HOW  A  PROPORTIONAL  DIVIDER  WORKS
( PART  ONE )


THE  H A F F  PROPORTIONAL DIVIDER  (= Reduktions-Zirkel in German)  Model 195aE
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PURPOSE:  Engineers have used this tool to transfer proportions from one drawing
to a new drawing in bigger or smaller scale. Open-air painters used it to trans-
fer distances in the landscape or art painters took proportions of the beautiful
model to sketch it on paper - as base of the artwork in mind.

USAGE:  With closed legs loosen the  TOP SCREW.   With the  OPPOSITE SCREW  move
the axle up or down to the  SCALE-MARK  of your choice.  Tighten the  TOP-SCREW.
Open the  LEGS, sample the original distance with  ONE SIDE - the OPPOSITE SIDE
will be in the desired  SCALE-PROPORTION.

There are 2 SCALE-MARKS:

  o  In  »LINES«  the sampled  DISTANCES  are set in proportion.
     ( 1:1 ;  4:3 ;  3:2 ;  5:3 ;  2 ;  2.5 ;  3 ;  4 ;  5 ;  ...  9 ;  10 )
  
  o  In  »CIRCLES«  when its  D I A M E T E R  IS TAKEN AS THE SAMPLE,
     the  OPPOSITE SIDE  will cut the circumference in equal sections:
     ( 4 ;  GS ;  5 ;  6 ;  7 ;  8 ;  9 ;  10 ;  ...     16 ;      ...  20 )

     E.g.:
     »4«  will make 4 Segments  ( = 90° Sectors = Corners of a Square ),
     »5«  will make 5 Segments  ( = 72° Sectors = Corners of a Pentagon),
         ...
     out of the full circle ...

     

           PART TWO - A DIFFERENT APPROACH - The BOWEN Model 770 ...


The Mark  »GS«  ( = "Goldener Schnitt" = "Golden Cut" )
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  (1)   Separates a  LINE  AB  in  C  to the well known proportion:

   |===============================|==================|
   A                               C                  B

        AC / AB  =  CB / AC   =   0.618   =   ( sqrt(5) - 1 ) / 2       (*)


  (2)   When the RADIUS is sampled, it divides its CIRCLE in 10 ( 36° ) SECTORS


R E M A R K   TO  THE  QUESTION            " How the FORMULA (*) came out of ? "
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  From (1)    we get   AC2 = AB * CB     and let       AB  =!=  1  =  AC + CB
                                         than                  CB  =   1 - AC   
              now substituted

                       AC2 = 1 - AC      follows     AC2 + AC - 1  =  0

              Using the well known formula to solve a squared equation ...

                       AC  =  - 1/2  -/+  sqrt( 1/4  + 1 )

                           =  - 1/2    +  sqrt( 5 ) / 2        { only + real }

         ...  we get       =  ( sqrt(5) - 1 ) / 2                    qed.

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impressum:
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© C.HAMANN           http://public.beuth-hochschule.de/~hamann          11/10/09