^^Geometrical algebra di Euclide.

Il libro2 di Euclide, dai commentatori viene titolato (dato che Euclide non da' titolo ai libri)

• Fundamentals of Geometric Algebra (Richard Fitzpatrick)
• Geometric algebra (David E. Joyce)
• Geometrical algebra (Heath)

Def2 gnomone

pag379 def2 Book2

Intro

the Pythagoreans and later Greek mathematicians exhibited different kinds of numbers as forming different geometrical figures.

Thus, says Theon of Smyrna (p. 36, 6—11), "plane numbers, triangular, square and solid numbers, and the rest, are not so called independently but in virtue of their similarity to the areas which they measure;

• 4, since it measures a square area, is called square by adaptation from it
• 6 is called oblong for the same reason."
• "plane number" is similarly described as a number obtained by multiplying two numbers together, which two numbers are sometimes spoken of as "sides," sometimes as the "length" and "breadth" respectively, of the number which is their product.

The product of two numbers was thus represented geometrically by the rectangle contained by the straight lines representing the two numbers respectively.

It only needed the discovery of incommensurable or irrational straight lines in order to represent geometrically by a rectangle the product of any two quantities whatever, rational or irrational; and it was possible to advance from a geometrical arithmetic to a geometrical algebra, which indeed by Euclid's time (and probably long before) had reached such a stage of development that it could solve the same problems as our algebra so far as they do not involve the manipulation of expressions of a degree higher than the second.

pag381 Heath vol1

Links

1. Numeri figurati; triangolari, rettangolari, quadrati. Storia.
2. Numeri greci.