^^Euclide. Definizioni.

Definizioni 23 | Postulati 5 | Nozioni comuni 5

ogni libro ha le sue definizioni

Book1  definitions

Note per chiarire le definizioni-concezione di Euclide

Per Euclide:

• "linea" e' in generale curva, cioe' il concetto di linea include le linee curve, per questo poi specifica le "linee rette".
• "linee rette"
  • sono i "segmenti (di retta)" della terminologia attuale,
  • non sono linee rette infinite, come oggi intese dalla terminologia attuale.

[Punto, linea, piano]

1. point is that which has no part. >>>
2. line is breadthless length.
3. ends of a line are points.
4. straight line is a line which lies evenly with the points on itself.
5. surface is that which has length and breadth only.
6. edges of a surface are lines.
7. plane surface is a surface which lies evenly with the straight lines on itself.

[Angoli] (Commento di Proclo)

8. plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
9. Rectilinear angle. And when the lines containing the angle are straight, the angle is called rectilinear.
10. Right angle. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
11. obtuse angle is an angle greater than a right angle.
12. acute angle is an angle less than a right angle.

[Figure]

13. boundary is that which is an extremity of anything.
14. figure is that which is contained by any boundary or boundaries.

[Cerchio]

15. circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.
16. And the point is called the center of the circle.
17. diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
18. semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

[Poligoni]

19. Rectilinear figures are those which are contained by straight lines,
    • trilateral figures being those contained by three,
    • quadrilateral those contained by four, and
    • multilateral those contained by more than four straight lines.
       
20. trilateral figures
    • equilateral triangle  3 sides equal
    • isosceles triangle    2 sides alone equal
    • scalene triangle       3 sides unequal.
21. trilateral figures, further
    • right-angled triangle  has a right angle
    • obtuse-angled triangle has an obtuse angle
    • acute-angled triangle has its 3 angles acute.
       
22. quadrilateral figures 
    • square is both equilateral and right-angled
    • oblong is right-angled but not equilateral
    • rhombus is equilateral but not right-angled
    • rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled.
    • trapezia quadrilaterals other than these.

     

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Links

1. Euclide.
2. mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI
3. Euclide. Definizioni angolo. Commento di Proclo.
4. https://www.maths.tcd.ie/~dwilkins/Euclid/
   1. Heath's Commentary on Euclid's Elements, Book I, Definitions 8 and 9
   2. Heath's Commentary on Euclid's Elements, Book I, Definitions 10, 11 and 12

Book3  definitions

1. Equal circles are those whose diameters are equal, or whose radii are equal.
2. A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
3. Circles are said to touch one another which meet one another but do not cut one another.
4. Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal.
5. And that straight line is said to be at a greater distance on which the greater perpendicular falls.
6. A segment of a circle is the figure contained by a straight line and a circumference of a circle.
7. An angle of a segment is that contained by a straight line and a circumference of a circle.
8. An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the ends of the straight line which is the base of the segment, is contained by the straight lines so joined.
9. And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.
10. A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
11. Similar segments of circles are those which admit equal angles, or in which the angles equal one another.