; all this simply means cheapness. And who gains by the cheapness of products? The man who buys them.
It is not in their power to annihilate any part whatever of our national labour. All they can do is to render it superfluous in the production of what is acquired by exchange, to furnish us with air without the aid of the pump, to enlarge in this way our disposable forces, and so render their alleged domination as much more impossible as their superiority becomes more incontestable.
# Economic Sophisms/250

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# Economic Sophisms/243

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"sovereign, having conquered the privilege of feeding and clothing the population she has ruined?"
It would not be difficult to demonstrate that these alarms are chimerical; that our alleged inferiority is much exaggerated; that our great branches of industry not only maintain their ground, but are actually developed under the action of external competition, and that tie infallible effect of such competition is to bring about an increase of general consumption, capable of absorbing both home and foreign products.
At present, I desire to make a direct answer to the objection, leaving it all the advantage of the ground chosen by the objectors. Keeping out of view for the present the special case of England and France, I shall inquire in a general way whether, when, by its superiority in one branch of industry, a nation comes to outrival and put down a similar branch of industry existing among another people, the former has advanced one step towards domination, or the latter towards dependence; in other words, whether both nations do not gain by the operation, and whether it is not the nation which is outrivalled that gains the most.
If we saw in a product nothing more than an opportunity of bestowing labour, the alarms of the protectionists would undoubtedly be well-founded. Were we to consider iron, for example, only in its relations with ironmasters, we might be led to fear that the competition of a country where it is the gratuitous gift of nature would extinguish the furnaces of another country where both ore and fuel are scarce.
But is this a complete view of the subject? Has iron relations only with those who make it? Has it no relations with those who use it? Is its sole and ultimate destination to be produced? And if it is useful, not on account of the labour to which it gives employment, but on account of the qualities it possesses, of the numerous purposes to which its durability and malleability adapt it, does it not follow that the foreigner cannot reduce its price, even so far as to render its production at home unprofitable, without doing us more good in this last respect, than harm in the other?
Pray consider how many things there are which foreigners,
by reason of the natural advantages by which they are
# Economic Sophisms/249

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"
Template:Rule
of Profits and Prices: Including an Examination of the Principles which determine the relative Value of Corn, Labour, and Currency.
Template:Fs90/s "We shall greatly mistake the rank or character of this treatise, if we but regard it as a successful exposition of its first and early rudiments for the benefit of those who are yet schoolboys on the subject. It is something far higher than an exposition, — ^it is a rectification of first principles ; for the author grapples with some of die first and fundamental doctrines of those who have been esteemed the greatest masters in the science, and, as we think, overturns them. Such are the changes which he advocates on the alphabetic characters of the science as lead to different readings and different lessons throughout, and will help onward to the establishment of a different school from any of those which now are most prized and most resorted to."—DR CHALMERS, IN NORTH BRITISH REVIEW. Template:Fs90/e
Template:Rule
of the Precious Metals: With Historical Notices of the Effects of the American Mines on European Prices in the Sixteenth, Seventeenth,
Template:Fs90/s "Toutes les questions de fait et de doctrine qui se rattachent à la découverte des nouvelles mines d'or ont été étudiées par M. Stirling avec sagesse et sobriété. Son livre est un des meilleurs recueils de faits et d'idées qui aient été publiés sur cette question."—JOURNAL DES ÉCONOMISTES.
M. Stirling fait remarqner que la dépréciation des métaux précieux est déterminée moins par les quantités extraites que par la réduction des prix de revient. Cette demonstration est l'originalite de son livre."—REVUE DES DEUX MONDES. Template:Fs90/e
Template:Rule
HARMONIES of POLITICAL ECONOMY. By FRÉDÉRIC BASTIAT. Translated from the French, with a Notice of his Life and Writings. Part I. 8vo, 7s. 6d.
HARMONIES of POLITICAL ECONOMY, Part II, with Notes and an Index to both Parts. 8vo, 68. 6d.
Template:Fs90/s "Bastiat speaks with the greatest force to the highest order of intellects. At the same time, he is almost the only Political Economist whose style is brilliant and fascinating, whilst his irresistible logic is relieved by sallies of wit and humour which make his Sophismes as amusing as a novel."—RICHARD COBDEN. Template:Fs90/e
Template:Rule
# Economic Sophisms/25

<pagequality level="4" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"
The obstacle mistaken for the cause,—scarcity mistaken for abundance,—this is the same sophism under another aspect; and it is well to study it in all its phases.
Man is originally destitute of everything.
Between this destitution and the satisfaction of his wants, there exist a multitude of obstacles which labour enables us to surmount. It is curious to inquire how and why these very obstacles to his material prosperity have come to be mistaken for the cause of that prosperity.
I want to travel a hundred miles. But between the starting-point and the place of my destination, mountains, rivers, marshes, impenetrable forests, brigands—in a word, obstacles—interpose themselves; and to overcome these obstacles, it is necessary for me to employ many efforts, or, what comes to the same thing, that others should employ many efforts for me, the price of which I must pay them. It is clear that I should have been in a better situation if these obstacles had not existed.
On his long journey through life, from the cradle to the grave, man has need to assimilate to himself a prodigious quantity of alimentary substances, to protect himself against the inclemency of the weather, to preserve himself from a
# Economic Sophisms/246

<pagequality level="3" user="Zoeannl" />style="background: #ececec; text-align: left; padding-left: 0.5em; font-weight: bold;" class="table-rh"Thus, by a rigorous and consoling demonstration, we arrive at this conclusion, that labour and violence, which are so opposite in their nature, are not less so in their effects.
All we are called upon to do is to distinguish between labour annihilated, and labour economized.
To have less iron because we work less, and to have less iron although we work less, are things not only different, but opposed to each other. The protectionists confound them; we do not. That is all.
We may be very certain of one thing, that if the English employ a large amount of activity, labour, capital, intelligence, and natural forces, it is not done for show. It is done in order to procure a multitude of enjoyments in exchange for their products. They most certainly expect to receive at least as much as they give. What they produce at home is destined to pay for what they purchase abroad. If they inundate us with their products, it is because they expect to be inundated with ours in return. That being so, the best means of having much for ourselves is to be free to choose between these two modes of acquisition, immediate production, and mediate production. British Machiavelism cannot force us to make a wrong choice.
Let us give up, then, the puerility of applying to industrial competition phrases applicable to war,—a way of speaking which is only, specious when applied to competition between two rival trades. The moment we come to take into account the effect produced on the general prosperity, the analogy disappears.
In a battle, every one who is killed diminishes by so much the strength of the army. In industry, a workshop is shut up only when what it produced is obtained by the public from another source and in greater abundance. Figure a state of things where for one man killed on the spot two should rise up full of life and vigour. Were such a state of things possible, war would no longer merit its name.
This, however, is the distinctive character of what is so absurdly called industrial war.
Let the Belgians and the English lower the price of their iron ever so much; let them, if they will, send it to us for nothing;
this might extinguish some of our blast-furnaces; but Template:Hws
# Engineering Analysis/MISO Transformations

Consider now a system with two inputs, both of which are random (or pseudorandom, in the case of non-deterministic data). For instance, let's consider a system with the following inputs and outputs:
* X: non-deterministic data input
* Y: disruptive noise
* Z: System output
Our system satisfies the following mathematical relationship:
    $Z=g(X,Y)$
Where g is the mathematical relationship between the system input, the disruptive noise, and the system output. By knowing information about the distributions of X and Y, we can determine the distribution of Z.
# Engineering Analysis/Matrix Exponentials
## Matrix Exponentials

If we have a matrix A, we can raise that matrix to a power of e as follows:
    $e^{A}$
It is important to note that this is not necessarily (not usually) equal to each individual element of A being raised to a power of e. Using taylor-series expansion of exponentials, we can show that:
    $e^{A}=I+A+{\frac {1}{2}}A^{2}+{\frac {1}{6}}A^{3}+...=\sum _{k=0}^{\infty }{1 \over k!}A^{k}$.
In other words, the matrix exponential can be reduced to a sum of powers of the matrix. This follows from both the taylor series expansion of the exponential function, and the cayley-hamilton theorem discussed previously.
However, this infinite sum is expensive to compute, and because the sequence is infinite, there is no good cut-off point where we can stop computing terms and call the answer a "good approximation". To alleviate this point, we can turn to the Cayley-Hamilton Theorem. Solving the Theorem for A$^n$, we get:
    $A^{n}=-c_{n-1}A^{n-1}-c_{n-2}A^{n-2}-\cdots -c_{1}A-c_{0}I$
Multiplying both sides of the equation by A, we get:
    $A^{n+1}=-c_{n-1}A^{n}-c_{n-2}A^{n-1}-\cdots -c_{1}A^{2}-c_{0}A$
We can substitute the first equation into the second equation, and the result will be A$^{n+1}$ in terms of the first n - 1 powers of A. In fact, we can repeat that process so that A$^m$, for any arbitrary high power of m can be expressed as a linear combination of the first n - 1 powers of A. Applying this result to our exponential problem:
    $e^{A}=\alpha _{0}I+\alpha _{1}A+\cdots +\alpha _{n-1}A^{n-1}$
Where we can solve for the α terms, and have a finite polynomial that expresses the exponential.

## Inverse

The inverse of a matrix exponential is given by:
    $(e^{A})^{-1}=e^{-A}$

## Derivative

The derivative of a matrix exponential is:
    ${\frac {d}{dx}}e^{Ax}=Ae^{Ax}=e^{Ax}A$
Notice that the exponential matrix is commutative with the matrix A. This is not the case with other functions, necessarily.

## Sum of Matrices

If we have a sum of matrices in the exponent, we cannot separate them:
    $e^{(A+B)x}\neq e^{Ax}e^{Bx}$

## Differential Equations

If we have a first-degree differential equation of the following form:
    $x'(t)=Ax(t)+f(t)$
With initial conditions
    $x(t_{0})=c$
Then the solution to that equation is given in terms of the matrix exponential:
    $x(t)=e^{A(t-t_{0})}c+\int _{t_{0}}^{t}e^{A(t-\tau )}f(\tau )d\tau$
This equation shows up frequently in control engineering.

## Laplace Transform

As a matter of some interest, we will show the Laplace Transform of a matrix exponential function:
    ${\mathcal {L}}[e^{At}]=(sI-A)^{-1}$
We will not use this result any further in this book, although other books on engineering might make use of it.
# Engineering Analysis/Matrices
## Derivatives

Consider the following set of linear equations:
    $a=bx_{1}+cx_{2}$
    $d=ex_{1}+fx_{2}$
We can define the matrix A to represent the coefficients, the vector B as the results, and the vector x as the variables:
    $A={\begin{bmatrix}b&c\\e&f\end{bmatrix}}$
    $B={\begin{bmatrix}a\\d\end{bmatrix}}$
    $x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}$
And rewriting the equation in terms of the matrices, we get:
    $B=Ax$
Now, let's say we want the derivative of this equation with respect to the vector x:
    ${\frac {d}{dx}}B={\frac {d}{dx}}Ax$
We know that the first term is constant, so the derivative of the left-hand side of the equation is zero. Analyzing the right side shows us:

## Pseudo-Inverses

There are special matrices known as pseudo-inverses, that satisfies some of the properties of an inverse, but not others. To recap, If we have two square matrices A and B, that are both n × n, then if the following equation is true, we say that A is the inverse of B, and B is the inverse of A:
    $AB=BA=I$

### Right Pseudo-Inverse

Consider the following matrix:
    $R=A^{T}[AA^{T}]^{-1}$
We call this matrix R the right pseudo-inverse of A, because:
    $AR=I$
but
    $RA\neq I$
We will denote the right pseudo-inverse of A as $A^{\dagger }$

### Left Pseudo-Inverse

Consider the following matrix:
    $L=[A^{T}A]^{-1}A^{T}$
We call L the left pseudo-inverse of A because
    $LA=I$
but
    $AL\neq I$
We will denote the left pseudo-inverse of A as $A^{\ddagger }$
# Engineering Analysis/Linear Spaces
## Linear Spaces

Linear Spaces are like Vector Spaces, but are more general. We will define Linear Spaces, and then use that definition later to define Function Spaces.
If we have a space X, elements in that space f and g, and scalars a and b, the following rules must hold for X to be a linear space:
* $f+g\in X$
* $f+g=g+f$
* There is a null element φ such that φ + f = f. $\phi \in X$
* $f\in X,-f\in X$
* f + (-f) = φ
# Engineering Analysis/Lyapunov Equation

[Lyapunov's Equation]
    $AM+MB=C$
Where A, B and C are constant square matrices, and M is the solution that we are trying to find. If A, B, and C are of the same order, and if A and B have no eigenvalues in common, then the solution can be given in terms of matrix exponentials:
    $M=-\int _{0}^{\infty }e^{Ax}Ce^{Bx}dx$
# Engineering Analysis/Function Spaces
## Function Space

A function space is a linear space where all the elements of the space are functions. A function space that has a norm operation is known as a normed function space. The spaces we consider will all be normed.

## Continuity

f(x) is continuous at x$_0$ if, for every ε > 0 there exists a δ(ε) > 0 such that |f(x) - f(x$_0$)| < ε when |x - x$_0$| < δ(ε).

## Common Function Spaces

Here is a listing of some common function spaces. This is not an exhaustive list.

### C Space

The C function space is the set of all functions that are continuous.
The metric for C space is defined as:
    $\rho (x,y)_{L_{2}}=\max |f(x)-g(x)|$
Consider the metric of sin(x) and cos(x):
    $\rho (sin(x),cos(x))_{L_{2}}={\sqrt {2}},x={\frac {3\pi }{4}}$

### C$^p$ Space

The C$^p$ is the set of all continuous functions for which the first p derivatives are also continuous. If $p=\infty$ the function is called "infinitely continuous. The set $C^{\infty }$ is the set of all such functions. Some examples of functions that are infinitely continuous are exponentials, sinusoids, and polynomials.

### L Space

The L space is the set of all functions that are finitely integrable over a given interval [a, b].
f(x) is in L(a, b) if:
    $\int _{a}^{b}|f(x)|dx<\infty$

### L $_p$ Space

The L$_p$ space is the set of all functions that are finitely integrable over a given interval [a, b] when raised to the power p:
    $\int _{a}^{b}|f(x)|^{p}dx<\infty$
Most importantly for engineering is the L$_2$ space, or the set of functions that are "square integrable".
# Engineering Analysis/Leibniz Rule

Leibniz' rule allows us to take the derivative of an integral, where the derivative and the integral are performed using different variables: