The following theorems are closely connected in various ways; one common thread that runs through them is that in each a critical stage of the proof consists in filling in a missing value in the range of values for which the result is known to hold. Another common feature, of no logical importance but none the less the raison d'être for this article, is that each serves to illustrate important fallacies and omissions in current accounts of elementary algebra and analysis.

This suggests to find a point P which is the centroid of its antipedal triangle as to ABC.

Let LMN be the antipedal triangle of P and A′, B′, C′ the inverse of A′, B′, C′ with respect to a circle Γ having its centre at P. If J is the intersection of AP and B′C′, the inverse J′ of J as to Γ is the point where the parallel to MN drawn through L meets again the circle BPCL. Hence the ratio PJ: A′J equals PA: J′A, and this means that the barycentric coordinates of P are the same in the triangle LMN as in the triangle A′B′C′ Therefore, if P is the centroid of its antipedal triangle, it is also the centroid of A′B′C′

In the development of the theory of complex numbers, it is important to give a definition of them dependent only upon real numbers. In the usual algebraic treatments the product is postulated in the form

or obtained from a matrix representation. The object of the present note is to show by elementary methods that the assumption that the multiplication of complex numbers is distributive, associative, and such that the vanishing of a product implies the vanishing of at least one factor, is sufficient to define the product. This is indeed merely a special case of results well known in the theory of linear algebras.

Introduction. De nombreuses analogies entre les propriétés du triangle et du tétraèdre ont déjà été signalées qui concernent des points, des droites, des cercles et des sphères remarquables associés à ces deux configurations.

Nous nous proposons de définir des points d’un tétraèdre qui possèdent des propriétés comparables é celles des points de Gergonne et de Nagel d’un triangle. Ceux-ci pouvant recevoir plus d’une définition, d’aprhs les propriétés qu’ils possèdent, plusieurs points pourront leur correspondre dans le tétraèdre comme nous le montrerons pour le point de Nagel.

It is well known that, if.

cos t>Cn(t) or cos t<Cn(t) holds for 0< t< ∞ according as n = 1, 3, 5, … or n = 2, 4, 6, …. It will be noted that this is not obvious in itself. In fact, the absolute values of the successive terms of the alternating series do not decrease from m = 0 onward, except when t is small.

There is a considerable literature on characteristic properties of the circle; a bibliography is listed in Bonnesen-Fenchel. It is proposed here to present some simple characteristic properties of types which are, in a sense, related to one another, as will be seen in what follows.

Principal Definitions and Theorems. The discussion will be restricted to smooth, convex, closed curves. Relative to such a curve, a point, P, will be called

(i) a π-point, if the product of the segments of all chords through P is constant;

(ii) an α-point, if, for every chord through P, the angles, on the same side of the chord, between the chord and the tangents to the curve at the extremities of the chord are equal ; and

(iii) a β-point, if the perpendicular bisectors of all chords through P are concurrent.

In the first part of this article we studied the relation between mathematics and the child and tried to suggest that a greater place could be given to mathematics in the schools by taking into account how the child learns. We tried to show that, to give the child a knowledge of mathematics, various approaches can be used, each of which can serve as a basis for the following one if we follow the development of logic, using the set of tools mastered at each stage in order to progress from one level to the next.

LET x be a real continuously variable quantity, and the functions Ft(t: 1, 2, .. ) be arbitrary functions of x which are assumed to have derivatives of sufficiently high order in a given interval of x. Let P≡d/dx so that P is skew-hermitian and satisfies

APRÈS avoir signalé un nouveau cercle associé à un quadrangle plan quelconque nous nous proposons d’étendre au quadrangle inscrit à un cercle des pro priétés du triangle et du tétraèdre signalées par G. de Longchamps, J. Griffith et par nous-même.

IT is well known that the identity

enables one to generate right-angled triangles with integral sides merely by putting x and y as integers (x>y). It is sufficient that x, y be integers, to give a right-angled triangle with hypotenuse x2+y2 and sides x2—y2, 2xy. Is the condition necessary? No, for the triangle with sides 9, 12, 15 (a multiple of the 3, 4, 5 case) cannot be expressed by the above identity without using surds. VVe will, however, prove the following proposition: