HOW  A  PROPORTIONAL  DIVIDER  WORKS
( PART  ONE )

 ```THE H A F F PROPORTIONAL DIVIDER (= Reduktions-Zirkel in German) Model 195aE ******************************************************************************** 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" ) ------------------------------------------------------- (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 ? " ******************************************************************************** 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. Back to the ENGINEERING TOOLS Main Page impressum: ******************************************************************************** © C.HAMANN http://public.beuth-hochschule.de/~hamann 11/10/09 ```