Some of the properties of the figure which, on account of its shape, the Greeks named the Shoemaker's Knife (ἄρβηλοσ) are given in the Lemmas attributed to Archimedes; others occur in the fourth book of Pappus's Mathematical Collection. The Lemmas (which are not extant in Greek, but have been translated from the Arabic) are generally considered to be spurious; it is, however, regarded as possible, if not probable, that the theorems among them relating to the Arbelos may be due to Archimedes. Whether they are or not, the figure and the principal proposition respecting it which Pappus gives are said by him to be “ancient.” It may be added that the Arbelos does not seem to have attracted much notice from geometers, few of them having treated of it, and fewer still having added to the properties known to the ancients. (See Steiner's Gesammelte Werke, Vol. I., pp. 47–76, and The Lady's and Gentleman's Diary for 1842 and 1845).

Fourier's Theorem, as usually stated, is

Most writers give this without limitation, but De Morgan (Diff. and Int. Calc. pp. 618 &c.) directs attention to what he calls the apparent neglect by previous writers of the limitation of the theorem to functions which satisfy the condition

In the part just issued of the American Journal of Mathematics (vii, pp. 3, 4) Proffessor Cayley has occasion to deal with the function

where stands for i (i – 1)…(i – j + 1). He Points out that

expresses {i, 1}, {i, 2},…{i, 5} each in descending powers of i, and tabulates the function from {1, 1} to {5, 5}.

This paper is an attempt (I) to deduce from first principles the number of conditions required to determine a plane polygon of n sides; (II) thence to deduce the numbers for special cases; and (III) to discuss the effects of a redundancy and a deficiency in the number of conditions. An investigation of this kind should form an important as well as interesting accompaniment to the ordinary study of elementary geometry.

If we consider the circle circumscribing any triangle ABC (see figures 11, 12), and diminish its radius still causing it to pass through A and B: then if ACB be an acute angle, C passes without the circle, but if ACB be an obtuse angle, C remains within the circle. If C be a right angle, the radius of the circle, being ½AB, cannot be farther diminished.

If the distances (12), (13), (14), (23), (24), (34) between four points 1, 2, 3, 4 on the circumference of a circle be denoted by a, b, c, d, e, f respectively, then a certain relation (A) is known to connect a, b, c, d, e, f. The same four points, however, being points in a plane, there subsists between their mutual distances another relation (B). Now, it occurs to one that from these two relations some deduction ought to be possible regarding the mutual distances of four points on a circumference, and the problem is suggested of making the said deduction.

A straight line KK′ meets the circumference of a circle at two real or two imaginary points K, K′, and H is the middle point of the real or imaginary chord KK′. If A, B, C, D be any four points on the circumference, and the pairs of straight lines AB, DC, AC, BD, AD, CB meet KK′ at the pairs of points E,E′, F,F′, G,G′; then if any one pair of points be equidistant from H, the two other pairs will also be equidistant.

The object of this note is to point out, by a few remarks on a single case, how well worth the attention of younger mathematicians is the full study of certain problems, suggested by physics, but limited (so far as that science is concerned) by properties of matter.

In my former paper on Fourier's double-integral I remarked that Poisson's form of the integral gave the same incorrect; result as Fourier's form in an example by which I tested it, and seemed subject to the same limitations.

It is evident that by an extension of the method of deriving from any triangle its polar triangle, it is possible to derive from any figure whatever another figure, the properties of which may be deduced at once from those of the first. This may be done either by imagining a point to move along the original figure and considering the envelope of the great circle of which the moving point is the pole; or by imagining a great circle to envelope the figure and considering the locus of its pole. In both cases two figures will be obtained; but these will be antipodal, and will therefore have like properties. Since the point of intersection of two great circular arcs is the pole of the great circular arc which joins the poles of these two arcs, the two methods of derivation mentioned above will lead to the same derived figure. These methods of transformation of figures are evidently closely analogous to that of reciprocal polars.

The method of treating tangents to confocal conicoids, of which I propose to give an account, is discussed in the number for December 1867 of the Nouvelles Annales de Mathématiques. The writer of the paper referred to is M. Ph. Gilbert, Professor at the University of Louvain. The results arrived at are in nearly every case already well known, but the method of reaching them is somewhat novel, and I have thought that it might interest the members of this society if I were to give a statement of the chief methods and results of Gilbert's paper. In one or two cases I have altered the proofs, and I have added two or three propositions that seemed to follow naturally from the equations dealt with. Gilbert deals only with central conicoids; but I have put in an equation for the paraboloids that corresponds to Gilbert's fundamental one.

Let f(t), ø(t), be two functions of t, such that they both vanish with t, that is, f(0) = 0, ø(0) = 0; and put

This paper is an attempt to collect and arrange some of the propositions regarding the so-called Simson line, contained in various Mathematical Treatises and Journals. Proofs have been altered, or new ones substituted to suit the arrangement.

In many questions of analysis relating to the theory of plane curves, it is convenient to be able to obtain quickly the expansions of and of These may be obtained as follows:—

By an obvious extension of the theorem of Leibnitz we have

where d1,d2,…dn differentiate y1, y2,…yn respectively.

1. Consider the conic represented by the general equation

Differentiating three with respect to a we get

where (y) stands for .