The graphic methods in use are open to the objection of requiring an improvised graph, depending on the constants of the equation. To obtain even a 5 % approximation, the construction of such a graph demands calculations far in excess of those involved in Horner’s method.

The following notes are a development of subject-matter which is treated in the writer’s book on Cross-Ratio Geometry, chiefly in Chap. XIV., and the reader is referred to articles in that work for the demonstration of any property that is given here without proof.

Convergents to tan nθ.

If the numerator and denominator of the fraction are both convergent series, one value of tan nθ is

While gathering the materials for something in the nature of a ‘Companion’ to De Morgan’s “Budget of Paradoxes,” I came across a considerable number of his notes, short papers, and other contributions to the famous English periodical known as “Notes and Queries.” The first number appeared on Nov. 3, 1849, and from that date “N. & Q.” has been a household word among those who are afflicted with the insatiable desire to penetrate into the bye-ways of Literature, Genealogy, Heraldry, Antiquarian Lore, the Science of History and the History of Science—all formed grist for those who came to that mill. By the kind permission of the Editor of this invaluable publication, I have been permitted to reproduce from its columns what is presented to the reader in the following pages. The early numbers of “N. & Q.” are exceedingly scarce, and a complete set is hardly to be obtained by those who live at a distance from the great libraries.