This site has been archived for historical purposes. These pages are no longer being updated.
A Sample of Saccheri's Contribution to the evolution of Non-Euclidean geometry
More can be read concerning Saccheri's contribution to non-Euclidean geometry by viewing
Saccheri's Solution to Euclid's BLEMISH
The Origins of Non-Euclidean Geometry
A sample of Saccheri's non-Euclidean geometry
Many of the theorems found in today's non-Euclidean geoemtry textbooks ultimately are derived from the theorems proven in Jerome Saccheri's 1633 book - and this usually without crediting Saccheri. Here are presented a few of his theorems illustrated by using the Poincaré model.
By use of similar triangles and congruent parts of similar triangles on the Saccheri quadrilateral, ABDC with AC = BD and A = B = p/2, he establishes his first 32 theorems. Most are too complicated to be treated in a short paper, but here some examples are merely stated, some are illustrated and some are proven. For those proofs which are brief enough to show here, the main steps are indicated and the reader is invited to fill in the missing details of the argument. A century after Saccheri, the geometers, Lobachevsky, Bolyai and Gauss would realize that, by substituting the acute case or the obtuse case for Euclid's postulate Number V, they could create two consistent geometries. In doing so they built on the progress made by Saccheri who had already proven so many of the needed theorems. They were able to create what we recognize today as the "elliptical" and "hyperbolic" non-Euclidean geometries. Most of Saccheri's first 32 theorems can be found in today's non-Euclidean textbooks. Saccheri's theorems are prefaced by "Sac."
Sac.1 for all cases O, E, A:
THE SUMMIT ANGLES ARE EQUAL: C = D
Sac.2 for all cases O, E, A: THE LINE EF JOINING THE MIDPOINTS E, F OF THE BASE AND SUMMIT IS PERPENDICULAR TO BOTH AB and to CD
1 The proof starts by constructing lines CF and DF.
2 triangles DCAF = DDBF
3 so that DCEF = DDEF
4 so that CEF = DEF.
5 But CEF + DEF =p
6 so that CEF = DEF =p/2.
Sac.3 for Obtuse case: IN THE OBTUSE CASE THE SUMMIT CD IS LESS THAN THE BASE: CD < AB
1 Construct line EF which connects the midpoints of the summit and base.
2 IN THE = CASE: CD = AB then CE = AF, so that AFEC is a Sac. Quadrilateral
then angles A = C = p/2 which contradicts the Obtuse case, C > p/2
3 IN THE > CASE: CD > AB then find point G so that EG = AF causing FAG < FAC (which is p/2) by construction.
4 Then AFEG is a Saccheri quadrilateral (by Sac.2)
5 Then FAG = EGA (by Sac.1).
6 But also EGA > ECA because it is an exterior angle.
7 But ECA is assumed to be obtuse (by the Obtuse hypothesis)
8 So EGA is both acute and obtuse which is impossible.
9 So the premise CD > AB is wrong and CD < AB.
Sac.9 ON THE HYPOTHESIS OF THE RIGHT ANGLE, THE OBTUSE ANGLE OR THE ACUTE ANGLE, THE SUM OF THE ANGLES OF A TRIANGLE IS ALWAYS EQUAL TO, GREATER THAN OR LESS THAN TWO RIGHT ANGLES. That is:
Obtuse case implies A + B + C > p
Euclid case implies A + B + C = p
Acute case implies A + B + C < p
Proof of the Obtuse case x+B+y =A+B+C> p
1 Construct a Saccheri quadrilateral with base AB.
2 So DC < AB. (by Sac.3)
3 Then CAD (z) < ACB (y) (angle opposite greater side)
4 So angles x + B + y > z +x + B = p
Sac.11 Obtuse and Euclid cases:
IF BC IS PERPENDICULAR TO AB AND BAD IS ACUTE THEN BC INTERSECTS AD.
This important theorem has a long complicated proof, but is used in S 13 in which Saccheri shows that the Obtuse case of postulate #V reduces to the Euclid case. (Here Saccheri makes the false the assumption that all lines are infinite in length).
S 13. EUCLID CASE OF POSTULATE #V FOLLOWS FROM BOTH O AND E POSTULATES.
1 Suppose A is acute and that A + ACD (C) < p
2 Construct CE perpendicular to AB
3 Then in the DACE: A + ACE (x) + E < p (by Sac.9)
4 But A + x + ECD (y) < p by supposition
5 Thus E > y by comparing steps 3 and 4
i.e. ECD is acute since E = p/2
6 So line AB intersects CD (by Sac.11)
7 And this is what is demanded of the Euclid case of postulate #V
Sac.15: IF THERE EXISTS A SINGLE TRIANGLE FOR WHICH THE SUM OF THE ANGLES IS EQUAL TO, GREATER THAN OR LESS THAN TWO RIGHT ANGLES, THEN FOLLOWS THE TRUTH OF THE HYPOTHESIS OF THE RIGHT ANGLE, THE OBTUSE ANGLE OR THE ACUTE ANGLE, RESPECTIVELY.
Sac.17: LINE AB IS PERPENDICULAR TO BASE BC: BY THE ACUTE CASE OF POSTULATE #V A LINE L CAN BE DRAWN FROM A MAKING AN ACUTE ANGLE WITH LINE AB WHICH DOES NOT INTERSECT BC.
Sac. 30 to 32:
BY ACUTE CASE OF POSTULATE #V THERE EXISTS IN THE PENCIL OF LINES THROUGH A TWO LINES p AND q ASYMPTOTIC TO B (ONE TO THE RIGHT AND ONE TO THE LEFT) WHICH DIVIDE THE PENCIL INTO TWO PARTS: THOSE LINES WHICH INTERSECT LINE B AND THOSE WHICH HAVE A COMMON PERPENDICULAR TO B.
In each figure #7 to #11 theorems are illustrated on two models. On the right is the proper model, sphere (#7, #8, # 9) or saddle (#10, #11) and on the left is our familiar Euclidean plane. It is clear from these figures how unsuitable the plane is to represent the Acute and Obtuse case, so it was a remarkable display of rigorous logic for Saccheri to work out the major non-Euclidean theorems without these models that fit the theorems. Instead of using drawings on the plane (which of little intuitive help), the theorems are clarified by seeing them in a modern two-dimensional model which were provided in later years.
Since the greater problem encountered by Saccheri was the acute case, we now demonstrate the meaning of the acute hypothesis by using another model introduced by Poincaré to illustrate the consistency of hyperbolic geometry. The latter can be shown to be consistent if Euclidean plane geometry is consistent by showing a specific model for hyperbolic geometry whose elements and relations are interpretations of the elements of Euclidean geometry. Henri Poincaré found a beautiful model within the Euclidean plane, which had been devised earlier from a steriographic projection of a sphere onto a plane. He chose a fixed disk "S" in the Euclidean plane which he called "the fundamental circle". Then he established the following representations of the primitive terms of hyperbolic plane geometry to match elements in the Euclidean plane. Here follows some explanation of the terms of the Lobachevskian plane which are in bold print in order to distinguish them from the corresponding terms of the Euclidean plane.* a Lobachevskian point P lies in the interior of SAB = log[(AT/TB)/(AS/SB)] = log [(AT/BT)(BS/AS)], where S and T are the W points at infinity in which the "circle" cuts disc S.
* a Lobachevskian line AB is the part interior to S of any circle which is orthogonal to S
* the Lobachevskian length of segment AB is a natural logarithm of a ratio of ratios:
So if a point C is between points A and B and then the length AB = AC + CB
The proof of this simply applies the definition of the metric.
So the length AB
= log[(AT/CT)(CS/AS)] + log[(CT/BT)(BS/CS)]
= length AC + length CB.
Now we are in a position to illustrate some of Saccheri's terms, postulates and theorems.
Figure # 12 Points, lines and triangles on the Poincaré's model
Figure # 13 The limit triangle and the Saccheri quadrilateral on in the Poincaré's model
Figure # 14 Some familiar theorems on the Poincaré's model
Figure # 15 Length depends on angles on the Poincaré's model
Saccheri's "flaw"Honest historians have praised Saccheri's book of 39 theorems: "The first 70 pages (theorems 1 to 32) is an ensemble of logic and geometric acumen which can be called perfect." But then, suddenly Saccheri abruptly swerves away from his carefully plotted course. In the 33rd theorem is found Saccheri's "flaw", where he breaks away from his rigorous logic and carefully crafted "perfect" structure and, as it were in disbelief, remarks ". . . but this is contrary to our intuitive knowledge of a straight line." Galileo had said something similar about infinite numbers: "I see it, but I do not believe it." Saccheri closes his book by admitting that he has not completely proven the "acute case" and for this reason is said to have withheld publication of the book during his lifetime.
I do not attain to proving the falsity of the other [acute angle] hypothesis without previously proving that the line, all of whose points are equidistant from an assumed straight line lying in the same plane with it, is equal to this straight line, which itself finally I do not appear to demonstrate from the viscera of the very hypothesis, as must be done for a perfect refutation. . . But this is now enough. (Halsted, 1920 p.233-234)
Alberto Dou, S.J. has studied Saccheri's Euclides for years and has written very insightful things about Saccheri's "flaws". Among these errors he includes the assumptions that all lines are infinite in length, that in every case the exterior angle is greater than an interior angle and that a point at infinity possess the same properties as an ordinary point. Dou also demonstrates that, contrary to an earlier theory, Riemann, Lobachevsky, Bolyai and Gauss not only had direct or indirect access to Saccheri's Euclides but they also used his method and his skillfully constructed set of theorems. The Euclides was transmitted through historians contemporary to Saccheri, J.C. Heilbronner (Leipzig, 1742) and Montucla (Paris, 1758), as well as successive geometers such as G.S. Kluegel, J.H. Lambert and A.G. Kaestner all of whose works were, in turn, available to Riemann, Lobachevsky, Bolyai and Gauss. These facts are well documented by the studies of C. Segre, P. Staeckel and R. Bonola.
Saccheri had produced a method that would solve the puzzle of the Euclid's "flaw" and also lead to the two major branches of non-Euclidean geometry. But, not satisfied with his own proofs, he realized that it was the best he could do. But, then, if he had some doubt about the rigor of these proofs, why did he put them in the Book? Perhaps otherwise there would have been no book, and that would have been much worse. Apparently he surrendered, like Lambert and many others, to the desire of giving a higher interpretation or value to his work. (Dou, p. 407)
In spite of his abrupt conclusion which is incompatible with his rigorously logical method, Saccheri's investigations were a crucial step in the evolving discovery of non-Euclidean geometries. His major achievement was to break ground for the later geometers who now were free to investigate these new geometries. Saccheri died too soon (while still reworking his manuscript) to discover that denying Euclid's postulate #V resulted not in a contradiction, but in two other completely different geometries which were just as consistent as Euclid's.
Why today's change in attitude toward the influence of Saccheri?One might ask: "Why has Saccheri's contribution been missed by past historians?" Unlike modern historians such as H. Eves and D. Struik, earlier writers viewed with contempt scholarly works of the members of the Jesuit order, that "Bete noire" of some European savants of the time. The Dictionary of Scientific Biography (vol. 6 p. 234) attributes "the public defamation of the Jesuits then in vogue" as the cause for valuable Jesuit scholarly work being attributed to other scholars. In his book The magic of Numbers Eric Temple Bell illustrates this contempt and spins an entertaining tale of sinister Jesuit intrigue in which Saccheri was silenced for advocating new geometries.
Saccheri was simply a natural genius at believing what he wished to believe. Though this is the simplest explanation of his twisted career it is not the only one possible; others will suggest themselves as we follow the devious misadventures of his masterpiece . . . . The Jesuits seem to sink their personalities in the discipline of their Order, and Saccheri was pretty thoroughly submerged.
In fact Bell seems intimidated by a loyal member of the Church who understood non-Euclidean geometry a century before any geometers had articulated it. In so doing Bell has volunteered himself as a candidate for his own dictum: "Facts will out; and whoever tries to conceal them sooner or later is shown up as a blundering incompetent."