Saccheri's Solution to Euclid's BLEMISH

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Girolamo Saccheri, S.J.
(1667 - 1733)
and his solution to Euclid's blemish

Saccheri's Flaw while eliminating Euclid's "Flaw"
The Evolution of Non-Euclidean Geometry


Non-Euclidean geometry is one of the marvels of mathematics and even more marvelous is how it gradually evolved through a process of eliminating flaws in logical reasoning. It is misleading to think that non-Euclidean geometry was similar to suddenly finding a precious jewel. Rather, it has may likened to the discovery of America by bold adventurers who would not be silenced by the complacent savants of the "known world". Complacent philosophies attempted to stifle mathematicians harboring suspicions concerning Euclid's Postulate #V. Nevertheless, America was discovered and so was non-Euclidean geometry. The story began with the Jesuit Girolamo (Jerome) Saccheri who undertook the fearsome task of removing Euclid's "flaw" concerning postulate #V. By introducing ingenious methods and rigorous logic (except for his own "flaw") he opened the way for geometers succeeding him to discover incredible geometries which provide a better model for our universe than Euclid ever dreamt of.

Proofs of some of Saccheri's theorems can be viewed by moving to the page Hyperbolic Geometry Theorems of Girolamo Saccheri, S.J.

What if . . .

Mathematicians have a persistent habit of asking the question: "What if . . .?" Dissatisfied with the mathematical status quo they seek to generalize a proposition as much as possible. An example concerns the Pythagorean triples ({3,4,5}, {5,12,13}, {x,y,z}): xn + yn = zn for n = 2. Three centuries ago a part-time mathematician, Pierre de Fermat ("the prince of amateurs") asked: "What if n > 2 ?" Mathematicians have been trying to answer his question ever since.

About three centuries ago the Jesuit geometer Jerome Saccheri asked another question: "What if the sum of the angles of a triangle were not equal to 180 degrees (or p radians)?" Suppose the sum of these angles was greater than or less than p. What would happen to the geometry we have come to depend on for so many things? What would happen to our buildings? to our technology? to our countries' boundaries? Also what two-dimensional surface could we ever find to illustrate these two peculiar premises (angle sum > p and angle sum p)? As a matter of fact we see such surfaces daily. On the surface of a sphere the angles of a triangle (seen on maps of crisscrossing airline flight paths) clearly the angle sum > p. Moreover, on the surface of a saddle the angle sum > p. Models of these three cases are seen in Figure 1: sphere, plane and saddle.

But Saccheri's question piqued the interest of geometers for over a century until new geometries eventually evolved into what are now called "non-Euclidean geometries". After Saccheri's courageous introduction to the problem most of the credit for creating these new geometries is given to the Transylvanian Janos Bolyai, the Russian Nicolai Lobachevsky and the Germans Bernhard Riemann and Carl Friedrich Gauss.
Two branches of non-Euclidean geometry are associated with Nicolai Lobachevsky and Bernhard Riemann, but other geometers involved included Eugenio Beltrami, Henri Poincaré Janos Bolyai and Carl Gauss. Sketches of these six are found in Marvin Greenberg's Euclidean and non-Euclidean Geometries

Here we will take a look at these two new geometries which challenge our unquestioning reliance on Euclid's geometry. To start with, the most reasonable definition of a line in Euclidean geometry is: "the path of the shortest distance between two points" which is called a geodesic. A geodesic is a curve C on a surface S such that at each point of C the principal normal of C coincides with the normal to S. A geodesic's curvature on a surface is zero. For example, the geodesic on a plane is a straight line, on a sphere is a great circle, on a cone is a helix. On different two-dimensional curved surface geodesics behave differently than they do on a plane surface. Figure 2 illustrates this difference if we consider the question: "How many lines can be drawn through a point P which are parallel to line L?"
NONE: Figure #2a On a positively curved surface, such as a sphere the shortest distance between two points (geodesic) follows a great circle. No geodesics can be drawn through a point parallel to line L since all these great circles intersect at the poles P, so in this geometry there are no parallel lines. Also on the sphere the angle sum > ¼, many lines can be drawn between two points, all perpendiculars to a line meet in a point and every line has the same finite length.
ONE: Figure #2b On a plane surface only one parallel to a given straight line L can be drawn through a point P not on that line. Here the angle sum = ¼
MANY: Figure #2c On a negatively curved surface, such as a saddle, many geodesics can be drawn through a point P without ever intersecting L (and so are parallel). On this surface the angle sum

Euclid's "flaw"

The discovery of these two geometries resulted from a 21-century search to correct what was considered a "flaw" in Euclid's geometry. Recall that around the third century BC Euclid took it upon himself to codify in 13 books all the mathematical knowledge amassed up to his time. His great achievement consisted in putting some order into the many geometrical discoveries that preceded him, presenting the material in a systematic form and treating geometry as an organic whole.

Before he could assemble this body of knowledge into a coherent arrangement of theorems he had to list terms which would remain undefined, such as points and lines. Then he had to articulate these unprovable premises which everyone took for granted. He found ten such premises. The first five, which he called postulates, were peculiar to geometry and could be constructed, such as his first postulate: "Two points determine one unique straight line." The other five, which he called axioms, were notions common to arithmetic and geometry, such as axiom #VI: "Things equal to the same thing are also equal to each other."

Nine of these premises were simple and compelling. One of them (postulate #V), however, was so long and was expressed in such a complicated way that it seemed more like a theorem. This postulate #V is called the parallel postulate, since it implies that through a given point P outside a line L passes at most one line parallel to L. This postulate #V has been called "the most famous single utterance in the history of science". It was also referred to as "the scandal of geometry" because it had cheated mathematicians of sleep for 21 centuries. Until two centuries ago it had been considered a serious "flaw" in Euclid's masterpiece because geometers felt that Euclid should have taken the trouble to prove it as a theorem. It is, after all, merely the converse of the 28th theorem in Book I and the first theorem to depend on it was the 29th. This "flaw" theory preceded our present appreciation of the three major properties postulates require:
CONSISTENCY so that it is impossible to derive two contradictory theorems from the postulates,
INDEPENDENCE so that no postulate can be derived from the others, and
COMPLETENESS so that everything that will be used to derive the theory is stated in the premises, leaving no tacit assumptions.

Postulate #V has been expressed in many different ways, perhaps in an effort to make it less intimidating. Three of them are shown in Figure #3: On the right : "The angles of a triangle sum to 180o." On the left: "Through a point in a plane one unique line can be drawn parallel to a given line." What is so complicated about that? These statements seem perfectly reasonable and it would be hard to imagine two such lines or no such line. It is much clearer than E's version shown in the middle: "If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles." This contrasts with the simplicity of the other nine premises and sounds like a theorem crying out for a proof. Many geometers tried to supply one.

We now know that Euclid was correct in not providing a proof for postulate #V because there is none: it is independent of the other four postulates as well as the first 28 theorems. Euclid did, however, have some other flaws which, with hindsight, we can identify today. These concern missing postulates which seemed so self-evident that they seemed unnecessary. Euclid did not have postulates of connectedness, of order, of continuity or of congruence. It was not until the last century that the geometer David Hilbert supplied a complete set of postulates needed for Euclidean geometry. Also about that time came the realization that two-dimensional physical space should be distinguished from geometrical space. Different geometries could represent differing behavior of points, angles and lines on two-dimensional surfaces other than the plane such as spheres, and saddles.

Saccheri's attempt to eliminate the "flaw"

The fact that over the years so many geniuses had tried unsuccessfully to prove postulate #V illustrates how baffling and elusive it was. In a time less concerned with rigorous logic, it was frequently considered more of an aesthetic problem than a logical one. One notable effort was made seven centuries ago by the Persian Nasir-Eddin using an isosceles quadrilateral ABDC shown in Figure 4, with right angles at A and B, having AC = BD. He claimed that the summit angles C and D must be right angles also, otherwise the summit side CD would approach AB. This would cause DB to be shorter than CA which was contrary to supposition. He could not prove this claim, but five centuries later his construction provided another geometer, the Italian Jesuit, Jerome Saccheri, with an excellent point of departure to begin a century-long evolution of ideas which would give birth to new geometries.

It is interesting to note that like Pierre Fermat, mentioned in the first paragraph, Jerome Saccheri, although possessing impressive intellectual mathematical abilities, was a part-time mathematician. He had other professorial duties and wrote 12 books on very disparate subjects. He was Professor of Grammar in Milan, and lectured on Philosophy in Turin, and on Mathematics and Theology in Pavia and most of his books concerned theology.

The geometer John Ceva, renowned for "Ceva's theorem" urged young Jerome Saccheri to study Euclid's Elements. He did, and in three years had published his first book Quaesita geometrica which led to correspondence with other mathematicians. He later wrote Logica Demonstrativa and then ended his illustrious mathematical career with the publication of his major work on Euclid. Saccheri eventually investigated geometries undreamed of before his time, thereby opening for his successors the path to amazing discoveries which enlarged greatly the bounds of philosophical and geometrical knowledge. He did not live long enough to continue his investigations which proved to be of great value in the history of mathematical research.

Saccheri was the first geometer to impose rigorous laws of logic in his attempt to eradicate Euclid's "flaw" and this approach makes him the first modern geometer to undertake the task. He introduced a postulate-free method and he formulated the problem in terms of three hypothesis, only one of which could be correct. Geometers who came after Saccheri initially followed his approach and then went on to establish non-Euclidean geometry as a respected mathematical structure. Today the two major divisions are "elliptical" whose model is the sphere and "hyperbolic" whose model is the saddle or the pseudosphere with its negative curvature.

A pseudosphere model

In 1733 Saccheri died, and that same year was published his book whose complete title reads like a short story:
"Euclides ab omni naevo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universae Geometriae principia." (Euclid Freed of Every Flaw: A geometrical work in which are established the fundamental principles of a universal geometry).
Saccheri was a very gifted mathematician whose phenomenal powers of concentration enabled him to vanquish three opponents in three distinct games of chess simultaneously while discussing abstruse philosophical topics with a fourth person. His distinction between nominal definitions and real definitions earned him a conspicuous place in the history of modern logic. When he grappled with our fearsome postulate #V he was first one to draw serious attention to the need for consistency among postulates. Saccheri employed what he calls "a most elegant form of proof" and in the process introduced ingenious procedures which would later evolve into completely new mathematical structures. He developed much of the universal geometry he promised in his title. The birectangular isosceles quadrilateral introduced five centuries earlier by Nasir-Eddin is now called the Saccheri Quadrilateral because Saccheri made such good use of it.

Saccheri's more elegant method of proof

Saccheri was fascinated by a particular method of proof because it is more direct than the usual "reductio ad absurdum" and it is free of postulates and extraneous arguments. Concerning any proposition "P", it is stated as follows"(not P implies P) implies P". In other words, supposition of the negation of the proposition leads directly to the proposition, without the intermediate step of bumping into some absurdity. This "nobler method" negates the negation of the proposition and consequently affirms the proposition. This elegant proof was introduced by Euclid in Book IX, Proposition 12 (referred to here as E IX/12), although modern histories of logic call it "Clavius' rule" after the Jesuit geometer Christopher Clavius. Saccheri was convinced that all fundamental laws of science are best proven by this process.

Now in fact we may conceive another way of proceeding, and as I think a beautiful way, by which I demonstrate the same truths without the assistance of the postulate. I shall proceed thus: I shall assume the contradictory of the proposition to be demonstrated and from it I shall elicit directly the proposition to be proved. This method of proof was used by Euclid in Book IX, Proposition 12. (E IX/12) (Halsted, 1920. p. xxii)

Saccheri's method was not a "reductio ad absurdum" argument which is reached by showing how a supposition leads to a conclusion whose falsity has already been proved. Whereas in the E IX/12 method the proof results from the fact that one obtains the very proposition which was to be proved: i.e. the proposition is a consequence of its very negation. It is worth showing Euclid's use of this method. In book IX Euclid proves the following proposition 12 (i.e. E IX/12).
If a series of numbers, starting with one (1, A, B, C, D), are in a geometric proportion and the last is a multiple of a prime number E then the second is also a multiple of E.


Suppose A is not a multiple of E; then A and E are prime to each other.
1. Since D is a multiple of E, then there must be an F so that EF = D
2. Since D is a multiple of A, there must be a C so that AC =D, then AC=D=EF or A/E=F/C
3. Now since A and E are supposed to be prime to each other, the equation demands that C be a multiple of E. (by Book VII Prop. 20)
4. Then there must be a constant G so that C = EG
5. But AB = C so AB=C=EG or A/E = G/B
6. Now since A and E are supposed to be prime to each other, the equation demands that B be a multiple of E. (by Book VII Prop. 20)
7. Then there must be an H so that EH = B
8. But AA = B so AA = EH or E/A = A/H
9. Then, since A and E are supposed to be prime to each other, the equation demands that A be a multiple of E. (by Book VII Prop. 20) Q.E.D.

This was the only time Euclid used this method of proof and he provides an example using the set {1, 4, 16, 64, 256} with E = 2.

Since Saccheri planned to use this method, he had to state postulate #V in a more convenient manner so that he could state the necessary contradiction. So he used Nasir-Eddin's isosceles birectangular quadrilateral which now bears his name (Saccheri quadrilateral) and expressed

Postulate #V in terms of its summit angles, allowing three cases.
Obtuse case: The summit angles of a Saccheri quadrilateral > p/2.
Euclid case: The summit angles of a Saccheri quadrilateral = p/2.
Acute case: The summit angles of a Saccheri quadrilateral p/2.

As he proceeded through the theorem, he treats the three cases O, E, A one after another hoping to reduce cases O and A to the Euclid case. Then with remarkable geometrical skill and fine logical precision, he established many theorems still in use today.

Some of Saccheri's theorems can be viewed in the context of Poincaré's model by moving to the page
Hyperbolic Geometry Theorems of Girolamo Saccheri, S.J.

Lines in the Poincaré's model are parts of circles orthogonal to the circle S

The significance of non-Euclidean geometry

Bernhard Riemann once claimed: "The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean." His prophesy was realized later with Einstein's general theory of relativity. It is futile to expect one "correct geometry" as is evident in the dispute as to whether elliptical, Euclidean or hyperbolic geometry is the "best" model for our universe. Henri Poincaré, in Science and Hypothesis (New York: Dover, 1952, pp. 49-50) expressed it this way.

If geometry were an experimental science, it would not be an exact science. It would be subjected to continual revision . . . The geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions and are only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. . . . What then are we to think of the question: Is Euclidean Geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates false. One geometry cannot be more true than another; it can only be more convenient.

On the other hand we know that since we live in an expanding universe, we have to ask: is the expansion slowing down, staying the same or speeding up? To answer this question requires a theoretical model of the universe. Any model based on Einstein's general theory of relativity must represent motion of the most distant galaxies. Let us consider parallel beams of light extending without limit. Figure 16 illustrates three possible outcomes: they can converge, they can remain parallel or they can diverge as they move across the universe.

In Riemannian space of positive curvature, parallel light beams eventually meet and the universe will eventually stop expanding and begin to contract. Figure 16.1 In Euclidean flat space of zero curvature, parallel light beams remain parallel and the universe will expand forever at a constant rate of change. Figure 16.2 In Lobachevskian space of negative curvature, parallel light beams eventually diverge and the universe will expand at an increasing rate. Figure 16.3 In short the future of the universe is related to the geometry of the universe. The physicist George Gamow also treated these three different geometries and their respective consequences. The basic cosmological notion of general relativity grew out of the work of great mathematicians of the nineteenth century. Suppose, now, we examine the properties of a two-dimensional geometry constructed not on a plane surface but on a curved surface. . . If space is negatively curved, the universe is infinite; if it is positively curved, the universe is finite. (Gamow, Pg. 59-76)

Figure #17 Escher's "Circle limit III" and "Circle limit IV" on the Poincaré model

Artists such as M. C. Escher have become fascinated with the Poincaré model of hyperbolic geometry and he composed a series of "Circle Limit" illustrations of a hyperbolic universe. In Figure 17.a he uses the backbones of the flying fish as "straight lines", being segments of circles orthogonal to his fundamental circle. In Figure 17.b he does the same with angels and devils. Besides artists and astronomers, many scholars have been shaken by non-Euclidean geometry. Euclidean geometry had been so universally accepted as an eternal and absolute truth that scholars believed they could also find absolute standards in human behavior, in law, ethics, government and economics. The discovery of non-Euclidean geometry shocked them into understanding their error in expecting to determine the "perfect state" by reasoning alone. On the other hand, non-Euclidean geometry showed us that the human mind can defy intuition, common sense and experience in order to experience what worlds reasoning could create. The historian of mathematics, Morris Kline, summed up the meaning of these new geometries. The importance of non-Euclidean geometry in the general history of thought cannot be exaggerated. Like Copernicus' heliocentric theory, Newton's law of gravitation, and Darwin's theory of evolution, non-Euclidean geometry has radically affected science, philosophy, and religion. It is fair to say that no more cataclysmic event has ever taken place in the history of all thought.

Kline then goes on to point out that the discovery of non-Euclidean geometry was a gradual and cooperative evolution. Those credited with the discovery of non-Euclidean geometry such as Riemann, Lobachevsky, Bolyai and Gauss all built on the work of the first geometer to treat the matter seriously and rigorously, Jerome Saccheri, who in uncovering the consequences of the acute angle hypothesis, directed subsequent geometers to a new geometry. Instead of leading to the discovery of a proof for postulate #V he lead to the discovery of a new creation. It is an application of Isaac Newton words to Robert Hooke: "If I have seen farther than Descartes, it is because I have stood on the shoulders of giants."


Bonola, R. Non-Euclidean geometry. New York: Dover, 1955 (Translated from Italian)
Bosmans, Henry "Le geométre Jèrome Saccheri S. J. (1667-1733)", Revue des Questions Scientifiques, 4, 1925, pp. 401-430.
Coolidge, Julian A. A History of Geometrical Methods. New York: Oxford University Press, 1940. pp. 12-13
Dou, Alberto "Logical and Historical Remarks on Saccheri's Geometry," Notre Dame Journal of Formal Logic, volume XI, 4, October 1970.
Gambarana, P. S.J., "An Account of the Life of Girolamo Saccheri," Rome: Archives Societatis Jesu, Ad Rom. 187, ff. 191 ss.
Gamow, George The Universe , Ch. 4 "The Evolutionary Universe" New York: Scientific American, 1956 pp. 59-76.
Gillispie, Charles. C. ed., Dictionary of Scientific biography. 16 vols. New York: Charles Scribner and Sons, 1970 Simon & Schuster, New York 1956 pp. 59-76.
Gray, Jeremy Non-Euclidean geometry: a re-interpretation. Historia-Mathematica Vol. 6 (1979), 3, pp. 236-258.
Halstead, G. B. Saccheri's Euclides Vindicates. Chicago: Open Court Publishing Co., 1920.

Theorems of Saccheri, S.J. - 1733: and his non Euclidean Geometry

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