1

PASCAL’S MYSTIC HEXAGRAM, AND A CONJECTURAL RESTORATION OF HIS LOST

TREATISE ON CONIC SECTIONS.

ANDREA DEL CENTINA*

ABSTRACT. Through an in-depth analysis of the notes that Leibniz made while reading Pascal’s

manuscript treatise on conic sections, we aim to show the real extension of what he called

“hexagrammum mysticum”, and to highlight the main results he achieved in this field, as well as

proposing plausible proofs of them according to the methods he seems to have developed.

1. INTRODUCTION

Geometry, in particular the theory of conic sections, played an important role in the scientific life of

Blaise Pascal, but very little remains of his writings on the subject: the Essay pour les coniques

(Pascal 1640), that he wrote at the age of sixteen under the influence of Desargues, and some extracts

from his never published treatise on conic sections, the Conicorum opus completum, on which Pascal

worked for a large part of his short life, manuscript of which unfortunately went missing a few

decades after his death.

Pascal, himself, gave some indication of the content of his treatise on conic sections in his address

to the Celeberrima Matheseos Academia Parisiensis in 1654,1 and other pieces of information come

from Leibniz, who, in 1676, had the opportunity to read the manuscript of Pascal’s treatise. Besides

a new and more detailed description of its content – which he enclosed in the letter addressed to Florin

Périer on the occasion of returning the manuscript to Pascal’s heir – Leibniz also left a copy of the

first part of the treatise, the Generatio conisectionum, and some explanatory notes, which turned out

to be very useful for an understanding of the real nature of the main theorem on which Pascal

developed the Conicorum opus completum, the “hexagrammum mysticum” (Mystic hexagram).

The Essay pour les coniques, the address to the Academia Parisiensis, and a copy of Leibniz’s

letter to Périer (but with no indication of the origin), were published in the Oeuvres de Pascal, (Pascal

1779, IV, 1-7, 408-411; V, 459-462), edited in five volumes by Abbot Charles Bossut.2 Thanks to

this work, some of Pascal’s forgotten geometrical achievements were brought to light and captured

the attention of French mathematicians.

Charles-Julien Brianchon was most likely alluding to the fourth proposition in the Essay pour les

coniques, when he wrote that Pascal “had preserved for us a fragment of Desargues” (Brianchon

1817). In the bibliography of that work, Brianchon also referred to “the third treatise of his [Pascal]

manuscript on conic sections bearing the inscription De quatuor tangentibus…” (see later on), clearly

referring to what Bossut had published in the fifth volume of the Oeuvres de Pascal.

* cen@unife.it; Dipartimento di Matematica e Informatica, Università di Ferrara, via Machiavelli 30, 44100 Ferrara, Italy. 1 The society of scientists founded by Marin Mersenne, also known as “Mersenne’s Academy”, not to be confused with

the Parisian Academy of Sciences founded in 1666. 2 Bossut tried in vain to retrieve the manuscript of the Conicorum opus completum, when he was preparing the edition of

Pascal’s works.

2

Following on from Brianchon, on the basis of the available documentation in (Bossut 1779), both

Jean-Victor Poncelet in the historical introduction to his treatise on the projective properties of figures

(Poncelet 1822, xvii-xxix), and Michel Chasles in his historical essay (Chasles 1837, 69-74), not only

underlined the importance of the mystic hexagram, but also provided an insight into what the

Conicorum opus completum was. However, the theorem commonly ascribed to Pascal was the so

called “hexagon theorem”: The pairs of opposite sides of any hexagon inscribed into a conic section

meet in three collinear points (Fig.1). This theorem expresses only one aspect of the mystic hexagram.

Fig. 1 Illustrates Pascal’s hexagon theorem. The pairs of opposite sides AB, DE; BC, EF; CD, AF,

of any hexagon ABCDEF inscribed into a conic section, meet in collinear points L, M, N.

It was not until some-time later that Carl Immanuel Gerhardt found, among Leibniz’s manuscripts

preserved in Hannover, a draft of the letter that Leibniz had addressed to Périer, the copy he had made

of the Generatio conisectionum, and the notes he had taken during the reading of Pascal’s treatise.3

Gerhardt published the Generatio conisectionum, a large part of the letter to Périer, and some extracts

from the notes translated into German (Gerhardt 1892).4

However, a critical edition of Leibniz’s letter to Périer was only published in 1963, by Jean

Mesnard and René Taton, who also considered the phrases that Leibniz had erased from the draft

when writing out the letter, to be of highly important documentary value (Mesnard and Taton 1963).

Although noticeable contributions to the geometrical work of Pascal appeared in the past century,

see (Taton 1955, 1962), (Costabel 1962), (Itard 1962), not all features of Pascal’s mystic hexagon

have been clarified, and, above all, the use that Pascal made of it in developing his treatise has not

been fully explained. In this paper, through an in-depth analysis of what Pascal meant for

“hexagrammum mysticum”, we aim to show the real extension of this concept, and we will attempt

to highlight the main results that Pascal achieved, as well as proposing plausible proofs according to

his method. We believe this will enable us to fully perceive the depth, breadth, and novelty of Pascal’s

Conicorum opus completum, the loss of which we cannot but regret.

A short biographical account will allow to place Pascal's geometrical achievements over time.

Blaise Pascal was born in Clermont (now Clermont-Ferrand), in France’s Auvergne region, the

19th of June 1623, third of four children of Étienne Pascal, a nobleman, judge and distinguished

amateur mathematician, and Antoinette Bégon, who died in 1626 a few months after giving birth to

3 These documents are presently held at the G.W. Leibniz Bibliothek Niedersächsische Landesbibliothek, Hannover,

Leibniz Handschriften XXXV, XV, I. They have been published in various editions of Pascal’s works, see (Taton 1962). 4 The copy Leibniz made of Generatio coni sectionsum has been also printed translated into French in (Pascal 1954,

1382—1387). We stress that the notes and the Generatio conisectionum were not available either to Poncelet or to

Chasles.

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her daughter Jaqueline.5 Five years after his wife’s death, Étienne Pascal left Clermont for Paris,

where the returns of his investment afforded him a comfortable life. In Paris, Étienne Pascal entered

the Academia found by Mersenne, and at the age of fourteen Blaise started to accompany his father

to the weekly meetings. Here, besides Mersenne, the young Pascal met some of the most outstanding

Parisian scientists, such as Gassendi, Roberval, Carcavi, Mydorge, and Desargues from whom he

learned a great deal.

When Desargues' Brouillon project (Desargues 1639) appeared in print, Blaise quickly mastered

its content, and the following year he presented his brief Essay pour les coniques to Mersenne’s

Academy. It is very likely that Desargues encouraged the young man to publish it. In his essay, Blaise

Pascal briefly outlined the programme he intended to develop toward a new and complete treatise on

conic sections, to which he referred to as Conicorum opus completum.

In 1639, Étienne Pascal was appointed the King's tax collector for Upper Normandy, thus the

Pascal family left Paris and settled in Rouen. Here Blaise continued to work on his planned treatise

on conic sections. Occasionally he went back to Paris with his father, and during his short stays in the

capital he was able to present his discoveries to Mersenne’s academy. In 1642, Desargues was eager

to see the proof of an important theorem, that he called “la Pascale”,6 by which, according to him,

“the four books of Apollonius are an immediate consequence” (Curabelle 1644, 70-71). Two years

later, in the preface of his Cogitata physico-mathemaica (1644), Mersenne drew attention to “a single

very general proposition by which Pascal could establish four hundred corollaries covering every

field of Apollonius's treatise”. On 17th of March 1648, Mersenne announced to Constantijn Huygens

that Pascal had finished writing an important treatise on conic sections by saying:

If your Archimedes will come with you,7 we will show him one of the finest geometrical treatises he has

ever seen, which has been completed by the young Pascal. It is the solution to the locus of Pappus’ 3 and 4

lines, that is here said not to have been solved in all its generality by Descartes. It took red lines, green, and

black, etc., to distinguish the variety of considerations...8

This suggests that Mersenne had to hand at least part of Pascal’s manuscript treatise, most likely

the sixth part that, it would seem, Pascal composed in the form of an autonomous treatise to be

circulated among his friends (see section 5.2). Pascal worked on the Conicorum opus completum

again in 1653 and in 1654, the year in which he presented a summary of his works to Mersenne’s

academy.

In 1659 Pascal fell seriously ill, and when in 1661 his sister Jacqueline died his emotional

condition greatly worsened. Pascal’ s lifestyle became more and more ascetic as he came to believe

that suffering and sickness was a natural state for Christians. On the 18th of August 1662, Pascal went

into convulsions, and the following morning he died.

5 The first source of information on the life of Blaise Pascal is the short biography by his sister Gilberte, see (Périer 1684).

More extensive biographies are (Davidson 1983), (Adamson 1995).

6 Likely the mystic hexagram, see later on. 7 Mersenne was undoubtedly referring to Christiaan Huygens, son of Constantijn. 8 “Si votre Archimède vient avec vous, nous lui ferons voir un des plus beaux traités de géométrie qu'il ait jamais vu, qui

vient d'être achevé par le jeune Pascal. C’est la solution du lieu de Pappus ad 3 et 4 lineas qu'on prétend ici n'avoir pas

été résolu par M. Descartes en toute son étendue. Il a fallu des lignes rouges, vertes et noires, etc., pour distinguer la

grande multitude des considérations…”, (Huygens 1889), also quoted in (Taton 1962, 226).

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2. CONIC SECTIONS IN PASCAL’S TIME, AND AN OVERVIEW OF DESARGUES’ RESULTS

Pascal’s project for a complete treatise on conic sections required a good, if not excellent, knowledge

of what was already known on the subject, thus we can safely argue that he was familiar with the first

four books of Apollonius’ Conics − the only ones known at that time −, and of Pappus’ Collection;

both in Commandino’s edition (Apollonius 1566), (Pappus 1588). He may also have read Apollonius

through Apollonii Pergaei conicorum et Sereni de sectione coni et cylindri libri, enclosed in

Mersenne’s work Synopsi mathematica (1626), a book that was almost certainly available to Pascal

when he started to attend the meeting of Mersenne’s Academy in the late 1630s.

The Matheseos Academia Parisiensis had been founded by the minim friar, philosopher, musician

and mathematician, Marin Mersenne in 1635. Claude Mydorge, Gilles Personne de Roberval, Pierre

de Carcavi, and Girard Desargues among others participated in the weekly meetings. The academy

was also animated by several eminent correspondents of Mersenne such as, René Descartes, Nicolas-

Claude Fabri de Peiresc, Pierre Gassendi, Pierre de Fermat, and other French, Italian, English and

Flemish scholars.

The 1630s were important years for French mathematics. In 1636 Mersenne published his most

influential work, the Harmonie universelle (Mersenne 1636). In it, Mersenne discussed the optical

and acoustical properties of conic sections, and, in this context, he resorted to Kepler’s plane system

of conic sections, illustrated in the section De coni sectionibus of (Kepler 1604), to explain “certain

analogies common to all conic sections”.

In this work Johannes Kepler introduced two concepts that profoundly changed geometry: that of

“analogy”, and archetype of Poncelet’s principle of continuity; and that of “point at infinity” of a

straight line (Field 1987). With the former, Kepler introduced a sort of “kinematic vision” in the study

of conic sections, by varying one curve into another (possibly degenerate), in a continuous way. Then,

by stressing the idea of conic sections as projective images of the circle (Del Centina 2016b), Kepler

demonstrated this by constructing a continuous plane system of conic sections in which all types of

such curves appear. He put the parabola in the “middle”, between ellipses and hyperbolas, and

introduced the second focus of the parabola – which he called the blind focus –, that appears when

one focus of an ellipse, or of a hyperbola, recedes infinitely far away from the other, that is at

“infinity”, along the major axis. In fact, it was the conception of the second focus of the parabola that

led him to the “invention” of the point at infinity of a straight line. By this concept, parallel straight

lines become convergent in their point at infinity.

In 1637, Mydorge published the last two books of Prodromi catoptricorum et dioptricorum sive

conicarum operis …libri tertius et quartus priores (the first two books of the work had already

appeared in 1631), in which the content of Apollonius’ Conics was presented under a new light, and

with simplified proofs. Mydorge’s work was well received, and was reprinted in a single volume in

1639.

In the fourth book, Mydorge re-brought to light what is known today as the “chords theorem” for

conic sections, which amounts to propositions 16-23 of the third book of Apollonius’ Conics. This

theorem can be stated as follows: If AB and DE are two chords of a conic section which meet in the

point C, then the ratio AC×CB : DC×CE does not change if the chords move parallel to themselves.

Mydorge used it in order to determine the conic section passing through five given points. If A, B,

D, E, A', are five points in a plane (no three of which are collinear), and A'B' is drawn parallel to AB,

the chords theorem clearly expresses the condition for the point B' to belong to the conic section

passing through A, B, A', D, E, (Mydorge 1639, 297). In this way, Mydorge unearthed the long

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forgotten problem for the ellipse discussed in the eighth book of Pappus’ Collection,9 but which

Apollonius had not treated in the Conics: to determine, given five points, the section of a circular

column of unknown diameter. By using the chords theorem, Pappus provided the construction for

points of the sought-for ellipse, and showed how to determine a diameter and the relative latus rectum

(erect side). Mydorge in (1639, 298-300) extended these results to any conic section. Apollonius, in the main introduction to the Conics, presenting it to Eudemus,10 alluded to the

chords theorem as the key for the synthesis of the locus of the three and four lines. The problem to

which Apollonius referred – which dated back to Euclid and was later known as “the Pappus three

and four lines problems” as it was recalled in his Collection −, concerned the determination of the

locus of points whose distances 𝑑1, 𝑑2, 𝑑3, 𝑑4 (each taken under an arbitrary but fixed direction) from

three, or four, given coplanar straight lines, satisfy the condition 𝑑1 𝑑2 = 𝑘𝑑32, or 𝑑1 𝑑2 = 𝑘𝑑3𝑑4, k

being a fixed constant. It is clear that the first problem is a particular case of the second, if one makes

two lines coincide. According to John J. Milne (1927, 108), Apollonius worked out for himself a

complete solution to these problems, but, in the Conics, he deliberately gave only half the solution

(namely the converse) of the three lines problem as a separate proposition, leaving the rest as a

challenge to the reader.

In 1637, La Géométrie also appeared, the work that Descartes appended to his Discours de la

méthode. In this epoch-making work, Descartes presented a solution to the Pappus problem of three

and four lines (Descartes 1637, 324-334). If x, y are the coordinates of a point P in the plane of the

given lines, he showed that the distances (in assigned directions) of P from the given lines are linear

functions of the coordinates, so that the relation which has to hold among them is always expressed

by an equation of second degree in x, y; hence the locus sought is a conic section. From the equation,

Descartes derived the axes, the vertices, the parameter, and was also able to construct the curve by

points. (Bos 2001, 313-323). However, his solution was not complete, because, as he recognized, he

failed to treat the case in which the coefficient of y2 is zero. Fifty years later, Isaac Newton provided

a pure geometrical solution to that problem, chiefly by resorting to the chords theorem (Newton 1687,

book I, sect. 5), (Guicciardini 2009).

Desargues played a fundamental role in the development of the geometrical ideas of the young

Pascal, and in the Essay pour les coniques he acknowledged that:

[the fourth proposition] which is due to M. Desargues of Lyon, one of the great spirits of these times, and

one of the most versed in mathematics, and Conics in particular, as his writings on the subject, although

small in number, have demonstrated to the attentive readers, and I confess that the little I have found on

this subject is thanks to his writings, and that I have tried to imitate as far as I could, what he has treated

without making use of the triangle through the axis.11

Pascal was referring to Desargues’ Brouillon project (1639), and to the projective method therein

adopted to treat conic sections. In his work, Desargues also made Kepler’s ideas on points at infinity

and on analogy his own. To understand the genesis of Pascal’s geometrical ideas it is useful to focus

9 See (Pappus 1588, VIII, propositions 13,14). where the problem of finding, given five points, the section of a circular

column of unknown diameter is posed. 10 See (Apollonius 1566, 4r, v), (Heath 1896, lxx-lxxi). 11 “Dont le premier inventeur est M. Desargues Lyonnois, un des grands esprits de ce temps, et des plus verséz aux

Mathématiques, et entr’autres aux Coniques, dont les escripts sur cette matiere, quoi qu’en petit nombre, en ont donné un

ample tésmoignage à ceux qui en auront voulu recevoir l'intelligence : et veux bien advoüer que ie doibs le peu que j'ay

trouvé sur cette matiere à ses escrits, et que j'ay taché d'imiter autant qu'il m’a ésté possible sa methode sur ce subjet,

qu’il a traitté sans se servir du triangle par l’axe”, (Pascal 1640).

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on some of Desargues’ definitions and major achievements, but first of all we may recall a theorem

that Desargues and Pascal used many times in their proofs: the theorem of Menelaus.12

This theorem appeared in the third book of Menelaus’ Spherics, a work which was edited for the

first time in (Maurolico 1558): If AB, AG are two straight lines and BE, GD are other two straight

lines intersecting each other in F, and AB, AG in D, E respectively, then

𝐺𝐸

𝐸𝐴=

𝐺𝐹

𝐷𝐹×

𝐵𝐷

𝐵𝐴·

Similar relations hold true if one exchanges E with A, and F with D; or A with D, and E with F.

We stress that its converse, in the form of the equation

𝐴𝐸

𝐸𝐺×

𝐺𝐹

𝐹𝐷×

𝐷𝐵

𝐵𝐴= 1

expresses the condition for E, F, B, to be collinear.

2.1 Desargues’ Brouillon project

Desargues’ Brouillon project can be divided into four parts. The first, which takes into account the

mutual position of straight lines and planes, is mainly devoted to the introduction of the concept of

points at infinity of a straight line; the second treats the “involution of six points”, the main tool

Desargues introduced in the study of conic sections; the third concerns Menelaus’ theorem, and other

results of plane geometry; the fourth, is finally devoted to the theory of conic sections.

Desargues defined an ordinance of straight lines to be any set of converging, or parallel, straight

lines.13 The straight lines in an ordinance have the same butt (that is the point at finite, or infinite,

distance they have in common). All points at infinite distance in a plane belong to the same straight

line, the line at infinity in that plane, which is common to all planes parallel to that plane.

Six points, or better three pairs of points, say B, H; C, G; D, F, on a straight line are said to be in

involution if the following relation holds true

𝐺𝐵

𝐶𝐵×

𝐺𝐻

𝐶𝐻=

𝐷𝐺

𝐶𝐹×

𝐺𝐹

𝐶𝐷 (2.1)

or any other similar relation obtained from it by permuting the pairs, holds true.

Desargues applied Menelaus’ theorem to show that the involution of six points is preserved under

central projection. Precisely he showed that: Let B, H; C, G; D, F, be six points in involution belonging

to a straight line r, and K be a point not on r; any straight line, s, not passing through K, cuts the

straight lines KB, KH; KD, KF; KC, KG, at six points b, h; d, f; c, g, which are in involution.

Desargues defined cones and cylinders as surfaces generated by a moving straight line. He

considered a circle C and a point P on it, then he took a point V, possibly at infinite distance away.14

The surface described by the straight lines VP (infinitely produced) when P slides on C, is a cone of

base C and vertex V (a cylinder if V is at infinity). By sectioning the cone with a plane he obtained the

various types of conic: a pair of straight lines (possibly coinciding) if the plane passes through the

12 This is also known as the Menelaus−Ptolemy theorem. 13 For the terms used by Desargues we will adopt the English translation provided by J.V. Field and J. J. Gray (1987). 14 The point V can be either outside or inside the plane of the circle.

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vertex, or an ellipse, a parabola, or a hyperbola, according to whether the plane cuts all the lines

generators VP, all but one, all but two. Desargues promoted a kinematical vision of conic sections

similar to that of Kepler, and clearly conceived conic sections as perspective images of the circle.

Fig. 2 Reproduces figure 13 as it appears in La Hire’s manuscript copy of (Desargues 1639), see

also (Taton 1951, 142). The conic section is drawn as a circle. The straight line NG is the

transversal (polar) of butt (pole) F; FT is the transversal (polar) of the ordinance of butt (pole) N;

NF is the transversal (polar) of the ordinance of butt (pole) G.

Desargues introduced the important concept of “transversal”. He considered a conic section Γ (in

its plane), and an ordinance of butt F; that is, as we would say today, a pencil of lines through the

point F (Fig. 2). Then he took a straight line FB intersecting Γ at the points B and C, and denoted by

O the fourth harmonic of F with respect to B, C. He claimed that when FB moves in the ordinance,

the fourth harmonic of F, with respect to the two intersection points with the conic section, moves on

a straight line GHR. Desargues called such a line transversal (“traversale”) of the ordinance. It is clear

that the concept of transversal is equivalent to that (modern) of polar of F with respect to Γ.

Next he proved the following theorem, known as Desargues’ involution theorem (see Fig. 3): Let

B, C, D, E be the vertices of a quadrilateral inscribed in a conic section, whose opposite sides BC,

ED intersect in N; BE, CD intersect in F, and BD, CE in R. Then these pairs of straight lines meet any

straight line l in the plane of the conic section in three pairs of points in involution. Moreover, any

other conic section passing through the points B, C, D, E, intersects l at a pair of points belonging to

the same involution.

The first part asserts that the following relation holds true

𝐼𝑄

𝐼𝑃×

𝐾𝑄

𝐾𝑃=

𝐺𝑄

𝐺𝑃×

𝑄𝐻

𝑃𝐻

Afterwards, Desargues proved that if B, C, D, E, are four points on a conic section, the straight

line NR (see Fig. 3) is the transversal (polar) of the point F with respect to the conic section. He also

proved what we would call today the “dual result”, that is: The transversals of points lying on a

straight line NR, all pass through the same point F. He first proved the result for the case of the circle,

and then extended it by projection to any conic section. This proposition provided him with the way

to draw the tangents to a conic section from any point P. Therefore, we can safely say as it was pointed

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out in (Taton 1951), that Desargues discovered the pole-polar properties with respect to a conic

section.15

Fig. 3 Illustrates Desargues’ involution theorem, and the involution defined by a conic section

on a straight line.

Desargues was also led to the concept of involution defined on a straight line l by a conic section

Γ. To any point A on l (see Fig. 3), Desargues associated the point A' on l in which the straight line

joining the contact points of the tangents to Γ issuing from A meets l. He recognized three kind of

involutions according to whether l does not intersect the conic section, it is tangent to it, or properly

intersects it. In the first case the involution (as a correspondence between the points of l) has no fixed

points (that is points which correspond to themselves), in the second it has one, and in the third it has

two.

2.2 Desargues’ theorem on perspective triangles

In 1648, Abraham Bosse, the engraver and friend to Desargues, published Manière universelle de Mr

Desargues pour pratiquer la perspective (Bosse 1648), to which he appended three geometrical

propositions belonging to Desargues, likely obtained long before. The first contains the famous

theorem on triangles in perspective, or, as we say today, “on homological triangles”. It is divided into

three parts, the first of which is (Fig. 4a): 16 When straight lines HDa, HEb, cED, lga, lfb, HlK, DgK,

EfK, [cab], which either lie in different planes or in the same one, cut one another in any order or

inclination at such points [as those implied in the lettering]; the points c, f, g lie on a straight line cfg.

To prove this, Desargues applied Menelaus’ theorem to show that

𝑐𝐸

𝑐𝐷×

𝑔𝐷

𝑔𝐾×

𝑓𝐾

𝑓𝐸= 1,

which implies that the points c, f, g, are collinear.

The second part is the converse of this proposition, that is (Fig. 4a): If the straight lines abc, HDa,

HEb, HK, DKg, KEf meet one another in any manner and at any angles, in points such as those [are

given], the lines lying either in different planes or in the same one; the lines agl, bfl will always meet

in the butt l on the line HK.

15 See also (Anglade and Briend, 2019). 16 We adopt the English translation in (Field and Gray 1987, 161-169).

9

(a) (b)

Fig. 4 (a) Reproduces the figure for the first geometrical proposition in (Bosse 1648), with the

only exception that the two triangles in perspective with respect to H, abl and DEK, are drawn in

bold. (b) Is a re-drawing of Bosse’s original figure simulating a 3-dimensional view. From this

diagram it clearly appears that the straight line cfg is the intersection of the plane cutting the

triangle DEK on the pyramid of base abl and vertex H, with the plan of the base. This figure

suggests the spatial origin of Desargues’ theorem.

In other words (see Fig. 4a,b): If the vertices a, b of a given triangle abl are the perspective

images of the vertices D, E of the triangle DEK, from a point H, and if the sides ab and DE, bl and

EK, al and DK, meet respectively at the points c, f, g which are collinear, then the vertex l of abl lies

on the straight line HK; that is, the triangles abl and DEK are in perspective from H.

In the third part, Desargues gave a spatial proof of the first one, assuming the given straight lines

not on the same plane.

3. THE ESSAY POUR LES CONIQUES

The Essay pour les coniques appeared in the form of a hand sheet, like a small manifesto 35×43cm

in size (Fig. 5), whose fifty copies were circulated among the members of Mersenne’s academy, their

correspondents, and friends. Today only two copies are known to exist, one of which is preserved at

the National Library in Paris, and the other can be found in the G. W. Leibniz Library of Hannover.

It consists of three definitions, three lemmas, five propositions, all without proofs, and it is illustrated

by three figures.

The first definition is that of order (“ordre”) of straight lines, by which Pascal meant any set of

straight lines having the same butt. Thus this concept coincided with that of ordinance as defined by

Desargues. In the second, he defined the conic sections exactly in the same way as Desargues. Hence,

according to the position of the plane of section with respect to the base and the line-generators, they

are: circles, ellipsis, hyperbolas, parabolas, and “angle rectilignes”, as Pascal called the sections

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formed by two straight lines. In the third, he simply pointed out that he would use the term “droit”

instead of “ligne droite”.

Fig. 5. Essay pour les coniques, LH XXXV, XV, I, Bl. 10r. Courtesy of the G.W. Leibniz

Bibliothek –Niedersächsische Landesbibliothek, Hannover.

Then Pascal stated the three lemmas. To have an idea of his style, we state the first in its original

form (Fig. 6): Figure 1. If in the plane M, S, Q, two straight lines MK, MV are issued from the point

M, two straight lines SK, SV are issued from S, and K is the intersection of the straight lines MK, SK,

and V, is the intersection of the straight lines MV, SK, and A, is the intersection of the straight lines

MK, SV, and μ, is the intersection of the straight lines MV, SK, and through two of the 4 points AKμV

not lying on the same straight lines with the points M, S, as it is for the points K, V, there passes a

circle intersecting the straight lines MV, MK, SV, SK, at the points O,P,Q,N, I say that the straight

lines MS, NO, PQ belong to the same order.17

17 “Figure 1. Si dans le plan M,S,Q du point M partent les deux droites MK, MV, et du point S, partent les deux droites

SK, SV, et que K, soit le concours des droites MK, SK, et V, le concours des droits MV, SV, et A, le concours des droites

MA[K], SV, et μ, le concours des droites MV, SK, et que par deux des quatre points AKμV, qui ne soient point en mesme

droite avec les points M,S, comme par les points K, V, passe la circonference d'un cercle coupante les droites MV, MP[K],

SV, SK, és points O, P, Q, N, ie dis que les droites MS, NO, PQ, sont de mesme ordre”, (Pascal 1640)

11

Fig. 6 Enhanced version of figure 1 in (Pascal 1640). The two dashed lines PQ, NO (not present

in that original figure) and the line MS are parallel, so their point of intersection R is at infinity

(see also Fig. 7b).

It is useful to point out that in the set {A, K, μ, V} there are only two pairs of points not collinear

with M and S, namely K, V and A, μ, but for this last pair the lemma does not hold (Fig. 7a).

Moreover, the first lemma is clearly equivalent to the hexagon theorem in the particular case of a

circle. In fact, we remark that the lemma can be stated in the form: If in a plane we join two points

M, S to other two points K, V on a circle, lying in the plane but not passing through any of M, S, and

the circle intersects the straight lines MV, MK, SV, SK, at the points O, P, Q, N, respectively, the two

straight lines NO and PQ meet at a point lying on the straight line MS.

We also remark that this means (Fig. 7b): (♯) The opposite sides of the hexagon KPQVONK (which

is inscribed in the circle intersect in three collinear points, precisely M, S, R.

(a) (b)

Fig. 7 a) Shows that the first lemma does not hold for a circle passing through A, μ instead of

passing through K, V. b) Illustrates the hexagon theorem for the case of the circle: the opposite

sides of the hexagon KPQVONK meet at the three collinear points M, S, R.

It is worth noticing that in (Pascal 1640) Pascal never refers to any “inscribed hexagon”.

The second lemma, which has a clear projective character, substantially claims that: If several

planes, all passing through a same straight line as axle, are cut by a plane not passing through the

axle, then the straight lines in the section all belong to the same order with the axle of the given

planes, i.e. either all meet the axle at the same point, or are all parallel to it.

12

Next Pascal stated the third lemma, which, he affirmed to be an easy consequence of the first two

(Fig. 6): Figure I. Let a configuration of straight lines as in the first lemma be given, if a conic section

which passes through the points K, V cuts the straight lines MK, MV, SV, SK, it the points P, O, N,

Q, respectively, the straight lines MS, NO, PQ, belong to the same order.

With this Pascal extended the hexagon theorem from the circle (♯) to any conic section.

Pascal later claimed that from the lemmas, and some consequences easily deducible from them,

flow the “complete elements” of conic sections:18 that is, all properties of diameters and their

parameters, of tangents, and so on; but also – as he pointed out – the restitution of almost all data of

the cone, the description of conic sections by points, etc. According to Taton, by “restitution” Pascal

meant the determination of the vertices of the cones passing through a conic section defined by certain

conditions (Taton 1955, 13), and by “description” he meant the construction, by points, of the conic

sections passing through 5 − n points and tangent to n straight lines given in position (Taton 1955,

13), (Taton 1962, 231).

At this point Pascal stated the first proposition (Fig. 6): Figure I. If in the plane MSQ, of the conic

section PKV, the straight lines AK, AV intersecting the conic section respectively at the points P, K

and Q, V are drawn, and if for two of these four points not on the same straight line with A, as it is

for K, V, and for two points N, O, on the conic, four straight lines KN, KO, VN, VO, are drawn cutting

the straight lines AV, AP at the points S, T, L, M, the composite ratio of PM to MA, and of AS to SQ,

it is equal to the composite ratio of PL to LA, and of AT to TQ. In modern notation, this means

𝑃𝑀

𝑀𝐴×

𝐴𝑆

𝑆𝑄=

𝑃𝐿

𝐿𝐴×

𝐴𝑇

𝑇𝑄· (3.1)

We remark that the (3.1) is equivalent to affirming the equality of the two cross-ratios (PAML)

and (AQTS), although this notion was completely extraneous to Pascal.

For sake of brevity we state the other propositions in modern terms.

The second proposition is (see Fig. 8): Figure I. If the three coplanar straight lines DE, DG, DH

are cut by the straight lines AP and AR at the points F, G, H, and C, γ, B, respectively, and on the

straight line DC is fixed a point E, then

𝐸𝐹 × 𝐹𝐺

𝐸𝐶 × 𝐶𝛾×

𝐴𝛾

𝐴𝐺=

𝐸𝐹 × 𝐹𝐻

𝐸𝐶 × 𝐶𝐵×

𝐴𝐵

𝐴𝐻=

𝐸𝐹 × 𝐹𝐷

𝐸𝐶 × 𝐶𝐷· (3.2)

Moreover, if a conic section passing through the points E, D intersects the straight lines AH, AB, at

the points P, K, R, ψ, respectively, one has

𝐸𝐹 × 𝐹𝐺

𝐸𝐶 × 𝐶𝛾×

𝐴𝛾

𝐴𝐺=

𝐹𝐾 × 𝐹𝑃

𝐶𝑅 × 𝐶𝜓×

𝐴𝑅 × 𝐴𝜓

𝐴𝐾 × 𝐴𝑃 · (3.3)

It is not hard to see that in (3.2) the equality between the first and third member follows by applying

Menelaus' theorem to the triangle AFC intersected by the transversal GγD, whereas the equality

between the second and third member follows by applying the same theorem to the same triangle cut

by the transversal HBD.

18 For “complete elements” it was commonly intended the “content of the first four books of Apollonius's Conics”.

13

Fig. 8 This diagram, extracted from figure 1 in (Pascal 1640), illustrates Pascal’s second

proposition.

To understand the value of the second proposition, we note that the first member in (3.3) is equal

to 𝐹𝐸×𝐹𝐷

𝐶𝐸×𝐶𝐷 and thus from that equation it follows that

𝐸𝐹 × 𝐹𝐷

𝐸𝐶 × 𝐶𝐷=

𝐹𝐾 × 𝐹𝑃

𝐶𝑅 × 𝐶𝜓×

𝐴𝑅 × 𝐴𝜓

𝐴𝐾 × 𝐴𝑃

which is equivalent to the well-known theorem of Lazare Carnot (1753-1823):19 Let ABC be a triangle

in the plane and let Γ a be a conic lying in the plane of the triangle; if the sides AB, BC and CA cut

the conic Γ respectively at the points P, P', Q, Q' and R, R', then

𝐴𝑃 × 𝐴𝑃′ × 𝐵𝑄 × 𝐵𝑄′ × 𝐶𝑅 × 𝐶𝑅′ = 𝐴𝑅′ × 𝐴𝑅 × 𝐶𝑄′ × 𝐶𝑄 × 𝐵𝑃′ × 𝐵𝑃 (3.4)

The statement of the third proposition can be put in the following form (see Fig. 9a): Figure III.

If the four straight lines AC, AF, EH, EL, intersect at the points N, P, M, O, and a conic section

intersects that same straight line at the points C, B, F, D, H, G, L, K, then

𝑀𝐶 × 𝑀𝐵

𝑃𝐹 × 𝑃𝐷×

𝐴𝐷 × 𝐴𝐹

𝐴𝐵 × 𝐴𝐶=

𝑀𝐿 × 𝑀𝐾

𝑃𝐻 × 𝑃𝐺×

𝐸𝐻 × 𝐸𝐺

𝐸𝐾 × 𝐸𝐿

(a) (b)

Fig. 9 a) Reproduces figure III, the top right figure in (Pascal 1640), see Fig. 5. b) Reproduces

figure II, the second figure on the top left in (Pascal 1640).

We point out that this proposition is a particular case of the theorem in (Carnot 1806, No. 379).

The fourth proposition can be stated as follows (Fig. 6): Figure I. Let in the plane MSQ a conic

section PQV be given, if four points K, N, O, V are taken on it, and the straight lines KN, KO, VN,

VO, are drawn in such a way that through each of those four points there pass only two of them, and

19 See (Carnot 1806, No. 378).

14

if a another straight line intersects the conic section at the points R, ψ, and the four lines above at

the points x, y, Z, δ, respectively, then:20

𝑍𝑅 × 𝑦𝑅

𝑍𝜓 × 𝑦𝜓=

𝛿𝑅 × 𝑥𝑅

𝛿𝜓 × 𝑥𝜓·

Pascal called the fourth proposition “propriété merveilleuse” (wonderful property), that, as already

said in section 1, he credited to Desargues. In fact, the above relation expresses the condition for the

six points R, ψ, x, y, Z, δ to be in involution.

The fifth, and last proposition, regards the equation of central conics, precisely (Fig. 9b): If in the

plane of a hyperbola, or of an ellipse, or of a circle, AGE, whose centre is C, one draws the straight

line AB, tangent to the conic section at A and whose square is equal to one fourth of the rectangle of

the figure,21 and the diameters AC, BC are also drawn, then for any straight line parallel to AB, as

DE, cutting the conic section in E, and the straight lines AC, CB at the points D, F respectively, the

sum, or the difference, of the squares of DE, DF will be equal to the square of AB, according to

whether the conic section is an ellipse or a circle, or a hyperbola.

It is readily seen that this proposition leads to the equation of a central conic section, referred to a

diameter and its conjugate as coordinates axes.22 It is worthy of note that similar statements appeared

in the last part of Desargues’ Brouillon project.

Pascal claimed that some problems could be solved by means of the previous propositions. For

instance: to draw the tangents to a conic from a given point, or to find two conjugate diameters

forming a given angle, and other similar questions. Then he concluded by saying:

We have several other problems and theorems, and consequences of the previous ones, but the little

confidence I have in myself for my lack of experience and ability, does not allow me to advance any further

before it is examined by skilful people, who will oblige us to take the trouble; afterwards if we judge it

deserves to be continued, we will try to push it up to where God will give us the strength to lead it.23

4. THE ADDRESS TO THE ACADEMIA PARISIENSIS

After the death of Mersenne, in 1648, the meeting of the Academia Parisiensis continued all

Saturdays at Jacques Le Pailleur's home (Mesnard 1963). In 1654, likely wishing to resume contact,

Pascal addressed a report on his past and present research to the academy.24 In that report, after a brief

excursus on his arithmetical achievements, Pascal listed, and briefly commented, a series of seven

geometrical works he said to have completed.

Promotus Apollonius Gallus (Generalization of the French Apollonius). This concerned circular

contacts. According to Pascal, in this work he had widely generalized the restitution by François Viète

20 In the original text appears

𝑍𝑅×𝑍𝜓

𝑦𝑅×𝑦𝜓=

𝛿𝑅×𝛿𝜓

𝑥𝑅×𝑥𝜓 which, as was remarked by Taton (1955, 16), is clearly incorrect.

21 For Pascal, and as was customary at that time, “rectangle of the figure” meant “rectangle whose sides are a diameter

and the erect side relative to it, see (Taton 1955, 17). 22 See (Taton 1955, 17) for details. 23 “Nous avons plusieurs autres Problèmes et Theoremes et plusieurs consequences des precedents, mais la defiance que

i’ay de mon peu d’experience et de capacité ne me permet pas d’en avancer davantage advant qu’il ait passé à l’examen

des habiles gens, qui voudront nous obliger d’en prendre la peine; apres quoy si l’on iuge que la chose merite d’estre

continuée, nous essayrons de la pousser jusques où Dieu nous donnera la force de la conduire” (Pascal 1640). 24 See (Pascal 1779, 408-411). A French translation appeared in (Pascal 1954), an extract of which has been published in

(Taton 1962, 211--212).

15

(1540-1603) of what was known to the ancients in this field. Viète, in Apollonius gallus (1600), had

solved the problem of finding the circles subject to the condition of being tangent to three given

circles. This problem, posed by Apollonius in his lost work on Contacts, had been recorded in the

seventh book of Pappus’ Collection. Viète reached the solution of the problem by studying limiting

cases when any given circle can be shrunk to a point, or infinitely enlarged to become a straight line.

This method was considered a plausible reconstruction of what Apollonius himself had adopted. It is

worth noticing that the same problem was also considered by Newton (1687, book I, IV, lemma16),

and some-time later by Poncelet (1811) and Joseph Diez Gergonne in (1814).

Tactiones spherica (Spherical contacts). Here classical questions about spherical contacts were

treated by generalizing the method applied in the case of circular contacts. “In fact”, Pascal wrote,

“the two methods are originated by a remarkable property of conic sections, which is useful in solving

many other difficult problems, whose proof is contained in less than one page”.

Tactiones etiam conicae (Conical contacts either). In which Pascal tackled the question of

determining the conic sections satisfying five conditions “among [to pass through] five given points

and [to be tangent to] five given straight lines”.25

Loci solidi (Solid loci).26 According to Pascal, it was comprehensive of all cases, and complete

from any point of view.

Loci plani (Plane loci).27 Here were treated those known to the ancients, and those that “the most

famous geometer of our times has mastered”.28 Pascal was referring to Fermat, whose achievements

on the subject, though published in 1679, were known to the members of the academy (Taton 1962,

222).

Conicorum opus completum (Complete work on conic sections). According to Pascal, it included

Apollonius’ Conics, and countless other results developed by almost a single proposition “which I

found when I was not yet aged sixteen, and that later I perfected”;29 that is, the hexagon theorem as

expressed in the second lemma of the Essay, which he later perfected in the mystic hexagram.

Perspectiva methodus (A perspective method). Pascal defined it to be “the shortest and most

advantageous of those already known, and of all that can be invented, because it provides the points

of drawing as intersection of only two straight lines”.30

We briefly comment on three of these arguments, postponing the discussions on the Conicorum

opus completum to the next section.

Conical contacts. Here Pascal studied the problem of finding the conic sections passing through

5 − 𝑛 points and which are tangent to n straight lines given in position, 0 ≤ 𝑛 ≤ 5. The case of five

points was easily achieved by applying the mystic hexagram. The special cases of four points, when

one of them is the contact point with the tangent, and of three points, when two of them are the contact

points of the two tangents, can also be achieved by resorting to the same theorem; precisely by making

25 “ubi ex quinque punctis et quinque rectis datis, quinque quibuslibet, etc.”, (Pascal 1779, 409). 26 That is determination of geometrical loci connected with conic sections. This is likely the treatise that Pascal made to

circulate among the members of the academy since 1648. See later on. 27 That is the geometrical loci formed only by straight lines and circles. 28 “nec solùm illi quos his restitutis perillustris hujus aevi Geometra subjunxit”, (Pascal 1779, 410). 29 “quod quidem nondum sex decimum aetatis annum assecutus excogitavi, et deindè in ordinem congessi”, Idem. 30 “Quâ nec inter inventas, nec inter inventu, possibiles ulla compendiosior esse videntur; quippe quae puncta

ichnographiae per duarum solummodò rectarum intersectionem praeset”, Idem.

16

one, or more, sides of the hexagon shrink to a point, so that the side itself becomes the tangent to the

conic section at that point (see later on). However, the problem of finding the conic section tangent

to certain straight lines when the contact points are not given, no longer admits a single solution, and

is more difficult. The case of five tangents was in fact proposed to René François de Sluse, as we

learn from a letter that Sluse addressed to Christiaan Huygens, on 23rd October 1657, in the form:

“Given five straight lines AG, BF, CK, DL, EH, determine the conic section touching those straight

lines”.31 We do not know how Pascal dealt with these problems, however, as will be shown later on,

reasonable conjectures can be made.

Solid loci. Here Pascal gave a geometrical solution of the Pappus four lines problem by means of

the properties of the mystic hexagram. The route through which Pascal reached the solution is not

known, but also in this case some plausible conjectures can be made.

Perspective method. This treatise confirms Pascal’s interest in perspective, which in turn shows

the strong influence that Desargues exerted on him. In agreement with Taton (1962, 224), such an

interest was resulted from his method for studying conic sections, based on those transformations of

figures to which Desargues devoted several works. As is well-known, Desargues’ work on

perspective and the Brouillon project were harshly criticized, both for the abstruse terms adopted and

for lack of proofs. In his response of 16th December 1642 to such criticisms he wrote that he had

provided, “the key [to his methods], when the proof of that great proposition called the Pascale will

see the light of the day” (Curabelle 1644, 70-71). This suggests a link between Desargues’ unknown

propositions (the key to his methods) and the mystic hexagram.

Given the few indications Pascal provided in the address to the Academia Parisiensis it is

impossible to recover it. However, according to Taton, it is plausible that Pascal’s perspective

method, like some modern procedure, resorted to a homological transformation, which links the

perspective of a given plane figure to its overturning in the plane of picture, similar to that shown in

(La Hire 1673). This opinion is corroborated by the close connection between the hexagon theorem

and Desargues' theorem on two triangles in perspective (see section 7.3).

5. AN OUTSTANDING TREATISE NEVER PUBLISHED

In 1673, Henry Oldenburg, Secretary of the Royal Society of London, was informed of the existence

of Pascal’s manuscript. On April 6th, Oldenburg, always eager to follow the developments of French

mathematics, wrote to Gottfried Wilhelm Leibniz, at that time in Paris, asking him to make enquiries

about that manuscript, so that it could be brought to light:

We have come to hear of this hitherto unpublished treatise [Pascal’s], based on the Desarguesian method

(that the disciple of that man fortuitously conceived), and we have been informed by the Parisian bookseller

de Prex that the manuscript is in the hands of a brother of his in Auvergne. I pray to heaven that it may be

brought to light!32

Leibniz took action, and finally he got permission from the heirs to view Pascal’s geometrical

manuscripts, which he received in two tranches. The manuscript of the treatise on conic sections was

31 “Datis quinque rectas AG, BF, CK, DL, EA, invenire conisectionem quae datas quinque rectas contingat”, (Huygens

1889). 32 “Inaudivimus, hunc Tractatum hactenus esse ineditum; insistere autem methodo Des-Argueanae (quam forte ceu viri

illius discipulus imbiberat) edoctique fuimus a Bibliopola Parisiensi de Prex, manuscriptum id esse penes fratem quendam

suum (Prexii) in Auvernia. Utinam id protrahi in lucem possit!”, (Oldenburg 1973, 559).

17

not in Leibniz’s hands until the latter part of 1675.33 At that time, also E. Walther von Tschirnhaus

was also in Paris, and, being in touch with Leibniz, he too had access to Pascal’s writings.

Leibniz examined Pascal’s treatise, taking some notes on separate sheets, and copied the first part

of it. On August 30, 1676, returning the manuscripts to the heirs, he wrote the already mentioned

cover letter to Florin Périer, husband of Pascal’s older sister Gilberte. From his first words we gather

that he did not read the manuscript very thoroughly, “I would however like to have been able to read

them with a little more application”, he wrote, “but the many distractions, which did not let me fully

dispose of my time, did not allow it”. “Nevertheless”, he continued, “I believe I have read them

enough to be able to satisfy your request, and to tell you I deem the work finished and ready for

publication”.

It seems the heirs wanted to print Pascal’s manuscript.

Next, Leibniz briefly described the six parts, or chapters, in which he thought the manuscript might

be arranged for the printing. Let us follow his description.

5.1 The “Generatio conisectionum”

In first place Leibniz put the one bearing the inscription “Generatio conisectionum tangentium et

secantium; seu projectio peripheriae, tangentium, et secantium circuli, in quibuscumque oculi, plani

ac tabellae positionibus” (Generation of conic sections, tangents and secants; that is projection of the

periphery, of tangents and secants of the circle, in any position of the eye and of the plane of picture).

According to Leibniz this part was the foundation of the whole work, and he remarked that figures

were included on two separate sheets.34

The Generatio conisectionum begins with the definitions of conic surface, or cone, with circular

basis, and of its elements vertex, slopes, generators; substantially the same definitions that Desargues

gave in the Brouillon project (1639). Then there are four corollaries that we may synthesize as

follows: 1) any straight line joining the vertex with any point of a cone is a generator; 2) any straight

line joining any two points of a cone is either a generator, in which case it passes through the vertex,

or does not pass through the vertex, in which case it does not contain other points of the cone; 3) three

generators of a cone cannot belong to the same plane; 4) a cone is necessarily intersected by any

plane.

In the subsequent scholium, Pascal introduced the six types of conic sections (see Fig. 10). If a

plane of section passes through the vertex of the cone there are three possible cases: the section

reduces to one point, coinciding with the vertex; or to a (double) straight line, if the plane is tangent

to the cone; or to a pair of straight lines intersecting each other in the vertex, a figure that he named

angolus rectilineus (rectilinear angle). If the plane of section does not pass through the vertex of the

cone, there are another three cases: if the plane of section is not parallel to any generator, the conic

section is called Antobola (ellipse), which, wrote Pascal, “is closed in itself”; if the plane of section

is parallel to only one generator, the conic section is called Parabola; if the plane of section is parallel

to two generators, the section is called Hyperbola.

Then Pascal pointed out the following.

33 For a detailed account see (Taton 2000). 34 Our description use the copy made by Leibniz, published in (Gerhardt 1892).

18

A straight line goes to a point of another straight line at finite, or infinite, distance; in the latter

case the first straight line is parallel to the second.

Two straight lines in the same plane always intersect each other, at finite or infinite distance, and

in the latter case they are parallel.

A straight line which intersects a conic section only in one point is said to be mono-secant. A

straight line in the plane of a conic section which intersects the conic section only at infinite distance,

and is parallel to certain mono-secants, is called Asymptote.

A straight line in a plane of a circle which touches, or intersects, the circle is said to the circle.

Fig. 10 Enhanced version of the top figure in document B (Fig. 13), that Leibniz likely copied

from Pascal’s original. AB, AD, AT, etc. are generators; the plains HK, FG (parallel to the

generator AC), RD (parallel to the generators AS and AT) give respectively an ellipse, a parabola,

and a hyperbola, respectively. If AT and AS merge together (for instance in AC), then the

hyperbola cut by the plane RD becomes a parabola cut by the plane FG. Similarly, the ellipse

becomes a parabola when K is infinitely far from A, so that it becomes parallel to AC. The plane

CAS cuts the pair of generators AC, AS; a plane through A which does not contain other points of

the cone, cuts a section which reduces to only one point, the vertex.

Next there are several of corollaries. To have an idea of Pascal’s style of writing we state the first

in a literal translation:

1) It is evident that, if the eye is placed in the vertex of a cone, having a circumference of a circle

as is basis, and a screen or a plane intersect the conic surface, then the conic section produced on

the surface, whether it be a point, a straight line, an angle, an Antobola (ellipse), a parabola, a

hyperbola, is the appearance of the circumference of the circle.35

This means that, since the eye (as the vertex of the cone) can be placed at will, every conic section

is projection of the circle.

By introducing symbols and wording extraneous to the original text, but for the sake of simplicity,

we synthesize the other corollaries as follows.

2) If the plane of section π does not pass through the vertex of the cone, and is not parallel to any

generator, the conic section Γ is an ellipse which has all its point at finite distance.

35 “Hic patet, quod si oculus sit in vertice coni, sitque objectum peripheria circuli qui est coni basis, et tabella sit planum

utrimque occurrens superficiei conicae, tunc conisectio quae ab ipso plano in superficie conica producetur, sive sit

punctum, sive sit Recta, sive sit Angulus, sive sit Antobola, sive Parabola, sive hyperbola, erit apparentia ipsius

Peripheriae circuli.”, (Gerhardt 1892, 199).

19

3) If π is parallel to only one generator, the section is a parabola, and all points of the base circle

C have their images at finite distance on Γ except one, that in which the generator, to which π is

parallel, meets C, which is at infinite distance.

4) If π is parallel to two generators, all points of C have their image at finite distance, except two

that, for the parallelism, have their images at infinity.

Pascal called points without image, the points of C whose image is infinitely far away in the plane

of section; and missing points of the conic section the points at infinity corresponding to the points

without image.

In the subsequent three scholia, Pascal pointed out: a) the ellipse is closed in itself and surrounds a

finite space; b) the parabola is infinitely extended, and although it is the image of a circle, which

surrounds a finite space, it surrounds an infinite space; c) the hyperbola, is extended to infinity, and

is composed of two parts (semi-hyperbolas), each of which surrounds an infinite space, and is the

image of one of the two arcs in which the two points without image divide the base circle.

Therefore, there are two missing points on the hyperbola, one on the parabola, and none on the

ellipse.

Afterwards Pascal gave three corollaries regarding the images of secants of C into the plane of

section, which we summarize as follows.

A) If Γ is an ellipse, every secant s of C has for its image a secant σ of Γ.

B) If Γ is a parabola, every secant s of C has its image σ in the plane of section π, and if s does not

pass through the point of C without image, then σ is a secant of Γ, otherwise σ is parallel to a generator

and meets Γ in only one point.

C) If Γ is a hyperbola, every secant s of C, which does not pass through any of the points of C

without image, has for its image a straight line which is a secant of Γ, if s passes through only one

point without image, σ cuts the hyperbola in only one point (i.e. is a mono-secant), if s passes through

both points without image, its image σ is not at finite distance in the plane of section.

It seems that this vision greatly appealed Leibniz, who felt the need to write a note (see section

5.3, and Fig. 19, 20).

Pascal also examined what happens to the tangents to the base circle.

D) In the case of the ellipse Γ, every tangent t to C has as its image a tangent τ to Γ.

E) In the case Γ is a parabola, every tangent t to C has as its image a tangent τ to Γ, except when t

is tangent to C at the point without image.

Then, in the following scholium, he added, “Hence in the parabola there is a missing straight line,

which actually play the role of a tangent, being in fact the image of a tangent”.36

Thus the line at infinity of the plane of section is tangent to any parabola at its point at infinity.

F) In the case of a hyperbola, every tangent t to the C has its image τ on π and if the point of contact

is not a point without image, then τ is tangent to Γ at a point at finite distance, otherwise they do not

36 “Est ergo in parabola recta deficiens, quae quidem vice fungitur tangentis, eum tangentis sit apparentia”, (Gerhardt

1892, 201).

20

touch the hyperbola up to an infinite distance, and each of them is parallel to one, or the other, of the

two generators to which the plane of section is parallel.

Therefore, the asymptotes are seen as the tangents to the hyperbola at its points at infinity. In fact,

in the first of the three scholium with which he concluded the first part, Pascal pointed out “The

asymptotes play the role of tangents at infinite distance, and they have to be considered as such”.37

In the second scholium Pascal described the mono-secants of the parabola; and, echoing Kepler,

concluded with the words “The parabola is in the middle between ellipse and hyperbola”.38 In the

third he discussed the mono-secants of the hyperbola, showing that there are two sets of parallel

mono-secants, in each of which there is an asymptote.

We can safely say that Pascal's spatial projective thought clearly emerges from the Generatio

conisectionum.

5.2 Leibniz’s description of parts II-VI

From the letter that Leibniz addressed to Périer we learn that the former gave the title “De

hexagrammo mystico et conico” to the second part of Pascal's treatise. In fact, after discussing the

optical generation of conic sections as projection of a circle into a plane (the plane of section of the

cone), Pascal introduced, in this very part, the remarkable figure composed of six straight lines, a

particular configuration of which he called Hexagrammum Mysticum. According to Leibniz, it was

by using projection that Pascal showed that every mystic hexagram “belongs” to a conic section, and

that, conversely, any conic section gives rise to a mystic hexagram (see next section). Leibniz also

remarked that the lack of figures in this part was compensated by the figures inserted in the sixth.

Leibniz thought that the third part should be the one bearing the inscription “De quatuor

tangentibus, et rectis puncta tactuum jungentibus, unde rectarum harmonice sectarum et

diametrorum proprietates oriuntur” (Of the four tangents, and the straight lines joining the points of

contact from which the properties of the line harmonically cut and diameters are deduced). Leibniz

remarked that it was here that Pascal explained the use of the mystic hexagram in order to find the

properties of centres and diameters of conic sections. Leibniz also pointed out that nothing seemed to

be missing from it.

Chasles wrote that in this part Pascal most likely expounded the theorems constituting the theory

of poles and polars (Chasles 1837, 70), a belief already expressed by Poncelet in (1822, 101). The

reason for this can be found in the hexagon theorem; in fact, when two opposite sides each shrink to

a point, the theorem gives (see Fig. 11): (⁕) The tangents to a conic section at the opposite vertices

of an inscribed quadrilateral, meet each other on the straight line joining the intersection points of

the opposite sides.

A statement that fits very well with the title “Of the four tangents, and straight lines joining the

contact points…”. Now, since this last result implies the whole theory of poles and polars, Chasles

claimed that such theory was contained among the applications that Pascal made of the mystic

hexagram. This hypothesis was decisively confirmed by Taton. In fact, a careful analysis of the draft

of Leibniz’s letter to Périer allowed Taton to see that Leibniz, before changing his mind, had written

“III and when it happens that some of these six straight lines which constitute the hexagram are

37 “Colligendum hinc, asymtotos censeri et sumi pro tangentibus ad distantiam infinitam”, (Gerhardt 1892, 202). 38 “Parabolam tenere medium inter Antobolam et hyperbolam”, (Gerhardt 1892, 202).

21

infinitely small, it is from there that the explanation of the properties of tangents come”.39 Therefore

the theory of poles and polars was included in this part, and, in particular, it seems that the diameters,

and the centre, of a conic section were respectively defined as “polars” of points at infinity, and “pole”

of the line at infinity, respectively (Taton 1962, 244). On the other hand, in the Brouillon project, of

which Pascal had a thorough knowledge, Desargues had already outlined a comprehensive theory of

poles, polars, centres and diameters, via his theorem on the inscribed quadrilateral, see (Taton 1951)

and (Anglade and Briend 2019).

Fig. 11 This diagram illustrates statement (⁕), from which the theory of poles and polars can be

deduced. The straight line ML is the polar of the point intersection of the straight lines joining the

contact points A, C and B, D, of the four tangents constituting the circumscribed quadrilateral.

We underline that the argument applied suggests the use of the principle of geometrical continuity:

when the two opposite sides become infinitesimal, their prolongations still meet on the straight line

joining the intersection points of the other two pairs of opposite sides.

According to Leibniz, the fourth part had to be the one entitled “De propositionibus segmentorum

secantium et tangentium”, which dealt with ordinates, and ratios of segments intercepted on a straight

line by a conic section. Of this part Leibniz just said that figures were included, and that nothing

seemed to be missing from it.

Chasles, in describing what he thought Pascal's treatise should have contained (Chasles 1837, 71),

suggested that the fourth part was related to the study of properties of the segments intercepted by a

conic section on lines parallel to two fixed straight lines (the chords theorem), and the properties of

the foci. Taton, in (1962, 246-247), hypothesized that this part dealt with the several segmental

properties stated Apollonius’ Conics, and also those stated in (Carnot 1806); the condition for six

points to be on a conic section; the general definition of conics by the focus-directrix property.

In this regard we stress that the relation (3.4) becomes

𝐵𝑄 × 𝐵𝑄′ × 𝐶𝑅 × 𝐶𝑅′ = 𝐵𝑃 × 𝐵𝑃′ × 𝐶𝑄 × 𝐶𝑄′,

when the sides AB and AC become parallel, that is A goes to infinity, since in this case the

corresponding segments can be considered as equal. Thus we get

39 “III Et comme il arrive que quelques-unes de ces six droites qui font l'Hexagramme sont infinitament petites, c'est de

là que viennent les proprietez des touchantes des sections du Cone expli[qué]”, see (Taton 1962, 213--215).

22

𝐵𝑃 × 𝐵𝑃′

𝐵𝑄 × 𝐵𝑄′=

𝐶𝑅 × 𝐶𝑅′

𝐶𝑄 × 𝐶𝑄′

which expresses the chords theorem (Poncelet 1822, 18).

In Leibniz’s view, the fifth part had to be that entitled “De tactionibus conicis”, that is, he

explained, “(so that the title does not mislead), of contact points and tangents to a conic section”.40

He also pointed out that there were no figures enclosed. According to Taton (1962, 231), this fifth

part dealt with the six classical problems concerning the construction of conics passing through 5 −

n points and tangent to n straight lines given in position, for 0 ≤ 𝑛 ≤ 5. It is highly likely that Pascal

tackled these questions, also because the construction of the conic section passing through five given

points had been already treated in (Mydorge 1637). We will return on this question in section 7.1.

To the sixth part, that Pascal left untitled, Leibniz gave the name “De loco solido”, pointing out

that it dealt with the same subject on which Descartes and Fermat had already worked, that is the

Pappus problem of three and four lines. Leibniz remarked that this last part included many definitions,

and results, already explained in the second part. In particular, it contained the definition and the

properties of the Hexagrammum conicum mysticum, illustrated by several large coloured figures.

Leibniz thought that the first four, or five, parts constituted the whole of Pascal’s treatise on conic

sections, and that the sixth part was an autonomous treatise, a “compendium”, chiefly devoted to the

mystic hexagram and the four lines problem, to show the excellence and power of this tool, which

was to be circulated among Pascal’s friends; it was probably the treatise to which Mersenne referred

in writing to Constantijn Huygens in 1648.

Leibniz ended his letter to Périer confirming that Pascal’s work was ready to be published and that

it would not be appropriate to delay the printing for fear that the treatise might lose its novelty.

According to Taton, Leibniz had in mind La Hire’s Nouvelle méthode en géométrie, printed three

years before, in which the method of projection was largely used (Taton 1962, 215). We will come

back to this later on.

6. LEIBNIZ’S NOTES EXPLAINED

The notes that Leibniz made while examining Pascal’s treatise, have been re-published, also

translated into French, and commented in detail by Pierre Costabel (1962). Leibniz wrote these notes

on four pieces of paper,41 that Costabel denoted with the letters A, B, C, D; a labelling we believe

useful to maintain.

Document A has the characteristics of a slip of paper used by Leibniz to record his remarks at the

front of the folder containing Pascal’s manuscript, which had been entrusted to him for some time. It

is headed “Coniques; Excerpta / MS. de M. Pascal --Voyez ma lettre à M. Périer”, and it is about the

content of the folder itself, and the way in which Leibniz intended to organize the various parts, and

as he wrote in his letter to Périer.

Document B, headed “Conica pascaliana”, is here reproduced in Fig. 12. It consists of three

distinct parts: 1) a summary, in Leibniz own hand, of the first part of the Generatio conisectionum,

40 “(a fin que le titre ne trompe pas), de punctis et rectis quas sectio conica attingit”, Idem. 41Leibniz Handschriften, XXXV, XV, I, Bl 11r, 1r, 12r, 27r.

23

containing a figure (at the top, which Costabel credited to Tschirnhaus); 2) some remarks in Leibniz’s

hand; 3) the statement of the four lines problem, illustrated by a figure (in the lower right hand corner).

In a note on the left Leibniz wrote:

Every method of discovery in geometry of situation, and hence without computation, lies in

simultaneously grasping several [things] in the same situation, this happens either by means of a figure

which includes several [others], where appears the use of solids, or by means of movement or mutation.

In particular, among movements and mutations, one sees that the mutation of appearance or optical

transformation of figures is applied very usefully.42

This very interesting remark asserts that Pascal not only used perspective very profitably, but also

a sort of geometrical principle of continuity, a refined form of Kepler’s analogy, and moreover, the

possible use of certain transformations in the study of conic sections.

Fig. 12 Reproduces the manuscript LH. XXXV, XV, I, Bl. 1r (document B in accordance with

Costabel’s labelling). Courtesy of the G.W. Leibniz Bibliothek –Niedersächsische

Landesbibliothek, Hannover.

42“Omnis in Geometricis ope situs inveniendi ratio, ideoque sine calculo, in eo constat ut plura simul eodem situ

complectamur; quod fit, tum ope figurae cujusdam plures includentis, ubi usus solidorum patet; tum ope motus, sive

mutationis. Porro, ex motibus et mutationibus, utilissime videtur adhiberi mutatio apparentiae, seu optica figurarum

transformatio; videndum an ejus ope possimus ultra conum ad altiora quoque assurgere”, (Costabel 1962, 259)}

24

At the end of the statement of the four lines problem, written on the right of the corresponding

figure, Leibniz added, “This is a problem of Pappus, that Pascal easily reduces to his hexagram and

by this to the cone”.43 A quite cryptic sentence to which we will return later on.

Document C, headed “Hexagrammum pascalianum” and dated January 1676, is here reproduced

in Fig. 13. This document is in our opinion the most important, and is the main subject of this section.

It presents four figures, placed at the four corners of the sheet, with notes written by Leibniz and

Tschirnhaus. Beneath the title there are two figures representing hexagrams, whose sides are

numbered from 1 to 6 (see also Fig. 14a, b). From a note we learn that Pascal named contiguous the

sides such as 1 and 2, 2 and 3 etc.; paired the sides such as 1 and 3, 2 and 4, etc.; opposite the sides

such as 1 and 4, 2 and 5, etc.

Fig. 13 Reproduces the manuscript LH. XXXV, XV, I, Bl. 12r (document C in accordance with

Costabel’s labelling). Courtesy of the G.W. Leibniz Bibliothek –Niedersächsische

Landesbibliothek, Hannover.

This suggests that a general hexagram is the figure which arises by taking six straight lines in

general position in a plane, that is no three of them intersect at the same point, numbered from 1 to 6,

and their fifteen points of intersection, that we agree to denote by ij, 1 ≤ i < j ≤ 6.

43 Est problema Pappi, quod Paschalius facile reducit ad suum hexagrammum et ejus ope ad conum'', Idem.

25

(a) (b)

Fig. 14 Enhanced version of the two top diagrams in document C. a) The straight line LRM is the

directrix. b) illustrates the case in which the straight lines 3, 6 and the directrix, are parallel to one

another.

We observe that the fifteen points of intersection can, in a natural way, be grouped as follows: the

six corresponding to the intersection of the contiguous lines, 12, 23, 34, 45, 56, 16; the six

corresponding to the intersection of the paired lines, 13, 24, 35, 46, 15, 26, which are the vertices of

the two triangles given by the lines 1, 3, 5, and 2, 4, 6, respectively; the three corresponding to the

intersection of the opposite lines, that is 14, 25, 36, which, in general, are not collinear (Fig. 15).

Fig. 15 Illustrates a general hexagram, in which the points 14, 25, 36, intersection of the

opposite lines, are not collinear. The different kind of points are distinguished: ●

(intersection of contiguous lines), ♦ (intersection of paired lines), ○ (intersection of

opposite lines).

Beneath the heading of document C, Leibniz wrote, “Mysticum ut vocat idemque semper conicum”

(Mysticum as he calls it, and which is always conic). Hence, it was Pascal himself who attributed the

adjective “mystic” to the hexagram when the points 14, 25, 36 are collinear, and in this case Pascal

called directrix the straight line on which these points lie. Moreover, with that phrase, Leibniz wanted

to point out that the points 14, 25, 36 are collinear if, and only if, the six points 12, 23, 34, 45, 56, 16

26

lie on a conic section (Fig. 16).44 Thus, Pascal was aware that also the converse of the hexagon

theorem holds true; that is: If the opposite sides of a hexagon intersect in three collinear points, then

its vertices lie on a conic section. This theorem is today known as theorem of Braikenridge and

Maclaurin, see section 8.1.

We will further comment on the mystic hexagram later on, but for now we want to point out its

abstract and combinatorial nature. Combinatorics was certainly not extraneous to Pascal.

Fig. 16 Illustrates a mystic hexagram. The directrix, the horizontal line, and the conic section

passing through the points 12, 23, 34, 45, 56, 16, are drawn. The two triangles of vertices (13, 15,

35), (46, 24, 26) are also drawn, which result in perspective with respect to the point h.

The figure drawn in the lower left hand corner in document C, here reproduced in an enhanced

version in Fig.17, represents a hyperbola with its asymptotes, and an inscribed hexagon having three

sides infinitely extended, one of which lies on the line at infinity. The figure requires some

explanation. Side 5 lies on the line at infinity, and does not appear in the diagram. We have also drawn

the intersections of two pairs of opposite sides (the points denoted by M, the intersection of the straight

lines 1, 4, and N, the intersection of the straight lines 3, 6), and the line joining them (i.e. the directrix),

which is parallel to BC, because the intersection point of the other two opposite sides 2, 5 (this last

lying on the line at infinity), is the point at infinity of side 2.

In the note beside the figure recalled above (here reproduced in Fig. 17), Leibniz remarked that

the asymptote FE and the straight line issuing from D parallel to it, meet at infinity in the point E. In

the original figure this fact is stressed by a curly bracket grouping the extremities of these two lines

(see Fig. 13).

44 An immediate proof of this fact (but not available to Pascal) is as follows. The two triples of paired straight lines 1, 3,

5 and 2, 4, 6 form two cubic curves which intersect in nine points. By the Cayley--Bacharach theorem (see for instance

(Eisenbud, Green, Harris 1996)), any other cubic curve passing through eight of the nine intersection points, necessarily

passes through the ninth. Therefore, if six points are on a conic (possibly degenerate) the other three must lie on a straight

line, and vice-versa.

27

Fig. 17 This diagram is an enhanced version of the drawing in the lower left hand corner in

document C, where only the letters A, B, C, D, F appear and the sides of the hexagon are not

numbered. In the present figure, the letters E, E′ denote the two points at infinity of the hyperbola.

The figure in the lower right hand corner in document C is more difficult to interpret (see Fig. 18a).

It seems to refer to Desargues’ involution theorem, as is suggested by the note “in omnia conica: Bf :

BE = fL : LE; FB : BD = FK : DK”. But, since the tangent to the conic section at point f is also drawn,

and admitting that the tangent at point E was omitted in the drawing, it could be that the figure

illustrates the well-known “pole-polar” property.

(a) (b)

Fig. 18 a) Is an enhanced version of the drawing in the lower right corner in document C. (b)

Enhanced version of the drawing on the left in document D (see Fig. 20), concerning the

generation of the two branches of a hyperbola, these are the projection on the plane of the section

of the two arcs EDC and EFC.

Document D, headed “A Pascalio” and dated “April 1676” (Fig. 19).

This document contains a brief note explaining the generation of the two opposite branches of the

hyperbola, a question that, according to Costabel, seemed to raise some concern for Leibniz (Costabel

1962, 268): the two opposite branches are the projective image from the vertex of the cone of the two

arcs of the base circle, whose extremities are the two points intercepted by the plane of section on

that same circle; that is, the two points without image (Fig. 18b).

28

Fig. 19 Reproduces manuscript LH, XXXV, XV, I, Bl. 27r (document D according to Costabel’s

labelling). On the left it is explained the generation of the two branches of the hyperbola is

explained as the projection of two arcs of the base circle. Courtesy of the G.W. Leibniz Bibliothek

–Niedersächsische Landesbibliothek, Hannover.

This document presents other drawings (without accompanying notes) that we have been unable to

interpret with certainty.

7. THE MYSTIC HEXAGRAM

In the Essay pour les coniques, Pascal had already said of the prominent role that the “hexagon

theorem” would play in the construction of his treatise on conic sections. One wonders what

connection of ideas led him to his famous theorem, and moreover how he arrived at the

“hexagrammum mysticum”, why it is always “conic”, and why he called it “mystic”. These questions

are not easy to answer, but some conjectures may be made.

According to Taton (1962, 209), it may have occurred by looking at lemma XIII of the seventh

book of Pappus’ Collection, known as “Pappus’ theorem” (Fig. 20a): If A, E, B are three collinear

points, and C, F, D are other three collinear points, then the intersection points G, M, K, of the line

pairs AF and EC, AD and BC, ED and BF are collinear.

(a) (b)

Fig. 20 a) Illustrates lemma XIII (prop. 139) of book VII in (Pappus 1588). b) The diagram shows

that the same result holds even if the six points are differently joined.

29

In fact, the configuration of lines connecting the two triples of points on the straight lines AB and

CD, as indicated in the lemma, coincides with the hexagon AFBCED inscribed in the degenerate

conic section constituted by that pair of straight lines, and thus the lemma affirms that the opposite

sides of that hexagon meet in three collinear points.

The collinearity of points G, M, K does not depend on the inscribed hexagon having its vertices in

the points A, B, C, D, E, F (Fig. 20b). It can be also observed that the collinearity of G, M, K is a

necessary and sufficient condition so that A, E, B, and C, F, D, are two triples of collinear points.

If six points A, B, C, D, E, F are taken on a circle, the inscribed hexagon still enjoys the same

property; that is, the points L, M, N, intersections of the opposite sides, are collinear (Fig. 21a,b).

In agreement with Taton (1962, 208), we think that an in-depth analysis of Desargues’ Brouillon

project convinced Pascal that a direct study of conic sections could be founded on the simple idea

that five points (in general position) in a plane, determine a unique conic, and that the condition for a

sixth point to lie on that curve gives a relation that may be used as a common definition for all types

of conic section. This was the case of the chords theorem in Mydorge (1637), and Desargues’

involution theorem in the Brouillon project. But, since the relation is connected with the latter result

was too difficult to apply, Pascal sought for a simpler relation that could also be expressed graphically,

aiming to apply the method of projection. Thus, seeking for such a condition, Pascal first thought of

a circle and six points regularly placed on it, and realized that the three pairs of opposite sides intersect

at three collinear points; in fact, they lie on the line at infinity. Pascal re-applied the construction for

other hexagons inscribed in the circle, and the drawings confirmed the intuition. Now, if accurate

diagrams do not constitute a proof, they may constitute sufficient reason to look for a proof.

(a) (b)

Fig. 21 a) Diagram for a Pappus-type theorem in the circle. The intersection points of the opposite

sides 1,4; 2,5, and 3,6 of the inscribed hexagon AECFBD, L, M, N, are collinear. b) This diagram

shows that joining differently the same points, one always gets other hexagons, as ACBEDF,

whose opposite sides 1,4; 2,5; 3,6 intersect at the three collinear points.

7.1 Pascal possible proofs of the hexagon theorem

In any event, Pascal was first led to prove the theorem for a hexagon inscribed in a circle, and then to

extend it to any conic section by projection, being aware of the projective character of that result. In

this regard it is significant that the first proposition in the Essay pour les coniques, which holds for

30

the circle, was extended to any conic section by projection. Likely, as suggested in (Taton 1955),

following Desargues’ footsteps –who had so successfully applied it –, Pascal used Menelaus’ theorem

in order to prove the hexagon theorem in the case of the circle.

If ABCDEF is a hexagon inscribed in a circle (Fig. 22), by Menelaus’ theorem applied to the

triangle LMN cut by the transversals ABR, FES, and CDT, we get respectively

𝑁𝐴

𝐴𝑀×

𝑀𝐵

𝐵𝐿×

𝐿𝑅

𝑅𝑁= 1,

𝑁𝐹

𝐹𝑀×

𝑀𝑆

𝑆𝐿×

𝐿𝐸

𝐸𝑁= 1,

𝑀𝐶

𝐶𝐿×

𝐿𝐷

𝐷𝑁×

𝑁𝑇

𝑇𝑀= 1,

then multiplying, and re-arranging, we have

𝑀𝑆

𝑆𝐿×

𝐿𝑅

𝑅𝑁×

𝑁𝑇

𝑇𝑀× (

𝑁𝐴

𝐴𝑀×

𝑁𝐹

𝐹𝑀×

𝑀𝐶

𝐶𝐿×

𝑀𝐵

𝐵𝐿×

𝐿𝐷

𝐷𝑁×

𝐿𝐸

𝐸𝑁) = 1.

Fig. 22 diagram for the proof of the hexagon theorem for the circle, via Menelaus’ theorem.

On the other hand, by the intersecting secants theorem it follows that

𝑁𝐴 × 𝑁𝐹 = 𝑁𝐷 × 𝑁𝐸, 𝑀𝐶 × 𝑀𝐵 = 𝑀𝐴 × 𝑀𝐹, 𝐿𝐷 × 𝐿𝐸 = 𝐿𝐶 × 𝐿𝐵,

Hence

𝑀𝑆

𝑆𝐿×

𝐿𝑅

𝑅𝑁×

𝑁𝑇

𝑇𝑀= 1

which, by the converse of Menelaus’ theorem, implies that R, S, T are collinear.

Next, to conclude the proof, Pascal passed into space. If a hexagon inscribed into the conic section

Γ is given (Fig. 23), and we project it from the vertex V of the cone into the plane of the base, we

obtain a hexagon inscribed into the base circle C. In accordance with what is shown above, the

opposite sides of the projected hexagon intersect at the three points R, S, T which belong to a straight

line d. Then the opposite sides of the given hexagon intersect at three points R', S', T', which are

31

collinear, because they belong to the straight line d' along which the plane of section cuts the plane

through the straight line d and the vertex V.

The converse of the hexagon theorem also holds true: If the opposite sides of a hexagon meet at

points which lie on a same straight line, then the vertices of the hexagon lie on a conic section.

To achieve this, Pascal most likely reasoned by reductio ad absurdum. Suppose that the opposite

sides, AB and DE, BC and EF, CD and AF, of the hexagon ABCDEF meet at the three collinear points

L, M, N, but that the six vertices do not lie on a conic section. In this case, F does not belong to the

conic section γ passing through the other five vertices. Then F', intersection of NA with γ, cannot

coincide with F. On the other side, F' lies on ME, since ABCDEF' is inscribed in γ, so F and F' being

both intersection of NA and ME have to coincide; a contradiction, thus F lies on the conic section

passing through A, B, C, D, E.

Fig. 23 Illustrates how to extend Pascal’s theorem from the circle to a conic section. The straight

line l in the intersection between the plane α and the plane of the base circle C.

7.2 How Pascal arrived at the hexagram, and why he called it “mystic”

Since Pascal never spoke of any inscribed hexagon in the Essay pour les coniques, and he was looking

for a condition for six points to lie on a conic section, he most likely considered the six straight lines

PK, KN, NO, OV, VQ, QP, (see Fig. 6) independently of the conic section. In this way he was led to

tackle the question from a combinatorial point of view, we have seen in section 6. We do not know

when Pascal took this step toward the “mystic hexagram”, but likely it was unlikely before 1643, if

that was the year Desargues was waiting for the proof of “la Pascale” before revealing the details of

his method for drawing in perspective.

The reason why Pascal used the adjective “mystic” for his hexagram is not known. Gerhardt (1892,

195), wrote that perhaps it was owing to its resemblance with the “mystic pentagram”, the five-

pointed star used in the Kabala; but Taton (1962, 241) did not agree with this explanation.

A complete understanding of Pascal’s theorem on the inscribed hexagon requires a knowledge of

the properties of elements at infinity, points and lines; that is, of the “projective plane”. For instance,

32

as noticed above, the opposite sides of a regular hexagon (which is necessarily inscribed in a circle),

meet in points at infinity, and hence they are collinear, lying on the line at infinity. In this regard it is

useful to look at the second figure in Document C, which illustrates an inscribed hexagon with two

parallel sides (Fig. 14b).

In some respects, it may be useful to recall how Poncelet proved the hexagon theorem in (1822,

No. 201). He considered a hexagon ABCDEF inscribed into a conic section, and supposed that the

opposite sides AB and DE, BC and EF, CD and AF, meet at the points L, K, I respectively. By suitably

projecting the figure on a new plane, he transformed the given curve in a circle, while the straight line

LK is sent at infinity. Thus, he transformed the given hexagon into another one inscribed into a circle,

having two pairs of parallel sides. He observed that this implies that also the remaining two sides

have to be parallel to each other. From this he concluded that the point I must lie on the straight line

LK. We observe that to apply this reasoning Poncelet needed the “4th principle of projection”

(Poncelet 1822): A conic section and a straight line in its plane, can always be projected in a circle

and the line at infinity of the plane of the circle. Poncelet proved this statement in the case in which

the given straight line and the conic section meet each other, and extended it to the general case by

applying the “principle of continuity”.45

To prove this was certainly within Pascal’s reach, and we wonder whether Pascal applied it.

(a) (b)

Fig. 24 a) Shows an inspiring mystic hexagram. b) Diagram for theorem (⁎⁎).

By producing the sides of a regular hexagon until they meet, we get the “David star”, or six-pointed

star, and this is what one obtains by intersecting paired sides in the sense intended by Pascal (Fig.

24a). The six-pointed star, is a figure also used in Christianity, and often reproduced on the facade of

churches. Was this the reason why Pascal used the adjective “mystic” for his hexagram? We do not

know, and, after all, answering this question is not so important. In fact, what is important is what

such a figure may have suggested to Pascal. Looking at it, we see that: the opposite sides of a regular

hexagon meet in three collinear points at infinity; the two triangles, of which this figure is composed,

are in perspective from the centre of the circle in which the hexagon is inscribed; through the centre

45 We know that this principle holds true in the extended complex plane, thus, as remarked in (Bos et altri, 1987), it led

Poncelet to correct results.

33

of this circle there pass the straight lines joining the opposite vertices of the hexagon; the hexagon is

circumscribed about another circle, concentric with the first, and the contact points are placed where

the sides are intersected by the straight lines joining the corresponding vertices of the two triangles,

and these same straight lines join the centre with the points of intersection of the straight lines joining

paired vertices (in accordance with Pascal) of the hexagon.

Now, the following theorem holds true (Fig. 24b): (⁎⁎) Suppose ABCDEF be any hexagon

inscribed in a circle. Produce the sides AB, CD, EF, until they meet in L, M, N, respectively, and

produce BC, DE, FA, until they meet in P, Q, R, respectively. Then the straight lines LQ, MR, NP

intersect in the same point O. If H, I, K, are the points where LQ, MR, NP intersect RP, PQ, QR

respectively, to show that LQ, MR, NP are concurrent in O, it is enough to prove that

𝐾𝑄

𝐾𝑅×

𝐻𝑅

𝐻𝑃×

𝐼𝑃

𝐼𝑄= 1

according to Ceva’s theorem.46 Pascal most likely proved this by a proceeding similar to the one he

applied to prove the first proposition in the Essay pour les coniques, and then, passing to the cone, he

extended this property to a hexagon inscribed into any conic section.

In this respect, we remark that if we consider the hexagon abcdef circumscribed about the conic

section at the vertices of the inscribed hexagon ABCDEF, by projecting simultaneously the conic

section and the straight line where the opposite sides meet, on a circle and the line at infinity (as

mentioned above when commenting Poncelet’s proof of the hexagon theorem), and maintaining the

same notation for the projected points, we see that the opposite sides AF and CD become parallel and

the vertices a and d belong to a diameter of the circle (Fig. 25a). Since the same happens for the other

two pairs of opposite vertices of the given hexagon, the three diagonals ad, be, cf meet in the centre

O of the circle (Poncelet 1822, No. 208). On the other hand, we see that the opposite vertices a and

d (on the original figure) have as polars the opposite sides AF and CD respectively, hence the diagonal

ad is the polar of the point I, where AF and CD meet, therefore it contains the pole O of the straight

line ILK. It follows that O is the intersection of the three diagonal which join the opposite vertices of

the circumscribed hexagon.

We believe that Fig. 25a somehow suggests the possibility of “changing lines with points and vice-

versa”; that is, to consider what is today called “duality”.Instead of starting from six numbered

straight lines in a plane (no three of which are concurrent), Pascal may also have considered six points

in a plane, no three of which collinear, numbered from 1 to 6. As in the case of lines he may have

called “contiguous” the points as 1 and 2, 2 and 3 and so on; “paired” those like 1 and 3, 2 and 4 etc.;

and “opposite” those like 1 and 4, 2 and 5 etc. Thus, joining two by two the given points, we get

fifteen straight lines: the six joining contiguous points; the six joining paired points; and the three

joining opposite points, which, in general, are not concurrent. Pascal may have wondered what

happens if the last three meet in a same point.

Pascal studied the properties of a quadrilateral circumscribed around a conic section (Fig.25b), as

is confirmed by part three of his treatise, entitled De quatuor tangentibus et rectis puncta tactuum

jungentibus, and most likely he found its main feature: The pole of the straight line joining two

opposite vertices, as E and F, is the point intersection G of the two straight lines, AB and CD, joining

46 This theorem, although published for the first time by Giovanni Ceva in De lineis rectis (Ceva 1678), had been known

since the eleventh century. It can be proved in several ways, for instance by resorting to Menelaus’s theorem (Irving

1902).

34

the contact points of the tangents from those vertices. For the pole of AB is E, and the pole of CD is

F, it follows that the pole of EF has to belong to the intersection of AB and CD.

(a) (b)

Fig. 25 a) Illustrates a hexagon abcdef circumscribed about a circle at the vertices of an inscribed

hexagon ABCDEF having the opposite sides parallel between them; the straight line joining the

opposite vertices of the hexagon are diameters of the circle. b) This diagram illustrates the

property of a quadrilateral circumscribed to a conic section according to the pole of the straight

line joining two opposite vertices of the quadrilateral (EF), is the point (G) in which the straight

lines joining the two pairs of corresponding contact points A, B and D, C, meet.

Now let ABCDEF be a hexagon circumscribed about a conic section, and let a, b, c, d, e, f be the

points where it touches the curve. The hexagon of vertices abcdef is inscribed in the conic section,

thus its opposite sides intersect at three collinear point L, M, N (Fig. 26). From the above the pole of

AD is the point L, and similarly the poles of BE and CF are M and N; and therefore the pole S of the

straight line LMN has to belong to the three lines AD, BE and CF.

Fig. 26 Illustrates the proof of Brianchon’s theorem deduced from the hexagon theorem, as

Pascal may have done.

35

This means: The straight lines joining the opposite vertices of a hexagon circumscribed about a

conic section are concurrent. We have the theorem that today bears the name of Brianchon, see

(Brianchon 1810, Art. XV). We believe that Pascal knew this result, and he probably used it to

determine the conic section tangent to five given straight lines (see section 8.1).

7.3 The mystic hexagram, and Desargues’ triangles

In a hexagram, as Pascal intended it, there are two underlying structures: one associated to the six

points 12, 23, 34, 45, 56, 16, and the other associated to the two sets of three points, 13, 15, 35, and

24, 26, 46. It is clear that to give the straight lines 1, 2, 3, 4, 5, 6, joining the consecutive points 12,

23, 34 etc., and to give the infinitely extended sides of the two triangles of vertices {13,15,35}, and

{24, 26, 46}, is the same; that is, one gets the same configuration of six straight lines. We agree to

call the first a Pascal configuration, and the second a Desargues configuration.47

In case of a mystic hexagram, that is when 14, 25, 36 are collinear, the sides of the triangles {13,

15, 35} and {24, 26, 46} are associated so that the corresponding ones intersect in the three collinear

points 14, 25, 36. Thus, by the converse of Desargues’ theorem on homological triangles, the straight

lines joining the corresponding vertices are concurrent in a point h (see Fig. 16, 27); that is, the two

triangles are in perspective with respect to h. Conversely, if one starts from a Desargues configuration,

i.e. from two triangles in perspective, the opposite sides of the hexagon, in the Pascal configuration

associated with it, intersect in three collinear points.

From the above, and what we have said in sections 7.1 and 7.2, we are convinced that Pascal was

aware that Desargues’ theorem on homological triangles and the mystic hexagram imply each other;

and that to show this he proceeded as above.

Fig. 27 Illustrates the two structures underlying a mystic hexagram. The two triangles {13, 35,

15} and {46, 26, 24} are in perspective with respect to h, and the corresponding sides intersect in

the three collinear points 14, 25, 36. The hexagon of vertices 12, 23, 34, 45, 56, 16 inscribed in

the hyperbola, has its opposite sides intersecting each other in the same points 14, 25, 36.

The consideration of limiting cases, that is when one or more sides of a hexagon collapse to a

point, led Pascal to discover new theorems. We recall that theorems on inscribed and circumscribed

47 We presume that Pascal was aware of Desargues three geometrical propositions long before they were published by

Bosse in (1648.

36

pentagons to a conic section, were rediscovered by the Scottish mathematicians Robert Simson and

Colin Maclaurin, and much later by Brianchon, by using such a method. We end this section by

observing that if the points 26 and 12, 45 and 46, 24 and 34, coincide in pairs (that is, three opposite

sides each reduces to one point), from the previous results it follows (Fig. 28): A triangle inscribed

into a conic section, and the triangle circumscribed about the same curve at vertices of the first, are

in perspective with respect to the intersection point of the straight lines joining one vertex of the

circumscribed triangle with the contact point of its opposite side. Moreover, the corresponding sides

of the two triangle meet at three collinear points.48

Fig. 28 Illustrates the theorem on the circumscribed triangle at the vertices of an inscribed triangle.

It goes without saying that this result shows the deep connection between Desargues’ and Pascal’s

theorems once again.

8. APPLICATIONS OF THE MYSTIC HEXAGRAM

Now we come to some questions we have so far left pending: the construction of conic sections

passing through 5 − n and tangent to n straight lines given in position (0 ≤ 𝑛 ≤ 5); the solution to the

four lines problem.

8.1 Construction of conic sections subject to five conditions

The conic section through five points. Given the points A, B, C, D, E, no three of which not on the

same straight line (Fig. 29), one draws the straight lines AB, BC, CD, DE, then from the point M, the

intersection of AB and DE, one issues a straight line g intersecting CD and BC in L and N, respectively,

and finally one draws the straight lines LA and NE. The point F, intersection of LA and NE necessarily

belongs to the (unique) conic section passing through the five given points. If one makes g rotate

around M, the point F sweeps out the sough-for curve.

48 See for instance (Enriques 1897, No. 64).

37

Let us remark that this is actually an organic construction of the conic section passing through five.

This construction, as already remarked, is known as the theorem of Braikenridge and Maclaurin, who

proved it independently from each other (Braikenridge 1733), (Maclaurin 1720, 1735).

Fig. 29 Illustrates the construction of the conic section through five given points by using Pascal's

hexagon theorem.

Following in the footsteps of Newton, Simson and Colin Maclaurin, pushed forward Newton’s

method for the organic construction of conic section. In doing so, both rediscovered Pascal’s hexagon

theorems, of which they were not aware.

Simson proved the hexagon theorem in (1735, prop. 47), which he stated it in the following form:

Let AB, CD be two chords of a conic section which meet in K, and E, F two points on that curve. If

AE, DF are produced until they meet in G, and CE, BF are produced until they meet in H, the three

points H, G, K, are collinear.

In the 1730s, Maclaurin composed De linearum geometricarum proprietatibus generalibus

tractatus (Treatise concerning the general properties of general lines), which was published

posthumously in 1748 as an appendix to A Treatise of Algebra (Maclaurin 1748). To develop part of

the Treatise of Fluxions (1742), Maclaurin inserted in it some results from the De linearum, gathered

in chapter XIV under the heading “General properties of conic sections transferred briefly from the

circle”. In particular, in Art. 623, Maclaurin gave a proof of the hexagon theorem, which he first

enunciated for the circle: Let the five points C, S, E, A, and B be in the circumference of the circle;

produce BC and ES till they meet in D; let CP and SP be drawn from C and S to any point P in the

circle; let CP meet EA in N, and SP meet BA in Q; D, Q and N will be always in a right line.

To show this he applied similarity of triangles and a result he had previously obtained.

Next, Maclaurin extended the theorem to conic sections by means of the method of projection, and

claimed (his own words): If C, S, E, B and A be five point in a conic section and [we join] any

two[pairs] of them [as] BC and ES [and these straight lines] intersect each other in D, CP and SP be

drawn to any point P in the section, CP meet EA in N, and SP meet AB in Q, then D, Q and N will be

always in a right line. A quite explicit statement of the hexagon theorem.

The case of four points and a tangent. To find the conic section passing through three given points

K, C, Q, say, and tangent to a straight line given in position at the given point B, Pascal surely applied

a particular case of the hexagon theorem, by making one side of the inscribed hexagon to collapse in

a point and substituting the “side” with the tangent to the conic section at that point. In this case things

go as follows (see Fig. 30a).

38

(a) (b)

Fig. 30 a) Diagram for the construction of the conic section passing through three points and

tangent to a given straight line at a given point. b) Illustrates the case of the conic sections passing

through four given points and tangent to a straight line given in position.

We produce CQ until it meets the given tangent in A, then we join K with Q, and from A we issue

a straight line AN which intersects KQ in S and BK in N. Now, the straight lines BS and CN intersect

at a point O which necessarily belongs to the conic section sought; by rotating the straight line AN

around A, the point O describes the curve.

When the fourth point B does not lie on the tangent, the problem is not solvable by directly

resorting to the hexagon theorem, and most likely Pascal appealed to Desargues’ involution theorem

(which he deduced from the hexagon theorem). In this case, if A, B, C, D are the given points (Fig.

30b), he produced the straight lines AB, BC, CD and DA, until they intersect the given tangent t at the

points E, F, G and H, respectively; he noticed that a point X where a conic section through A, B, C,

D touches t, is in involution with the pairs of points E, G and H, F, thus a relation like (2.1) has to be

satisfied, precisely:

(𝐺𝑋

𝐸𝑋)

2

=𝐻𝐺

𝐻𝐸×

𝐺𝐹

𝐸𝐹

which allows us to determine the position of the point X. Clearly there are two solutions to this

problem.

We may argue that in the cases of three points and two tangents, and one point and four tangents,

Pascal proceeded analogously.

The conic section tangent to five straight lines given in position. In this case it is sufficient to find

the five points of contact, and Pascal surely resorted to the “circumscribed hexagon”, whose

properties, as said above, he knew.

Thus, if a, b, c, d, e are the given straight lines (Fig. 31a), to find the point H where a touches the

conic section we are looking for, it is enough to determine the point T, the intersection of AD with

BE, and intersect CT with a. The others points of contact are similarly determined.

39

An intriguing question is whether Pascal was aware of a sort of “principle of duality”, but we will

probably never know. However, if five straight lines are given, and we carry out the dual construction

of that used for determining the conic section passing through five given points (see Fig. 31b and its

caption), we get the envelope of the conic section tangent to those five straight line.

(a) (b)

Fig. 31 a) Illustrates the construction of the conic section tangent to five give straight lines a, b,

c, d, e, by the determination of the five contact points. b) a, b, c, d, e are the given straight lines,

A is the intersection point of a and b, B is the intersection point of b and c, and so on. On the

straight line AD we take a point G, and we draw the straight lines BG and CG, which intersect a

and e in the points K and N respectively. When the point G moves on AD, the straight line KN

envelops the conic section tangent to a, b, c, d, e.

8.2 The four lines problem

As we have seen, Pascal devoted the sixth part of his treatise on conic sections, that Leibniz entitled

De loco solido, to the problem that, as he wrote, “Mrs. Descartes and Fermat have worked out, when

they provided the composition of the solid locus, each in his own way, Pappus gave them the

opportunity”.49 Leibniz pointed out that the several large coloured figures belonged to this part. In a

note in document B, that headed “Conica Pascaliana”, Leibniz simply wrote (Fig.12, and Fig. 32),

“Let 4 straight lines, AB, CD, EF, GH be given in position, issue from a given point L the straight

lines LB, LD, LF, LH making with the first given angles and such that the rectangles constructed on

BL, LD and on LF, LH be equal or in a given ratio. Find the point L or its locus that is a conic”.

That is, find the locus of points L such that

𝐵𝐿 × 𝐿𝐷 = 𝑘𝐿𝐹 × 𝐿𝐻, (8.1)

k being a constant.

Leibniz did not give many indications of the strategy Pascal applied for solving the problem, he

just remarked, “Pascal easily brings back [the question] to his hexagram and through it to the cone”.50

49 “Messieurs Descartes et Fermat ont travaillé, quand ils ont donné la composition du lieu solide, chacun à sa mode,

Pappus leur ayant donné l’occasion”, Leibniz letter to Périer. 50 See section 6, and footnote 43.

40

This sentence suggests that Pascal used his theorem on the inscribed hexagon to solve the problem,

but the reference to the cone is quite intriguing; what did Leibniz mean?

Fig. 32 Enhanced version of the drawing in Tschirnhaus’ hand for the Pappus problem of four

lines, see Fig. 12 lower right hand corner. The drawing bears handwritten notes in Tschirnhaus’

hand: “Direct. Correctas”, “Caput proportionalium” which are of not of easy interpretation, see

(Costabel 1962).

In the Principia, as we have already said in section 2, Newton presented a geometrical complete

solution of the locus of four lines, which constituted the departing point for the geometrical

achievements he presented in section five, book I. We stress that the complete solution of the problem

requires, once it is known that the locus sought is a conic section passing through the vertices of a

quadrilateral, the construction of the curve itself; which means the determination of a fifth point

through which the curve has to pass, the construction by points of the conic section passing through

five given points, the determination of one diameter of the curve and its latus rectum.

To understand how Pascal may have proceeded to solve the four lines problem, it is useful to recall

what steps Newton took to carry it out.

1) (Lemma 17) If ABCD is a quadrilateral inscribed into a conic section, and from any point P on

it straight lines PQ, PR, PS, PT are drawn making given angles with the sides AB, BC, CD, DE, then

PQ×PR is to PS×PT in a given ratio. It is here that Newton applied the chords theorem, and this

lemma is today known as Newton’s theorem.

2) (Lemma 18) If a point P moves in such way that PQ×PR : PS×PT in a given ratio, the locus of

P is a conic section passing through A, B, C, D. (Lemma 19) Determine a point P which if four

straight lines PQ, PR, PS, PT are drawn to as many other straight lines AB, BC, CD, DE ad give

angles, PQ×PR will be to PS×PT in a given ratio.

Then Newton gave two corollaries, in the first of which he showed how to construct the tangent

at any point of the curve, and in the second he showed how to determine a diameter and its erect side,

and to distinguish to which type of conic section the curve belongs. In the subsequent lemmas 20 and

21, Newton provided his famous organic construction of conic sections by means of two rotating

angles: If two angles DBM, DCM, of given sizes, rotate around their fixed vertices B, C, so that BM

and CM intersect along a fixed straight line, then the intersection point of the other two sides BD,

CD will describe a conic section passing through B, C. And conversely. Next, Newton showed how

to construct a conic section subject to the five conditions (as listed above).

41

Going back to the first proposition stated in the Essay pour les coniques, and looking at Fig. 33,

we see that the relation (3.1) says that the two sets of four straight lines KP, KQ, KO, KN, and VP,

VQ, VO, VN, have the same cross-ratio.

Fig. 33 Is a variation of figure 1 in (Pascal 1640), in which are drawn the two projective pencils

KP, KQ, KO, KN and VP, VQ, VO, VN.

This also means that if the first four lines meet a transversal in four points, and the second four

lines meet another transversal in four points corresponding one by one to the first four points, the

cross-ratio of the first four is equal to the cross-ratio of the second four. In fact, if the first transversal

coincide with the straight line KP, and the second transversal is the straight line VQ, we get exactly

the first proposition in (Pascal 1640).

We also observe that, if the two transversals coincide with a side of the inscribed hexagon,

KPQVON, we have the theorem of Desargues on six points in involution, the fourth proposition in

(Pascal 1640). If, in this theorem, we substitute the segments intercepted on the transversal by the

conic section and the sides of the quadrilateral, the expressions of these segments as functions of the

oblique segments issued from any point of the conic section, and making given angles with the four

sides, we get that for any point of the conic section a relation like (8.1) holds true. This is the first

step toward the solution to the four lines problem. See also (Chasles 1837, 338).

Next Pascal proved the converse (point 2 above) above, most likely in the same way Newton later

did, and completed the proofs of the other steps. According to Taton, the complexity of the coloured

figures mentioned by Mersenne, and which Leibniz found in the sixth part of Pascal’s treatise, is

easily explained by the fact that a complete analysis of the four lines problem is equivalent to making

a precise distinction of the different kinds of conic sections.

If this was the way Pascal proceeded to solve the four lines problem, it does not explain why,

according to Leibniz, he resorted to the cone. Leibniz’s assertion seems to say that Pascal solved the

first step (point 1 above) in the case of the circle, and then he extended it to any conic section by

projection. In this case he would have needed a transformation preserving the parallelism, and thus

he would be limited to the case of the ellipse.

The essential point is, however, that Pascal knew how to bend the solution of Pappus’ celebrated

problem to his method for the projective study of conics, and that this problem, that had served as

Descartes’ main argument to claim the superiority of his analytic geometry, turned out perfectly

accessible to the geometrical methods. It was basically an astonishing demonstration of the power,

and of the fertility of Pascal’s discoveries.

42

9. FINAL REMARKS AND CONCLUSION

Pascal’s main source of inspiration was undoubtedly Desargues’ Brouillon project, as he

acknowledged in the Essay pour les coniques. If Pascal, as was remarked, began to attend the

meetings of Mersenne’s academy in the late 1630s, it is clear that he grasped Desargues’ methods

and results very quickly. We believe this is definitely down to his genius, but also to the fact he was

young and his mind not yet been bent to ancient schemes, but was open to new ideas and suggestions.

We can say that, in writing his treatise on conic sections, the aim Pascal proposed was clear: to

establish the equivalence of the various ways of defining conics so that the easiest or the most suitable

one could be chosen in relation to the situation.

Leibniz found a certain similarity between the Pascalian treatise and La Hire’s Nouvelle méthode.

Indeed, both adopted the projective method, and did not escape Leibniz’s notice that La Hire

recognized how at each point of the base circle there corresponds a point of the ellipse, and that to

tangents correspond tangents, but that for the parabola and the hyperbola there are exceptions. La

Hire, specifically in the Planiconiques appended to the Nouvelle méthode, developed a homological

transformation to study conic sections, and it would be an interesting question to ascertain whether

Pascal did something similar.

La Hire was not, in fact, the possible beneficiary of Pascal’s ideas as Leibniz seemed to have

thought. In the Sectiones conicae (1685), La Hire applied the methods developed in (1673), and

harmonic division, to simplify and unify Apollonius’ proofs; and he seemed more interested in

comparing his work with that of Apollonius (including books V-VII, which had come to light in

1661), rather than producing new results of a projective character (Del Centina and Fiocca 2020).

Unfortunately, Leibniz's exhortation to publish Pascal’s work was not taken up by his heirs, and

the manuscript he returned to them was not preserved as it should have been.

It is surprising that Leibniz was not very interested in spreading the new geometrical methods and

results contained in Pascal’s treatise, except perhaps for some methodological aspects he followed in

the Characteristica geometrica (1677). It should be underlined that Leibniz copied only the first part

of Pascal's manuscript, and his notes reveal an interest in the “limiting geometrical situations”, and

the principle of continuity (Debuiche 2013).

The manuscript of Pascal’s treatise on conic sections, at least the sixth part – which was composed

for this purpose –, circulated among his friends. Carcavi, a good friend of his, was certainly among

those who benefitted from it. In a letter to Christiaan Huygens of 1656, Carcavi illustrated the merits

of Pascal’s work, and showed he was well informed about, “Mister Pascal has also given the

asymptotes of the ellipse, of the circle and of the parabola”.51 In his response, dated May 1st, Huygens

said he had seen nothing of that work, but he also observed that he did not see how it was possible

that these curves had asymptotes.52 On June 22nd, answering to Huygens, Carcavi specified “he

[Pascal] is also the one that noticed the two lines which in the circle and in the ellipse have the same

properties that the asymptotes have in the hyperbola, and whose construction is very similar”, and

added (Fig. 34a), “Let AEB be an ellipse of diameter ACB, and let the tangent at B, DBE, be drawn,

51 “Monsieur Pascal ayant aussi donné les asymptotes de l’ellipse, du cercle et de la parabole”, letter of 20 May 1656

(Huygens 1888, 418-419). 52 Idem, p. 429.

43

where the points D and E are so that the rectangle DB×BE be equal to one fourth of the figure,53 the

straight lines CD, CE will have in these sections the same properties that asymptotes have in the

hyperbola”.54

(a) (b)

Fig. 34 a) Corresponds to Carcavi’s diagram; he drew AB as the major axis. b) It shows an ellipse

and its supplementary hyperbola, whose asymptotes we presume Pascal defined as asymptotes of

the ellipse relative to a diameter.

It is clear that the straight lines to which Carcavi referred are “real”, so we must not confuse them

with the “imaginary” asymptotes of the ellipse as they are intended today. From what Carcavi wrote

it seems that: any diameter (or direction) PP′ being fixed, Pascal associated to it a pair of straight

lines through the centre C of the given ellipse, CD and CE say, with D, E on the tangent to the ellipse

at B, such that P′D×P′E = (QQ′)2 /4, being QQ′ the conjugate diameter of PP′. These straight lines

are exactly the asymptotes of the supplementary conic section,55 a hyperbola in this case (see Fig.

34b), of the given ellipse associated to the diameter PP′ (Poncelet 1822, No. 54). It is worth

mentioning that any supplementary conic section of a circle is an equilateral hyperbola, and that it

was while studying this situation that Poncelet discovered that all circles have a common ideal secant

at infinity (Poncelet 1822, No. 95). We wonder whether Pascal adopted more general transformations

than simple projections. If, as it was, Pascal conceived the asymptotes of the ellipse as above, he

could do so only by bearing in mind the principle of continuity, and a concept of homologous figures

like that introduced by Poncelet in (1822). We shall not go any further with such speculations, except

to say that all this shows that in his treatise on conic sections Pascal could have anticipated topics that

Poncelet treated in (1822).

As was observed by Taton (1963, 248), Leibniz came back to the question of the publication of

Pascal’s treatise in a letter to the scholar Gilles Filleau des Billets in 1692, and in 1696. In both letters

Leibniz, reiterating his very positive judgement on the value of Pascal’s work, he complains that it

53 For any diameter PP′, that the rectangle BD×DE “is one fourth of the figure” means that it is equal to the square of half

the conjugate diameter QQ′, that is equal to CQ2. See (Heath 1896, clxiii). 54 “Soit l’Ellipse AEB, dont le diamètre soit ACB, et soit menée la touchante DBE, coupée en D et E, en sorte que le

rectangle DBE, soit esgal au quart de la figure, ayant mené les lignes CD, CE, du centre C elles auront les mesmes

proprietes dans ces deux sections que les asymptotes dans l’hyperbole”, (Huygens 1888, 433). 55 For more details on these arguments see (Del Centina 2016a, 22-26), (Del Centina, Fiocca 2018, 170-171).

44

has not yet been published, for “these ideas [of Pascal] which have already been communicated to

others will not fail to be published by others”.56

Thus, it is certain, Pascal’s work on conic section was read by others besides Leibniz and

Tschirnhaus, before and after 1676. However, nobody seems to have benefitted from it at least in

France. If, as it seems, Tschirnhaus sent a detailed analysis of Pascal’s treatise to Oldenburg (Gerhardt

1892, 195), also English mathematicians may have benefitted of Pascal’s ideas, and it would be an

interesting question to ascertain if this was really true. What it certain is that it was in Great Britain

before anywhere else, that the hexagon theorem, and the organic construction in Pascal’s style of the

conic section passing through five given points, resurfaced. But this seems to be due mostly to

Newton’s influence rather than Pascal’s.

In the historical introduction to his treatise of 1822, Poncelet gave some information on the

geometrical achievements of Pascal, extracting from what Bossut had written when editing Pascal’s

works. Poncelet quoted the Essay pour le coniques, “notice trés-courte”, but very remarkable for the

use Pascal made of central projection. “This essay”, wrote Poncelet, “contains the property of the

hexagon inscribed into a conic section, ascribed to Desargues by Descartes (sic), and that later Pascal

used under the name of mystic hexagon, in an unedited treatise on conic sections, that Leibniz had in

his hands”. Then Poncelet continued by saying that this work must have been much more extensive

than that of Desargues on the same subject, and have contained “the most beautiful projective

properties of conic sections, today known to geometers… as simple corollaries of the mystic

hexagram, which is just an extension of the porism of Pappus, or rather of Euclid, on the hexagon

inscribed into the angle formed by two straight lines”.

Poncelet remarked that only at the beginning of the following century, from the 1720 to 1750,

while resuming the work of La Hire (sic), did Maclaurin re-discover, without knowing the works of

Desargues and Pascal, the property of the mystic hexagram.

Poncelet also ascribed to Brianchon (1817) the determination of a conic section subject to be

tangent to certain straight lines and to pass through given points, a problem to which, he said,

“Maclaurin, Braikenridge, and Simson, had given only partial solutions, but that it seemed Pascal had

completely solved in his writings”.

According to Chasles (1837, 69-75), the most salient of Pascal’s discoveries, which he masterfully

used, was the beautiful theorem of the mystic hexagram, “that property of every hexagon inscribed

in a conic section to have the three intersection points of the opposite sides always lying on a straight

line”. Since five points determine a conic section, observed Chasles, this theorem links the position

of any sixth point on the curve to the position of the other five; therefore, this is a fundamental

property of the conic sections.

In fact, in relation to the third proposition of the Essay pour les coniques, Chasles wrote (1837,

72):

The proposition, expressed by a double equality of ratios, contains two different propositions. The first is

the 129th [Lemma XIII] of book VII of Pappus’ Mathematical collections, which gave us the opportunity

to introduce the notion of cross-ratio, which, as we have already said, can be placed at the basis of a large

part of recent geometry; the second is the theorem of Ptolemy on the triangle cut by a transversal.

56 Draft of letter dated 2-13 July 1692, Library of Hanover, Leibniz’s Correspondence; quoted in (Taton 1963, 248,

251).

45

Next, on page 73, Chasles added:

We remark that each of these different main theorems [the propositions of the Essay] expresses a certain

property of six points on a conic section; this explains how Pascal was able to deduce them from his mystic

hexagram, which is itself a general property of these five points. But each of these theorems has taken a

different form, making it suitable to particular uses which include a huge number of properties of conic

sections. It is this useful art of deducing from a single principle a large number of truths, of which the

writings of the ancients offer no examples, which makes the advantage of our methods over theirs.

In the Aperçu historique, Chasles showed the equivalence among his main proposition (the

projective invariance of the cross-ratio) and the theorems of Desargues and Pascal, and also showed

how they could be applied to solve the Pappus four lines problem (Chasles 1837, Note XV).

Although our analysis is based on fragmentary information, we hope to have contributed some

new element by demonstrating the novelty, breadth, and depth of Pascal's geometrical work on conic

sections. We are deeply convinced that if the Conicorum opus completum had been published during

Pascal’s lifetime, or even in 1676 as Leibniz whished, it would have changed the history of projective

geometry, allowing its birth a century and a half earlier.

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Andrea Del Centina, Dipartimento di Matematica e Informatica, Università di Ferrara, via

Machiavelli 30, 44100 Ferrara, Italy, e-mail: cen@unife.it