Michel Chasles

How to use this course

Michel Chasles belongs to the generation that turned the projective ideas of Jean-Victor Poncelet into a systematic geometry. He did it as a historian as much as a mathematician: his Aperçu historique of 1837 explained what modern geometry was by telling where its methods came from. His work introduced terms that are still standard, gave the first clean statement of the screw theorem for rigid motions, and began enumerative geometry. Late in life he was the victim of one of the largest forgery frauds in the history of science.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows his career: the Brussels memoir on duality (1829–1837), rigid motion (1831), attraction (from 1837), the Traité de géométrie supérieure (1852), the Porisms (1860) and the method of characteristics (1864).

Timeline

Épernon, Chartres and the Polytechnique, 1793–1815

Michel Chasles was born on 15 November 1793 at Épernon, a small town in the Eure-et-Loir, in the year of the Terror. The family was comfortable and Catholic. His father, Charles-Henri Chasles, was a timber merchant who later became president of the chamber of commerce at Chartres. According to his MacTutor biographers he was registered with the Republican name Floréal, and a court order changed his given name to Michel shortly after his sixteenth birthday.

He attended the Lycée Impérial and entered the École polytechnique in 1812. There he was taught by Siméon Denis Poisson and absorbed the descriptive geometry of Gaspard Monge, whose school shaped the whole French revival of pure geometry. In early 1814 the allied armies approached Paris and the polytechnicians were mobilised to defend the city. After Napoleon’s abdication on 6 April Chasles returned to his studies.

On graduating he was offered a place in the corps of engineers. He gave it up in favour of a classmate whose family needed the income, an act often cited as typical of his generosity.

The years in Chartres, 1815–1837

At his father’s insistence Chasles joined a stockbroking firm in Paris. He did badly at it and went home to Chartres. He had private means, and for about twenty years he lived there as an independent scholar, reading the history of mathematics and working on geometry. Like Monge’s other heirs he published short notes in Joseph Diez Gergonne’s Annales de mathématiques and in the Correspondance sur l’École polytechnique.

This independent life shaped his work in two ways. First, he became a collector of books and manuscripts, a passion that would later ruin his reputation. Second, he approached geometry historically. He read the Greeks, Desargues and Pascal closely, and he believed that the modern methods of Poncelet and his contemporaries continued theirs. A short note of 1831, on two similar bodies placed arbitrarily in space, already contained one of his lasting results: every displacement of a rigid body is a screw motion (Lesson 06).

The Aperçu historique, 1829–1837

In 1829 the Royal Academy of Brussels set a prize question asking for a philosophical examination of the different methods of modern geometry, in particular the method of reciprocal polars. Poncelet and Gergonne had argued bitterly over the origin of the principle of duality. Chasles’s answer was crowned in 1830. It argued that duality and homography were two instances of a general theory of transformations that preserve the cross-ratio, which he called the anharmonic ratio.

The memoir was published by the Brussels Academy in 1837 together with a long historical introduction, the Aperçu historique sur l’origine et le développement des méthodes en géométrie. The Aperçu traces geometry from the Greeks to his own day in five epochs and adds extensive notes, among them treatments of the cross-ratio and of involution. It made his name. Ludwig Adolph Sohncke translated it into German in 1839, and it was reprinted in Paris in 1875 and 1888. Its weakness was that Chasles read no German, so the work of August Ferdinand Möbius, Jakob Steiner and Julius Plücker was underused. Möbius had introduced the cross-ratio in his Barycentrische Calcul of 1827.

The book won Chasles a corresponding membership of the Académie des sciences in 1839 and, in 1841, a professorship at the École polytechnique. Lessons 04 and 05 present its mathematics.

Duality and homography

For the general reader. In the projective plane, two points determine a line and two lines determine a point. Swap the words “point” and “line” in any true theorem, and you get another true theorem. Chasles explained this principle of duality as one case of a larger idea: geometry studies the properties that survive certain transformations of the plane or of space.

Two kinds of transformation

Points of the real projective plane are nonzero vectors \(x\in\mathbb R^3\) up to scale, and lines are nonzero covectors \(u\) up to scale, with incidence \(u\cdot x=0\). Let \(A\) be an invertible \(3\times3\) matrix.

• A homography (collineation) sends points to points by \(x\mapsto Ax\). It sends lines to lines by \(u\mapsto A^{-\top}u\) and preserves incidence.
• A correlation sends points to lines by \(x\mapsto Ax\) (read as a line). It sends lines to points, collinear points to concurrent lines, and preserves incidence in the reversed sense.

If \(A\) is symmetric, the correlation is a polarity: the point \(x\) goes to its polar line \(Ax\), and applying it twice returns to the start. With \(Q(x)=x^\top Ax\), the polar of a point \(p\) with respect to the conic \(Q=0\) is the line \(\{x: p^\top Ax=0\}\). If \(p\) lies on the conic, its polar is the tangent at \(p\).

Proof. \(q\) is on the polar of \(p\) when \(p^\top Aq=0\), and \(p\) is on the polar of \(q\) when \(q^\top Ap=0\). These are equal since \(A^\top=A\).

Gergonne justified duality by the symmetry of the axioms; Poncelet by reciprocal polars. Chasles’s contribution was to put both inside a single theory of transformations, homographic and correlative, and to show that what they preserve is the cross-ratio. In space the same holds for points and planes, and Chasles studied the correlations of space in detail.

Cross-ratio and involution

For the general reader. Projection distorts lengths and ratios of lengths. One combination of four points on a line survives every projection: the cross-ratio. Chasles built his Traité de géométrie supérieure (1852) on it, together with the related idea of an involution, a way of pairing up points on a line that is unchanged by projection.

The cross-ratio

For four distinct points with coordinates \(a,b,c,d\) on a line, define

extended to \(\infty\) by continuity. Chasles called it the rapport anharmonique. When \((a,b;c,d)=-1\) the four points are harmonic.

Proof sketch. If \(y_i=(\alpha x_i+\beta)/(\gamma x_i+\delta)\), then \(y_i-y_j=(\alpha\delta-\beta\gamma)(x_i-x_j)/((\gamma x_i+\delta)(\gamma x_j+\delta))\). In the cross-ratio the determinant cancels and each denominator factor appears once above and once below. For the converse, the map \(x\mapsto(a,b;x,d)\) is itself fractional linear and sends \(a,b,d\) to \(0,\infty,1\), so preserving cross-ratios forces the map to be a composition of such maps.

Chasles called two ranges of points (or pencils of lines) homographic when they are related by such a map. A central projection from one line to another is homographic, so the cross-ratio of four collinear points equals that of the four lines joining them to any external point.

For a circle the proof is immediate: by the inscribed angle theorem, the angles between the lines \(PA,\dots,PD\) do not depend on \(P\), and the cross-ratio of four concurrent lines is a function of their angles, \(\dfrac{\sin\angle(\ell_1,\ell_3)\,\sin\angle(\ell_2,\ell_4)}{\sin\angle(\ell_2,\ell_3)\,\sin\angle(\ell_1,\ell_4)}\). A homography sends any conic to a circle.

Involutions

An involution of a line is a homography \(\sigma\ne\mathrm{id}\) with \(\sigma^2=\mathrm{id}\). Equivalently, it pairs points by a symmetric relation

An involution is determined by two pairs, and it has either two fixed points or none (over \(\mathbb R\)); when there are two fixed points \(e,f\), every pair \(x,x'\) is harmonic with them: \((e,f;x,x')=-1\). Girard Desargues introduced the notion in 1639 in his work on conics; Chasles developed it as a working tool.

Proof: write the pencil as \(C_0+\lambda C_1\); restricted to the line, \(C_0\) and \(C_1\) are quadratic forms \(q_0(x),q_1(x)\), and \(\lambda\) determines the pair of roots of \(q_0+\lambda q_1\). Both roots \(x,x'\) satisfy \(q_0(x)q_1(x')-q_0(x')q_1(x)=0\) after dividing by \(x-x'\), which is a symmetric relation of the form above.

Rigid motions are screws

For the general reader. Pick up a book and put it down somewhere else in any orientation. However you moved it, the same final position can be reached by a single screw motion: turning about one fixed line while sliding along it. Giulio Mozzi found this in 1763; Chasles rediscovered it in 1831, and the result is usually called Chasles’s theorem.

Proof

Let \(u\) be a unit vector with \(Ru=u\), and split \(t=t_\parallel+t_\perp\) with \(t_\parallel=(t\cdot u)u\). On the plane \(u^\perp\), \(R\) is a rotation by an angle \(\theta\ne0\), so \(I-R\) is invertible there. Choose \(p\in u^\perp\) with \((I-R)p=t_\perp\). Then

since \(R(x-p)+p+t_\parallel=Rx+(I-R)p+t_\parallel=Rx+t\). This is rotation about the line \(\ell=p+\mathbb R u\), followed by translation along it. The axis is unique: it is the set of points moved by \(f\) the least distance, namely \(|t\cdot u|\).

The ratio of translation to rotation angle is the pitch. Pure rotations have pitch 0; translations are screws of infinite pitch. Chasles’s note in Férussac’s Bulletin gave the theorem in the language of two similar bodies. Louis Poinsot’s theory of couples (1803) is the static counterpart, and Robert Stawell Ball built the full screw theory of mechanisms from both in 1876.

Confocal quadrics and the attraction of ellipsoids

For the general reader. How strongly does a solid ellipsoid, like a flattened planet, pull on a nearby object? Isaac Newton, Colin Maclaurin, Pierre-Simon Laplace and James Ivory all worked on the question. From 1837 Chasles gave purely geometric solutions organised around confocal surfaces, families of ellipsoids and hyperboloids sharing the same foci.

Confocal quadrics

Fix \(a>b>c>0\). The confocal family is

For \(\lambda>-c^2\) it is an ellipsoid; for \(-c^2>\lambda>-b^2\) a hyperboloid of one sheet; for \(-b^2>\lambda>-a^2\) a hyperboloid of two sheets.

Clearing denominators gives a cubic equation in \(\lambda\) whose sign changes at \(-a^2,-b^2,-c^2\) and at \(+\infty\) place one root in each interval. Orthogonality is proved in Exercise 7.2 for the plane case; the argument in space is identical.

Attraction

A homoeoid is the shell between two similar, similarly placed concentric ellipsoids, \(E\) and \((1+\varepsilon)E\). Take the density uniform and the force Newtonian.

Chasles derived such results with geometric arguments. He decomposed a solid ellipsoid into thin homoeoids, and for each one he compared the field outside with that of a thin shell on a confocal ellipsoid through the attracted point. His 1845 general theory of attraction was largely anticipated by George Green and Carl Friedrich Gauss, whose potential theory he did not know.

Characteristics and the number 3264

For the general reader. How many conics touch five given conics? Steiner said 7776 in 1848. In 1864 Chasles showed that the right answer is 3264, by a method that counts conics through two simple numbers, the number passing through a point and the number touching a line. It launched enumerative geometry, which Hermann Schubert and Hieronymus Zeuthen carried much further.

Characteristics

Conics in the plane form a 5-dimensional family. Imposing four conditions leaves a one-parameter system. Chasles defined its two characteristics: \(\mu\), the number of conics of the system through a general point, and \(\nu\), the number tangent to a general line.

The formula fails for systems containing too many degenerate conics, a point raised by Georges Henri Halphen, who gave the needed corrections. With the intersection theory of the twentieth century, Chasles’s formula became the statement that a condition is a class \(\alpha P+\beta L\) in the cohomology of the space of complete conics, where \(P\) and \(L\) are the classes of “through a point” and “tangent to a line”.

Correspondence principle

Proof: the correspondence is a polynomial \(F(x,y)\) of degree \(\alpha\) in \(x\) and \(\beta\) in \(y\) (in homogeneous terms). The united points are the roots of \(F(x,x)\), a binary form of degree \(\alpha+\beta\). Arthur Cayley and Alexander von Brill extended the principle to curves of positive genus.

Counting with characteristics

Write \(N(p^{5-k}\ell^{k})\) for the number of conics through \(5-k\) general points and tangent to \(k\) general lines. These are \(1,2,4,4,2,1\) for \(k=0,\dots,5\); the symmetry comes from duality (Lesson 04). Chasles’s formula lets one treat “tangent to a conic” as the formal combination \(2p+2\ell\) and expand.

Signed segments and the Porisms

For the general reader. French students learn “Chasles’s relation” \(\overline{AB}+\overline{BC}=\overline{AC}\) for directed lengths, which made sign rules automatic in geometry. Chasles also tried to recover a lost book of Euclid, the Porisms, known only from a summary by Pappus. He believed it contained the cross-ratio in disguise.

Signed segments

On an oriented line, let \(\overline{AB}=b-a\) where \(a,b\) are coordinates. Then \(\overline{AB}=-\overline{BA}\) and, for any three points,

Chasles used signed segments systematically in the Traité de géométrie supérieure, so that a single formula covers every arrangement of points. The cross-ratio of Lesson 05 is \(\dfrac{\overline{AC}\cdot\overline{BD}}{\overline{BC}\cdot\overline{AD}}\).

Projective ranges

The first claim follows by intersecting the line \(OX\) with \(h\): solving a linear system gives the coordinate of the image as a ratio of linear functions of \(x\). The last claim is Poncelet’s; with it, “homographic” and “obtained by projections” mean the same.

Euclid’s Porisms

Pappus, in Book VII of his Collection, summarised Euclid’s three lost books of Porisms, propositions asserting that something can be found satisfying given conditions. Robert Simson had attempted a reconstruction in the eighteenth century. Chasles’s Les trois livres de Porismes d’Euclide (1860) argued that many porisms are statements about homographic ranges and that the Greeks knew the cross-ratio in substance. Historians did not accept his broader reading. Many porisms concern lines through fixed points that meet on a fixed line; Exercise 9.2 shows the projective core of such statements.

Nine points on a cubic

For the general reader. Two cubic curves in the plane usually cross in nine points. Those nine points are not independent: any cubic through eight of them must pass through the ninth. This observation, usually attributed to Chasles, contains Pascal’s theorem on conics and explains why the chord-and-tangent addition of points on a cubic is associative.

Counting conditions

A plane curve of degree \(d\) is given by a homogeneous polynomial with \(\binom{d+2}{2}\) coefficients, so curves of degree \(d\) form a projective space of dimension \(\binom{d+2}{2}-1=d(d+3)/2\). Passing through a point is one linear condition. Conics form a \(\mathbb P^5\) and cubics a \(\mathbb P^9\): five general points determine a conic, nine general points a cubic.

Proof sketch

Let \(F_1,F_2\) be equations of \(C_1,C_2\). It suffices to show that the cubics through \(p_1,\dots,p_8\) form exactly the pencil \(\lambda F_1+\mu F_2\), since every member of the pencil vanishes at \(p_9\). Those cubics form a linear space of dimension at least \(10-8=2\); we show it is exactly 2, that is, the eight points impose independent conditions on cubics. A classical lemma states that eight distinct points fail to impose independent conditions on cubics only if four of them lie on a line or seven lie on a conic. In the first case the line meets \(C_1\) in four points, so it is a component of \(C_1\), and similarly of \(C_2\), contradicting the hypothesis. The second case is excluded the same way: by Bézout a conic meeting a cubic in seven points shares a component with it, and that component (the conic or one of its lines) would be common to \(C_1\) and \(C_2\).

Arthur Cayley and later Isaak Bacharach (1886) extended the statement to curves of degrees \(d\) and \(e\), and the modern Cayley–Bacharach theorems belong to the theory of complete intersections.

Pascal from Chasles

Let \(A_1,\dots,A_6\) be distinct points on a non-degenerate conic \(K\). Let \(L_1=A_1A_2\cup A_3A_4\cup A_5A_6\) and \(L_2=A_2A_3\cup A_4A_5\cup A_6A_1\), two cubics made of three lines each. They meet in the six points \(A_i\) and in the three points \(X=A_1A_2\cap A_4A_5\), \(Y=A_2A_3\cap A_5A_6\), \(Z=A_3A_4\cap A_6A_1\). The cubic \(K\cup XY\) passes through the six \(A_i\) and through \(X,Y\): eight of the nine points. By the theorem it passes through \(Z\). A line meets \(K\) in at most two points, and the line \(A_3A_4\) already meets it at \(A_3,A_4\), so \(Z\notin K\). Hence \(Z\) lies on the line \(XY\). So \(X,Y,Z\) are collinear.

The Sorbonne and the Académie, 1841–1880

In 1841 Chasles became professor at the École polytechnique, where he taught geodesy, mechanics and astronomy until he resigned in 1851. In 1846 the Faculty of Sciences in Paris created a chair of higher geometry for him, and he held it for the rest of his life. His lectures there grew into the Traité de géométrie supérieure (1852) and the Traité des sections coniques (1865), whose planned second part never appeared. Through these books his methods entered the teaching of geometry in France, Germany and Britain. His doctoral students included Gaston Darboux, and the American Hubert Anson Newton studied with him in Paris.

He was elected a full member of the Académie des sciences in 1851. He was elected a Foreign Member of the Royal Society in 1854 and received its Copley Medal in 1865. In 1864 he became a Foreign Honorary Member of the American Academy of Arts and Sciences, and in 1867 the first foreign member of the young London Mathematical Society. He also belonged to the academies of Brussels, Copenhagen, Naples, Stockholm and St Petersburg. When the Société mathématique de France was founded in 1872, he was chosen its first president, for 1873.

Chasles also wrote history. His Histoire de l’arithmétique (1843) argued, on the evidence of an abacus described by Boethius and Gerbert, for a Pythagorean origin of positional numerals; scholars did not accept it. His Rapport sur les progrès de la géométrie (1870), commissioned by the minister of public instruction in 1867, remains a valuable survey of French geometry from 1800 to 1866, though thin on foreign work. He never married; outside mathematics his interests seem to have been the Académie and charitable work.

The Vrain-Lucas affair and last years

In 1861 a man named Denis Vrain-Lucas called on Chasles. Born in 1816 near Châteaudun and trained as a lawyer’s clerk, he had been forging documents since about 1854. He presented himself as agent for an aristocratic family that had to sell an autograph collection, and he offered letters of Pascal, Newton and Robert Boyle. Chasles, an avid collector, bought them, and kept buying for eight years.

Vrain-Lucas wrote the documents himself, on old paper aged with smoke, in an imitation of old French. Besides scientists, his “authors” included Alexander the Great writing to Aristotle, Cleopatra writing to Julius Caesar, Mary Magdalene, Lazarus, Joan of Arc and Rabelais, all in French. In total Chasles acquired about 27,000 pieces and paid, by his own testimony, some 140,000 francs.

In 1867 Chasles presented to the Académie des sciences letters in which Pascal appeared to communicate the law of universal gravitation to Boyle, and to the young Newton, decades before the Principia. Critics pointed out at once that the handwriting did not match genuine Pascal autographs and that the letters contained anachronisms. One alleged exchange between Pascal and Galileo was dated after Galileo had gone blind. In Britain, David Brewster, Augustus De Morgan, Robert Grant and Thomas Archer Hirst joined the debate in defence of Newton. Each time an objection was raised, Vrain-Lucas supplied new “documents” to answer it, and Chasles presented them. The controversy ran until 1869.

In 1869 Chasles finally had the dealer watched by the police and arrested. Experts examined the collection and Vrain-Lucas confessed. At his trial in February 1870 before the correctional tribunal in Paris, the prosecutor read examples aloud, and Chasles had to testify that he had bought them. Vrain-Lucas was sentenced to two years in prison, a fine of 500 francs and costs. Chasles recovered nothing. Most of the forgeries were destroyed; a few survive in a bound volume at the Bibliothèque nationale de France.

The affair damaged Chasles’s reputation as a historian but not his mathematics. He kept publishing on enumerative geometry and on curves into the 1870s, and his biographer in the Dictionary of Scientific Biography notes that his original work continued to his last years. He died in Paris on 18 December 1880, aged 87. In 1889 his name was among the 72 inscribed on the Eiffel Tower.

Legacy

• Projective geometry: cross-ratio, homography, correlation and involution, as the working vocabulary of synthetic projective geometry.
• Mechanics: the screw theorem, foundation of screw theory, robot kinematics and rigid-body dynamics.
• Enumerative geometry: characteristics and the correspondence principle, developed by Schubert and Zeuthen, made rigorous in modern intersection theory, and still the standard illustration via 3264.
• History of mathematics: the Aperçu historique and the Rapport of 1870 remain primary sources for nineteenth-century geometry.

Connections

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

Teachers and advisors

• Siméon Denis Poisson

Students

• Gaston Darboux
• Hubert Anson Newton

Influenced by

• Gaspard Monge
• Girard Desargues
• Jean-Victor Poncelet
• Lazare Carnot
• Pappus of Alexandria

Influenced

• Georges Henri Halphen
• Hermann Schubert
• Hieronymus Georg Zeuthen

Colleagues and collaborators

• Arthur Cayley
• August Ferdinand Möbius
• Jakob Steiner
• Ernest de Jonquières
• Thomas Archer Hirst

Rivals

• Augustus De Morgan
• David Brewster
• Urbain Le Verrier

Friends

• Sadi Carnot

Also mentioned in this life

• Aristotle
• Robert Boyle
• Euclid
• Leonhard Euler
• Galileo Galilei
• Carl Friedrich Gauss
• George Green
• Pierre-Simon Laplace
• Colin Maclaurin
• Isaac Newton
• Blaise Pascal
• Julius Plücker

Fields

• projective geometry 7 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Michel Chasles, MacTutor History of Mathematics.
2. E. Koppelman, Chasles, Michel, Dictionary of Scientific Biography; PDF at MacTutor.
3. H. Strick, Michel Chasles (stamp biography), MacTutor.
4. Michel Chasles and Denis Vrain-Lucas, Wikipedia; French article on the trial and the surviving documents.
5. R. Higgitt, ‘Newton dépossédé!’ The British response to the Pascal forgeries of 1867, British Journal for the History of Science 36 (2003), 437–453.
6. Linda Hall Library, Scientist of the Day: Michel Chasles.

Mathematics

1. M. Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie, Brussels, 1837; 2nd ed. Paris, 1875.
2. M. Chasles, Traité de géométrie supérieure, Paris, 1852; 2nd ed. 1880.
3. M. Chasles, Les trois livres de Porismes d’Euclide, Paris, 1860.
4. M. Chasles, Détermination du nombre des sections coniques qui doivent toucher cinq courbes d’ordre quelconque, Comptes rendus 58 (1864).
5. M. Chasles, Traité des sections coniques, Paris, 1865.
6. M. Chasles, Rapport sur les progrès de la géométrie, Paris, 1870.
7. N. Michel, The values of simplicity and generality in Chasles’s geometrical theory of attraction, Journal for General Philosophy of Science 51 (2020).
8. Screw theory and Steiner’s conic problem, Wikipedia.
9. D. Eisenbud and J. Harris, 3264 and All That, Cambridge, 2016.