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classical geometry
In these chapters are presented in an informal way some topics of classical geometry: the geometry of that which concerns the development made possible by using only ruler and compass in a plane or a Euclidean space.
Summary:
- Euclid's Elements
Point and Line
Circle
ruler and compass
Applying a distance?
Prima Proposizione
Proposizioni Seconda e Terza
Scorciatoia!
+ Euclide: il "Primo" Teorema
Gli Elementi
Il merito principale di Euclide (vissuto intorno al 300 a.C.) è stato di codificare in modo organico tutte le conoscenze di geometria (piana e solida), aritmetica (numeri razionali e irrazionali) e altre belle cose note nel mondo antico... insomma ha compilato the encyclopedia of all mathematical knowledge of his time: we are just talking of his "Elements". Here, each topic is covered from a number of definitions and postulates, and then continue with the demonstration of a remarkable number of theorems absolutely valid to this day (his proof of the theorem of Pythagoras is exquisite elegance) .
The following paragraphs are not meant to be exhaustive of the subject, but provide a few tips that will intrigue and possibly to appreciate the greatness of Euclid. ▲
The Point and the line
recap: According to this definition, a line not only "has no width, but is made by points" that have no parties. " These points are so small, very small, even infinitesimal say: If you could draw a perfect line, it would be absolutely invisible. This must make us reflect on the level of abstraction attained by the ancient Greek geometry: their buildings are just perfect, as the "figures" that can be done with the means we use coarse only serve to represent things in order to make them more easily understood.
As for the straight down Euclid also two "postulates", or statements that, although not proven, are considered true:
Circle
Definitions:
ruler and compass
We go to school we become immediately familiar tools called ruler and compass, which actually serve to draw the lines and circles defined by Euclid. In fact, given any two points, they can be joined with a straight segment, and you can even continue liking design extending to the segment over the data points:
regard the circle as a center and a point on its circumference, with a compass, you can track all the other points the same distance from the center:
Here, as Euclid did not appoint them ever, ruler and compass are the only two instruments admitted in his geometric constructions. In other words, are the tools and just enough to allow the graphic representation of all his demonstrations! ▲
Applying a distance?
nelle costruzioni geometriche capita di dover riportare (o, più esattamente, "applicare") la lunghezza di un segmento a un altro segmento. E qui viene fuori un problema non da poco, in quanto la riga di Euclide NON è graduata. "Misurare" un segmento con un righello e poi riportare quella misura da un’altra parte non è certo un sistema esatto, e infatti non viene mai preso in considerazione da Euclide.
L'unica altra possibilità dunque è quella di utilizzare il compasso. Visto che, come abbiamo visto, il compasso è in grado di determinare tutti i punti equidistanti da un punto dato (centro), non è che lo si potrebbe usare anche per applicare una lunghezza, come mostrato qui sotto?
The answer is ... Ni. In the sense that this is a practice used continuously, but is not directly justified by the postulates of Euclid, so in theory it is a prohibited transaction. We have to imagine that the compass has the particularity of Euclid from closing automatically when it is removed from the sheet, just like in the animation above, where it is shown that the design of the circle.
Now it is obvious that Euclid could not be stranded in the face of this difficulty, it devotes the first three propositions of Book elements of his own to overcome this obstacle. ▲
Before Proposition
The first proposition shows how to construct an equilateral triangle on a given segment (that is an equilateral triangle has three equal sides):
As you can see the animation, it draws two arcs, both with a radius equal to the given segment, and every time the compass is closed and reopened here is that we are not committing the ' offense to keep the compass open between drawing a bow to another. I do not think that it is necessary to show that the distances AB, AC and BC sono uguali: per completare il triangolo a questo punto è sufficiente tracciare i due lati mancanti. ▲
Proposizioni Seconda e Terza
Nelle proposizioni 2 e 3 del Libro Primo, che qui vengono unite insieme in un’unica animazione, Euclide riesce a trasportare (o, più precisamente, applicare) il segmento AB sul segmento CD:
Ecco il procedimento (più in basso l'animazione completa):
— Si traccia un segmento fra i punti A and C, and this segment is built on the ACE equilateral triangle, the sides AC and AE are extended accordingly (drawn in purple).
- pointing the compass to the length of AB is reported to determine the point F, the segment EF is then the sum of the two segments EA (side of the triangle) and AF = AB.
- pointing the compass at E, the length of EF is reported to determine the point G, so that EF = EG. But since EA = EC (same side of the equilateral triangle), CG can not be equal to AF = AB.
- Aiming the compass in C can finally bring the CG on the segment length CD here then CH = AB.
As promised, below shows the complete construction:
is shown as a length can be transported properly, from one point to another of the paper, without transgressing the "rules" of Euclid. The fact that this is feasible based on existing definitions and postulates is the reason why Euclid did not need to add, among its postulates, something that said the ability to carry a length. ▲
Shortcut!
Fortunately for us we can use the procedure that I had said "no" (see above) we can freely use that shortcut to simplify the work, but only after having demonstrated the feasibility of this method with "prescribed." The
which methods are prescribed: use lines not graduate and compasses which close when removed from the sheet. With the help of these two instruments as Euclid was able to build a castle incredible ... we'll start to "explore" a bit 'at a time, starting from the next chapter . ▲ 
In these chapters are presented in an informal way some topics of classical geometry: the geometry of that which concerns the development made possible by using only ruler and compass in a plane or a Euclidean space.
Summary:
- Euclid's Elements
Point and Line
Circle
ruler and compass
Applying a distance?
Prima Proposizione
Proposizioni Seconda e Terza
Scorciatoia!
+ Euclide: il "Primo" Teorema
Gli Elementi
Il merito principale di Euclide (vissuto intorno al 300 a.C.) è stato di codificare in modo organico tutte le conoscenze di geometria (piana e solida), aritmetica (numeri razionali e irrazionali) e altre belle cose note nel mondo antico... insomma ha compilato the encyclopedia of all mathematical knowledge of his time: we are just talking of his "Elements". Here, each topic is covered from a number of definitions and postulates, and then continue with the demonstration of a remarkable number of theorems absolutely valid to this day (his proof of the theorem of Pythagoras is exquisite elegance) .
The following paragraphs are not meant to be exhaustive of the subject, but provide a few tips that will intrigue and possibly to appreciate the greatness of Euclid. ▲
The Point and the line
- One point is what is without parts.I think it's the most famous definition of the geometry: the first is among those listed in the First Book of Euclid's "Elements." The next three are instead:
- A line is length without breadth.(If I were someone who knows nothing about it, I do not know if these definitions, especially the last one, I can imagine what a straight line ... thank goodness we all have at least Once you have drawn a line with a ruler and pencil!)
- The end of a line are points.
- The straight line is lying on her points in a uniform manner.
recap: According to this definition, a line not only "has no width, but is made by points" that have no parties. " These points are so small, very small, even infinitesimal say: If you could draw a perfect line, it would be absolutely invisible. This must make us reflect on the level of abstraction attained by the ancient Greek geometry: their buildings are just perfect, as the "figures" that can be done with the means we use coarse only serve to represent things in order to make them more easily understood.
As for the straight down Euclid also two "postulates", or statements that, although not proven, are considered true:
- Between any two points can be drawn one and only one segment.
- You can extend a segment more than two points indefinitely. ▲
Circle
Definitions:
- a circle is said plane figure bounded by one line such that all segments ending on it from the same point between the inner to the figure, are equal.Postulate:
-That point is called the center of the circle.
- Given a point and a length, you can describe a circle. ▲
ruler and compass
We go to school we become immediately familiar tools called ruler and compass, which actually serve to draw the lines and circles defined by Euclid. In fact, given any two points, they can be joined with a straight segment, and you can even continue liking design extending to the segment over the data points:
regard the circle as a center and a point on its circumference, with a compass, you can track all the other points the same distance from the center:
Here, as Euclid did not appoint them ever, ruler and compass are the only two instruments admitted in his geometric constructions. In other words, are the tools and just enough to allow the graphic representation of all his demonstrations! ▲
Applying a distance?
nelle costruzioni geometriche capita di dover riportare (o, più esattamente, "applicare") la lunghezza di un segmento a un altro segmento. E qui viene fuori un problema non da poco, in quanto la riga di Euclide NON è graduata. "Misurare" un segmento con un righello e poi riportare quella misura da un’altra parte non è certo un sistema esatto, e infatti non viene mai preso in considerazione da Euclide.
L'unica altra possibilità dunque è quella di utilizzare il compasso. Visto che, come abbiamo visto, il compasso è in grado di determinare tutti i punti equidistanti da un punto dato (centro), non è che lo si potrebbe usare anche per applicare una lunghezza, come mostrato qui sotto?
The answer is ... Ni. In the sense that this is a practice used continuously, but is not directly justified by the postulates of Euclid, so in theory it is a prohibited transaction. We have to imagine that the compass has the particularity of Euclid from closing automatically when it is removed from the sheet, just like in the animation above, where it is shown that the design of the circle.
Now it is obvious that Euclid could not be stranded in the face of this difficulty, it devotes the first three propositions of Book elements of his own to overcome this obstacle. ▲
Before Proposition
The first proposition shows how to construct an equilateral triangle on a given segment (that is an equilateral triangle has three equal sides):
As you can see the animation, it draws two arcs, both with a radius equal to the given segment, and every time the compass is closed and reopened here is that we are not committing the ' offense to keep the compass open between drawing a bow to another. I do not think that it is necessary to show that the distances AB, AC and BC sono uguali: per completare il triangolo a questo punto è sufficiente tracciare i due lati mancanti. ▲
Proposizioni Seconda e Terza
Nelle proposizioni 2 e 3 del Libro Primo, che qui vengono unite insieme in un’unica animazione, Euclide riesce a trasportare (o, più precisamente, applicare) il segmento AB sul segmento CD:
Ecco il procedimento (più in basso l'animazione completa):
— Si traccia un segmento fra i punti A and C, and this segment is built on the ACE equilateral triangle, the sides AC and AE are extended accordingly (drawn in purple).
- pointing the compass to the length of AB is reported to determine the point F, the segment EF is then the sum of the two segments EA (side of the triangle) and AF = AB.
- pointing the compass at E, the length of EF is reported to determine the point G, so that EF = EG. But since EA = EC (same side of the equilateral triangle), CG can not be equal to AF = AB.
- Aiming the compass in C can finally bring the CG on the segment length CD here then CH = AB.
As promised, below shows the complete construction:
is shown as a length can be transported properly, from one point to another of the paper, without transgressing the "rules" of Euclid. The fact that this is feasible based on existing definitions and postulates is the reason why Euclid did not need to add, among its postulates, something that said the ability to carry a length. ▲
Shortcut!
Fortunately for us we can use the procedure that I had said "no" (see above) we can freely use that shortcut to simplify the work, but only after having demonstrated the feasibility of this method with "prescribed." The
which methods are prescribed: use lines not graduate and compasses which close when removed from the sheet. With the help of these two instruments as Euclid was able to build a castle incredible ... we'll start to "explore" a bit 'at a time, starting from the next chapter . ▲
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