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each and its intended character； and secondly the principle we laid
down above that no Sanguineous creature can move itself at more than
four points。 Granting this it is evident that Sanguineous animals like
snakes； whose length is out of proportion to the rest of their
dimensions； cannot possibly have limbs； for they cannot have more than
four （or they would be bloodless）； and if they had two or four they
would be practically stationary； so slow and unprofitable would
their movement necessarily be。
But every limbed animal has necessarily an even number of such
limbs。 For those which only jump and so move from place to place do
not need limbs for this movement at least； but those which not only
jump but also need to walk； finding that movement not sufficient for
their purposes； evidently either are better able to progress with even
limbs or cannot otherwise progress at all every animal which has limbs
must have an even us for as this kind of movement is effected by
part of the body at a time； and not by the whole at once as in the
movement of leaping； some of the limbs must in turn remain at rest；
and others be moved； and the animal must act in each of these cases
with opposite limbs； shifting the weight from the limbs that are being
moved to those at rest。 And so nothing can walk on three limbs or on
one； in the latter case it has no support at all on which to rest
the body's weight； in the former only in respect of one pair of
opposites； and so it must necessarily fall in endeavouring so to move。
Polypods however； like the Centipede； can indeed make progress on an
odd number of limbs； as may be seen by the experiment of wounding
one of their limbs； for then the mutilation of one row of limbs is
corrected by the number of limbs which remain on either side。 Such
mutilated creatures； however； drag the wounded limb after them with
the remainder； and do not properly speaking walk。 Moreover； it is
plain that they； too； would make the change of place better if they
had an even number； in fact if none were missing and they had the
limbs which correspond to one another。 In this way they could equalize
their own weight； and not oscillate to one side； if they had
corresponding supports instead of one section of the opposite sides
being unoccupied by a limb。 A walking creature advances from each of
its members alternately； for in this way it recovers the same figure
that it had at first。
9
The fact that all animals have an even number of feet； and the
reasons for the fact have been set forth。 What follows will explain
that if there were no point at rest flexion and straightening would be
impossible。 Flexion is a change from a right line to an arc or an
angle； straightening a change from either of these to a right line。
Now in all such changes the flexion or the straightening must be
relative to one point。 Moreover； without flexion there could not be
walking or swimming or flying。 For since limbed creatures stand and
take their weight alternately on one or other of the opposite legs； if
one be thrust forward the other of necessity must be bent。 For the
opposite limbs are naturally of equal length； and the one which is
under the weight must be a kind of perpendicular at right angles to
the ground。
When then one leg is advanced it becomes the hypotenuse of a
right…angled triangle。 Its square then is equal to the square on the
other side together with the square on the base。 As the legs then
are equal； the one at rest must bend either at the knee or； if there
were any kneeless animal which walked； at some other articulation。 The
following experiment exhibits the fact。 If a man were to walk parallel
to a wall in sunshine； the line described （by the shadow of his
head》 would be not straight but zigzag； becoming lower as he bends；
and higher when he stands and lifts himself up。
It is； indeed； possible to move oneself even if the leg be not bent；
in the way in which children crawl。 This was the old though
erroneous account of the movement of elephants。 But these kinds of
movements involve a flexion in the shoulders or in the hips。 Nothing
at any rate could walk upright continuously and securely without
flexions at the knee； but would have to move like men in the wrestling
schools who crawl forward through the sand on their knees。 For the
upper part of the upright creature is long so that its leg has to be
correspondingly long； in consequence there must be flexion。 For
since a stationary position is perpendicular； if that which moves
cannot bend it will either fall forward as the right angle becomes
acute or will not be able to progress。 For if one leg is at right
angles to the ground and the other is advanced； the latter will be
at once equal and greater。 For it will be equal to the stationary
leg and also equivalent to the hypotenuse of a right…angled
triangle。 That which goes forward therefore must bend； and while
bending one； extend the other leg simultaneously； so as to incline
forward and make a stride and still remain above the perpendicular；
for the legs form an isosceles triangle； and the head sinks lower when
it is perpendicularly above the base on which it stands。
Of limbless animals； some progress by undulations （and this
happens in two ways； either they undulate on the ground； like
snakes； or up and down； like caterpillars）； and undulation is a
flexion； others by a telescopic action； like what are called
earthworms and leeches。 These go forward； first one part leading and
then drawing the whole of the rest of the body up to this； and so they
change from place to place。 It is plain too that if the two curves
were not greater than the one line which subtends them undulating
animals could not move themselves； when the flexure is extended they
would not have moved forward at all if the flexure or arc were equal
to the chord subtended； as it is； it reaches further when it is
straightened out； and then this part stays still and it draws up
what is left behind。
In all the changes described that which moves now extends itself
in a straight line to progress； and now is hooped； it straightens
itself in its leading part； and is hooped in what follows behind。 Even
jumping animals all make a flexion in the part of the body which is
underneath； and after this fashion make their leaps。 So too flying and
swimming things progress； the one straightening and bending their
wings to fly； the other their fins to swim。 Of the latter some have
four fins； others which are rather long； for example eels； have only
two。 These swim by substituting a flexion of the rest of their body
for the （missing） pair of fins to complete the movement； as we have
said before。 Flat fish use two fins； and the flat of their body as a
substitute for the absent pair of fins。 Quite flat fish； like the Ray；
produce their swimming movement with the actual fins and with the
two extremes or semicircles of their body； bending and straightening
themselves alternately。
10
A difficulty might perhaps be raised about birds。 How； it may be
said； can they； either when they fly or when they walk； be said to
move at four points？ Now we did not say that all Sanguinea move at
four points； but merely at not more than four。 Moreover； they cannot
as a fact fly if their legs be removed； nor walk without their
wings。 Even a man does not walk without moving his shoulders。
Everything indeed； as we have said； makes a change of place by flexion
and straightening； for all things progress by pressing upon what being
beneath them up to a point gives way as it were gradually；
accordingly； even if there be no flexion in another member； there must
be at least in the point whence motion begins； is in feathered
（flying） insects at the base of the 'scale…wing'； in birds at the base
of the wing； in others at the base of the corresponding member； the
fins； for instance； in fish。 In others； for example snakes； the
flexion begins in the joints of the body。
In winged creatures the tail serves； like a ship's rudder； to keep
the flying thing in its course。 The tail then must like other limbs be
able to bend at the point of attachment。 And so flying insects； and
birds （Schizoptera） whose tails are ill…adapted for the use in
question； for example peacocks； and domestic cocks； and generally
birds that hardly fly； cannot steer a straight course。 Flying
insects have absolutely no tail； and so drift along like a
rudderless vessel； and beat against anything they happen upon； and
this applies equally to sharded insects； like the scarab…beetle and
the chafer； and to unsharded； like bees and wa