The purpose of the present note is to show how the student who is acquainted with the methods only of Using and transforming rectangular axes is already in a position to determine the geodesic curvature and torsion of any curve traced on a surface, and in particular to prove Gauss’ formula for the product of the principal curvatures in terms of the geodesic curvaturesof the lines of curvature.

I should like at the outset to disclaim any intention of laying down the law dogmatically. The opinions expressed are stated definitely, but they are submitted with deference for your consideration.

The tripolar coordinates of a point P are defined as the squares of its distances from the angular points of the triangle of reference.

The equation of the first degree in x, y, z (x = PA2, etc.) has been considered by E. Lucas (Mathesis, 1889) and by de Poulain (Journal de Mathématiques spéciales, 1889). I am also indebted to Mr. R. F. Davis for a note in accordance with which a portion of this paper has been rewritten.

In what follows x, y, z will denote the tripolar coordinates; α, β, γ the trilinear coordinates of a point with respect to a triangle of reference ABC; while ξ and η will denote Cartesian coordinates referred to B as origin and BC as the axis of x.

Having accepted with some mental reservation the assurance of the Editor of this Gazette that the Slide Rule, although invented by Gunter as far back as 1624, is still unknown to many mathematicians, the writer has ventured to make this article explanatory and descriptive. To such then as are unenlightened he offers the good news that half-a-guinea will purchase an instrument capable of multiplying and dividing, of squaring and cubing, of extracting square and cube roots, of solving triangles and performing rapidly many similar combinations of arithmetic and trigonometric operations, all without any mental effort and to an accuracy of from ⅕ to ⅓ %. To examiners having to reduce marks to percentages the Slide Rule is invaluable, to physicists it is indispensable, to engineers it is both invaluable and indispensable.

An inhabitant of Flatland would not be in the same position with regard to elementary geometry as a three-dimensioned being. He could only use superposition for figures of the same aspect. I have not met with any discussion of the way in which he would prove, say, Euc. I. 5; the following seems complicated, but at least it shews that the scope of the usual assumption as to superposition can be made narrower.

The equation of the curve of order 14, on which lie the points of contact of the 28 bitangents of a quartic curve, was first given by Hesse (Crelle’s Journal, Vols. 36, 40, 41). A very concise form of the equation in the Clebsch-Aronhold symbols was obtained by Dersch (Math. Ann., Bd. 7). The method adopted in his Memoir was practically that given by Salmon (Phil. Mag., Oct. 1858), and followed by Cayley (Phil. Trans., 1859).