*CSHPM Notes brings scholarly work on the history and philosophy of mathematics to the broader mathematics community. Authors are members of the Canadian Society for History and Philosophy of Mathematics (CSHPM). Comments and suggestions are welcome; they may be directed to the column’s co-editors:*

**Amy Ackerberg-Hastings**, *independent scholar (aackerbe@verizon.net)*

**Hardy Grant***, York University **[posthumous]*

The nature of geometry changed dramatically between 1800 and 1855. My interest in that topic and time period began a number of years ago, as my enthusiasm for geometry, particularly of the synthetic type, merged with an even older enthusiasm for history. Following geometry through the centuries, I saw an interesting story line emerging. Ancient Greeks envisioned geometric figures as stationary, and they showed almost no interest in propositions about collinear points or concurrent lines. The exceptions are Menelaus’s Theorem and various propositions from Pappus, but those ideas emerged only in the late ancient period.

Coherent work involving movement of a sort, by plane-to-plane projection, together with incipient projective geometry and propositions about collinear points and concurrent lines, appeared in a burst in the 17th century, with Girard Desargues [6; 7], Blaise Pascal [15], and Philippe de La Hire [9]. (It is worth noting that Desargues made great use of Menelaus’s Theorem.) But, for lack of interest, the publications by these authors that were the most projective in approach were soon lost or forgotten.

Around 1800, the atmosphere changed. After the publication of Adrien-Marie Legendre’s *Éléments de géométrie* of 1794 [10], which would have felt familiar to Greek geometers of two millennia earlier, Gaspard Monge [14] and Lazare Carnot [4; 5] presented a variety of propositions involving collinear points and concurrent lines. Forgotten propositions, including Desargues’s Theorem and Pascal’s Hexagon Theorem, were rediscovered. They appeared at first without credit to their discoverers, and proofs were not projective, but in 1822 J. V. Poncelet [20] would prove them by projective methods and properly note their discoverers. Of particular importance to Poncelet was an 1810 article by Charles Julien Brianchon [3] in which he solved several problems by a plane-to-plane projection that let one line of points “pass to infinity,” as Poncelet put it.

**Figure 1.** The first two pages of Brianchon’s article [3]. Gallica.

This began a flood of geometric transformations and relationships, called *Verwandtschaften* by A. F. Moebius in his *Der Barycentrische Calcul* [11]. That 1827 work gave the first overview of transformations. Moebius examined four different geometric relationships, starting with *equality*, *similitude*, and *affinity*. *Similitude*, now called *dilation*, the topic of Euler’s E693 [8], was thoroughly developed by Poncelet in 1813, 1820, and 1822 [18; 19; 20]. Moebius’s fourth transformation was new and the most general, the *collineation*. He set his transformations in a plane whose elements were real triples, except \((0, 0, 0)\), with the homogeneity property, namely, that for \(k \ne 0\) triples \((a, b, c)\) and \((ka, kb, kc)\) represent the same point. Moebius demonstrated a remarkable property, that given any four points \(A, B, C, D\), no three collinear, then any point \(P\) could be represented as \(aA + bB +cC + dD\), and for any choice of four points \(A’, B’, C’, D’\), no three collinear, then \(P’ = aA’ + bB’ +cC’ + dD’\) defines a collineation, and any collineation could be defined this way. (He characterized an *affinity*, a collineation mapping parallel lines to parallel lines, in a corresponding way, with three non-collinear finite points.) With modification to Moebius’s homogeneous coordinate system by Julius Plücker in 1831 [17], the triples form what we now recognize as the *real projective plane*. In that plane, a *collineation* is the same as a *projectivity*.

Plane-to-plane projection was the basis for *homology*, as developed by Poncelet in his 1822 *Traité des Propriétés Projectives des Figures* [20]. The line of intersection of the two planes, the *axis*, is a line of fixed points. Poncelet showed that a homology can operate in a single plane when one of the two planes is rotated about the axis to coincide with the other, but still with a line of fixed points.

The *projective transformation*, in synthetic geometry, was created by Jacob Steiner [23]. He began with *elementary forms*, in particular the *line* of points and the *pencil*, *Strahlbüschel*, of concurrent lines; then related the elementary forms by *perspectivities*; and, finally, defined a *projectivity* as a composition of perspectivities.

Where Poncelet, in 1813, had declared that a conic section was the plane-to-plane projection of a circle, Steiner defined a point conic to be the set of points of intersection of corresponding lines in projectively related pencils. (Poncelet’s *Notebooks*, written as a prisoner in Russia in 1813–1814, were only published in 1862 [21].)

Another transformation, *inversion* (over a circle), anticipated in work of the 1820s, was developed into modern form by Giusto Bellavitis in 1836 [2].

**Figure 2.** Moebius, Poncelet, and Steiner. *Convergence* Portrait Gallery.

Why do I end this period of transformations in 1855? First, I wished to include important works in the line of development sketched above, especially K. G. C. von Staudt’s 1847 *Geometrie der Lage* [24]. Von Staudt placed projective geometry in a more abstract setting; his book, for example, has no figures. The book includes two proofs of Moebius’s theorem that a projectivity is a collineation determined by four points, no three collinear, and their images. Von Staudt committed himself to removing any dependence on length from his geometry. Significantly, the cross-ratio of four collinear points, whose invariance was Steiner’s main identifying property of a projective relation, depends on length, so von Staudt chose an alternative, invariance of the harmonic relation. That works because the *complete quadrilateral*, introduced by Carnot, was a non-metric concept, and it determined a harmonic set of four points. In later years, geometers tended to follow von Staudt’s penchant for abstraction, and many would require that a truly projective geometry be non-metric.

A second important transitional work was Moebius’s second paper [12] on what has come to be called the *Moebius transformation*. Without using the function definition of the *Kreisverwandtschaft*, as he called it, Moebius derived the familiar properties, including the invariance of the cross-ratio among four complex points, not necessarily collinear. He was a pioneer in setting his work in the complex plane.

**Figure 3.** Illustration of inversion described by G. Bellavitis [1836, Art. 20], “Construct a circle which passes through two given points **A B** and is tangent to a given circle \(Db\).” Figure created by the author.

Two notable works signaled a new direction in geometry. The first was Bernhard Riemann’s inaugural lecture at Göttingen in 1854, on the bases of geometry [22]. The second involved von Staudt. While his 1847 *Geometrie der Lage* followed the trajectory of contemporary work in projective geometry, by contrast his *Beiträge zur Geometrie der Lage*, which appeared in three volumes from 1856 to 1859 [25], introduced a complex projective space and an algebra derived from geometric axioms, and it greatly influenced geometry in the coming years.

And where does my work with this material stand? A narrowly focused paper, “Poncelet’s discovery of homology,” was recently published [1]. My current project is a book that combines history with the exposition of the mathematics involved, written with an undergraduate mathematics major in mind. It is the sort of project that retirement makes possible.

*Christopher Baltus earned a PhD at the University of Colorado in 1984, and taught 33 years at SUNY Oswego. He and his wife retired in 2019 and moved to Poughkeepsie, NY. In retirement he still rides a bicycle, reads history and mathematics, and is a volunteer mathematics aide in an elementary school. He notes that this article was written under guidance from Hardy Grant, whose wisdom and kindness will be greatly missed.*

**References**

[1] Baltus, Christopher. (2023) Poncelet’s discovery of homology. *Historia Mathematica *63, 1–20.

[2] Bellavitis, Giusto. (1836) Teoria delle figure inverse, e loro uso nella Geometria elementare. *Annali della Scienze del Regno Lombardo-Veneto *6, 126–141.

[3] Brianchon, Charles Julien. (1810) Solution de plusieurs problémes de géométrie. *Journal de l’École polytechnique*, Cahier 10, Tome 4, 1–15.

[4] Carnot, Lazare. (1803) *Géométrie de Position*. Paris: Duprat.

[5] Carnot, Lazare. (1806) *Essai sur la Théorie des Transversales*. Paris: Courcier.

[6] Desargues, Girard. (1639) *Brouillon project d’une atteinte aux événements des rencontres d’un cône avec un plan*. In [16], translated by Field. Original published in Paris, 1639. http://catalogue.bnf.fr/ark:/12148/cb35944713p.

[7] Desargues, Girard. (1648) Note: Extrait de la perspective de Bosse 1648, et faisant suite à la perspective de Desargues de 1636. In *A Source Book in Mathematics*, edited by David Eugene Smith, ii:307–309. New York: Dover, 1959.

[8] Euler, Leonhard. (1791) De centro similitudinis [E693] (original1777). In *Leonhardi Euleri Opera Omnia *sub ausp. Soc. Scient, Nat. Helv. 1911- Series I, vol. 26, 276–285.

[9] Hire, Philippe de La. (1673) *Nouvelle Méthode en Géométrie pour les Sections des Superficies coniques et Cylindriques*. Paris.

[10] Legendre, A-M. (1794) *Éléments de géométrie*. Paris: Fermin Didot.

[11] Moebius, August Ferdinand. (1827) *Der Barycentrische Calcul: ein neues Hülfsmittel zur analytischen Behandlung der Geometrie*. Leipzig: Barth.

[12] Moebius, August Ferdinand. (1855) Die Theorie der Kreisverwandtschaft in rein geometrischer Darstellung. In [13], vol. II, 243–314.

[13] Moebius, August Ferdinand. (1855–1887) *Gesammelte Werke*. Königlich sächsischen Gesellschaft der Wissenschaften. Leipzig: S. Hirzel.

[14] Monge, Gaspard. (1799) *Géométrie descriptive. Leçons donnée aux Écoles normales, l’an 3 de la République *(1797). Paris: Baudouin.

[15] Pascal, Blaise. (1640) *Essay pour les coniques*, English translation in [16], 180–184.

[16] Field, J. V., and J. J. Gray. (1987) *The Geometrical Work of Girard Desargues*. New York: Springer.

[17] Plücker, Julius. (1831) *Analytisch-Geometrische Entwicklungen*. Vol. 2. Essen: Baedeker.

[18] Poncelet, J. V. (1813) 7 cahiers de 1813–14. In [21], i: 1– 441.

[19] Poncelet, J. V. (1820) Essai sur les propriétés projectives des sections coniques. In [21], ii: 365–441.

[20] Poncelet. J. V. (1822) *Traité des Propriétés Projectives des Figures*. Paris: Bachelier.

[21] Poncelet, J. V. (1862/4) *Applications d’analyse et de géométrie qui ont servi, en 1822, de principal fondement au traité des propriétés projectives des figures, etc**.* 2 vol. Paris: Mallet-Bachelier.

[22] Riemann, Bernhard (1854) Über die Hypothesen welche der Geometrie zu Grunde liegen, lecture of 1854 at the University of Göttingen. In *Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen*, xiii:133–150. 1868.

[23] Steiner, Jacob. (1832) *Systematische Entwicklung der Abhängigkeit geometrischer Gestalten*. Vol. 1. Berlin: G. Fincke. In *Jacob Steiner’s Gesammelte Werke*, edited by K. Weierstrass, i:229–460. Berlin: G. Reimer, 1881.

[24] von Staudt, K. G. C. (1847) *Geometrie der Lage*. Nürnberg: Bauer und Raspe.

[25] von Staudt, K. G. C. (1856) *Beiträge zur Geometrie der Lage*. 3 vol. Nürnberg: Bauer und Raspe.