Ranked as follows according to the Ps that enter into their construction and the points that are occupied by those Ps

First salient = QKtP at Q Kt 2, Q B P at Q B 3, QPatQl

First auxiliary salient = KKtPatKKt2. K B P at K B 3, K P at K 4.

Second salient = KBPat K B 2, K P at K 3, Q P at Q 4.

Second auxiliary salient = QRP at Q R 2, Q Kt P at Q Kt 3, Q B P at Q B 4.

Third salient = Q B P at Q B 2, Q P at Q 3, kJ> at K 4.

Fourth salient = Q P at Q2, KPat K 3, KBPatKBl

The functions of the first salient have already been stated (p. 94). The first auxiliary salient is useful in conjunction with the first salient as a prime defensive measure for covering the developing manoeuvres of the superior pieces in the opening ; its most important feature is KB P at K B 3, which supports K P and prevents the posting of an adverse piece at KKt 4. Both these salients belong to what is called the open game, in which the first move is 1. K P-K 4, and the forces are developed with the primary objects of attack and defence on the K's side.

The second salient and the second auxiliary salient, on the other hand, are characteristic of the close game, in which the opening move is either 1. Q P-Q 4, 1. K P-K 3, 1.QBP-QB4, i.QKtP-QKt3 or i.KKt-KB3, and the forces are developed with the primary objects of attack and defence on the Q's side. These two formations, unlike the first and the first auxiliary salients, are most frequently associated with each other ; the first salient may be established without the addition of the first auxiliary, but the second salient should be accompanied by its auxiliary. Again, the protection of K from attack is not a direct object of the formation of the second and the second auxiliary salients, as it is of the first salient and its auxil iary; their purpose is rather to protect the Q's side during the developing operations of the opening.

The third salient marks a purely defensive variety of the open game; it is properly established only by the second player, and its virtues as compared with those of the first salient are so few that it must be regarded as greatly inferior to that formation. Still, it is frequently necessary to adopt this class of salient when the establishment of a better one is impracticable, and for that reason it plays an important part in minor tactics. In its composition Q P at Q 3, supporting K P at K 4, is the only element worthy of consideration.

The fourth salient is rarely employed, and when it is pressed into service it is consistently accompanied by the second auxiliary salient; the class of game into which it enters is essentially close, but possesses also some characteristics of the open game, inasmuch as it contemplates a counter-attack on the K's side in reply to the adversary's close attack on the Q's side. The fourth salient is generally adopted only by the second player.

The supporting parallels are four in number, ranked as follows : —

First supporting parallel = QP at Q 4, K P at K 4.

First composite supporting parallel = QP at Q 4, K P at K 4, KBPatKBi

Second supporting parallel = Q P at Q 4, Q B P at Q B 4.

Third supporting parallel = K P at K 4, K B P at K B 4.

As the salients repel adverse attacks along diagonals, so the supporting parallels oppose radii of offence directed along verticals ; but as the Ps exert no active force in vertical directions, their opposition to adverse force along verticals is solely passive, and exercised by the process of intercepting, rather than repelling, adverse radii of offence. For this reason the supporting parallels are chiefly subsidiary to the salients, in conjunction with which they form the angles of resistance.

Our former definition of angles of resistance (p. 95) was incomplete, in that it failed to state that the salient and the supporting parallel of which an angle of resistance is composed are of the same rank. Bearing this fact in mind, the student will have no difficulty in perceiving that all the possible angles of resistance are five in number, formed as follows : —

First angle of resistance (first salient together with first supporting parallel) == Q Kt P at Q Kt 2, QBP at Q B 3, Q P at Q 4, K P at K 4.

First composite angle of resistance (first salient together with first composite supporting parallel) = Q Kt P at Q Kt 2, QBP at Q B 3. Q P at Q 4, K P at K 4, K B P at KB 4.

First auxiliary angle of resistance (first auxiliary salient together with first supporting parallel) = K Kt Pat K Kt 2, K B F at K B 3, K P at K 4, Q P at Q 4.

Second angle of resistance (second salient together with second supporting parallel) = KBP at KB 2, KP at K3, QF at Q 4, Q B P at Q B 4.

Second auxiliary angle of resistance (second auxiliary salient together with second supporting parallel) = Q R P at Q R 2, Q Kt P at Q Kt 3, QBP at Q B 4, Q P at Q 4.

Third angle of resistance (third salient together with third supporting parallel) = QBP at Q B 2, Q P at Q 3, K P at K 4, K B P at K B 4.

Of the fourth variety of P units — elements of the normal P base — no classification is deemed necessary, and their consideration may be dismissed in a few brief observations : First, in every P formation of the open game both RPs (QRP and KRP) and both KtPs (QKtP and K Kt P) remain unmoved; secondly, in some P formations of the open game either QBP or KBP, or both QBP and KBP, remain unmoved, together with R Ps and Kt Ps constituting elements of the normal P base ; thirdly, in the close game the only P on the Q's wing that remains unmoved, thus forming an element of the normal P base, is QR P, with a single exception (see p. 165).

104 thr minor tactics of chess.

We now proceed to the examination of P integrals, in which all that has been said of the various P units will find its application. In the ensuing diagrams the positions are represented by white Ps when the integrals properly belong to the first player, and by black Ps when they properly belong to the second player. It is to be understood, however, that no hard and fast rule is set down, restricting the use of any integral to either player : but, in general, if the second player is able to establish an integral that properly belongs to the first player, so much the better for the second player; and if the first player is compelled to adopt an integral that properly belongs to the second player, so much the worse for the first player. This may seem to the student like an instance of " Heads> I win; tails you lose: " and so it really is; for we may right here enunciate the broad principle that —

Any subversion of P integrals — that is, the adoption by one player of a formation that properly belongs to the other — gives an advantage in minor tactics to the second or defending player.

The P integrals in the following classifications are divided into those which arise in the open game and those which arise in the close game, and in each division are numbered in order from the most to the least advantageous:

First Open Pawn Integral.

The first open P integral (Figs. 25, 33, 34,

35) is composed of the first salient and the first composite supporting parallel (together constituting first the composite angle of resistance), and two elements of the normal P base, with K in its position after castling (KR) ; it is a feature of the attack in the open game, and is formed by the moves KP -

Maxims : KP-K4 is* the initial and distinctive move of the open game, and in the open attack it should be followed as early as practicable byQP-Q4. KPis best posted atK4, and QP is better posted at Q4 than at Q3. The point QB 3 is the key to the general P position in the open game, and it should be occupied as early as practicable by a P, by means of the move QBP-QB3; QBP should not, therefore, without urgent-necessity, be allowed to remain unmoved, nor be advanced two steps. The first salient should be formed in the surest and speediest manner, as the fundamental unit in the composition of the first open P integral ; and this integral, having been established, should thereafter be maintained intact save for the most cogent reasons, either because victory is assured and may immediately be achieved, or because defeat may be averted only by the disturbance of this invaluable disposition of the P line. The first open P integral is the perfect P line of the open game, and any other P line is scientifically correct in direct proportion to the facilities it affords for establishing this integral. If, in the formation of the first, second or third open P integral, "he move Q B P - Q B 3 be made before Q P-Q4 ;^and either before or after QP - Q3, a risk is involved in the lack of support of Q P by another P ; moreover, the occupation of the point Q4 by an adverse P would prevent the advance of QPto Q4, and thus perpetuate the defect existing in the P line by reason of the weakness of the position of Q P : this manœuvre, therefore, if it be ventured at all, should be adopted with the greatest caution, and only when the adversary is manifestly unable to prevent the subsequent advance of QP to Q4.

Second Open Pawn Integral.

The second open P integral (Figs. 26, 36, 37, 38) is composed of the first salient and the first supporting parallel (together constituting the first angle of resistance), and two elements of the normal P base, with K either in its normal position or in its position after castling (KR) ; it is a feature of the attack in the open game, and it is formed by the moves KP-K4, QP-Q4, QBP - QB3.

Maxim: The second open P integral having been established, and K having been castled (KR), the position should be converted as early as practicable into the first open P integral, by the move KBP - KB4. All the maxims enunciated with reference to the formation of the first open P integral apply equally to the second open P integral.

Third Open Pawn Integral.

The third open P integral (Figs. 27, 39) is composed of the first salient, the first auxiliary salient and the first supporting parallel (together constituting the first and the first auxiliary angles of resistance), with K in its position after castling (KR) ; it is a feature of the attack in the open game, and it is formed by the moves KP-K4, Q P -Q4, QBP-QB3, KBP-KB3.

Maxim : KB P being posted at KB3 for defence and at KB4 for offence to co-oper-ate with KP for offence, the third open P integral is the most efficient disposition of the P line for purely defensive purposes, and it is readily susceptible of development into the first open P integral, the perfect offensive P formation, by the further advance of KBP from KB3 to KB4.

Fourth Open Pawn Integral.

The fourth open P integral (Figs. 28, 40, 41) is composed of the third salient and the third supporting parallel (together constituting the third angle of resistance), and two elements of the normal P base, with K in its position after castling (KR) ; it is a feature of the defence in the open game, and it is formed by the moves K P - K 4, Q P -Q3, and KBP- KB4.

Maxims: The fourth open P integral is an important element of a strong and accurate counter-attack. The fundamental unit in its composition is the third salient.

Fifth Open Pawn Integral.

The fifth open P integral (Figs. 29, 42, 43) is composed of the third salient and two elements of the normal P base, with K either in its normal position or in its position after castling (KR) ; it is a feature of the defence in the open game, and it is formed by the moves KP - K4 and QP - Q3.

Maxims : The fifth open P integral is the method of disposition of the Ps which is usually adopted by the second player for purely defensive purposes. It is the most readily established, but also the least effective, of all the P integrals.

First Close Pawn Integral.

The first close P integral (Figs. 30, 44) is composed of the second salient, the second auxiliary salient and the second supporting parallel (together constituting the second and the second auxiliary angles of resistance), and two elements of the normal P base, with K either in its normal position or in its position after castling (KR) ; it is a feature both of the attack and of the defence in the close game, and it is formed by the moves QP - Q4, KP-K3, QBP-QB4 and QKtP- QKt3.

Maxims : Q P - Q 4 is the initial and distinctive move of the close game, and it should be followed as early as practicable by K P -K3, except in the close defence to the open attack (second close P integral), when the move K P - K 3 is made first, and then followed as early as practicable by QP — Q4; in any case K P is posted and maintained at K3, and QPat Q4. In the close game QBP is properly posted and maintained at QB4. The second salient, as the fundamental unit in the composition of the first close P integral, should be formed in the speediest and surest manner. QKtP, whenever moved at all, is posted and maintained at Q Kt3, in order to form the second auxiliary salient in conjunction with QBP at QB4. The first close P integral is the perfect P line of the close game, and any other P line is scientifically correct in direct proportion to the facilities it affords for establishing this integral.

Second Close Pawn Integral.

The second close P integral (Figs. 31, 45) is composed of the second salient and the second supporting parallel (together constituting the second angle of resistance), and two elements of the normal P base, with K either in its normal position or in its position after castling (KR) : it is a feature of the close defence to the open attack, and it is formed by the moves K P -K 3, QP-Q4, and QBP-QB4.

Maxim : The second close P integral is an important element of a strong and accurate close counter-attack, when the first player has initiated the open attack. The maxims enunciated with reference to the formation of the first close P integral apply equally to the second close P integral.

Third Close Pawn Integral.

The third close P integral (Figs. 32, 46, 47) is composed of the fourth salient, the second auxiliary salient and three elements of the normal P base, with K in its position after castling (KR) ; it is a feature of the defence in the close game, and it is formed bythe moves KP-K3, KBP-KB4, QBP-QB4, and QKtP - QKt3.

Maxims : The fundamental unit in the composition of the third close P integral is the fourth salient, which should be formed in the surest and speediest manner when this class of defence is adopted. The third close P integral is inferior both to the first and to the second close P integrals ; its chief defect is the weak position of QP, and its chief merit is that it affords means for an early counter-attack, when the first player has initiated the close attack.

There are other inferior P formations that possess some good qualities, but of them all it may be stated, that in whatever details they differ from the foregoing models, they are to that extent fundamentally defective and inadvisable.

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