**From Euclidss Elements** (see here)

__Common Notions__ (CN) _{ }

1 | Things equal to the same thing are also equal to one another |

2 | And if equal things are added to equal things then the wholes are equal |

3 | And if equal things are subtracted from equal things then the remainders are equal |

4 | And things coinciding with one another are equal to one another |

5 | And the whole [is] greater than the part |

1 | Let it have been postulated to draw a straight-line from any point to any point |

2 | And to produce a finite straight-line continuously in a straight-line |

3 | And to draw a circle with any center and radius |

4 | And that all right-angles are equal to one another |

5 | And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side) |

__Proposition 1__

*To construct an equilateral triangle on a given finite straight-line.*_{ }

__Proof__:_{ }

**0**) Let AB be the given finite straight-line_{ }

**1**) I use Post3 by considering A as center and the finite straight-line AB as radius_{ }

**2**) I use Post3 again, by considering B as center and the finite straight-line AB as radius_{ }

**3**) Let C be a point where the circles cut one another. I use Post1 by considering A and C as points_{ }

**4**) I use Post1 again, by considering B and C as points [click the image if you want to enlarge it]_{ }

• The existence of common points between straight lines is guaranteed by Post5, but, without continuity,who guarantees me the existence [asserted in 3)] of common points between "curves" that bypass ? |

• [In 1)] an operation of "transport" of figures whose possibility is not postulated is performed.Moreover: • From the fact that there is an M1 movement carrying ∠A to ∠D, an M2 movement carryingthe AB segment in the DE segment and an M3 movement carrying the AC segment in the DF segment, I cannot deduce that there is a "movement" (M1 or another) that he does all three things. And then, given two points, Post1 ensures the existence of a segment that has them for extremes, not its uniqueness. |