Part 7

Furthermore, it is to be noted as equally fundamental, that:

1. A piece exerts no force against that point upon which it is posted;

2. That whatever power a piece exerts, always is exerted against some other point than the point upon which it stands; and that;

3. In order to exert such power, it is an all-essential that the piece move from the point which it occupies to the point at which its power is to be exerted.

Hence, it is obvious and may be mathematically demonstrated, that,

1. A piece which cannot move, cannot capture.

2. A piece which cannot capture, does not exercise any threat of capture; and

3. Consequently, a piece deprived of its right to move; which cannot capture nor exercise any threat to capture, obviously and by mathematical demonstration, cannot give check, inasmuch, as “check” merely is the threat by a piece to move and capture the adverse King.

Therefore, whatever may be the normal area of movement belonging to a piece, whenever from any cause such piece loses its power of movement, then,

It no longer can capture, nor exercise any threat of capture, upon the points contained within said area; and consequently such points so far as said immovable piece is concerned, may be occupied in safety by any adverse piece including the adverse King, for the reason that:

An immovable piece cannot move; and not being able to move it cannot capture, and not being able to capture, it does not exercise any threat of capture, and consequently it cannot give check.

This incongruity of permitting an immovable piece to give check constitutes the second mathematical blemish in the game of Chess, as at present constructed.

This fallacy, the correction of which any schoolboy may mathematically demonstrate, is defended, even by many who would know better, if they merely would take time for reflection; by the inane assumption, that:

A piece which admittedly is disqualified and rendered dormant by all the fundamentals of the science of Chess, and which therefore cannot legally move and consequently cannot legally capture anything; by some hocus-pocus may be made to move and to capture that _most_ valuable of _all_ prizes, the adverse King; and this at a time and under circumstances when, as is universally allowed, it cannot legally move against, nor legally capture _any other_ adverse piece.

The basis of this illogical, illegal, and untenable assumption is:

The pinned piece, belonging to that force which has the privilege of moving, can abandon its post, and capture the adverse King; this stroke ends the game and the game being ended, the pinning piece never can avail of the abandonment of the covering post by the pinned piece to capture the King thus exposed.

The insufficiency of this subterfuge is clear to the mathematical mind. Its subtlety lies in confounding together things which have no connection, viz.:

Admittedly the given body of Chess-pieces has the right to move, but it is of the utmost importance to note that this privilege of moving extends only to a single piece and from this privilege of moving the pinned piece is debarred by a specific and fundamental law of the game, which declares that:

“A piece shall not by removing itself uncover the kindred King to the attack of a hostile piece.”

Thus, it is clear, that a pinned piece is a disqualified piece; its powers are dormant and by the laws of the game it is temporarily reduced to an inert mass, and deprived of every faculty normally appertaining to it as a Chess-piece. On the other hand, as is equally obvious, the pinning piece is in full possession of its normal powers and is qualified to perform every function.

To hold that a piece disqualified by the laws of the game can nullify the activities of a piece in full possession of its powers, is to assert that black is white, that the moon is made of green cheese, that the tail can wag the dog, or any other of those things which have led the German to promulgate his caustic formula on the Anglo-Saxon.

Hence, artificially to nullify the normal powers of an active and potential piece which is operating in conformity to the laws of the game, and artificially to revivify the dormant powers of a piece disqualified by the same laws; to debar the former from exercising its legitimate functions and to permit the latter to exercise functions from which by law, it specifically is debarred, is a self-evident incongruity and any argument whereby such procedure is upheld, necessarily and obviously, is sophistry.

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No less interesting than instructive and conclusive, is reference of this question to those intellectual principles which give birth to the game of Chess, _per se_, viz.:

As a primary fundamental, with the power to give check, is associated concurrently the obligation upon the King thus checked, not to remain in check.

Secondly: The totality of powers assigned to the Chess-pieces is the ability to move, provided the King be free from check. This totality of powers may be denoted by the indefinite symbol, X.

The play thus has for its object:

The reduction to zero of the adverse X, by the operation of the kindred X.

This result is checkmate in its generalized form. In effect, it is the destruction of the power of the adverse pieces to move, by means of check made permanent.

By the law of continuity it is self-evident that:

The power to move appertaining either to White or to Black, runs from full power to move any piece (a power due to freedom from check), down to total inability to move any piece, due to his King being permanently checked, _i.e._, checkmated.

This series cannot be interrupted without obvious violation of the ethics of the game; because, so long as any part of X remains, the principle from which the series emanated still operates, and this without regard to quantity of X remaining unexpended.

Thus, a game of Chess is a procedure from total ability to total disability; _i.e._, from one logical whole to another; otherwise, from X to zero.

Checkmate, furnishes the limit to the series; the game and X vanish together.

This is in perfect keeping with the law of continuity, which acts and dominates from beginning to end of the series, and so long as any part of X remains.

Hence to permit either White or Black to move any piece, leaving his King in check, is an anomaly.

* * * * *

Denial to the Pawn of ability to move to the rear is an accurate interpretation of military ethics.

Of those puerile hypotheses common to the man who does not know, one of the most entrancing to the popular mind, is the notion that Corps d’armee properly are of equal numbers and of the same composition.

This supposition is due to ignorance of the fact that the multifarious duties of applied Strategetics, require for their execution like variety of instruments, which diversity of means is strikingly illustrated by the differing movements of the Chess-pieces.

The inability of the Pawn to move backward strategically harmonizes with its functions as a Corps of Position, in contradiction to the movements of the pieces, which latter are Corps of Evolution.

This restriction in the move of the Pawn is in exact harmony with the inability of the Queen to move on obliques, of the Rook to move on obliques or on diagonals, of the Bishop to move on obliques, verticals and horizontals, of the Knight to move on diagonals, verticals, and horizontals, and of the King to move like any other piece.

* * * * *

Possessed of the invaluable privilege of making the first move in the game, knowing that no move should be made without an object, understanding that the true object of every move is to minimize the adverse power for resistance and comprehending that all power for resistance is derived from facility of movement, the student easily deduces the true object of White’s initial move in every game of Chess, viz.:

PRINCIPLE

_To make the first of a series of movements, each of which shall increase the mobility of the kindred pieces and correspondingly decrease the mobility of the adverse pieces._

As the effect of such policy, the power for resistance appertaining to Black, ultimately must become so insufficient that he no longer will be able adequately to defend:

1. His base of operations.

2. The communications of his army with its base.

3. The communications of his corps d’armee with each other, or,

4. To prevent the White hypothetical force penetrating to its Logistic Horizon.

To produce this fatal weakness in the Black position by the advantage of the first move is much easier for White than commonly is supposed.

The process consists in making only those movements by means of which the kindred corps d’armee, progressively occupying specified objectives, are advanced, viz.:

I. _To the Strategetic Objective, when acting against the communications of the adverse Determinate Force and its Base of Operations._

II. _To the Logistic Horizon, when acting against the communications between the adverse Determinate and the adverse Hypothetical Forces._

III. _To the Strategic Vertices, when acting against the communications of the hostile corps d’armee with each other._

To bring about either of these results against an opponent equally equipped and capable, of course is a much more difficult task than to checkmate an enemy incapable of movement.

Yet such achievement is possible to White and with exact play it seemingly is a certainty that he succeeds in one or the other, owing to his inestimable privilege of first move.

For the normal advantage that attaches to the first move in a game of Chess is vastly enhanced by a peculiarity in the mathematical make-up of the surface of the Chess-board, whereby, he who makes the first move may secure to himself the advantage in mobility, and conversely may inflict upon the second player a corresponding disadvantage in mobility.

This peculiar property emanates from this fact:

_The sixty-four points, i.e., the sixty-four centres of the squares into which the surface of the Chess-board is divided, constitute, when taken collectively, the quadrant of a circle, whose radius is eight points in length._

Hence, in Chessic mathematics, the sides of the Chessboard do not form a square, but the segment of a circumference.

To prove the truth of this, one has but to count the points contained in the verticals and horizontals and in the hypothenuse of each corresponding angle, and in every instance it will be found that the number of points contained in the base, perpendicular, and hypothenuse, is the same.

For example:

Let the eight points of the King’s Rook’s file form the perpendicular of a right angle triangle, of which the kindred first horizontal forms the base; then, the hypothenuse of the given angle, will be that diagonal which extends from QR1 to KR8. Now, merely by the processes of simple arithmetic, it may be shown that there are,

1. Eight points in the base.

2. Eight points in the perpendicular.

3. Eight points in the hypothenuse.

Consequently the _three_ sides of this given right angled triangle are _equal_ to each other, which is a geometric _impossibility_.

Therefore, it is self-evident that there exists a mathematical incongruity in the surface of the Chess-board.

That is, what to the eye _seems_ a right angled triangle, is in its relations to the _movements_ of the Chess-pieces, an equilateral triangle. Hence, the Chess-board, in its relations to the pieces when the latter are at _rest_, properly may be regarded as a great _square_ sub-divided into sixty-four smaller squares; but on the contrary, in those calculations relating to the Chess-pieces in _motion_, the Chess-board must be regarded as the _quadrant_ of a circle of eight points radius. The demonstration follows, viz.:

Connect by a straight line the points KR8 and QR8. Connect by another straight line the points QR8 and QR1. Connect each of the fifteen points through which these lines pass with the point KR1, by means of lines passing through the least number of points intervening.

Then the line KR8 and QR8 will represent the segment of a circle of which latter the point KR1 is the center. The lines KR1-KR8 and KR1-QR1 will represent the sides of a quadrant contained in the given circle and bounded by the given segment, and the lines drawn from KR1 to the fifteen points contained in the given segment of the given circumference, will be found to be fifteen equal radii each eight points in length.

* * * * *

Having noted the form of the Static or positional surface of the Chess-board and its relations to the pieces at rest, and having established the configuration of the Dynamic surface upon which the pieces move, it is next in sequence to deduce that fundamental fact and to give it that geometric expression which shall mathematically harmonize these conflicting geometric figures in their relations to Chess-play.

As the basic fact of applied Chessic forces, it is to be noted, that:

PRINCIPLE

_The King is the SOURCE from whence the Chess-pieces derive all power of movement; and from his ability to move, emanates ALL power for attack and for defence possessed by a Chessic army._

This faculty of mobility, derived from the existence of the kindred King, is the all essential element in Chess-play, and to increase the mobility of the kindred pieces and to reduce that of the adverse pieces is the simple, sure and only scientific road to victory; and by comparison of the Static with the Dynamic surface of the Chess-board, the desired principle readily is discovered, viz.,

The Static surface of the Chess-board being a square, its least division is into two great right angled triangles having a common hypothenuse.

The Dynamic surface being the quadrant of a circle, its least division also is into two great sections, one of which is a right angled triangle and the other a semi-circle.

Comparing the two surfaces of the Chess-board thus divided, it will be seen that these three great right angled triangles are equal, each containing thirty-six points; and having for their common vertices, the points KR1, QR1 and R8.

Furthermore, it will be seen that the hypothenuse common to these triangles, also is the chord of that semi-circle which appertains to the Dynamic surface.

Again, it will be perceived that this semi-circle, like the three right angled triangles, is composed of thirty-six points, and consequently that all of the four sub-divisions of the Static and Dynamic surfaces of the Chess-board are equal.

Thus it obviously follows, that:

1. The great central diagonal, always is one side of each of the four chief geometric figures into which the Chess board is divided; that:

2. It mathematically perfects each of these figures and harmonizes each to all, and that:

3. By means of it each figure becomes possessed of eight more points than it otherwise would contain.

Hence, the following is self-evident:

PRINCIPLE

_That Chessic army which can possess itself of the great central diagonal, thereby acquires the larger number of points upon which to act and consequently greater facilities for movement; and conversely:_

_By the loss of the great central diagonal, the mobility of the opposing army is correspondingly decreased._

It therefore is clear that the object of any series of movements by a Chessic army acting otherwise than on Line of Operations, should be:

PRINCIPLE

_Form the kindred army upon the hypothenuse of the right angled triangle which is contained within the Dynamic surface of the Chess-board; and conversely,_

_Compel the adverse army to act exclusively within that semi-circle which appertains to the same surface._

Under these circumstances, the kindred corps will be possessed of facilities for movement represented by thirty-six squares; while the logistic area of the opposing army will be restricted to twenty-eight squares.

There are, of course, two great central diagonals of the Chess-board; but as the student is fully informed that great central diagonal always is to be selected, which extends towards the Objective Plane.

* * * * *

Mobility, _per se_, increases or decreases with the number of squares open to occupation.

But in all situations there will be points of no value, while other points are of value inestimable; for the reason that the occupation of the former will not favorably affect the play, or may even lose the game; while by the occupation of the latter, victory is at once secured.

But it is not the province of Mobility to pass on the values of points; this latter is the duty of Strategy. It is sufficient for Mobility that it provide superior facilities for movement; it is for Strategy to define the Line of Movement; for Logistics, by means of this Line of Movement, to bring into action in proper times and sequence, the required force, and for Tactics, with this force, to execute the proper evolutions.

Mobility derives its importance from three things which may occur severally or in combination, viz.:

1. All power for offense or for defense is eliminated from a Chess-piece the instant it loses its ability to move.

2. The superiority possessed by corps acting offensively over adverse corps acting defensively, resides in that the attack of a piece is valid at every point which it menaces; while the defensive effort of a piece, as a rule, is valid only at a single point. Consequently:

PRINCIPLE

_Increased facilities for movement enhance the power of attacking pieces in a much greater degree than like facilities enhance the power of defending pieces._

Such increasing facilities for movement ultimately render an attacking force irresistible, for the reason that it finally becomes a physical impossibility for the opposing equal force to provide valid defences for the numerous tactical keys, which at a given time become simultaneously assailed. Hence:

PRINCIPLE

_Superior facilities for occupying any point at any time and with any force, always ensure the superior force at a given point, at a given time._

The relative advantage in mobility possessed by one army over an opposing army always can be determined by the following, viz.:

RULE

1. That army whose strategic front of operations is established upon the Strategetic Center has the relative advantage in Mobility.

2. To utilize the advantage in Mobility extend the Strategic Front in the direction of the objective plane.

3. To neutralize the relative disadvantage in Mobility eliminate that adverse Corps d’armee which tactically expresses such adverse advantage; or so post the Prime Strategetic Point as to vitiate the adverse Strategic front.

Advantage in Mobility is divided into two classes, viz.:

I. General Advantage in Mobility.

II. Special Advantage in Mobility.

A General Advantage in Mobility consists in the ability to act simultaneously against two or more vital points by means of interior logistic radii due to position between:--

1. The adverse army and its Base of Operations.

2. Two or more adverse Grand Columns.

3. The wings of a hostile Grand Column.

4. Two or more isolated adverse Corps d’armee.

Such position upon interior lines of movement is secured by occupying either of the Prime Offensive Origins, _i.e._:

1. Strategic Center _vs._ Adverse Formation in Mass.

2. Logistic Center _vs._ Adverse Formation by Grand Columns.

3. Tactical Center _vs._ Adverse Formation by Wings.

4. Logistic Triune _vs._ Adverse Formation by Corps.

Special Advantage in Mobility consists in the ability of a corps d’armee to traverse greater or equal distances in lesser times than opposing corps.

MILITARY EXAMPLES

_“Never interrupt your enemy when he is making a false movement.”--Napoleon._

In the year (366 B.C.) the King of Sparta, with an army of 30,000 men marched to the aid of the Mantineans against Thebes. Epaminondas took up a post with his army from whence he equally threatened Mantinea and Sparta. Agesilaus incautiously moved too far towards the coast, whereupon Epaminondas, with 70,000 men precipitated himself upon Lacedaemonia, laying waste the country with fire and sword, all but taking by storm the city of Sparta and showing the women of Lacedaemonia the campfire of an enemy for the first time in six hundred years.

* * * * *

Flaminius advancing incautiously to oppose Hannibal, the latter took up a post with his army from whence he equally threatened the city of Rome and the army of the Consul. In the endeavor to rectify his error, the Roman general committed a worse and was destroyed with his entire army.

* * * * *

At Thapsus, April 6, 46 B.C., Caesar took up a post with his army from whence he equally threatened the Roman army under Scipio and the African army under Juba. Scipio having marched off with his troops to a better camp some miles distant, Caesar attacked and annihilated Juba’s army.

* * * * *

At Pirna, Frederic the Great, captured the Saxon army entire, and at Rossbach, Leuthern and Zorndorf destroyed successively a French, an Austrian and a Russian army merely by occupying a post from whence he equally threatened two or more vital points, awaiting the time when one would become inadequately defended.

* * * * *

Washington won the Revolutionary War merely by occupying a post from whence he equally threatened the British armies at New York and Philadelphia; refusing battle and building up an army of Continental regular troops enlisted for the war and trained by the Baron von Steuben in the system of Frederic the Great.

* * * * *

Bonaparte won at Montenotte, Castiglione, Arcola, Rivoli and Austerlitz his most perfect exhibitions of generalship, merely by passively threatening two vital points and in his own words: “By never interrupting an enemy when he is making a false movement.”

* * * * *

Perfection in Mobility is attained whenever the kindred army is able to act unrestrainedly in any and all directions, while the movements of the hostile army are restricted.

NUMBERS

_“In warfare the advantage in numbers never is to be despised.”--Von Moltke._

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_“Arguments avail but little against him whose opinion is voiced by thirty legions.”--Roman Proverb._

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_“That king who has the most iron is master of those who merely have the more gold.”--Solon._

* * * * *

_“It never troubles the wolf how many sheep there are.”--Agesilaus._

NUMBERS

_“A handful of troops inured to Warfare proceed to certain victory; while on the contrary, numerous hordes of raw and undisciplined men are but a multitude of victims dragged to slaughter.”--Vegetius._

* * * * *

_“Turenne always was victorious with armies infinitely inferior in numbers to those of his enemies; because he moved with expedition, knew how to secure himself from being attacked in every situation and always kept near his enemy.”--Count de Saxe._

* * * * *

_“Numbers are of no significance when troops are once thrown into confusion.”--Prince Eugene._

Humanity is divisible into two groups, one of which relatively is small and the other, by comparison, very large.

The first of these groups is made up comparatively of but a few persons, who, by virtue of circumstances are possessed of everything except adequate physical strength; and the second group consists of those vast multitudes of mankind, which are destitute of everything except of incalculable prowess, due to their overwhelming numbers.