The following are the principles on which this equality of wear depends: The rapidity of wear of a surface measured in an oblique direction is to the rapidity of wear measured normally as the **secant** of the obliquity is to unity.

Fermat and Descartes agreed in regarding the tangent to a curve as a **secant** of that curve with the two points of intersection coinciding, while Roberval regarded it as the direction of the composite movement by which the curve can be described.

4) that Bessel had indicated, by notes in his handbooks, the following points which should be kept in mind in the construction of future heliometers: (I) The segments should move in cylindrical slides; b (2) the screw should be protected from dust; 6 (3) the zero of the position circle should not be so liable to change; 7 (4) the distance of the optical centres of the segments should not change in different position angles or otherwise; 8 (5) the points of the micrometer screws should rest on ivory plates; 9 (6) there should be an apparatus for changing the screen.'° Wilhelm Struve, in describing the Pulkowa heliometer,' 1 made The distances of the optical centres of the segments from the eye-piece are in this method as I; **secant** of the angle under measurement.

As far as the circlesquaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the **secant** in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries 1 See Euler, ” Annotationes in locum quendam Cartesii," in Nov.

129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the **secant** of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional.

DEF, is termed a " **secant** "; if it touches the circle, e.g.