Squaring the Circle

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Three Famous Problems
Greek Circle Squarer

### Greek Circle Squarers

#### Quadratrix-Hippias

Dinostratus (about 390 BC- about 320 BC),
a brother of Menaechmus (about 380 BC - about 320 BC),
gave a proof for the following proposition using
"reductio ad absurdum".

(arc PB) : PO = PO : OR

Or OR = (2/p) PO

For detail of the proof, see ref.1.

The modern derivation of this relation is given in the next section .

********* Hippias_circle_squarer_desc.dwg** *******

#### How to Draw a Quadratrix

OBQP is a square.
Divide both OP and BQ in equal parts N.

Divide a quarter circle BP into the same number N.
Point C,D & E arer such points.

Horizontal line CD(yellow) and polar line OE(red) intersect .

The locus of such points(cyan color) is the "Quadratrix".

You can see the process in **animation**.

************* quadratrix_curve_10_div.dwg** ***********

**To create this drawing and animation: **

** Load qd_trix.lsp (load "qd_trix")**

Then from command line, type **quadratrix_2 **for drawing quadratrix for 1000 division.

test_1 & test_2 for drawing manually.

If OP & OB are x & y axis respectively, the curve is expressed as
**y = x tan(p*y/2)**

Using identity tan(a) = sin(a)/cos(a), and replacing (p/2)y = h

x = (2/p)*cos(h)*(h/sin(h))

As h approaches zero, both cos(h) and (h/sin(h)) approach 1.

So the x-coordinate of the point R,where the Quadratrix
intersect X-axis is (2/p).

This means that length OR is used to get p value, and we now have succeeded in "Squaring the Circle".

This is the reason why this curve is named "Quadratrix", i.e. curve for circle quadrature.

#### References

- Heath,Thomas L. :"History of Greek Mathematics Vol. I From Thales to Euclid" Dover 1981

- Heath,Thomas L. :"A Manual of Greek Mathematics" Dover 1963 original 1931

- Heath,Thomas L. :"A History of Greek Mathematics Vol. II" Dover 1981 original in 1921

- Heath,Thomas L. :"The Works of Archimedes Dover" 2002 original in 1912

- Knorr,Wilber Richard :"The Ancient Tradition of Geometric Problems" Dover 1993

- Knorr,Wilber Richard :"The Textual Studies in Ancient and Medieval Geometry",Birkhauser, 1989

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Last Updated Jan 22, 2007

Copyright 2006 Takaya Iwamoto All rights reserved.
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