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Chapter II: The First Non-Euclidean Geometry

The obvious nature of the fifth postulate of Euclid made some people suspect the veracity of it.

One of the first notable case is that of the Jesuit mathematician Giovanni Girolamo Saccheri (1667-1733), who to see there was no obvious way to get the fifth postulate from the other nine principles, chose to change its strategy and began trying to prove the validity of Euclid's fifth postulate using the oldest trick in the bag of tricks math: given a proposition that assumed to be true, you start assuming that the statement is false and you start working with that assumption in mind, until we invariably reach a point where we get an absurd (understandably this method is known in logic as the method of reductio ad absurdum), which demonstrated that the statement was not false but true, because for elementary logic (developed by a contemporary of Euclid, Aristotle) \u200b\u200bis not possible to obtain a false conclusion based on a true proposition. Thus, in his attempt to vindicate Euclid, produced a book entitled Euclides ab omni vindicatus Naeve, which translates as "free from any failure Euclid, where he attempted to prove the fifth postulate by demonstrating the use of all other alternatives as absurd, for which he built what is now known as the "quadrilateral Saccheri:

rectaGH
starting from a base which is the the ring, to which are added two lines perpendicular AB and CD, through which a fourth line is drawn so that the length of the arms the ring, the lines that run from the base to the summit , are equal. After that, Saccheri made three different assumptions about the sum of the interior angles of the ring (angles m, n , or and p): the sum of the angles was less than, equal to and greater than four right angles (360 degrees). If he could show that the first and third scenarios leading to logical absurdities, then the hypothesis would have shown that intermediate, equivalent to the fifth postulate given by Euclid, was the only geometry consistent, and therefore the only real geometry, it being severely tested as true the fifth postulate. Saccheri quickly rejected the third hypothesis because almost immediately began to surface contradictions. However, the first hypothesis does not lead to any logical conflict, and indeed could prove Saccheri theorem after theorem using the new alternating premise, building to his astonishment the first non-Euclidean geometry.

It is instructive to reproduce here some of the Saccheri reasoning that led him to discover a geometry as valid as Euclid's geometry which was not satisfied the fifth postulate.

Let us begin, as did Saccheri, with a quadrilateral formed by two pairs of parallel lines that represent where we have designated the vertices of the quadrilateral with different letters:

Note that, as a game, we have assumed that the angles at the vertices and A B are right angles. This always can be carried out by construction, and requires no further explanation. Note, however, that the angles at the vertices C D and are left as questions. There are then three possibilities:

1) The angles at the corners and C D are equal both at a right angle (90 degrees), which must be so whether the fifth postulate of parallels is valid.

2) The angles at the vertices C D and both are greater than a right angle (this is the hypothesis of obtuse angle ).

3) The angles at the vertices C D and both are less than a right angle (this is the hypothesis of obtuse angle ).

Without using the fifth postulate, it is easy show that if the angles at the vertices and A B are right angles, then the angles C D and can not be different, must be equal. If we assume the fifth postulate of Euclid as fair, then the angles at the corners and C D have to be right angles, is the conclusion which is reached by applying the fifth postulate. If one denies that the angles at the vertices and C D are right angles, then you are denying the fifth postulate. But if these angles are not right angles, in any case have to be equal, so both have to be obtuse or acute. Assume that the fifth postulate is not fulfilled and that both angles are acute. Then the situation at hand is:

This combination of internal angles is not possible in Euclidean geometry, in which the fifth postulate requires that all angles are right angles. But we are assuming that the fifth postulate is not valid.

The problem that presented here is that Saccheri, using precisely this ring, was prepared theorem after theorem, always looking for some absurd. But he was unable to find any. He had discovered what is now known as hyperbolic geometry without realizing it. Several of the theorems he discovered and rejected "as absurd" are in fact fully valid theorems in hyperbolic geometry. Unable to understand and accept the scope of discovery, having rejected the third hypothesis Saccheri also dismissed the first (the sum of the angles of a quadrilateral is less than four right angles) and not on logical grounds but on theological grounds.

then try to reproduce the thinking of Saccheri used by him for the demonstration of some of the theorems he discovered in his new geometry. There have been some slight modifications to make it understandable procedures, adapted to the present day. All demonstrations will take place with only one rule (no graduations or marks) and a compass, as did the geometers of antiquity, will and assuming the hypothesis that the internal angles at the vertices of the summit the quadrilateral are acute.

begin by first constructing a Saccheri quadrilateral . We broke the ring starting with a line AB, the base the ring, and trace the ends of two perpendicular base, the lines AC and BD, so that the interior angles at vertices A and B are at right angles, making it also the bar that lines AC and BD are equal to equal in length. After that, we join the points C and D with a line, completing the ring:

now show the first theorem of our non-Euclidean geometry for this ring:

Theorem: Angles in vertices C and D of the top of the Saccheri quadrilateral are equal . Demonstration

: Then draw two to unite diagonal opposite corners of the quadrilateral:

triangles BAC and ABD are congruent (equal geometric figures) by having two equal sides (common side AB and AC equal to the side by side BD construction) and an interior angle (the angle) as well. Because congruent triangles, then the lines AD and BC should be the same length. This intermediate result can be expressed as a theorem : The diagonals of a Saccheri quadrilateral are equal . And this allows us to find in the ring two congruent triangles, the triangles ACD and CBD, which are consistent this time by having its three sides equal (the same common base line CD, the side AD is equal to the side BC, and the side AC is equal to the BD side by construction). And if you have three sides equal, to be consistent then their corresponding angles at the vertices C and D must also be equal, which completes the proof.

If the angles at vertices C and D are equal, as just demonstrated, can say something more about them?. Actually this is all I can say about these angles. Do not know yet if they are right angles, acute or obtuse. Although it is easy to yield to the temptation to proclaim as angles straight, it does not prove that they are. In fact, unless you let us use the fifth postulate of Euclid, there is no way to prove they are right angles. If Euclid's fifth postulate is valid, then one can easily prove to be right angles. But if we use the fifth postulate, then we have the doubt that they can be acute angles or obtuse angles. All we know is that they must be equal. Work with the assumption that the angles at vertices C and D are acute . This necessarily implies that the fifth postulate of Euclid should be taken as false. At this time, Euclidean geometry begins to crumble before our eyes.

The hypothesis that the internal angles at the corners of the top of the Saccheri quadrilateral are acute, which is just to prove they are the same, together with the fact that the internal angles at the corners of the base are straight by construction equivalent to saying that within a Saccheri quadrilateral the sum of the interior angles is less than 360 degrees (four right angles) . This statement is unprovable, just as Euclid's fifth postulate is not. We have to accept as a starting point, like a postulate or axiom.

Working on the assumption that the angles C and D are acute, theorems we can continue to demonstrate more fully valid in our first non-Euclidean geometry. Then we have another theorem: Theorem

: The line joining the midpoints of the base and summit Saccheri quadrilateral is a line that is perpendicular to both lines . Demonstration

: Then we draw a line joining the midpoints of the quadrilateral, which we denote as E and F. Now we draw lines from point E to the vertices C and D of the ring.

Again, we formed two triangles congruent. The triangles AEC and BED are consistent to have two sides equal (by construction, the side AE \u200b\u200bis equal to the side EB, and the side AC is equal to the side BD) and a right angle (the angle at the vertex A and angle the vertex B are both straight and construction). Then the CE straight line is equal to the ED by the relation of congruence. But this in turn implies that we have two congruent triangles, the triangles CEF and DEF, having three sides equal (CF side is equal to the FD side being the midpoint of the line at the top of the ring). Then the angle q must be equal to the angle r, which is only possible if both are right angles. This already tells us that the line EF is perpendicular to line CD. On the other hand, for the same congruency of triangles, the angle r must be equal to the angle t, so that:

o + s = p + t

This can only be possible if both members of equality is a right angle. Therefore, the line EF is also perpendicular to the line AB, and is therefore perpendicular to both lines, which concludes the proof.

This theorem is interesting because it tells us that in our non-Euclidean geometry is always possible to draw a line that is perpendicular to both "parallel."

Then we have another theorem for our geometry non-Euclidean:

Theorem: If a quadrilateral Saccheri quadrilateral arms are unequal, so are the angles at the top, and vice versa . Demonstration

: Be the quadrilateral ABCD in which BD has drawn the line with a length greater than the length of the line AC. In the longer side, take a point E such that the line segment BE is equal in length to the side AC:

In the first of our theorems, we had shown that the angles at the vertices of the top are equal. If here the lines AC and BE are equal, then the ACE angle theorem is equal to the angle CEB:

ang (ACE) = ang (BEC)

Since the EC line subdivides the angle of the ring ACD necessarily the angle ACD is greater than the angle ACE:

ang (ACD)> ang (ACE)

Since BED angle is an angle outside the triangle CED, then the angle is greater than the angle BDC:

ang (BEC)> ang (BDC)

Of the three relations we obtain that the angle ACD is greater than the angle BDC through the following steps:

ang (ACD)> ang (ACE)

ang (ACD)> ang (BEC) using the relationship of equality

ang (ACD)> ang (BEC)> ang (BDC)

ang (ACD)> ang (BDC)

O ie, the angles at the summit will be uneven, having been the arms of uneven ring, which is proved the theorem.

Using this last theorem with the second theorem, we are able to demonstrate another interesting theorem: Theorem

: Sacchieri In a quadrilateral, the summit is longer than the base . Demonstration

: To demonstrate this theorem, again divided with a perpendicular ring that joins the midpoints of the base and summit of the ring:

This line actually divides the original ring in two Saccheri quadrilateral, the quadrilateral AEFC and the ring BEFD. Since the angles at the vertices D and C are sharp, by the theorem just demonstrated the AC side will be greater than the EF hand in the ring AEFC, and the side BD is greater than the EF hand in the ring BEFD. AEFC now sleeping on his side ring EF, which allows us to see the side EF as a base and sides AE and CF as the "arms" of a quadrilateral, we have the CF side is greater than the side AE. Doing the same with the other ring, we have something similar: FD side is greater than the EB side, which can represent relationships:

CF> AE

FD> EB

Adding the respective members inequalities, we have a new inequality:

CF + DF> AE + EB

is the same in the figure:

CD> AB

So the top of the ring has a length greater than its base!, as a direct consequence of the hypothesis of the acute angles at the vertices C and D. This finding begins to seem absurd and not consistent with what we said our "intuition", which suggests our everyday experience. However, all the steps we have undertaken are fully justified, we have not committed any logical contradiction, we have never violated the rules. Same as discovered Saccheri, before our eyes is developing an entirely new geometry, a non-Euclidean geometry, which forces us to see things from a different perspective.

now define, within the geometry Saccheri, if two lines have a common perpendicular , as the line EF shown above in the Saccheri quadrilateral, then these lines are parallel . Obviously, if two lines AB and CD have a common perpendicular, then they can not have another common perpendicular than this, certainly not in the Saccheri quadrilateral where the internal angles at the corners and C D were defined as acute. This can be expressed as a simple theorem just demonstrated using the following keywords: If two lines in a Saccheri quadrilateral have a common perpendicular, then they can not get a second .

The existence of a common perpendicular offers the interesting possibility to construct a Saccheri quadrilateral modified so that one side of the ring is precisely this common perpendicular . Thus, we have a ring in which not just two but three its three internal angles are right angles as shown below:

This ring is best known as the Lambert quadrilateral to have been used in the 18 th century by the mathematician Johann Lambert in his studies of non-Euclidean geometries, but in fact was previously used in the 11th century by noted Muslim scientist Ibn al-Haytham, a pioneer of modern scientific method. Although the Lambert quadrilateral we can have a non-Euclidean ring in which three internal angles are right (unlike the Saccheri quadrilateral in which only two internal angles are right), the Lambert quadrilateral vertical sides of the ring AC and BD longer be the same as they were in the Saccheri quadrilateral, which on the one hand what is gained by the other party is lost. In fact, the Lambert quadrilateral is no more helpful to try to prove the "postulate parallel "of Euclid that Saccheri quadrilateral, for the simple reason that it is something that can not be shown .

now proceed to prove the following:

Theorem: Two lines are parallel with a common perpendicular to both, if there is a straight cross cut these lines so as to form equal alternate interior angles, corresponding angles equal or .

This will be demonstrating that the assumption that the alternate interior angles formed by transverse cuts two lines are equal implies that the lines crossed are parallel in the sense which gives the word on the geometry of Saccheri. First, take the following lines AB and CD so as to be cut in sections and P Q cross the line PQ :

so that the m angle is equal to the angle n . If the angles and m n are right angles (90 degrees), then the line GH must be a common perpendicular lines AB and CD , and the demonstration becomes trivial. If angles m and n are acute angles and select the M within the PQ straight so that is the midpoint of that line. E is the projection of point M on the line AB . At the top of the ring, in line CD, take F point to the left of Q so that QF segment is the same length as the segment PE . Then the triangles MEP and MFQ be congruent (similar equal) by having two equal sides and equal angles (the angles of the vertices played on M point are equal). Being congruent triangles, then the angle QFM must be equal to the angle PEM the same consistency (similarity) of the triangles. As the angle PEM a right angle as the point E projection (perpendicular) on the line AB M point, then the angle QFM must also be, and concludes that the line EF is the common perpendicular to both lines AB and CD , which is also concluded that E points, M and F part of the same line (are collinear ). Then the equality of alternate interior angles and m n leads to the result that the line EMF is a common perpendicular to both lines AB and CD , which can only occur if both are parallel.

We are now able to deduce, by Saccheri quadrilateral, the following theorem

: In geometry based on the Saccheri quadrilateral, the sum of the angles of any triangle is less than 180 degrees ( two right angles) . We will make

demonstration in two parts. We will do first for the case of triangles (a triangle with one internal angle equal to 90 degrees). Consider the following right triangle ABC enrolled in the Saccheri quadrilateral, which means that the internal angle at the vertex p A is a right angle:

Being a quadrilateral Saccheri, the angle at the vertex C the ring, which is equal to the sum of the angles or n and must be acute, less than 90 degrees:

90 °> o + n

As we saw in the above proof, if the line EF is the common perpendicular to the parallel AB and CD , then alternate interior angles m and n should be equal, thus the previous inequality becomes:

90 °> o + m

Adding now the angle p both members of inequality, we have :

p + 90 °> p + o + m

90 ° + 90 °> p + o + m

180 °> p + o + m

This tells us clearly that Saccheri's geometry, the sum of the angles of any triangle rectangle be less than 180 degrees.

Now, we must prove the theorem for any triangle in general other than a triangle. This is done to trick relieved divide a triangle into two triangles by drawing the height of a vertex to the base. Take the next triangle PQR and lie height from vertex Q to its base so that the line QS is perpendicular to the line PR :

By being divided the triangle into two triangles, so we just have to show for a triangle then the following inequalities must be true:

180 °> m + o + p

180 °> n + q + r

memberwise Adding both inequalities, we obtain a new inequality:

360 °> m + o + p + n + q + r

But the sum of the angles p q and should be 180 degrees because they both share the same straight line. Rearranging and simplifying:

360 °> m + o + n + r + 180 °

180 °> (m + n) + o + r

But m + n is the inner corner of the triangle PQR at the top Q . This tells us that the geometry Saccheri the sum of the angles of any triangle must be less than 180 degrees.

And as we show the above theorems, we can continue deriving more theorems. The conclusion that this geometry the sum of the angles of any triangle always be less than 180 degrees, or less than two right angles, collide directly with the Euclidean result tells us that in every triangle the sum of the interior angles is always equal to two right angles. Today we find it easier to digest all this because our thinking has evolved and we are more willing to accept ideas that go against our intuition, as has happened with quantum mechanics, the Heisenberg uncertainty principle, with the Theory Einstein's general relativity, and Gödel's theorem. But in the Middle Ages and even in the times of the Renaissance, we have seen the same thing would have been viewed almost as a heresy.

Gradually, the picture begins to emerge on Saccheri's geometry is two perpendicular parallel with a common and are taking out of it like this:

A critical observer might may object as follows: "The line of two parallel lines is that these are straight, not curved lines. By having two curved lines can no longer speak of parallel lines. This suggests that the fifth postulate of Euclid is still quite valid and no other reality than that shown by this assumption. " To which you may respond as follows:

First, the curve has been greatly exaggerated for pedagogical purposes. Much less pronounced curvature in terms of terrestrial distances but not astronomical distances, seem to show falsely look like a straight line is not. But actually, the length of another perpendicular drawn from one of the "parallel" does not match the common perpendicular will increase as we get further and further away from the common perpendicular. However, it completely lost the perspective behind the construction of the Saccheri quadrilateral. To build a Saccheri quadrilateral, we start from a line AB, the base of the ring, drawn in the most "straight" that we can conceive (perhaps using a laser beam) without the slightest hint of a curve, making the assumption that the angles at the base of the ring are angles and the angles at the top are sharp. For the geometer who is "down", his vision of what happens is this:

However, a surveyor who lives "up" and trace its Saccheri quadrilateral top down , starting with the line "straighter" as possible for the base map of ring, perhaps with the help of a laser beam, their view of what happens is as follows:

Thus, each of them swear by what is most sacred to them based on their Saccheri quadrilateral is completely straight while the top in the ring "the other" is one that shows a curve that makes the interior angles acute summit, which will reportedly both baffled when they exchange notes. But both will be even more amazed when someone can jump out of this peculiar universe watching from above (we) tell them the straight both exhibit a curvature. To make matters worse, no one has come from the far edge of the universe, located beyond the infinite, to tell us how space behaves in such a region. Perhaps the only valid geometry there is the Saccheri, taken to extremes.

Assuming the base of the Saccheri quadrilateral as a perfectly straight line, then it is now more than obvious that from a given point P to a given line can draw an infinite number of parallel to the given line all which have their common perpendicular to the same external point P, as shown in the following figure in which they have drawn three lines parallel passing through point P:

The difference between the sum of the angles of a triangle and 180 degrees is what is commonly known as the triangle defect, which be represented here by the Greek letter delta (δ ), and is an important quantity to carry out measurements related to a triangle, then relations will be discussed in Chapter IV of this blog that deal with area of \u200b\u200ba triangle defined within this type of geometry.

The definition of "default" a triangle (this in no way implies that something in our new geometry or the triangle is defective, take it as a completely new use of the word without any fault connotation) allows us to establish another theorem

: The defect δ of a triangle equals the sum of the two subtriangles defects resulting from the stroke of a simple cross within the triangle .

To demonstrate this, consider the following triangle ABC , subtriangles divided in two by the line from vertex C point to the point D :

By definition of the defect of a triangle, the shortcomings of each of the two subtriangles be given by the following relations:

δ (ADC) = 180 - (m + o + p )

δ (BCD) = 180 - (n + q + r)

Adding both expressions
limb from limb and grouping:

δ (ADC) + δ (BCD) = 360 - [ m + o + (p + q) + n + r]

But the sum of the angles p q and 180 degrees will be supplementary angles on the same line, and vertex angle C equals the sum of the angles and m n . Simplifying the above to these facts, it comes down to:

δ (ADC) + δ (BCD) = 360 - 180 - [(m + n) + o + r]

δ ( ADC) + δ (BCD) = 180 - [ang (C) + o + r]

δ (ADC) + δ (BCD) = δ (ABC)

and the theorem is demonstrated.

This additive property later found to be very useful when it comes to defining the concept of the area contained inside a triangle Saccheriano .

Saccheri

Had promoted his findings, possibly in the long run it would have earned undying fame as a mathematician revolutionary. The problem is that in the times he lived Saccheri was not so easy to try to challenge the geometry developed by Euclid, was not so easy to challenge it as the only possible geometry. Euclid was held in such high esteem by the classic authority of that time, that question it was considered almost heretical. And what I found not only crumbling Saccheri Euclidean geometry as the only possible geometry, but brought with it a new set of theorems that came into direct conflict with the theorems proved by Euclid. We are talking about something capable of stopping the head Euclidean geometry. And what I found was a geometry Saccheri completely new, consistently, without contradiction.

But Saccheri was not the only one who did not dare to recognize the magnitude of their discovery, much less to challenge some established and accepted as true only for hundreds of years. None other than the German mathematician Carl Gauss (1777-1855), prince of mathematics, he met also with the fact that the fifth postulate of Euclid was not an absolute truth as many believed, could be replaced by an alternative assumption, after which they could build a whole new geometry is perfectly consistent, without contradictions, with their own theories. Gauss came to these conclusions following a completely different path. One of the most important contributors to differential geometry, he discovered the theorem that the "curvature" of a non-planar surface was related to the metric used to measure the curvature (the metric is defined as the mathematical expression used to measure the distance between any two points on a surface, and if the surface is not flat, then the line over short drawn on that surface from one point to another is given a name more elaborate: the geodesic . Gauss was able to establish in the Theorema Egregium that the curvature of a surface is independent of the space within which there is such a surface, this being an intrinsic property of the sum of the angles of a triangle built on the surface, which led directly to the logical conclusion that not only the sum of the angles of a triangle built on a particular area could be 180 degrees different from that obtained Euclidean geometry, but even that amount could be used to calculate the curvature of the surface. In other words, to determine the curvature of the Earth's surface, it is necessary to carry out an external measurement, just carefully draw a large triangle on the surface, and adding the internal angles we know if the Earth is "flat" as it assumed many contemporaries of Christopher Columbus (in which case the measurement would yield 180 degrees) or can we know if the Earth's surface is a curved surface. And he repeated that to obtain this information is not necessary to go outside the triangle, simply measure the interior angles of a triangle, which are an intrinsic property of this figure geometric.

Before Gauss, Johann Heinrich Lambert (1728-1777) who has been mentioned previously in connection with the Lambert quadrilateral had come back to the construction of non-Euclidean geometries in his book "Theory of parallel lines" published in 1776. Using a methodology similar to that of Saccheri, found that the three hypotheses of Saccheri was equivalent to the assertion that the sum of the angles of a triangle be equal, greater or less than two right angles (180 degrees) and showed also that spherical geometry was similar to the third case, speculating that first geometry may correspond to a geometry drawn on a sphere with imaginary radius (where the base unit is the square root of negative 1, or -1, denoted in mathematics as i ). Replacing a real radio for a radio imagery led to what can be considered as the first hyperbolic geometry in which the usual trigonometric formulas sin (x) and cosine (x) are replaced by the hyperbolic sine or sinh (x) and or hyperbolic cosine cosh (x) . So a question arises: why is one of the alternative hypothesis of Saccheri what led to logical contradictions? The answer turns out to be much easier than it looks. So that in this "alternate geometry" Saccheri not come with contradictions, not enough to change the parallel postulate, also had to modify the second postulate which says that a line can be extended infinitely in both directions, which did not consider possibility that a line extended to infinity in one of its sides could return by a curvature of space in the universe the starting point, in which case we would not have an open line but a line closed.

Gauss, like Saccheri, realized risk running if he published his findings. In addition to the huge scandal that surely would raise the stop to the Euclidean geometry of the head, was exposed to public ridicule those who would not understand the magnitude of his discovery (which he called "Boeotian.") Gauss is why he preferred silence and keep their well-earned reputation. He went to his death when the rummage among his papers were found manuscripts which were found to Gauss, by way of differential geometry, had confirmed its earlier findings of a different geometry to Euclidean geometry. However, despite its enormous size, not given to Gauss the credit it deserves for this discovery because in science the credit goes not to those who discover something for the first time but for anyone who publishes the results of its first discovery (which is why the inventorship of differential and integral calculus was always source of bitter arguments and claims between the two people who claimed to the end of his days the merit of being the first to develop, first Sir Isaac Newton and Gottfried Leibniz the other side).