Why the  P Y T H A G O R A S  can be calculated with the P2, P1 & Q-Scales ???
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                                                                c2  =  a2 + b2

Starting with a scale of lenght 2p ...

  0       1/4      1/2      3/4       1       1.25     1.5      1.75      2
  |--------+--------+--------+--------+--------+--------+--------+--------|

Take the  SQUAREROOT  of this scale ...

  0       0.5      0.707    0.866     1       1.118    1.225    1.323    1.414
  |--------+--------+--------+--------+--------+--------+--------+--------|

  |<=============== P1 ==============>|<=============== P2 ==============>|

  |<=============== Q ===============>|         P2  =  sqrt( 1 + P12 )

  1       0.866    0.707    0.5       0          Q  =  sqrt( 1 - P12 )
  |--------+--------+--------+--------|             =  P1 inverted


Proof:        P2 = sqrt( 1 + 0.52 )   = 1.118
======             sqrt( 1 + 0.7072 ) = 1.225
                                     ...
               Q = sqrt( 1 - 0.52 )   = 0.866
                   sqrt( 1 - 0.7072 ) = 0.707
                                     ...

Now the scales  P1, P2, & Q  can be used for  ADDING SQUARES  ( see EX.A ) ...
                                                                ========
                                a=0.4
                             0     v 0.5      0.707    0.866     1       1.118 
                             |<====|--+--------+--------+--------+--------+---
  |--------+--------+--------+-----|=>|
  1       0.866    0.707    0.5    ^  0        c2   =   a2  +  b2
                                b=0.3         0.52  =  0.42 + 0.32


The Hyperbolic Functions of the BACK SIDE, based on scales P1 & P2 ...

  o  have an extended domain and resolution
  o  AND  use  P2  for calculating  CosH(X)  directly
  o  AND  calculate directly  HYPERBOLIC FUNCTIONS OF COMPLEX QUANTITIES

  Back to the SIC-1580 Main Page

Papers presented at the 3rd BERLIN-BRANDENBURGER SAMMLER-TREFFEN (BBST) in Berlin

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© C.HAMANN               http://public.BHT-Berlin.de/hamann              03/15/09