of a Debt that Euler does: namely, it is Money in the possession of the Debtor owed and pledged to the Creditor, and therefore affected with Negative Sign: and that the Release of a Debt is the change of the sign of affection of Money owed into Money possessed

Now this is exactly the same error as Euler has fallen into ; and is exactly the error which we have already shown is so carefully provided against in the Digest, and by Pothier, Austin, and many other Jurists

If these distinguished mathematicians had reflected, they would have seen that their interpretation could not be correct. Because the signs and always refer to similar Quantities, but of opposite Qualities. Now the Creditor's Right is +, and the inverse of a simple Right cannot be a simple quantity of Money it must be something which is the Inverse of a Right: and the Inverse of a Right is a Duty. Besides, releasing an insolvent Debtor from a Debt does not put him in possession of any actual Money: it is only equivalent to it: but not identical with it

:

The fact is that the Debt is not Money in the possession of the Debtor, owed or pledged to the Creditor: but the abstract Duty to pay Money: and the Negative Sign denotes the Cancelling of the Duty, or Releasing the Debtor from the Duty to pay

Hence the result is not produced in the way in which Peacock says it is but exactly in the way he says it is not

On the Application of the Theory of Algebraical Signs to Economics

19. The perplexities of the Theory of Credit, which have baffled all the the Economists in the world to explain, can only be unraveled by the great modern doctrine of the Separation of the Signs of Affection or Distinction and Operation

As the introduction of this great Doctrine into Economics is perfectly novel, we shall have to treat of it somewhat fully: especially as there may be students of Economics who are not very familiar with it in other sciences

It is a remarkable example of the almost universal truth that

practice has always preceded theory, that even the Practice of Science long preceded the Theory of Science. Thus from the days of Diophantus it was perfectly well understood as an empirical rule in Algebra that X - gives +

Sixteen hundred years ago Diophantus said

“ λειψις ἐπὶ λείψιν πολλαπλασιασθεῖσα ποιεῖ ὕπαρξιν”

"Defect multiplied into Defect gives Existence," which is expressed in common language as two Negatives make an Affirmative

When the great pioneers of Algebra in modern times, Harriot, Fermat, Vieta, Des Cartes, Cardan, Tartaglia, translated their reasonings into general symbols, they found that they had created a machine whose working they were unable fully to understand. They found among other things that many problems produced Negative answers. Unable at first to apprehend the meaning of Negative answers, they believed that they had no real meaning and they called Positive Roots true (vera radices) and Negative Roots fictitious (ficto radices)

In the progress of Natural Philosophy the Negative Sign was used to a vast variety of Quantities: but no general Theory of Signs was devised, and the progress of Mathematics was much impeded by the want of the generalisation. The rule that X

gives was universally adopted in practice, because no other produced right results. But Algrebraists were wholly unable to explain it it was wholly unknown to Newton: and when he tried to explain it, the great Euler babbled like a child

Even so late as 1813, a distinguished mathematician at Cambridge denied the existence and ridiculed the idea of there being any such things as "Negative" Quantities

Many centuries ago, at least about 1100 A.D., the Hindoo Algebraists had made considerable advances in explaining the Theory of Signs: but nothing was done in Europe until nearly the close of the last century. Since then a new spirit of philosophy has been breathed into the old science: and a number of distinguished Algebraists, Arbogast, Argand, Buée, Armand, Carnot, Warren, Peacock, De Morgan, and others, have completely established the Theory of Signs: and their labours

have resulted in the Doctrine of the Separation of the Signs of Affection or Distinction and Operation

In most of the common books on Algebra we are told that the sign means addition, and the sign means subtraction We are also told that + x + gives +: and that X also gives a doctrine which, without further explanation, is simply an inscrutable mystery, not to say an absurdity

Writers who are not versed in Natural Philosophy have no conception of the signs and meaning anything but addition and subtraction. It is no doubt perfectly true that in some cases they have that meaning: but that is only one of their meanings. But every one who has any knowledge of Natural Philosophy knows perfectly well that they have in reality an immense variety of meanings, according to the particular circumstances out of which they arise or the body of facts to which they relate and that it is wholly impossible to determine their meaning until we know the particular circumstances under which they occur

We must now explain the general use of these signs in Natural Philosophy, and show how they may be applied by analogy to the particular facts of Economics

All Sciences deal with Quantities and Operations

20. In order to explain the matter in the simplest way possible it may be said that all Sciences deal with Quantities and Operations: and that throughout all Nature there is Opposition or Contrariety or Inverseness - both Opposition or Contrariety of Quality, and Opposition or Contrariety of Operation

Quantities that are endowed with Opposite or Inverse Qualities are universally distinguished in Mathematics and Natural Philosophy by the Signs + and

These Signs so used in Natural Philosophy to denote Opposite or Inverse Qualities in Quantities of a similar nature, no matter what the Opposition or Inverseness may consist in, are usually termed Signs of Affection or Position: or we may with equal propriety, term them Signs of Distinction

But also Opposite or Inverse Operations may be performed on these Quantities endowed with Opposite or Inverse Qualities: and these Operations of Opposite, Inverse, or Contrary natures are also designated by the same Signs and And any Operations whatever of an Opposite, Contrary, or Inverse Nature, no matter what the Opposition, Contrariety or Inverseness may consist in, may be denoted by these signs

They are then termed Signs of Operation

So in every new body of facts which are brought under scientific control and in every new science whatever, Opposition or Contrariety or Inverseness is sure to appear: and consequently the Signs and receive new applications of meaning in every new science and it is quite impossible to determine their meaning, unless we know the Quantities they refer to, and the Operations they denote

As each one of the Physical Sciences has been brought in succession under the control of Mathematics, these Signs have received new meanings according to the Quantities and Operations they denote. Consequently they have already received an immense variety of meanings: and they will receive new applications of meaning according as every new body of facts is brought under scientific control: and we have now to investigate and determine what is their true meaning and application in the body of facts which we denominate the Science of Economics

And the Combination of these Opposite Signs, denoting Opposite or Contrary Qualities with the same signs denoting Opposite or Inverse Operations performed upon them that is, the Combination of the Signs of Distinction with the Signs of Operation, gives rise to the well known Algebraical Rules

+ + gives +

These Laws, which are universally applicable in Natural

Philosophy, are equally applicable in Economics and among other things are alone capable, by a due adaptation of their general meaning to the particular facts of Economics, of giving the complete solution of the Theory of Credit, which has hitherto been the opprobrium of the science.

There are Economic Quantities of Inverse or Opposite Qualities or Properties: and therefore following the strictest analogy of Physical Science, we shall distinguish them by Opposite Signs. And also Opposite Operations may be performed on these Opposite Quantities: bringing into play the well known Algebraical Rules; which will lead to consequences which may surprise some students

Examples of the Algebraical Signs applied to Quantities

21. We will now give some examples of the Signs + and applied to Quantities of a similar nature, but of Opposite Qualities to furnish us with analogies to guide us to their application in Economics

If we take the meridian of Greenwich as 0, degrees of Longitude East and West of Greenwich are Opposite to each other if therefore the ones are called +, the others may be called. So also taking the Equator as 0, degrees of North and South Latitude are Opposite to each other ones be denoted by +, the others will be denoted by

:

and if the

So in Algebraical Geometry in which it is necessary to fix the position of the lines, if any given fixed point be taken, lines drawn in Opposite directions from it, either to the Right or to the Left or Upward or Downward from it: are distinguished by the Signs and

So if a line revolving in one direction be denoted by +, then when it revolves in the Opposite direction, it is denoted by

If two Mechanical Forces act in Opposite directions they are distinguished by the Opposite Signs + and

If 1 be multiplied by powers of a, the results are termed Positive powers of a: if 1 be divided by powers of a, the results are termed Negative powers of a

In modern Kinematics an accelerating Force is one which causes a body to change its Rate of Velocity: if it Increases the