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[PEP Editor’s Note: Due to the use of symbols in this article, a PDF Original is available showing the original formatting.]
The ‘point’ and the ‘line’ represent visual images which remain invariant under a wide range of conditions. The same is true of the visual images associated with the propositions of Euclid; hence the propositions themselves are communicated over long periods of time and between widely separated cultures and races. This may appear to be true of any visual representation, such as cave paintings, but although the column 3 component, record or notation, is strong, the artistic representation does not appear to be so rich in ideational content as the geometrical representation. The geometer Bhāskara1 drew four right-angled triangles inwards, one on each side of the square of the hypotenuse, and left it with the comment, “See!”2
There are two dissimilar observations that can be made. One is mathematical and can be stated: C2 = 4 (ab/2) + (a – b)2 = a2 – b2. The other is mythical and can be stated: of the 3, 4, 5 triangle, 3 is the first odd number and is perfect, 4 is the square on an even side 2, while 5 partly resembles the father and partly the mother, being the sum of 3 and 2, “and we must then liken the perpendicular to the male, the base to the female and the hypotenuse to the offspring of both”.3
The mathematical statement of the theme has been profitable and has led to development. The mythological statement, though widely
[Bhāskara (1114 ad–1185 bc) (also known as Bhāskarācārya) was the leading mathematician of the twelfth century, and the first known to have made full and systematic use of the decimal number system. Ed.]
T. Heath (1956), The Thirteen Books of Euclid's Elements, Vol. I, Book I, Proposition 47, p. 355.
[T. Heath (1956), The Thirteen Books of Euclid's Elements, Vol. I, Book I, Proposition 47, p. 417. Ed.]
spread, has not. Is it possible to release it from its fetters by reverting to point and line?
If we use the point alone to c