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 ...

     

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

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

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

How this FORMULA (*) came out of ???
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[1st]  A GEOMETRICAL CONSTRUCTION ...

     

IF you want to find the »GS«-POINT (C) in a given LINE-AB =!= 2, THAN ...
  o  Draw a rectangular 1/2 LENGTH-OF-LINE-AB =!= 1 in POINT-B to POINT-X
  o  Draw the HYPOTENUSE of this RECTANGLE from POINT-A to POINT-X
  o  With PYTHAGORAS's RULE the lenth of the HYPOTENUSE is SQRT(5)
  o  Put the COMPASS in X: (a) Mark the LENGHT = 1 on the HYPOTENUSE
  o  Put the COMPASS in A: (b) Transfer THIS POINT to LINE-AB as POINT-C
  o  POINT-C is »GS«, the "Golden Cut" of LINE-AB

The GEOMETRICAL CONSTRUCTION showed the relation AC/AB = 0.618 ...
Now the COUNTER-PROOF:

  CB       2 - ( SQRT(5) - 1 )       3 - SQRT(5)                 AC
 ----  =  ---------------------  =  -------------  =  0.618  =  ----   qed.
  AC           SQRT(5) - 1           SQRT(5) - 1      =====      AB

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[2nd]  AN ANALYTICAL SOLUTION ...

  From (*)  AC / AB  =  CB / AC

            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 ...

                x2 + px + q = 0   ===>>   x1,2 = - p/2 +/- SQRT( [p/2]2 - q )

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

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

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

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REMARK:  As a "Rule-Of-Thumb"  60% / 40%  is a good guess of the  »GS«-POINT (C)

PART TWO = AN ENGINEERING APPROACH = BOWEN Model 770 ...

                     Back to the ENGINEERING TOOLS Main Page
impressum:
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© C.HAMANN              http://public.BHT-Berlin.de/hamann          (*) 09/10/11