Little moon of Hippocrates

The sum of the red "little moon" corresponds to the area of ​​the right triangle

With the little moons of Hippocrates , which are ascribed to the Greek mathematician Hippocrates of Chios (around 450 BC), it was already possible to prove in ancient Greece that even curved areas can be calculated using rational numbers .

proof

According to the Pythagorean theorem , the sum of the areas of the cathetus squares of a right triangle is equal to the area of ​​the hypotenuse square . According to the generalized Pythagorean theorem, this connection also applies to other similar figures . For semicircles this means: The sum of the area of ​​the semicircles above the cathetus corresponds to the area of ​​the semicircle above the hypotenuse (step 1).

Step 1

step 2

step 3

If you mirror the semicircle over the hypotenuse, it overlaps with the two cathetus semicircles, whereby the circular arc goes through point C according to Thales's theorem (step 2).

If you remove the overlapping circle segments (step 3), the triangle itself remains of the hypotenuse semicircle and the two crescent-shaped outer circle parts of the two cathetus semicircles, the moons.

The following applies:

${\displaystyle A_{\text{Blau}}+A_{\text{Gelb}}=A_{\text{Rot}}\,}$

and

${\displaystyle A_{\text{Blauer Mond}}+A_{\text{Gelber Mond}}=A_{\text{Blau}}+A_{\text{Gelb}}-A_{\text{Kreissegmente}}.\,}$

Out

${\displaystyle A_{\text{Kreissegmente}}=A_{\text{Rot}}-A_{\text{Dreieck}}\,}$

then follows:

${\displaystyle A_{\text{Blauer Mond}}+A_{\text{Gelber Mond}}=A_{\text{Dreieck}}.\,}$

variants

The sum of the red “little moon” corresponds to the area of ​​the square

There are many different variants and possibilities for generalizing the Pythagorean theorem and the little moon of Hippocrates. In addition to the right-angled triangle mentioned above, another example is the following square, each of which has a lunar over the four sides of the square.

See also

• Similar figures erected over the sides of the right triangle

literature

• Egmont Colerus: From multiplication tables to integral. Math for everyone . Rowohlt, Reinbek 1982 (Chapter: Problem of the Quadrature , p. 249 in the Paul Zsolnay Verlag edition, 1934). ISBN 3-499-16692-5
• Paul Karlson: On the magic of numbers. Fun math for everyone . Ullstein, Berlin 1954. p. 140
• Hans Wußing : 6000 years of mathematics . Springer, Berlin a. a. 2008. ISBN 978-3-540-77189-0 . P. 172 ff.

Web links

Commons : Little Moons of Hippocrates  - Collection of images, videos and audio files

• Meyer's large conversation lexicon from 1905: Lunŭlae Hippocrătis at Zeno.org .

Individual evidence

1. ^ Oskar Becker : The mathematical thinking of antiquity ; III. Mathematics of the 5th Century , 3. Lunulae Hyppocratis , Göttingen Vandenhoeck & Ruprecht, 1966, p. 58 ( limited preview in the Google book search), accessed on May 21, 2019
2. ↑ Thomas Heath : A History of Greek Mathematicus, (a) Hippocrates's quadrature of lunes. tape 1 . The Clarendon Press, Oxford 1921, pp. 183 ff. Fig. page 185 (English, wilbourhall.org [PDF]). accessed on May 21, 2019