Although some aspects of the theory of polarity for conics were known to mathematicians of Ancient Greece, the theory originates in projective geom- etry, in the works of G. Desargues, G. Monge and J. Poncelet. For Desargues the polar of a conic was a generalization of the diameter of a circle (when the pole is taken at infinity). He referred to a polar line as a“transversale de l’ordonnance”. According to the historical accounts found in [215], vol. II, and [135], p. 60, the name “polaire” was introduced by J. Gergonne. Apparently, the polars of curves of higher degree appear first in the works of E. Bobillier [45] and then, with introduction of projective coordinates, in the works of J. Pl¨ucker [448]. They were the first to realize the duality property of polars: if a pointxbelongs to thes-th polar of a pointywith respect to a curve of de- greed, then ybelongs to the (d−s)-th polar ofxwith respect to the same curve. Many properties of polar curves were stated in a purely geometric way by J. Steiner [543], as was customary for him, with no proofs. Good historical accounts can be found in [41] and [433], p.279.
The Hessian and the Steinerian curves with their relations to the theory of polars were first studied by J. Steiner [543] who called themconjugate Kern- curven. The current name for the Hessian curve was coined by J. Sylvester
[555] in honor of O. Hesse who was the first to study the Hessian of a ternary cubic [289] under the nameder Determinanteof the form. The current name of the Steinerian curve goes back to G. Salmon [493] and L. Cremona [142]. The Cayleyan curve was introduced by A. Cayley in [72] who called it the pippiana. The current name was proposed by L. Cremona. Most of the popular classical text-books in analytic geometry contain an exposition of the polarity theory (e.g. [114], [215], [493]).
The theory of dual varieties, generalization of Pl¨ucker formulae to arbitrary dimension is still a popular subject of modern algebraic geometry. It is well- documented in modern literature and for this reason this topic is barely touched here.
The theory of apolarity was a very popular topic of classical algebraic ge- ometry. It originates from the works of Rosanes [479] who called apolar forms of the same degreeconjugate formsand Reye [463]. who introduced the term “apolar”. The condition of polarityDψ(f) = 0was viewed as vanishing of the simultaneous bilinear invariant of a formf of degreedand a formψof class
d. It was called theharmonizant.. We refer for survey of classical results to [433] and to a modern exposition of some of these results to [184] which we followed here.
The Waring problem for homogeneous forms originates from a more gen- eral problem of finding a canonical form for a homogeneous form. Sylvester’s result about reducing a cubic form in four variables to the sum of 5 powers of linear forms is one of the earliest examples of solution of the Waring prob- lem. We will discuss this later in the book. F. Palatini was the first who recog- nized the problem as a problem about the secant variety of the Veronese variety [425], [426] and as a problem of the existence of envelopes with a given num- ber of singular points (in less general form the relationship was found earlier by J. E. Campbell [60]). The Alexander-Hirschowitz Theorem was claimed by J. Bronowski [57] in 1933, but citing C. Ciliberto [101], he had only a plausi- bility argument. The casen= 2was first established by F. Palatini [426] and the casen= 3was solved by A. Terracini [561]. Terracini was the first to rec- ognize the exceptional case of cubic hypersurfaces inP4 [560]. The original
proof of Terracini’s Lemma can be found in [562]. We also refer to [241] for a good modern survey of the problem. A good historical account and in depth theory of the Waring problems and the varieties associated to it can be found in the book of A. Iarrobino and V. Kanev [314].
The fact that a general plane quintic admits a unique polar 7-gon was first mentioned by D. Hilbert in his letter to C. Hermite [295]. The proofs were given later by Palatini [428] and H. Richmond [469], [471]
constants to assert that three general quadrics inP3admit a common polar pen-
tahedron. G. Darboux [154] was the fist to show that the counting of constants is wrong. W. Frahm [225] proved that the net of quadrics generated by three quadrics with a common polar pentahedron must be a net of polars of a cubic surface and also has the property that its discriminant curve is a L¨uroth quar- tic, a plane quartic which admits an inscribed pentagon. In [565] E.Toeplitz (the father of Otto Toeplitz) introduced the invariantΛ of three quadric sur- faces whose vanishing is necessary and sufficient for the existence of a com- mon polar pentahedron. The fact that two general plane cubics do not admit a common polar pentagon was first discovered by F. London [367]. The Waring Problem continues to attract attention of contemporary mathematicians. Some references to the modern literature can found in this chapter.