Symmetry has two meanings, a more usual particular,
geometric meaning, and a less usual, general, logical meaning.
     Elementary solid geometry considers two kinds of symmetry,
 symmetry with respect to a plane (called plane of 
symmetry) ,and symmetry with respect to a point (called 
centre of symmetry).  The human body appears to be 
fairly symmetrical but in fact it is not; many interior
organs are quite unsymmetrically disposed. A statue may
be completely symmetrical with respect to a vertical
plane so that its two hales appear completely "inter-
changeable."
     In a more general acceptance of the word, a whole is 
termed symmetric if it has interchangeable parts. There
are ma y kinds of symmetry; they differ in the number 
of interchangeable parts, and in the operations which 
exchange the parts. Thus, a cube has high symmetry; its
6 faces are interchangeable with each-other, and so are 
its 8 vertices and so are its 12 edges. The expression
yz+ zx+ xy
is symmetric; any two of the three letters x, y, z can be 
interchanged without changing the expression
     Symmetry in a general sense, is important for our 
subject.  If a problem is symmetric in some ways we may 
derive some profit from noticing its interchangeable
parts and it often pays to treat those parts which play
the same role in the same fashion (see AUXILIARY ELEMENTS,3).
     Try to treat symmetrically what is symmetrical, and do 
not destroy wantonly any natural symmetry.  However, 
we are sometimes compelled to treat unsymmetrically
what is naturally symmetrical.  A pair of gloves is certainly
symmetrical; nevertheless, nobody handles the pair
quite symmetrically, nobody puts on both gloves at the 
same time, but one after the other.
     Symmetry may also be useful in checking results; see
section 14.