Euclidean Geometry, (4) Text-books on Elementary Geometry. The first of these divisions has reference to modern synthetic methods of research, the second division refers to new theorems in elementary geometry, the third considers the modern conceptions of space and the several geometries resulting therefrom, the fourth discusses questions pertaining to geometrical teaching.

I. Modern Synthetic Geometry. It was reserved for the genius of Gaspard Monge (1746-1818) to bring modern synthetic geometry into the foreground, and to open up new avenues of progress. To avoid the long arithmetical computations in connection with plans of fortification, this gifted engineer substituted geometric methods and was thus led to the creation of descriptive geometry as a distinct branch of science. Monge was professor at the Normal School in Paris during the four months of its existence, in 1795; he then became connected with the newly established Polytechnic School, and later accompanied Napoleon on the Egyptian campaign. Among the pupils of Monge were Dupin, Servois, Brianchon, Hachette, Biot, and Poncelet. Charles Julien Brianchon, born in Sèvres in 1785, deduced the theorem, known by his name, from "Pascal's Theorem" by means of Desargues' properties of what are now called polars.1 Brianchon's theorem says: "The hexagon formed by any six tangents to a conic has its opposite vertices connecting concurrently." The point of meeting is sometimes called the "Brianchon point.

Lazare Nicholas Marguerite Carnot (1753–1823) was born at Nolay in Burgundy. At the breaking out of the Revolution he threw himself into politics, and when coalesced Europe, in 1793, launched against France a million soldiers, the gigantic

1 Brianchon's proof appeared in "Memoirs sur les Surfaces courbes du second Degré," in Journal de l'Ecole Polytechnique, T. VI., 297-311, 1806. It is reproduced by TAYLOR, op. cit., p. 290.

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task of organizing fourteen armies to meet the enemy was achieved by him. He was banished in 1796 for opposing Napoleon's coup d'état. His Geometry of Position, 1803, and his Essay on Transversals, 1806, are important contributions to modern plane geometry. By his effort to explain the meaning of the negative sign in geometry he established a "geometry of position" which, however, is different from Von Staudt's work of the same name. He invented a class of general theorems on projective properties of figures, which have since been studied more extensively by Poncelet, Chasles, and others.

Jean Victor Poncelet (1788–1867), a native of Metz, engaged in the Russian campaign, was abandoned as dead on the bloody field of Krasnoi, and from there taken as prisoner to Saratoff. Deprived of books, and reduced to the remembrance of what he had learned at the Lyceum at Metz and the Polytechnic School, he began to study mathematics from its elements. Like Bunyan, he produced in prison a famous work, Traité des Propriétés projectives des Figures, first published in 1822. Here he uses central projection, and gives the theory of "reciprocal polars." To him we owe the Law of Duality as a consequence of reciprocal polars. As an inde pendent principle it is due to Joseph Diaz Gergonne (17711859). We can here do no more than mention by name a few of the more recent investigators: Augustus Ferdinand Möbius (1790-1868), Jacob Steiner (1796-1863), Michel Chasles (1793– 1880), Karl Georg Christian von Staudt (1798-1867). Chasles introduced the bad term anharmonic ratio, corresponding to the German Doppelverhältniss and to Clifford's more desirable cross-ratio. Von Staudt cut loose from all algebraic formulæ and from metrical relations, particularly the metrically founded cross-ratio of Steiner and Chasles, and then created a geometry of position, which is a complete science in itself, independent of all measurement.

We can

II. Modern Geometry of the Triangle and Circle. not give a full history of this subject, but we hope by our remarks to interest a larger circle of American readers in the recently developed properties of the triangle and circle.1 Frequently quoted in recent elementary geometries is the "ninepoint circle." In the triangle ABC, let D, E, F be the middle points of the sides, let AL,

BM, CN be perpendiculars to the sides, let a, b, e be the middle points of AO, BO, CO, then a circle can be made to pass through the points, L, D, c, E, M, a, N, F, b; this circle is the "nine-point

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circle." By mistake, the earliest discovery of this circle has been attributed to Euler. There are several independent discoverers. In England, Benjamin Bevan proposed in Leybourn's Mathematical Repository, I., 18, 1804, a theorem for proof which practically gives us the nine-point circle.

1 A systematic treatise on this subject, which we commend to students is A. EMMERICH's Die Brocardschen Gebilde, Berlin, 1891. Our historical notes are taken from this book and from the following papers: JULIUS LANGE, Geschichte des Feuerbachschen Kreises, Berlin, 1894; J. S. MACKAY, History of the nine-point circle, pp. 19-57, Early history of the symmedian point, pp. 92-104, in the Proceed. of the Edinburgh Math. Soc., Vol. XI., 1892-93. See also MACKAY, The Wallace line and the Wallace point in the same journal, Vol. IX., 1891, pp. 83-91; E. LEMOINE's paper in Association française pour l'avancement des Sciences, Congrès de Grenoble, 1885; E. VIGARIÉ, in the same publication, Congrès de Paris, 1889. The progress in the geometry of the triangle is traced by VIGARIÉ for the year 1890 in Progreso mat. I., 101–106, 128-134, 187-190; for the year 1891 in Journ. de Math. élém., (4) 1. 7-10, 34-36. Consult also CASEY, Sequel to Euclid.

2 MACKAY, op. cit., Vol. XI., p. 19.

The proof was supplied to the Repository, Vol. I., Part 1, p. 143, by John Butterworth, who also proposed a problem, solved by himself and John Whitley, from the general tenor of which it appears that they knew the circle in question. to pass through all nine points.

These nine points are explic

itly mentioned by Brianchon and Poncelet in Gergonne's Annales de Mathématiques of 1821. In 1822, Karl Wilhelm Feuerbach (1800-1834) professor at the gymnasium in Erlangen, published a pamphlet in which he arrives at the nine-point circle, and proves that it touches the incircle and the excircles. The Germans called it "Feuerbach's Circle." Many demonstrations of its characteristic properties are given in the article above referred to. The last independent discoverer of this remarkable circle, so far as known, is F. S. Davies, in an article of 1827 in the Philosophical Magazine, II., 29-31.

In 1816 August Leopold Crelle (1780–1855), the founder of a mathematical journal bearing his name, published in Berlin a paper dealing with certain properties of plane triangles. He showed how to determine a point inside a triangle, so that the angles (taken in the same order) formed by the lines joining it to the vertices are equal.

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In the adjoining figure the three marked angles are equal. If the construction be made so as to give angle 'AC = 'CB = Q'BA, then a second point 'is obtained. The study of the properties of these new angles and new points led Crelle to exclaim: "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be!" Investigations were made also by C. F. A. Jacobi of Pforta and some of his pupils, but after his death, in 1855, the whole matter

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was forgotten. In 1875 the subject was again brought before the mathematical public by H. Brocard, who had taken up this study independently a few years earlier. The work of Brocard was soon followed up by a large number of investigators in France, England, and Germany. The new researches gave rise to an extended new vocabulary of technical terms. Unfortunately, the names of geometricians which have been attached to certain remarkable points, lines, and circles are not always the names of the men who first studied their properties. Thus, we speak of "Brocard points" and "Brocard angles,” but

historical research brought out the fact, in 1884, and 1886, that these were the points and lines which had been studied by Crelle and C. F. A. Jacobi. The "Brocard Circle," is Brocard's own creation. In the triangle ABC, let 2 and ' be the first and second "Brocard point." Let A' be the intersection of BQ and C''; B', of AQ' and CN; C', of BQ' and AN. The circle passing through A', B', C" is the "Brocard circle." A'B'C' is "Brocard's first triangle." Another like triangle, A"B"C" is called "Brocard's second triangle." The points A", B", C" together with 2, ', and two other points, lie in the circumference of the "Brocard circle."