Atticus Rare Books

On a New Geometry of Space [Memoir] in Philosophical Transactions of the Royal Society of London 155, Part II, Received December 22, 1864; Read February 2, 1865, pp. 725-791 WITH On a New Geometry of Space [Abstract] Proceedings of the Royal Society of London 14, No. 72, Received December 22, 1864; Read February 2, 1865, pp. 53-58

Plücker, Julius [Plucker; Pluecker]

1865. 1st Edition. FIRST EDITIONS, FIRST PRINTINGS, BOTH IN ORIGINAL WRAPS, OF TWO PAPERS BY PLÜCKER RECONSTRUCTING ANALYTICAL GEOMETRY, “HERALDING THE ERA OF LINE AND PROJECTIVE GEOMETRY” and introducing the Plücker coordinates which abandoned the use of Cartesian coordinates in a plane in favor of a system of three coordinates to identify a point — a geometry based on points as basic elements (DSB, XI, 46; MAA). Plücker coordinates created “a way to assign six homogeneous coordinates to each line in projective three dimensional space” (Wikipedia). Particularly gifted at obtaining the maximum amount of geometric information from his equations, Plücker’s work employed creative and unusually useful computational properties that ultimately made it exceptionally well-suited to solving a wide array of computational three dimensional problems, most especially those involving visibility and light transfer.

The papers, titled exactly alike, appeared in two different journals, each received on December 22, 1864; each read on February 22, 1865. They are not, however, the same. The 6 page paper in the Proceedings is known as the ‘abstract’; the 66 page paper in the Phil Trans is known as the ‘memoir’. Plücker’s work was later republished in 1868 in Neue Geometrie; these two papers, however, represent his first and most important publications on analytic and projective geometry.

Julius Plücker was a German mathematician and physicist. [Plücker also discovered cathode rays and we offer that work separately.]

Plücker's work in these papers bases “space geometry upon the self-dual straight line as element, rather than upon the point or in dual manner upon the plane as element” (DSB). He argues that the dimensionality of space depends on our choice of the elements (points, lines, planes, etc.) from which space is construed. Plücker also points out that a right line may be construed in two different ways and that, accordingly, there are two constructions of space: in one, space is traversed by lines determined by planes passing through them. The first is used in optics when luminous points are assumed to send rays in all directions, the second when instead of rays one considers wave-fronts and their consecutive intersections” (Giedymin, Science, 64; Plücker, 1865).

“The leading idea of Plücker’s memoir appears in the first words of his [Phil Trans paper]: “I. On linear Complexes of Right lines”. He works at first with the 4 coordinates of a line; as long as these are arbitrary, the line is any line whatever; but considering them as connected by a single equation, then he says ‘a Complex’, considering them as connected by two equations ‘a Congruency’, and considering them as connected by 3 equations ‘a Configuration’ or ruled surface” (Cayley, Papers, 618).

It is worth noting that because Plücker coordinates “satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in P3 and points on a quadric in P5 (projective 5-space)” (Wikipedia) Though debatable, in Plücker’s work “the dimensions of the space of lines is four and it is probably the first four-dimensional space that appeared in science” (Gindikin, Mathematicians and Physicists, 370).

Also in Phil Trans: Cayley “On the Sextactic Points of a Plane Curve” and a paper by Spottiswoode with the same title; Roscoe’s “On a Method of Meteorological Registration of the Chemical Action of Total Daylight”; Harley’s “On the Influence of Physical and Chemical Agents upon Blood”.

Also in Proceedings: De La Rue, “Researches in Solar Physics”. Item #970

CONDITION & DETAILS: The Proceedings paper is an individual issue in original wrappers. Some sunning to edges, slight edge wear. Overall, very good. The Phil Trans paper is also in original wrappers, Part II of Vol. 155. Slight spotting of edges and plates; slight sunning of spine. Small bend in lower front wrapper corner. Near fine.

Price: $1,200.00