^^Euclide. Nozioni comuni.

Definizioni 23 | Postulati 5 | Nozioni comuni 5

Versione1

  1. Things which equal the same thing also equal one another.
  2. If equals are added to equals, then the wholes are equal.
  3. If equals are subtracted from equals, then the remainders are equal.
  4. Things which coincide with one another equal one another.
  5. The whole is greater than the part.

ref: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html

Versione 2

  1. Things equal to the same thing are equal to one another.
  2. If equals are joined to equals, the wholes will be equal.
  3. If equals are taken from equals, what remains will be equal.
  4. Things that coincide with one another are equal to one another.
  5. The whole is greater than the part.
  6. Equal magnitudes have equal parts; equal halves, equal thirds, and so on.

ref: http://www.themathpage.com/aBookI/first.htm

 

Links

  1. Euclide.

 

 

How can we know when things are equal?

That is one of the main questions of geometry.

The definition (and existence) of a circle provides our first way of knowing that two straight lines could be equal. Because if we know that a figure is a circle, then we would know that any two radii are equal. (Definition 15.)

 

Implicit in these Axioms is our very understanding of equal versus unequal, which is:  Two magnitudes of the same kind are either equal or one of them is greater.

 

Length, area, and angle are the three kinds of magnitude we study in plane geometry.