1. Def. The reciprocal (1/N) of any number (N) will be termed a Binal, Tertial, Quintal, ... Decimal, ... r-mal fraction, when expressed in the scale whose radix is r = 2, 3, 4, ... 10, ... r respectively. In this paper the developments will be given for the most part in a general form in the r-scale; the examples will be in the scales of r = 3 and 5. The notation is similar to that of ordinary decimals. Thus (see Tables at end):

1. Giulio Carlo dei Toschi di Fagnano, Count and Marquis, was an Italian mathematician of the first half of the eighteenth century. He was one of the second generation of the followers of Leibniz, a generation rather pale and unimportant compared either with the first great founders of the Differential Calculus or with the third generation that dated from about the middle of the eighteenth century, of which Lagrange was the leading genius.

Several of the men of this second generation are known mainly for some one piece of work. This is particularly the case with Fagnano, who extended the application of the Integral Calculus to the rectification of curves in a way that may be regarded as the first germ from which afterwards sprang the Theory of Elliptic Functions.

Experiencing the well known difficulty of rectifying, by elementary integration, many curves whose equations seem, at first sight, easy to deal with, he showed that on some curves two arcs could be found whose difference was rectifiable ; although each separate arc presented the same difficulty as before.

1. The present paper is an attempt to sketch a mode of procedure for teaching Limits and Convergence to beginners. It is felt by many teachers that at present

(1) the subject is introduced too academically;

(2) that its Arithmetical foundation is not brought into sufficient prominence;

(3) that therefore pupils leave school with only a very hazy notion of what it is all about;

FEW branches of elementary mathematics have escaped reform in recent years; algebra, trigonometry, elementary geometry, statics, and the calculus have all been transformed to suit the practical needs of the time. But amid all this change, there have been few attempts to alter the methods employed in introducing students to the geometry of the conic. Dr. Filon, in his most interesting treatise on Projective Geometry published in 1909, points out that students learn the same facts about the conic practically three times over, (1) analytically, (2) through a course of what is called “geometrical conies” based on the focus-directrix definition, and (3) protectively ; and his book indicates a method of co-ordinating and uniting the last two systems of approach. Although it is undoubtedly the general custom in this country to start in the first place from the focus definition, it is worth while considering afresh whether this plan is really the most advantageous.

It is interesting to note that historically this was not the starting point of the early geometers. It is true that the focus-directrix property was known to Pappus, but so far from being regarded as fundamental, it was actually lost sight of altogether, until attention was called to it by Newton. The geometrical investigations of Apollonius were based on the conception of the conic as the plane section of a cone, the advantages of which plan may be enumerated roughly as follows: