The  quantity  — b,  taken  once,  is  — b ;  taken  twice  is  — 2b ;
taken  3  times,  is  — 36 ;  and  hence,  taken  a  times,  is  — ab ;  that
is,  a  negative  quantity,  multiplied  by  a  positive  quantity,  gives  a
mgative  product.  This  is  generally  expressed  by  saying,  that
minus f  multiplied  by  plus,  gives  minv^.
80  RAY'S  ALGEBRA,  PART  SECOND.
3d.  Let  it  be  required  to  multiply  h  by  — a.
Since,  when  two  quantities  are  to  be  multiplied  together,  either
may  be  made  the  multiplier  (Art.  53),  this  is  the  same  as  to  mul-
tiply — a  by  b,  which  gives  — ah.  That  is,  a  positive  quantity
multiplied  by  a  negative  quantity,  gives  a  negative  product ;  or
more  briefly,  plus  multiplied  by  minus,  gives  minus.
4th.  Let  it  be  required  to  multiply  — h  by  — a.
The  negative  multiplier  signifies  that  the  multiplicand  is  to  be
taken  positively  as  many  times  as  there  are  units  in  the  multi-
plier, and  then  subtracted.
The  product  of  — b  by  a,  is  — ab,  and  then  changing  the  sign  to
subtract,  it  becomes  -^ab.  Hence,  the  product  of  two  negative
quantities  is  positive ;  or,  more  briefly,  minus  multiplied  by
minus,  gives  plus.
Note.  —  The  following  proof  of  the  last  principle,  that  the  product
of  two  negative  quantities  is  positive,  is  generally  regarded  by  mathe-
maticians as  more  satisfactory  than  the  preceding,  though  it  is  not  quite
80  simple.     Either  method  may  be  used.
5th.  To  find  the  product  of  two  negative  quantities.
