The Beginning of Modern Synthetic Geometry

About the beginning of the seventeenth century the first decided advance, since the time of the ancient Greeks, was made in Geometry. Two lines of progress are noticeable: (1) the analytic path, marked out by the genius of Descartes, the inventor of Analytical Geometry; (2) the synthetic path, with the new principle of perspective and the theory of transversals. The early investigators in modern synthetic geometry are Desargues, Pascal, and De Lahire.

Girard Desargues (1593-1662), of Lyons, was an architect and engineer. Under Cardinal Richelieu he served in the siege of La Rochelle, in 1628. Soon after, he retired to Paris, where he made his researches in geometry. Esteemed by the ablest of his contemporaries, bitterly attacked by others unable to appreciate his genius, his works were neglected and forgotten, and his name fell into oblivion until, in the early part of the nineteenth century, it was rescued by Brianchon and Poncelet. Desargues, like Kepler and others, introduced the doctrine of infinity into geometry.1 He states that the straight line may be regarded as a circle whose centre is at infinity; hence, the two extremities of a straight line may be considered as meeting at infinity; parallels differ from other pairs of lines only in having their points of intersection at infinity. He gives the theory of involution of six points, but his definition of “involution" is not quite the same as the modern definition, first found in Fermat,' but really introduced into geometry by Chasles. On a line take the point A as origin (souche), take also the three pairs of points B and H, C and G, D and F; then, says Desargues, if AB. AH AC AG

=

1 CHARLES TAYLOR, Introduction to the Ancient and Modern Geometry CANTOR, II., 606, 620.

of Conics, Cambridge, 1881, p. 61.

2

3 Consult CHASLES, Note X.; MARIE, III., 214.

= AD · AF, the six points are in "involution." If a point falls on the origin, then its partner must be at an infinite distance from the origin. If from any point P lines be drawn through the six points, these lines cut any transversal MN in six other points, which are also in involution; that is, involution is a projective relation. Desargues also gives the theory of polar lines. What is called "Desargues' Theorem" in elementary works is as follows: If the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a

point, then their sides meet in three points lying on a line, and conversely. This theorem has been used since by Brianchon, Sturm, Gergonne, and others. Poncelet made it the basis of his beautiful theory of homological figures.

Although the papers of Desargues fell into neglect, his ideas were preserved by his disciples, Pascal and Philippe de Lahire. The latter, in 1679, made a complete copy of Desargues' principal research, published in 1639. Blaise Pascal (1623–1662) was one of the very few contemporaries who appreciated the worth of Desargues. He says in his Essais pour les coniques, "I wish to acknowledge that I owe the little that I have discovered on this subject to his writings." Pascal's genius for geometry showed itself when he was but twelve years old. His father wanted him to learn Latin and Greek before entering on mathematics. All mathematical books were hidden out of sight. In answer to a question, the boy was told by

his father that mathematics "was the method of making figures with exactness, and of finding out what proportions they relatively had to one another." He was at the same time forbidden to talk any more about it. But his genius could not be thus confined; meditating on the above definition, he drew figures with a piece of charcoal upon the tiles of the pavement. He gave names of his own to these figures, then formed axioms, and, in short, came to make perfect demonstrations. In this way he arrived, unaided, at the theorem that the angle-sum in a triangle is two right angles. His father caught him in the act of studying this theorem, and was so astonished at the sublimity and force of his genius as to weep for joy. The father now gave him Euclid's Elements, which he mastered easily. Such is the story of Pascal's early boyhood, as narrated by his devoted sister.1 While this narrative must be taken cum grano salis (for it is highly absurd to suppose that young Pascal or any one else could re-discover geometry as far as Euclid I., 32, following the same treatment and hitting upon the same sequence of propositions as found in the Elements), it is true that Pascal's extraordinary penetration enabled him at the age of sixteen to write a treatise on conics which passed for such a surprising effort of genius that it was said nothing equal to it in power had been produced since the time of Archimedes. Descartes refused to believe that it was written by one so young as Pascal. This treatise was never published, and is now lost. Leibniz saw it in Paris, recommended its publication, and reported on a portion of its contents. However, Pascal published in 1640, when he was sixteen years old, a small geometric treatise of six octavo

1 The Life of Mr. Paschal, by MADAM PERIER. Translated into English by W. A., London, 1744.

2 See letter written by Leibniz to Pascal's nephew, August 30, 1676, which is given in Oeuvres complètes de Blaise Pascal, Paris, 1866, Vol. III.,

pages, bearing the title, Essais pour les coniques. Constant application at a tender age greatly impaired Pascal's health. During his adult life he gave only a small part of his time to the study of mathematics.

Pascal's two treatises just noted contained the celebrated proposition on the mystic hexagon, known as "Pascal's Theorem," viz. that the opposite sides of a hexagon inscribed in a conic intersect in three points which are collinear. In our elementary text-books on modern geometry this beautiful theorem is given in connection with a very special type of a conic, namely, the circle. As, in one sense, any two straight lines may be looked upon as a special case of a conic, the theorem applies to hexagons whose first, third, and fifth vertices are on one line, and whose second, fourth, and sixth vertices are on the other. It is interesting to note that this special case of "Pascal's Theorem" occurs already in Pappus (Book VII., Prop. 139). Pascal said that from his theorem he deduced over 400 corollaries, embracing the conics of Apollonius and many other results. Pascal gave the theorem on the cross ratio, first found in Pappus. This wonderfully fruitful theorem may be stated as follows: Four lines in a plane, passing through one common point, cut off four segments on a transversal which have a fixed, constant ratio, in whatever manner the transversal may be drawn; that is, if the transversal cuts the rays in the points A, B, C, D, then the AC BC ratio formed by the four segments AC, AD, BC, AD BD'

:

1

BD, is the same for all transversals. The researches of Desargues and Pascal uncovered several of the rich treasures pp. 466-468.

The Essais pour les coniques is given in Vol. III., pp. 182-185, of the Oeuvres complètes, also in Oeuvres de Pascal (The Hague, 1779) and by H. WEISSENBORN in the preface to his book, Die Projection in der Ebene, Berlin, 1862.

1 Book VII., 129. Consult CHASLES, pp. 31, 32.

of modern synthetic geometry; but owing to th interest taken in the analytical geometry of Des later, in the differential calculus, the subject was tirely neglected until the close of the eighteenth ce

Synthetic geometry was advanced in England searches of Sir Isaac Newton, Roger Cotes (1682Colin Maclaurin, but their investigations do not c the scope of this history. Robert Simson and Matt (1717-1785) exerted themselves mainly to revive ometry. An Italian geometer, Giovanni Ceva (16 deserves mention here; a theorem in elementary bears his name. He was an hydraulic engineer, a was several times employed by the government His death took place during the siege of Mantua, in ranks as a remarkable author in economics, bein clear-sighted mathematical writer on this subject. published in Milan a work, De lineis rectis. Thi "Ceva's Theorem" with one static and two geomet

tilinear figures are proved by considering the pro the centre of inertia (gravity) of a system of points.

Modern Elementary Geometry

We find it convenient to consider this subject following four sub-heads: (1) Modern Synthetic (2) Modern Geometry of the Triangle and Circle, 1 PALGRAVE'S Dict. of Political Econ., London, 1894 2 CHASLES, Notes VI., VII.