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Title: An Attempt to Analyse the Automaton Chess Player of Mr. De Kempelen - To Which is Added, a Copious Collection of the Knight's - Moves over the Chess Board
Author: Willis, Robert
Language: English
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Transcriber’s Notes:

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                         AN ATTEMPT TO ANALYSE
                                  THE
                        AUTOMATON CHESS PLAYER.


                          HOWLETT and BRIMMER,
                   Printers, 10, Frith Street, Soho.

[Illustration: _Plate 1. To face the Title. Fig. 1._]

[Illustration: _Fig. 2._

_Drawn on Stone by the Author. Printed by C. Hullmandel._]

                         AN ATTEMPT TO ANALYSE
                                  THE
                        AUTOMATON CHESS PLAYER,
                                   OF
                            MR. DE KEMPELEN.

         WITH AN EASY METHOD OF IMITATING THE MOVEMENTS OF THAT
                           CELEBRATED FIGURE.

                   ILLUSTRATED BY ORIGINAL DRAWINGS.

               TO WHICH IS ADDED, A COPIOUS COLLECTION OF
                           THE KNIGHT’S MOVES
                         OVER THE CHESS BOARD.

                               _LONDON_:
                         PRINTED FOR J. BOOTH,
                      DUKE STREET, PORTLAND PLACE.
                                 1821.

      Mine eyes are made the fools o’ the other senses.
                                            SHAKSPEARE.

      I had not thought to have unlockt my lips
      In this unhallowed air, but that this juggler
      Would think to charm my judgment, as mine eyes.
                                              MILTON.



AN ATTEMPT TO ANALYSE, &c.


The Automaton Chess Player was first introduced into England by Mr. de
Kempelen, its inventer, about the year 1783. It was brought again into
this country two years ago, and exhibited under the direction of a very
ingenious gentleman, Mr. Maelzel.

The annexed drawings, (plate 1, figs. 1 and 2,) represent the general
appearance of the machine. It runs on castors, and is either seen
on the floor when the doors of the apartment are thrown open, or is
wheeled into the room at the commencement of the exhibition.

The exhibiter, in order to shew the mechanism, as he informs the
spectators, unlocks the door (A, fig. 1.) of the chest, which exposes
to view a small cupboard, lined with black or dark coloured cloth, and
containing different pieces of machinery, which seem to occupy the
whole space. He next opens the door (B, fig. 2.) at the back of the
same cupboard, and, holding a lighted candle at the opening, still
further exposes the machinery within. The candle being withdrawn, the
door (B) is then locked. The drawer (G G, fig. 1.) in the front of the
chest is next opened, and a set of chess men, a small box of counters,
and a cushion for the support of the Automaton’s arm, are taken out
of it. The exhibiter now opens the two front doors (C C, fig. 1.) of
the large cupboard, and the back door (D, fig. 2.) of the same, and
applies a candle as in the former case. This cupboard is lined with
cloth like the other, but it contains only a few pieces of machinery.
The chest is now wheeled round, the garments of the figure are lifted
up, and the door (E, fig. 2,) in the trunk, and another (F,) in the
thigh, are opened. But it must be observed that the doors (B and D) are
closed. The circumstance is mentioned, because Mr. de Windisch, in his
letters on this subject, has a passage which would seem to imply that
Mr. Maelzel’s mode of exhibiting the interior differs from that which
Mr. de Kempelen employed. “But do not imagine,” says De Windisch, “like
many others, that the inventer shuts one door as he opens another;
the entire Automaton is seen at the same time uncovered, his garments
turned up, and the drawer opened, as well as all the doors of the
chest.”

Now a reference to De Kempelen’s second drawing, published by Mechel,
and annexed to De Windisch’s letters, will shew that, when the chest
was turned round, the doors (B and D) were actually closed, as they
always have been under the direction of Mr. Maelzel. In the chest of
the latter gentleman, indeed, the doors in question are suspended by
hinges attached to the upper part, (as in fig. 2), and consequently
close by their own gravity. But the fact is, that the exhibiter never
fails to lock them, though he leaves the keys in one of the locks. The
other doors are allowed to swing about whilst the chest is wheeled
round.

The chest is now restored to its former position on the floor; the
doors in front, and the drawer, are closed and locked; and the
exhibiter, after he has occupied some time at the back of the chest, in
apparently adjusting the machinery, removes the pipe from the hand of
the figure, winds up the works, and the Automaton begins to move.

These movements, resulting as they appear to do, from mere mechanism,
yet strongly impressed with the distinctive character of an intellectual
guidance, have excited the admiration of the curious during a period
little short of forty years. In that time various conjectures have
been offered to the world as solutions of the problem; but no one, as
far as I know, have attempted to imitate the movements, it is fair to
conclude, either that the means proposed are inadequate to the end, or
that the description of them is too imperfect to enable a workman to
complete the machinery.

Automata may be divided into three classes—the simple—the compound—and
the spurious.

The first class comprises those insulated Automata whose movements
result from mechanism alone; by the aid of which they perform certain
actions, and continue them, so long as the moving force is kept in an
active state.

The second class includes those Automata, which, like the former, are
moved by machinery; but, possessing at the same time a communication,
not immediately apparent, with human agency, are enabled to change the
regular order and succession of their movements, according to existing
circumstances; and hence, in some measure, to assume the character of
living beings.

The third class contains those Automata which, under the semblance only
of mechanism, are wholly directed and controlled by a concealed human
agent.

The phenomena of the Chess Player are inconsistent with the effects
of mere mechanism; for, however great and surprising the powers of
mechanism may be, the movements that spring from it, are necessarily
limited and uniform: it cannot usurp and exercise the faculties of
mind; it cannot be made to vary its operations, so as to meet the
ever-varying circumstances of a game of chess. This is the province of
intellect alone; and the Chess Player must consequently relinquish all
claim to be admitted into the first division. Let us examine its title
to be ranked in the second class.

       *       *       *       *       *

The chess board contains sixty-four squares, and in order to execute
the movements of the Chess Player, distinct trains of machinery must be
formed, which shall be capable, when set in motion, of conveying the
hand of the Automaton to each, and to any, of these several squares.
Having arrived at a square, and taken up a chess man, it will be
requisite, either to withdraw the hand towards the side, and without
the limits of the board, for the purpose of letting drop the chess man
there, and thence to proceed to another square, and remove a chess man
to a third square; or it may be required to pass at once from the first
square to any other on the board, and there to deposit the man. These
movements must be promptly performed, and repeated as often as the
circumstances of the game may call for them.

Setting aside a great variety of minor details, it will be evident
to any person, even slightly acquainted with mechanics, that the
execution of these movements, so extensive, so complicated, and so
variable, would be attended with difficulties almost insurmountable;
but we will suppose for a moment that these obstacles are overcome;
let it be conceded that a machine has been constructed so perfect,
that, on giving motion to the respective trains, the required movement
shall be instantly performed. What then? The main object will be
still unattained! Where is the intelligence and the “promethean heat”
that can animate the Automaton and direct its operations? Not only
must an intellectual agent be provided, but between such an agent
and his deputy, the Automaton, a direct communication must be formed
and preserved, liable to no interruption, and yet so secret that the
penetrating eye of the most inquisitive observer may not be able to
detect it. Till this be done, the Chess Player’s title to be admitted
into the second division will, at any rate, continue in abeyance.

I am aware that on this part of the subject conjecture has been busy,
and different plans have been devised for the maintenance of the
intercourse alluded to. The task has been imposed on the exhibiter of
the machine, he being the only person on whom it could devolve with
even a shadow of probability; and to effect his purpose it has been
suggested that he might touch certain springs, or pull “a wire not
much thicker than a hair,” or be furnished with a powerful magnet. But
such conjectures are unworthy of serious refutation; for besides the
uncertainty and constant liability to interruption of such modes of
communication, they are actually at variance with the uniform conduct
of the exhibiter. Whoever has witnessed the exhibition will have seen
that the exhibiter is not confined to a particular spot in the room,
but, on the contrary, that he is frequently, during the progress of the
game, at a distance from the chest, far beyond the sphere of influence
of any of these proposed modes; and if, at such times, the Automaton
can move a single joint, it is proof decisive that its action springs
from another source.

Having now shewn how difficult, and perhaps impossible, it would be to
execute the movements of the Chess Player by mechanism, and maintain,
at the same time, a communication with the agent, who would be required
to give life, as it were, and intelligence to the operations, it
becomes necessary to inquire whether the prevailing opinion, which
attributes these movements to machinery, be, or be not, established
in fact; for, if this opinion should be found, on examination, to
originate merely in the artful management and display of some parts of
the apparatus, and to rest on no solid basis, there would be no longer
any embarrassment in appreciating the real value of the Chess Player,
nor in apportioning a proper station for it, considered as a work of
art.

At the commencement of the exhibition the spectators are gratuitously
made acquainted with the interior of the chest, which is divided into
two unequal compartments, and occupied by pieces of machinery, so
arranged, as apparently to render the concealment of any human being
impossible. When the movements of the Automaton begin, the beholders,
in the first moments of surprise, and in the absence of any ostensible
living cause, very naturally refer the effect to the mechanism, which
has been exhibited; and with likelihood enough, for the movements
immediately follow the familiar action and well known sound of winding
up clockwork, and are moreover very skillfully accompanied by the
grating noise of moving wheels. But, these indications excepted,
where is the evidence that the machinery moves, or that the slightest
influence is exerted by it on the arm of the Automaton? The whole is
excluded from view, and a moment’s reflection will convince any one
that no stress can be laid on the winding up, nor on the accompanying
sounds, which are imitable in various ways.

If, however, no proof can be given of the actual movement of the
machinery, the following considerations will tend to shew that it
remains quiescent, and is probably not formed for motion.

An artist, whose talents had enabled him to contrive machinery capable
of executing the varied and extensive movements displayed by the
Automaton, would surely be desirous of laying open to view as much of
the mechanism of his contrivance, while in actual motion, as he could
do, consistently with the reservation of his secret; if for no other
reason, at least to convince the lookers-on that deception formed no
part of his plan. Now it cannot be reasonably urged, in vindication
of the inventer’s forbearance, in the instance of the Chess Player,
that even a glance at any part of the machinery in motion would betray
the secret; for a question will immediately arise, Why then is the
machinery at rest so freely exposed? On that score no apprehension
seems to be entertained; the chest is ostentatiously opened, and the
semblance, at least, of wheels, and pullies, and levers, is submitted
to inspection without reserve: but when their reality should appear,
and their connection with the Automaton be made manifest, the doors
are carefully closed, and the spectators are required to pay large
drafts on their credulity, without any means of further examination.
The glaring contradiction between eager display on the one hand, and
studied concealment on the other, can only be reconciled by considering
the exhibition of the mechanism as a mere stratagem, calculated to
distract the attention, and mislead the judgment, of the spectators.

The truth of this opinion receives additional support from the regular
and undeviating mode of disclosing the interior of the chest. If the
mechanism were the real object in view, the whole being quiescent, it
would be matter of indifference which part was first laid open; and
accident alone, unless powerful reasons operated against it, would lead
occasionally to some variation. But no variation has ever been observed
to take place. One uniform order, or routine, is strictly adhered to,
and this circumstance alone is sufficient to awaken suspicion, for it
shews plainly that more is intended by the disclosure than is permitted
to meet the eye.

It has already been suggested, that little stress could be laid on the
winding up: indeed the simple act of turning round a key or winder
can offer no argument in proof of the efficiency of the machinery,
unless at the same time it could be shewn that the key, in turning,
either acted upon a spring, or pulled up a weight, for the purpose of
giving motion to the machinery in question. But unluckily for the Chess
Player, the phenomena afford positive proof that the axis turned by the
key is quite free, and unconnected, either with a spring, or a weight,
or any system of machinery.

In all machines requiring to be wound up, two consequences are
inseparable from their construction: the first is, that, in winding up
the machinery, the key is limited in the number of its revolutions;
and the second is, that some relative proportion must be constantly
maintained betwixt the winding up and the work performed, in order to
enable the machine to continue its movements. Now these results are
not observable in the Chess Player; for the Automaton will sometimes
execute sixty-three moves with only one winding up; at other times
the exhibiter has been observed to repeat the winding up after seven
moves, and even three moves; and once, probably from inadvertence,
without the intervention of a single move; whilst, in every instance
and the circumstance, though trifling, calls for particular attention,
(for, in these matters, be it remembered, “trifles light as air, are
confirmations strong,”) the key appeared to perform the same number of
revolutions; evincing thereby, that the revolving axis was unconnected
with machinery, except, perhaps, a ratchet-wheel and click, or some
similar apparatus, to enable it to produce the necessary sounds,
consequently that the key, like that of a child’s watch, might be
turned, whenever the purposes of the exhibition seemed to require it.

       *       *       *       *       *

I shall now pass on to the third division, and point out a method by
which any person, well skilled in the game, and not exceeding the
ordinary bulk or stature, may secretly animate the Automaton, and
successfully imitate the movements of Mr. De Kempelen’s Chess Player.

The general plan and dimensions of the chest will be understood by
inspecting the plates, but some particulars, relative to the interior,
will require further explanation.

The drawer (GG, plate 5,) when closed, does not reach to the back of
the chest; it leaves a space (O) behind it, about 1 foot 2 inches
broad, 8 inches high, and 3 feet 11 inches long. This space is never
exposed to view.

The small cupboard is divided into two parts by the door or screen (I,
fig. 6,) which is moveable on a hinge, and is so contrived that when B
is closed, this screen may be closed also. The machinery (H) occupies
the whole of the front division as far as I; the hinder division is
nearly empty, and communicates with the space behind the drawer, the
floor of this division being removed.

The back of the great cupboard is double, and the part (P Q,) to which
the quadrants, &c. are attached, moves on a joint (Q), at the upper
part, and forms, when raised, an opening (S) between the two cupboards,
by carrying with it part of the partition (R), which is composed of
cloth stretched tight. Fig. 10 shews the false back closed. Fig. 11
shews the same raised, forming the opening (S) between the chambers.

When the trunk of the figure is exposed by lifting up the dress, it
will be seen that a great part of it is occupied by an inner trunk (N),
which passes off towards the back in the form of an arch, (fig. 2), and
conceals a portion of the interior from the view of the spectators.
This inner trunk opens to the chest by an aperture (T, fig. 9), about 1
foot 3 inches high, by 1 foot broad.

When the false back is raised, the two chambers, the trunk, and the
space behind the drawer, are all connected together.

The player may be introduced into the chest through the sliding panel
(U, fig. 6), at the end. He will then elevate the false back of the
large cupboard, and assume the position represented by the dotted
lines in figs. 3 and 4. Every thing being thus prepared, “the charm’s
wound up,” and the exhibiter may begin his operations by opening the
door (A). From the crowded and very ingenious disposition of the
machinery in this cupboard, the eye is unable to penetrate far beyond
the opening, and the spectator is led to conclude that the whole space
is occupied with a similar apparatus. This illusion is strengthened
and confirmed by observing the glimmering light which plays among the
intricacies of the machinery, and occasionally meets the eye, when the
lighted candle is held at the door (B). A fact, too, is ascertained,
which is equally satisfactory, though indeed for opposite reasons,
to the spectator and the exhibiter, viz. that no opake body of any
magnitude is interposed between the light and the spectator’s eye.
The door (B) must now be locked, and the screen (I) closed, which
being done at the moment the light is withdrawn, will wholly escape
observation.

It has already been mentioned, that the door (B), from its
construction, closes by its own weight; but as the player’s head will
presently be very near it, the secret would be endangered, if, in
turning round the chest, this door were, by any accident, to fly open;
it becomes necessary, therefore, “to make assurance double sure,” and
turn the key. If the circumstance should be observed, it will probably
be considered as accidental, the keys being immediately wanted for the
other locks.

The opening (B) being once secured, and the screen (I) closed, the
success of the experiment may be deemed complete. The secret is no
longer exposed to hazard; and the exhibiter is at liberty to shape his
conduct in any way, he may think, most likely to secure the confidence
of the spectators, and lead them insensibly from the main object of
pursuit. The door (A) may be safely left open; and this will tend to
confirm the opinion, which the spectators probably formed on viewing
the candle through this cupboard, that no person was concealed within
it: it will further assure them that nothing can pass in the interior
without their knowledge, so long as this door continues open.

The drawer stands next in the order of succession: it is opened,
_apparently_, for the purpose of taking out the chess men, cushion, &c.
but _really_ to allow time for the player to change his position, (see
fig. 5.) and to replace the false back and the partition, preparatory
to the opening of the great cupboard.

The machinery is so thinly scattered over this cupboard, that the eye
surveys the whole space at one glance, and it might seem unnecessary
to open a door at the back, and to hold a lighted candle there, as in
the former instance; but the artifice is dictated by sound policy,
which teaches that the exhibiter cannot be too assiduous in affording
facilities to explore every corner and recess, which, he well knows,
contain nothing that he is desirous of concealing.

The chest may now be wheeled round for the purpose of shewing the trunk
of the figure; leaving, however, the front doors of the great chamber
open. The bunch of keys, too, should be suffered to remain in the door
(D); for the apparent carelessness of such a proceeding will serve
to allay any suspicion, which the circumstance of locking the door
(B) might have excited, more especially as the two doors resemble one
another in point of construction.

When the drapery has been lifted up, and the doors in the trunk and
thigh opened, the chest may be returned to its former situation, and
the doors be closed. In the mean-time the player should withdraw his
legs from behind the drawer, as he will not so easily effect this
movement after the drawer has been pushed in.

Here let us pause awhile, and compare the real state of the chest
at this time, with the impression which, at a similar period of an
exhibition of the Chess Player, has generally been left on the minds
of the spectators; the bulk of whom have concluded that each part of
the chest had been successively exposed; and that the whole was at that
time open to inspection: whereas, on the contrary, it is evident that
some parts had been entirely withheld from view, others but obscurely
shewn, and that nearly half of the chest was then excluded from their
sight. Hence we learn how easily, in matters of this sort, the judgment
may be led astray by an artful combination of circumstances, each
assisting the other towards the attainment of one object.

When the doors in front have been closed, the exhibiter may occupy as
much time, as he finds necessary, in apparently adjusting the machinery
at the back, whilst the player is taking the position described
in figs. 7 and 8. In this position he will find no difficulty in
executing every movement required of the Automaton: his head being
above the table, he will see the chess board through the waistcoat,
as easily as through a veil; and his left hand extending beyond the
elbow of the figure, he will be enabled to guide its hand to any part
of the board, and to take up and let go a chess man with no other
“delicate mechanism” than a string communicating with the fingers. His
right hand being within the chest, may serve to keep in motion the
contrivance for producing the noise, which is heard during the moves,
and to perform the other tricks of moving the head, tapping on the
chest, &c.

In order to facilitate the introduction of the player’s left arm into
the arm of the figure, the elbow of the latter is obliged to be drawn
backwards; and to account for, and conceal, this strained attitude, a
pipe is ingeniously placed in the Automaton’s hand. This pipe must not
be removed till the other arrangements are completed.

When all is ready, and the pipe removed, the exhibiter may turn round
the winder, or key, to give the impression to the spectators of
winding up a spring, or weight, and to serve as a signal to the player
to set the head of the Automaton in motion.

The above process is simple, feasible, and effective; shewing
indisputably that the phenomena may be produced without the aid of
machinery, and thereby rendering it probable that the Chess Player
belongs in reality to the third class of Automata, and derives its
merit solely from the very ingenious mode by which the concealment of a
living agent is effected.

In conducting this analysis, the author disclaims even the slightest
wish or intention to depreciate, or detract from, the real merits of
Mr. De Kempelen: those merits have long since received the stamp of
public approbation; indeed, a more than ordinary share of skill and
ingenuity must have fallen to his lot, who could imagine and execute a
machine (it matters not by what means the phenomena are brought about)
which has never failed to delight the spectators, by exciting and
maintaining, above all other contrivances of the kind, that pleasing
delusion in the mind, which the Roman poet has so happily denominated
“_Mentis gratissimus error_.”

    _December, 1820._



                 EXPLANATION OF THE PLATES.


                          PLATE I.
    Fig. 1.   A perspective view of the Automaton, seen in
                front, with all the doors thrown open.
    Fig. 2.   An elevation of the back of the Automaton.

                         PLATE II.
    Fig. 3.   An elevation of the front of the chest, the
                dotted lines representing the player in
                the first position.
    Fig. 4.   A side elevation, shewing the player in the
                same position.

                         PLATE III.
    Fig. 5.   A front elevation, shewing the second position.
    Fig. 6.   An horizontal section through the line WW. fig. 5.

                         PLATE IV.
    Fig. 7.   A front elevation, shewing the third position.
    Fig. 8.   A side elevation of the same position.

                          PLATE V.
    Fig. 9.   A vertical section through the line XX, fig. 8.
    Fig. 10.  A vertical section through the line YY, fig. 7,
                shewing the false back closed.
    Fig. 11. A similar section, shewing the false back raised.


               THE FOLLOWING LETTERS OF REFERENCE ARE
                    EMPLOYED IN ALL THE PLATES.

    A    Front door of the small cupboard.
    B    Back door of ditto.
    CC   Front doors of the large cupboard.
    D    Back door of ditto.
    E    Door in the trunk.
    F    Door in the thigh.
    GG   The drawer.
    H    Machinery in front of the small cupboard.
    I    Screen behind the machinery.
    K    Opening caused by the removal of part of the
           floor of the small cupboard.
    L    A box which serves to conceal an opening in the
           floor of the large cupboard, made to facilitate
           the first position; and which also serves as a
           seat for the player in the third position.
    M    A similar box to receive the toes of the player in
           the first position.
    N    The inner chest, filling up part of the trunk.
    O    The space behind the drawer.
    PQ   The false back, turning on a joint at Q.
    R    Part of the partition formed of cloth stretched
           tight, which is carried up by the false back,
           to form the opening between the chambers.
    S    The opening between the chambers.
    T    The opening connecting the trunk and chest,
           which is partly concealed by the false back.
    U    Panel which is slipped aside to admit the player.

[Illustration: _Plate 2. Fig. 3._]

[Illustration: _Fig. 4._

_Drawn on Stone by the Author. Printed by C. Hullmandel_.]

[Illustration: _Plate 3. Fig. 5._]

[Illustration: _Fig. 6._

_Drawn on Stone by the Author. Printed by C. Hullmandel._]

[Illustration: _Plate 4. Fig. 7._]

[Illustration: _Fig. 8._

_Drawn on Stone by the Author. Printed by C. Hullmandel._]

[Illustration: _Plate 5. Fig. 9._]

[Illustration: _Fig. 10._]

[Illustration:_Fig. 11._

_Drawn on Stone by the Author. Printed by C. Hullmandel._]



                     APPENDIX.


The Knight’s move over the chess board has engaged the attention of so
many scientific men, that I cannot doubt that a collection of different
solutions of the problem will prove acceptable to all admirers of chess.

The Knight’s path is of two kinds—terminable and interminable—it is
interminable, whenever the last, or concluding, move of a series be
made on a square, which lies within the Knight’s reach of that from
which he originally set out—and terminable in every other instance.

Euler published a paper in the Memoirs of the Academy of Berlin, 1759,
which contains a method of filling up all the squares, setting out from
one of the corners. It also contains an endless or interminable route;
and explains a principle by which these routes may be varied so as to
end upon any square. Montmort, Demoivre, and Mairan, have severally
given solutions of the same problem. These solutions will be found in
the following collection.

Observing that the Automaton, under the direction of Mr. Maelzel,
occasionally traversed half the board, I was induced to pursue
the subject, and I found that the move might be performed on any
_parallelogram_ consisting of _twelve_ squares and upwards, with the
exception of _fifteen_ and _eighteen_ squares. The whole board admits
of a great variety both in the terminable and interminable routes.

In describing the Knight’s path, I have preferred lines to figures;
the former giving a clearer idea of the plan pursued, and affording a
greater facility of comparing one route with another, than the latter.

DIRECTIONS FOR PLACING THE PLATES.

        _Plate_  1 _to face the Title._
        _Plates_ 2 _to_ 5 — _Page_ 36.
           ——    6 _to_ 10    ——   38.

[Illustration: _Plate 6._]

[Illustration: _Plate 7._]

[Illustration: _Plate 8._]

[Illustration: _Plate 9._]

[Illustration: _Plate 10._]



LIST OF THE KNIGHT’S MOVES

Contained in Plates 6, 7, 8, 9, and 10.


_Methods of performing the Move on Parallelograms less than the whole
Board._

           No. 1 the Move on 12 Squares
           —   2  -   -   -  20    -
           —   3  -   -   -  25    -
           —   4  -   -   -  21    -
           —   5  -   -   -  24    -
           —   6  -   -   -  24    -
           —   7  -   -   -  30    -
           —   8  -   -   -  36    -
           —   9  -   -   -  28    -
           —  10  -   -   -  32    -
           —  11  -   -   -  35    -
           —  12  -   -   -  40    -
           —  13  -   -   -  42    -
           —  14  -   -   -  48    -
           —  15  -   -   -  49    -
           —  16  -   -   -  56    -
           —  17 an Interminable Route on 48 Squares
           —  18       Do.         Do.    56    -

     _Terminable Routes over the whole Board._

                  No. 20  By Euler
                   —  21  -    Do.
                   —  22  -    Do.
                   —  23  -    Do.
                   —  24  By Demoivre
                   —  25  -    Do.
                   —  26  By Mairan
                   —  27  By Montmort
                   —  28  By the Author
                   —  29  -    Do.

    _Interminable Routes over the whole Board._

                  No. 30   By Euler
                   —  31   By Mons. W.
                   —  32   By the Author
                   —  33   -    Do.
                   —  34   -    Do.
                   —  35   -    Do.
                   —  36   -    Do.
                   —  37   -    Do.
                   —  38   -    Do.
                   —  39   -    Do.


HOWLETT AND BRIMMER,

Printers, 10, Frith Street, Soho.





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