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 ... The Mark »GS« ( = "Goldener Schnitt" = "Golden Cut" ) ------------------------------------------------------- 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 ??? ******************************************************************************** [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 ******************************************************************************** [2nd] AN ANALYTICAL SOLUTION ... From (*) AC / AB = CB / AC we get AC |