Giovanni Girolamo Saccheri

How to use this course

Giovanni Girolamo Saccheri spent his working life as a Jesuit teacher of philosophy, theology and mathematics in northern Italy. He is remembered for one book, Euclides ab omni naevo vindicatus (Euclid freed from every flaw), printed in Milan in 1733. Its aim was to prove Euclid’s parallel postulate by showing that its denial leads to absurdity. To do so he developed the consequences of the denial in more depth than anyone before him. Most of what he proved is correct, and much of it is now a chapter of hyperbolic geometry. The book was forgotten for a century and a half and then recognized as the most important work on parallels before Nikolai Lobachevsky and János Bolyai.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Proposition numbers (Prop. I, Prop. XXIII and so on) are Saccheri’s own, in Book I of Euclides vindicatus. Lessons 05 to 09 follow the order of his book. Lesson 10 steps outside it and computes his quadrilateral in the models that settled the question he could not.

The course on Omar Khayyam describes the same quadrilateral six centuries earlier. The course on Nikolai Lobachevsky continues the story after 1826.

Timeline

San Remo and Genoa, 1667–1690

Giovanni Girolamo Saccheri was born on 5 September 1667 in San Remo, a coastal town of the Republic of Genoa. His father, Giovanni Felice Saccheri, was a lawyer. Early accounts describe the boy as precocious and inquisitive, and little else is known of his childhood.

On 24 March 1685, aged seventeen, he entered the novitiate of the Society of Jesus in Genoa. The Jesuits ran the largest network of schools in Catholic Europe, and their training was long: a novitiate, then years of teaching in the lower classes of a college alternating with study of philosophy and theology. From 1687 Saccheri taught at the Jesuit college in Genoa, whose building, the Palazzo Balbi, now houses the University of Genoa.

Jesuit mathematics

Mathematics had an assured if modest place in the Jesuit curriculum. Its model was Christopher Clavius, whose Latin edition of Euclid’s Elements (first published 1574) was the standard school text across Europe for a century. Clavius had himself discussed the parallel postulate and proposed a proof of it. A young Jesuit with mathematical talent would meet Euclid through Clavius, and Saccheri did.

Brera and the Ceva brothers, 1690–1694

In 1690 his superiors sent Saccheri to the Collegio di Brera in Milan, the principal Jesuit college of Lombardy, to study philosophy and theology while teaching grammar. Brera’s professor of mathematics was Tommaso Ceva, a Jesuit poet and geometer who taught there for about forty years. Ceva recognized Saccheri’s ability, encouraged him to read Clavius’s Euclid closely, and urged him to devote himself to mathematics.

Through Tommaso he met his brother Giovanni Ceva, mathematician to the Gonzaga court at Mantua, whose 1678 book De lineis rectis contains the theorem on concurrent lines in a triangle that bears his name. Saccheri worked in close contact with both brothers. On Tommaso’s advice he also began a correspondence with Vincenzo Viviani in Florence, a pupil of Galileo.

Quaesita geometrica

His first book, Quaesita geometrica, appeared in Milan in 1693. It solves a set of elementary geometric problems that had been posed by the Sicilian mathematician Ruggero Ventimiglia, using the methods of Giovanni Ceva. Viviani responded by sending Saccheri his own solution of a problem he had circulated, the so-called window of Viviani. In March 1694 Saccheri was ordained priest at Como.

Turin and Pavia, 1694–1708

From 1694 to 1697 Saccheri taught philosophy at the Jesuit college in Turin. There he came to the notice of Victor Amadeus II, Duke of Savoy, who reportedly called on him when difficult calculations were needed. His philosophy course became his second book, Logica demonstrativa, printed in Turin in 1697. It was reissued in 1701 and again at Cologne in 1735. Lesson 04 describes its content.

In 1697 he moved to the Jesuit college at Pavia, where he taught philosophy and theology for the rest of his life. In 1699 the Senate of Milan, which governed the University of Pavia, appointed him to its chair of mathematics. He stayed in that chair for thirty-four years. In 1713 Victor Amadeus offered him the chair of mathematics at Turin, and later he was offered the chair at Padua that Galileo had held. He declined both.

Neo-statica

In 1708 he published Neo-statica, dedicated to the Senate of Milan. It is a treatise on the equilibrium of weights, partly a critique of Tommaso Ceva’s De natura gravium (1669). It stays within the Aristotelian tradition of statics and is of little interest today. His real mathematical project, slowly developed over the next quarter century, was the foundations of Euclid’s geometry.

Definitions and the logic of reductio

For the general reader. Before he wrote about parallels, Saccheri wrote a textbook of logic. Two ideas from it shaped everything he did later. A definition does not prove that the thing defined exists. And one powerful way to prove a statement is to assume it false and show that the assumption leads back to the statement itself.

Nominal and real definitions

The Logica demonstrativa follows the plan of Aristotle’s logical works in four parts. Its most original pages concern definitions. Saccheri separated a definitio quid nominis, which only fixes the meaning of a word, from a definitio quid rei, which in addition asserts that the thing exists. A real definition is a hidden postulate, and its use in a proof must be justified. He also asked whether the clauses of a complex definition are compatible with one another, a question close to the modern notion of consistency. Thomas Heath later observed that John Stuart Mill’s account of the same distinction had been fully anticipated by Saccheri.

Saccheri’s example was the definition of parallel lines as lines everywhere equidistant, used by Giovanni Alfonso Borelli and others. As a nominal definition it is harmless. Used as a real definition it assumes that the points at a fixed distance from a line form a straight line, and that assumption is equivalent to the parallel postulate (Exercise 4.2).

The rule of Euclid IX.12

Saccheri found the pattern in Euclid’s proof of Elements IX.12, where the denial of what is to be proved yields the proposition itself. It is stronger in feel than ordinary reductio, which derives some contradiction \(Q\wedge\neg Q\), though the two are equivalent in classical logic. This was his plan for the parallel postulate: assume it false, and derive it.

The quadrilateral and three hypotheses

For the general reader. Take a line segment, raise two equal perpendiculars at its ends, and join their tops. Without the parallel postulate, nobody can prove that the top angles are right angles. Saccheri’s idea was to split geometry into three cases according to these two angles, and to follow each case to the end.

All results in this lesson belong to absolute geometry: Euclid’s Book I up to Proposition 28, without the fifth postulate. Saccheri explicitly allowed himself the first 26 propositions of Euclid, together with the Archimedean axiom and a principle of continuity.

Definitions

A Saccheri quadrilateral \(ABDC\) has base \(AB\), legs \(AC\) and \(BD\) perpendicular to \(AB\) on the same side with \(AC=BD\), and summit \(CD\). Its summit angles are \(\angle ACD\) and \(\angle BDC\). The figure appears in the work of Omar Khayyam in the eleventh century. Whether Saccheri knew Khayyam’s text is not established; he did know the treatment in the school of Nasir al-Din al-Tusi through John Wallis, whose argument he criticizes.

For (I), triangles \(CAB\) and \(DBA\) are congruent by SAS, so \(BC=AD\); then triangles \(ACD\) and \(BDC\) are congruent by SSS. For (II), triangles \(AMC\) and \(BMD\) are congruent (SAS), so \(MC=MD\), and triangles \(CMN\) and \(DMN\) are congruent (SSS), giving right angles at \(N\); a similar argument with \(N\) gives right angles at \(M\).

Each half, such as \(AMNC\), has three right angles. Johann Heinrich Lambert later built his own theory on this figure, now called a Lambert quadrilateral.

Comparing sides and angles

Proof. If \(AC>BD\), choose \(E\) on \(AC\) with \(AE=BD\). Then \(ABDE\) is a Saccheri quadrilateral, so \(\angle AED=\angle BDE\) by Prop. I. The angle \(\angle AED\) is exterior to triangle \(EDC\), so it exceeds \(\angle ECD=\angle ACD\) by I.16. Since \(E\) lies inside \(\angle BDC\), \(\angle BDC>\angle BDE\). Hence \(\angle BDC>\angle ACD\). The converse follows by exchanging the roles of the legs and using Prop. I when \(AC=BD\). ∎

Apply the lemma to the Lambert quadrilateral \(MNCA\) with base \(MN\) and right angles at \(M\) and \(N\): the sides are \(MA=\tfrac12AB\) and \(NC=\tfrac12CD\), and the opposite angles are \(\angle NCA\) (the summit angle) and \(\angle MAC=\pi/2\). The summit angle is less than \(\pi/2\) exactly when \(NC>MA\), that is when \(CD>AB\). The other cases are identical.

Saccheri’s proofs combine Prop. III with a continuity argument: a summit angle that changed from acute to obtuse as the figure was varied would have to pass through a right angle, and a single right summit angle already forces right angles everywhere. The modern reason is simpler. Lesson 10 shows that the summit angle is governed by the curvature of the plane, and its sign is the same everywhere.

Angle sums of triangles

For the general reader. At school we learn that the angles of a triangle add up to 180 degrees. Saccheri showed that this fact is tied to his quadrilateral. Under each of his three hypotheses, every triangle has angle sum equal to, more than, or less than 180 degrees, and the three cases cannot mix.

Proof sketch

Take a right triangle \(ABC\) with right angle at \(A\). Erect the perpendicular \(BD\) at \(B\) with \(BD=AC\), on the same side as \(C\), to form a Saccheri quadrilateral \(ABDC\) with summit angle \(\varphi\). Under the right hypothesis triangles \(ABC\) and \(DCB\) are congruent, so the angles of \(ABC\) at \(B\) and \(C\) sum to \(\pi/2\). Under the obtuse hypothesis \(CD<AB\) (Prop. III); comparing triangles \(ABC\) and \(DCB\), which share \(BC\) and have \(AC=BD\), the angle opposite the longer side \(AB\) is larger, so \(\angle ACB>\angle DBC=\pi/2-\angle ABC\), giving an angle sum above \(\pi\). The acute case is the mirror image. Any triangle is split into two right triangles by an altitude from its largest angle, and the angle sums add up correctly.

For the converse, the three hypotheses are exhaustive and their conclusions about triangles exclusive, so each conclusion forces its own hypothesis. Saccheri also proved a statement on angles in a semicircle: they are right, obtuse or acute according to the hypothesis (Exercise 6.1).

Adrien-Marie Legendre rediscovered these facts around 1794, and they are often called the Saccheri–Legendre theorems. The course on Lobachevsky (Lesson 04) gives Legendre’s doubling proof that, in absolute geometry, no triangle has angle sum above \(\pi\).

Refuting the obtuse angle

For the general reader. On a globe, a quadrilateral with two equal sides along meridians standing on the equator has obtuse top angles. So the obtuse hypothesis is not absurd in itself. Saccheri ruled it out using a property of Euclid’s lines that great circles lack: they go on forever without returning.

The structure is the one of Lesson 04. From the obtuse hypothesis, Props. XI–XIII derive the fifth postulate. But the fifth postulate implies that every triangle has angle sum exactly \(\pi\) (Euclid I.32), which by Prop. XV is the right hypothesis. So the obtuse hypothesis implies its own negation. The proofs of XI and XII lay off equal steps along the oblique line and use Archimedes’ axiom to show it must eventually cross the perpendicular.

Where infinity enters

Saccheri’s argument depends on Euclid’s first 26 propositions, and these use the second postulate in a strong form: a segment can be extended indefinitely, and two lines meet at most once. The key case is I.16, the exterior angle theorem. In its proof one extends a median to twice its length, and the argument fails if the line returns on itself. On a sphere it does. In modern terms the obtuse hypothesis describes spherical or elliptic geometry, where lines are finite closed curves and I.16 fails for large triangles. The Dictionary of Scientific Biography remarks that Saccheri’s refutation is not cogent by modern lights for exactly this reason: he refuted the obtuse hypothesis only in the presence of infinite lines.

The acute angle: common perpendiculars and asymptotes

For the general reader. The acute hypothesis would not die. Saccheri pursued it through two dozen propositions, deducing stranger and stranger consequences in the hope that one would contradict the others. What he found is now a consistent geometry. In it, two lines that never meet either spread apart in both directions or approach each other ever more closely in one direction without touching.

In the quadrilateral \(ABDC\), the line \(CD\) makes the acute angle \(\varphi\) with \(CA\), and \(AB\) is perpendicular to \(CA\). Yet \(AB\) and \(CD\) never meet, since both are perpendicular to \(MN\) (Euclid I.28). So the fifth postulate fails.

Saccheri also proved that from a point outside a line, the lines through the point that meet the given line are separated from those that do not by two boundary lines, which do not meet it and are asymptotic to it. A century later Lobachevsky called these boundary lines the parallels and organized his geometry around the angle they make with the perpendicular.

“Repugnant to the nature of the straight line”

For the general reader. After thirty-two careful propositions, Saccheri declared the acute hypothesis false. His reason was not a contradiction but an appeal to what straight lines are like. He seems to have known this was weak, for he tried a second, longer refutation. It contains a mistake.

The first refutation

Saccheri considered two asymptotic lines, as in Prop. XXV. Extending to the infinitely distant region results proved for finite figures, he argued that the two lines would meet there and have a common perpendicular at that point at infinity. Two lines sharing a point and a perpendicular there would coincide.

The flaw is the passage to the limit. Asymptotic lines approach each other, but no point at infinity belongs to either of them, and properties of finite configurations need not survive the limit. Corrado Segre judged the work up to Prop. XXXII logically perfect; the break comes exactly here.

The second refutation

In the remaining propositions of Book I Saccheri tried again with the locus of points at fixed distance from a line, the curve that the advocates of “equidistant” parallels had assumed to be straight (Lesson 04). He sought to show that under the acute hypothesis an arc of this curve would have the same length as the segment of the base below it, which would make the curve a straight line and so force the right hypothesis. His own summary concedes the point: he could not reach his goal without first proving that the equidistant line equals its base. The argument he gave for that rests on a faulty limiting step about infinitesimal pieces of the curve.

The quadrilateral in modern geometry

For the general reader. Since Eugenio Beltrami’s work of 1868 we know that each of Saccheri’s hypotheses describes a real geometry, realized on a surface of constant curvature. In these models his quadrilateral can be computed exactly. The summit angle turns out to measure the curvature of the plane.

Derivation of the summit length

Use the hyperboloid model with \(k=1\): the plane is \(\{x:\langle x,x\rangle=-1,\ x_0>0\}\) with \(\langle x,y\rangle=-x_0y_0+x_1y_1+x_2y_2\), and distance given by \(\cosh d(x,y)=-\langle x,y\rangle\). Put the base on the geodesic \(x_2=0\) with \(A=(\cosh\frac b2,-\sinh\frac b2,0)\) and \(B=(\cosh\frac b2,\sinh\frac b2,0)\). The unit normal to the base along it is \(n=(0,0,1)\), so the tops of the legs are

Then \(\cosh s=-\langle C,D\rangle=\cosh^2a\,\cosh b-\sinh^2a\). Writing \(\cosh s=1+2\sinh^2\frac s2\) and \(\cosh b=1+2\sinh^2\frac b2\) gives \(\sinh^2\frac s2=\cosh^2a\,\sinh^2\frac b2\). The angle formula follows by applying the right-angled trigonometry of the Lambert quadrilateral \(AMNC\), and the area formula from the Gauss–Bonnet theorem, which says that area equals \(k^2\) times the angle defect \(2\pi-(\pi/2+\pi/2+2\varphi)\).

Every quantity here depends on the size of the figure. In Euclid’s plane the summit angle is \(\pi/2\) whatever the size. In Saccheri’s acute plane, a quadrilateral with given summit angle has a determined size, and this is the “absolute unit of length” that Saccheri found unacceptable and Lambert found intriguing.

The book of 1733 and its author’s death

Saccheri seems to have worked on the parallel problem for many years at Pavia, alongside his teaching of philosophy and theology. Euclides ab omni naevo vindicatus was dedicated to the Senate of Milan. As the work of a Jesuit it required the approval of the Inquisition, granted on 13 July 1733, and of his provincial, granted on 16 August. It was printed in Milan that year.

The book has two parts. Book I, in 39 propositions, treats the parallel postulate: Lessons 05 to 09 follow it. Book II defends Euclid’s theory of proportion in Book V of the Elements, in particular the definition of equal ratios, against critics who found it obscure. Halsted, the book’s English translator, praised this second part as evidence of Saccheri’s penetration and modernity. The title promises to remove “every blemish” from Euclid, and Saccheri in fact treated three, of which the parallel postulate is by far the most important.

Saccheri died in Milan on 25 October 1733, about two months after the second approval, aged 66. He did not see the book’s reception, and in his lifetime there was little to see.

Oblivion and rediscovery

The book was not entirely ignored. In 1763 Georg Simon Klügel, a student of Abraham Gotthelf Kästner at Göttingen, wrote a dissertation examining nearly thirty attempted proofs of the parallel postulate, Saccheri’s among them, and concluded that none succeeded. Through Klügel, Saccheri’s approach reached Johann Heinrich Lambert, whose Theorie der Parallellinien (written about 1766) pursued the acute hypothesis further using the three-right-angled quadrilateral. After that the book faded from view. Lobachevsky and Bolyai, who created hyperbolic geometry around 1830, apparently did not know it.

Rediscovery

In 1889 Eugenio Beltrami, who had given the first models of hyperbolic geometry in 1868, published a note drawing attention to Saccheri’s book and to how much of the new geometry it contained. Giovanni Vailati rediscovered the Logica demonstrativa in 1903 and placed Saccheri in the history of logic. George Bruce Halsted’s Latin–English edition appeared in 1920, and Roberto Bonola’s history of non-Euclidean geometry (1906, English 1912) gave Saccheri a central chapter. A modern edition with commentary, edited by Vincenzo De Risi, appeared in 2014.

Legacy

• Mathematics: the Saccheri quadrilateral, the three hypotheses, and the Saccheri–Legendre theorem are standard in every course on foundations of geometry. His Props. I–XXXII are a correct and systematic development of the first theorems of hyperbolic geometry, including asymptotic and ultraparallel lines.
• Method: he showed how far one can go in deducing consequences from a hypothesis one believes false, which is the method of independence proofs, even though he drew the wrong conclusion.
• Logic: his analysis of definitions and existence anticipated later work on consistency and on nominal and real definitions.

Connections

People linked to Giovanni Girolamo Saccheri across this collection. A name without a link has no page here yet. See Giovanni Girolamo Saccheri in the network of all 337 lives →

Teachers and advisors

• Tommaso Ceva 1690-1694

Influenced by

• Euclid
• John Wallis
• Nasir al-Din al-Tusi
• Omar Khayyam
• Christopher Clavius

Influenced

• Adrien-Marie Legendre
• Eugenio Beltrami
• Johann Heinrich Lambert
• Georg Simon Klügel

Colleagues and collaborators

• Giovanni Ceva
• Vincenzo Viviani

Also mentioned in this life

• Archimedes
• Aristotle
• János Bolyai
• Galileo Galilei
• Carl Friedrich Gauss
• David Hilbert
• Nikolai Lobachevsky
• Henri Poincaré

Fields

• geometry 32 lives
• foundations of geometry 4 lives
• logic 8 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Giovanni Girolamo Saccheri, MacTutor History of Mathematics.
2. A. Natucci, Saccheri, Girolamo, Dictionary of Scientific Biography.
3. Giovanni Girolamo Saccheri, Wikipedia.
4. J. J. O’Connor and E. F. Robertson, Georg Simon Klügel, MacTutor (on the 1763 dissertation).

Mathematics

1. G. Saccheri, Euclides ab omni naevo vindicatus, Milan, 1733. English translation by G. B. Halsted, Euclides Vindicatus, Open Court, Chicago, 1920; reprinted Chelsea, 1986.
2. G. G. Saccheri, Euclid Vindicated from Every Blemish, ed. V. De Risi, transl. G. B. Halsted and L. Allegri, Birkhäuser (Springer), 2014.
3. R. Bonola, Non-Euclidean Geometry: A Critical and Historical Study of its Development, 1912; Dover reprint. Italian original La geometria non-euclidea, 1906.
4. Saccheri quadrilateral, Wikipedia.
5. A. Papadopoulos and G. Théret, Hyperbolic geometry in the work of Johann Heinrich Lambert, arXiv:1503.01865.
6. M. J. Greenberg, Euclidean and Non-Euclidean Geometries, 4th ed., W. H. Freeman, 2008.
7. J. Gray, Ideas of Space: Euclidean, Non-Euclidean and Relativistic, 2nd ed., Oxford, 1989.
8. B. A. Rosenfeld, A History of Non-Euclidean Geometry, Springer, 1988.