^^Euclide. Book2 propositions
& definitions.
Euclide non ha coniato una parola singola per "rettangolo", ma lo qualifica
come particolare parallelogramma.
Rettangolo = parallelogramma retto, parallelogramma rettangolare.
Geometric algebra
Definitions
- Any rectangular parallelogram is said to be contained by the two
straight lines containing the right angle.
- And in any parallelogrammic area let any one whatever of the
parallelograms about its diameter with the two complements be called a
gnomon.
Propositions
- If there are two straight lines, and one of them is cut into any number
of segments whatever, then the rectangle contained by the two straight lines
equals the sum of the rectangles contained by the uncut straight line and
each of the segments.
- If a straight line is cut at random, then the sum of the rectangles
contained by the whole and each of the segments equals the square on the
whole.
- If a straight line is cut at random, then the rectangle contained by the
whole and one of the segments equals the sum of the rectangle contained by
the segments and the square on the aforesaid segment.
- If a straight line is cut at random, the square on the whole equals the
squares on the segments plus twice the rectangle contained by the segments.
- If a straight line is cut into equal and unequal segments, then the
rectangle contained by the unequal segments of the whole together with the
square on the straight line between the points of section equals the square
on the half.
- If a straight line is bisected and a straight line is added to it in a
straight line, then the rectangle contained by the whole with the added
straight line and the added straight line together with the square on the
half equals the square on the straight line made up of the half and the
added straight line.
- If a straight line is cut at random, then the sum of the square on the
whole and that on one of the segments equals twice the rectangle contained
by the whole and the said segment plus the square on the remaining segment.
- If a straight line is cut at random, then four times the rectangle
contained by the whole and one of the segments plus the square on the
remaining segment equals the square described on the whole and the aforesaid
segment as on one straight line.
- If a straight line is cut into equal and unequal segments, then the sum
of the squares on the unequal segments of the whole is double the sum of the
square on the half and the square on the straight line between the points of
section.
- If a straight line is bisected, and a straight line is added to it in a
straight line, then the square on the whole with the added straight line and
the square on the added straight line both together are double the sum of
the square on the half and the square described on the straight line made up
of the half and the added straight line as on one straight line.
- To cut a given straight line so that the rectangle contained by the
whole and one of the segments equals the square on the remaining segment.
- In obtuse-angled triangles the square on the side opposite the obtuse
angle is greater than the sum of the squares on the sides containing the
obtuse angle by twice the rectangle contained by one of the sides about the
obtuse angle, namely that on which the perpendicular falls, and the straight
line cut off outside by the perpendicular towards the obtuse angle.
- In acute-angled triangles the square on the side opposite the acute
angle is less than the sum of the squares on the sides containing the acute
angle by twice the rectangle contained by one of the sides about the acute
angle, namely that on which the perpendicular falls, and the straight line
cut off within by the perpendicular towards the acute angle.
- To construct a square equal to a given rectilinear figure.
credits: ©2002 David E. Joyce
Links
Book1 map.