Notes on the History of Geometrical Ideas

Summary

Homogeneous coordinates

It is agreed, even by those who disparage them (see [6], p. 712) that barycentric coordinates, first introduced by August Ferdinand Möbius [4] in 1827, were the first homogeneous coordinates systematically used in geometry. The Möbius idea, in plane geometry for example, is to attach masses p, q and r, respectively, to three non-collinear points A, B and C in the plane under consideration, and then to consider the centroid P = _pA + _q + _rC of the three masses. The point P necessarily lies in the plane, and varies as the ratios p : q : r vary. As Möbius points out:

And conversely, given any point P in the plane, the ratios p : q : r are always and uniquely determinable.

It will be noted that Möbius was using position vectors for his points in 1827, and reading of the text shows that he developed all the techniques of homogeneous coordinates known nowadays, changing the simplex of reference, if necessary, and so on. For a more accessible account, see Section 4.2 of [5]. Nobody has suggested that there is a better system of coordinates for projective geometry.

But new ideas are not always easily accepted. All the same, it is strange nowadays to read some of Cauchy's criticisms (XI of [4]). He says:

only by deeper study can one decide whether the advantages of this method outweigh its difficulties …