I fear that brevity and compression have been but too much studied in the following essay, but the necessity of comprising the whole matter in a small compass, and the pressure of other avocations, will plead, I hope, a sufficient apology.
From the same cause I have been obliged to omit altogether subjects which might have been with propriety introduced, for example, the general theory of shadows ; and have only touched upon others which would require perhaps further development.
Among other applications of the method, I trust that to the theory of reciprocal polars will be found simple and satisfactory.
My attention has been just directed by a friend to a letter from M Chasles, dated December 10, 1829, published in the Correspondence Mathématique of M Quetelet, tom. vi. p. 81, in which the writer asserts his claim to the invention of a system of coordinates, noticed by M Plucker in one of the livraisons of Crelle's Journal, to which work I have never had an opportunity of referring. After some preliminary observations, he states his system as follows:- "Pour cela, par trois points fixes A, B, C, je mène trois axes parallèles entre eux, un plan quelconque rencontre ces axes en trois points dont les distances aux points A, B, C, respectivement, sont les coordonnées x, y, z, du plan," &c.; and then goes on to apply his system to a few examples, using the principles and notation of the differential calculus. To any one consulting the letter from which the above extract is taken, it will be apparent that the method there proposed, however excellent and ingenious it may be, bears not the least resemblance to the one developed in the following pages.
Some valuable improvements in the notation I have adopted, have been suggested by the Reverend Charles Graves, Fellow Trinity College Dublin, of which I have thankfully availed myself.
Trinity College, Dublin,
March 25th, 1840.