Chasles in his Aperçu, Historique sur l'origine et le développement des Méthodes en Géométrie (seconde édition, 1875, pp. 214–215) makes the following statement:

“Essays of the same kind as the geometry of the rule and that of the compasses, and which hold, so to speak, the mean between the two, had long previously engaged the attention of famous mathematicians. Cardan first of all in his book De Subtilitate had resolved several of Euclid's problems by the straight line and a single aperture of the compasses, as if one had in practice only a rule and invariable compasses. Tartalea was not long in following his rival on this field, and extended this mode of treatment to some new problems. (General trattato di numeri et misure; 5ta parte, libra terzo; in-fol. Venise, 1560). Finally, a learned Piedmontese geometer, J.–B. de Benedictis, made it the object of a treatise entitled: Resolutio omnium Euclidis problematum, aliorumque ad hoc necessario inventorum, una tantummodo circini data apertura; in-4°. Venise, 1553.”

On donne un cercle et deux points P et Q situés sur un diamètre, on joint les points P et Q aux extrémités A et B d'ux diamètre du cercle par les droites I'A et QB qui se coupent au point M. On fait tourner le diamètre AB et on demande

I. D' étudier les variations du rappoet de construire la figure quand le rapport a unedonnée.

II. D' étudier les variations de l' angle AMB, et de construire la figure quand cet angle a une valeur donnée.

III. A′ et B′ étant les seconds points d' intersection des droites MA, MB avec la circonférence donnée, trouver le, lieu du centre du cercle circonscrit au triangle MA′B′.

Since an n-gon is determined by 2n − 3 conditions, and n − i conditions are involved in its being equilateral, there are still in the case of an equilateral n-gon n − 2 conditions to be determined. These n − 2 conditions cannot all be given in terms of the angles, since an infinite number of n-gons may always be described similar to, out not necessarily congruent with, any given equilateral n-gon. Hence only n − 3 of the angles of an equilateral n-gon may be assigned arbitrarily, and there must therefore be 3 independent relations connecting the angles of any equilateral n-gon. These three conditions may be obtained by projecting the perimeter of the n-gon on any three lines.

The problem of designing cams or centrodes to produce any given motion in one plane is one of some practical importance; and it seems worth while to illustrate by examples some simple methods by which the solution can in certain cases be arrived at. These methods are founded, for the most part, on the use of the so-called Pedal Equation (or p-r-equation), which has great advantages in the present investigation, inasmuch as it depends on the form but not on the position of the ourve which it represents.

The middle points of the diagonals of a complete quadrilateral are collinear.

Let ABCDEF be a complete quadrilateral (see fig. 70), and X, Y, Z, the middle points of its diagonals. To prove X, Y, Z, collinear.

Helmholtz and most, if not all, subsequent writers on vortex motions have, except in obtaining the fundamental equations, confined themselves to fluid of invariable density.

In the following paper are considered some simple systems of vortices in a compressible fluid. To show that such systems are of considerable importance it is sufficient to refer to the phenomena of cyclonic storms. It may be as well, however, to state that though the vortices are here treated as compressible, the circumstances are still so different from those found in nature that the results obtained could bear only a general resemblance at most to the phenomena of storms.

If we have given two equations φ(x) = 0 and ψ(y) = 0, it is possible to express in the form of a determinant the equation whose roots are f(x, y), where f is any given rational integral function.

The theorem is—

The distance between the circumscribed and the inscribed centres of a triangle it a mean proportional between the circumscribed radius and its excess above the inscribed diameter.

Some points of difference between polygons with an even number and polygons with an odd number of sides.

A polygon with an odd number of sides is determined when its angles are given and it is such that a circle of given radius may be circumscribed about it; while a polygon with an even number of sides is not determined by these conditions.

I had occasion, lately, to consider the following question connected with the Kinetic Theory of Gases:—

Given that there are 3.1020 particles in a cubic inch of air, and that each has on the average 1010 collisions per second; after what period of time is it even betting that any specified particle shall have collided, once at least, with each of the others?

Etant donnés une conique K, dont l'équation est K = 0, et un point P (α, β), l'équation générale des coniques qui passant par les points d'intersection de la conique K et du cercle P de rayon nul, qui a le point (α, β) pour centre, est

Comme les quatre points d'intersection du cercle P et de la conique K sont imaginaires, le système (1) comprend un seul couple de droites réelles Δ et Δ. Ces droites seront dites, par analogie avec une expression proposée par Chasles, les conjointes du point P et de la conique K.

I send a note on the following problem, a solution of which was requested of me by one of the tutors at King's College, Cambridge.

We are given a quadrilateral of four jointed bars ABCD (fig. 83). The bar CD being held fast, find the tangent to the locus of P, the intersection of DA, CB in any position; and verify the following construction for the radius of curvature of the path of P:—

Let PQ be the third diagonal, draw through P a perpendicular to PQ meeting BA, CD in L and L′; through L and L′ draw parallels to PQ meeting AD in M and M′; through M and M′ draw perpendiculars to AD meeting the normal at P in O and O′; then will

The course of geometry here referred to was given to pupils in George Heriot's School in the session preceding that in which they should begin the usual systematic study of geometry. The chief object of the course was to furnish their minds with a number of geometrical ideas before they should meet with these ideas as treated by Euclid. Subsidiary ends were also kept in view—such as to get them to make neat and accurate figures, and to enable them to solve various practical problems of construction and measurement.

This curious device is figured at p. 149 of De Morgan's Budget of Paradoxes, where it is described as a “hollow semi-cylinder, but not with a circular curve,” revolving on pivots. The form of the cylinder is such that, whatever quantity of oil it may contain, it turns itself till the oil is flush with the wick, which is placed at the edge.

What sort of an equation is

Write

and start with the equations

This last gives

and the system thus is

viz., this is a system of ordinary differential equations between the five variables θ, r, X, Y, Z: the system can therefore be integrated with 4 arbitrary constants, and these may be so determined that for the value β of θ, X, Y, Z shall be each = 0; and r shall have the value r0.

In reducing some experiments, I noticed that the logarithm of 237 is about 2.37 …. Hence it occurred to me to find in what cases the figures of a number and of its common logarithm are identical:—i.e., to solve the equation

where m is any positive integer.