Motorola M500 User Manual download pdf (Page 17)

M500 180 Page 15

The full construction is as follows.

Draw a diameter of the circle AB and name the centre O. Choose a

point P on the circle. Draw a line parallel to AOB through P using the

given construction. Let this meet the circle the second time at Q. Draw AP

and BQ to meet at R. Draw and extend OR to meet the circle at F and G.

By symmetry, OR is perpendicular to AB.

Construct a line through F parallel to AOB and a line through A parallel

to FOG crossing at J. Then JFOA is the required square.

ADF writes—Observe that the ruler is used only to draw straight lines

through well-deﬁned points. As a couple of readers demonstrated, the con-

struction becomes much easier if it is permitted to use the ruler in an imag-

inative manner. Notice also that signiﬁcant use is made of the curved part

of the circle-and-its-centre. This is expected. A unit line segment by itself

is not suﬃcient for constructing the irrational length

√

2, the diagonal of

the square.

Problem 178.2 and Problem 176.5 (Given a unit line segment, construct

a unit square using only a pair of compasses—see M500 176 28 and p.

25 of this issue) are special cases of a more general result: None of the

traditional ‘Euclidean’ ruler-and-compasses constructions actually require

both instruments.

As suggested above, it seems that we can dispense with the compasses

if there is a unit circle already on the page. This is indeed the case, as was

proved by Jakob Steiner in Die geometrischen Konstruktionen ausgef¨uhrt

mittels der gareden Linie und Eines festen Kreises (Berlin 1833). On the

other hand, L. Mascheroni (La geometria del compasso, Pavia 1797) proved

that Euclidean constructions do not require the ruler if one has access to

a ﬁxed unit line segment. Heinrich Dorrie discusses these two theorems

(as well as 98 others) in 100 Great Problems of Elementary Mathematics

(Dover, New York 1965).

1 2 ... 12 13 14 15 16 17 18 19 20 21 22 ... 31 32

No comments

Motorola M500 User Manual Page 17