Synthetic Geometry

The conflict between geometry and analysis which arose near the close of the last century and the beginning of the present has now come to an end. Neither side has come out victorious. The greatest strength is found to lie, not in the suppression of either, but in the friendly rivalry between the two, and in the stimulating influence of the one upon the other. Lagrange prided himself that in his Mécanique Analytique

he had succeeded in avoiding all figures; but since his time mechanics has received much help from geometry.

Modern synthetic geometry was created by several investigators about the same time. It seemed to be the outgrowth of a desire for general methods which should serve as threads of Ariadne to guide the student through the labyrinth of theorems, corollaries, porisms, and problems. Synthetic geometry was first cultivated by Monge, Carnot, and Poncelet in France;

it then bore rich fruits at the hands of Möbius and Steiner in

Germany and Switzerland, and was finally developed to still

higher perfection by Chasles in France, von Staudt in Germany,

and Cremona in Italy.

Augustus Ferdinand Möbius (1790–1868) was a native of

Schulpforta in Prussia. He studied at Göttingen under Gauss, also at Leipzig and Halle. In Leipzig he became, in

1815, privat-docent, the next year extraordinary professor of astronomy, and in 1844 ordinary professor. This position he held till his death. The most important of his researches are on geometry. They appeared in Crelle's Journal, and in his celebrated work entitled Der Barycentrische Calcul, Leipzig, 1827. As the name indicates, this calculus is based upon properties of the centre of gravity.58 Thus, that the point $S$ is the centre of gravity of weights $a$, $b$, $c$, $d$ placed at the points $A$, $B$, $C$, $D$ respectively, is expressed by the equation $(a+b+c+d)S=aA+bB+cC+dD.$ His calculus is the beginning of a quadruple algebra, and contains the germs of Grassmann's marvellous system. In designating

segments of lines we find throughout this work for the first time consistency in the distinction of positive and negative by the order of letters $AB$, $BA$. Similarly for triangles and tetrahedra. The remark that it is always possible to give three points $A$, $B$, $C$ such weights $\alpha$, $\beta$, $\gamma$ that any fourth point $M$ in their plane will become a centre of mass, led Möbius to a new system of co-ordinates in which the position of a

point was indicated by an equation, and that of a line by co-ordinates. By this algorithm he found by algebra many geometric theorems expressing mainly invariantal properties,–-for example, the theorems on the anharmonic relation. Möbius wrote also on statics and astronomy. He generalised spherical trigonometry by letting the sides or angles of triangles

exceed $180°$.