The following apparatus for verifying Boyle's Law in an elementary manner is of interest from the point of view of pure mathematics as well as that of the laboratory. The method of construction, which I have described in the July (1910) number of the School World, presents a little difficulty in the first attempt, but is easily mastered in the second.

The formula for the sum of n terms of the geometric series

where T1 is the first term and Tn+1 the term immediately succeeding the last to be summed This form is useful for finding the sum of a closed geometric series without first finding the number of terms. Thus

since the terms of the series which succeed 32 and a19 are respectively 64 and a21.

The reciprocal polar figure of a circle s, with regard to another circle c, is a conic, one of whose foci is the centre of the reciprocating circle. The polar reciprocal of a with regard to c is obtained by taking the polars of points on s with regard to c, and then constructing the envelope of these lines.

The approximate rules given in this note may perhaps be new to some readers.

If an observer be at a height h above the Earth's surface, the distance of the horizon as seen by him is given by the formula

where r is the radius of the Earth.

The equation of the hyperbola usually presents itself to the student in the form in the elements of graphs, before the canonical form is reached in Analytical Geometry. Some of the properties of the curve are easily obtained from the above equation by the methods of elementary geometry; the object of this note is to illustrate the process. We shall use the equation in the form xy = c2, and it will be convenient to arrange the discussion in numbered Theorems. The axes may be rectangular or oblique.

The object of this note is to suggest how Newton's Laws of Motion may be approached explicitly by means of rate of change of momentum. It will be seen that the term “force” is not made prominent at the expense of the notion of rate of change of momentum. Once the student believes that rate of change of momentum is a vital aspect of stopping or starting the motion of a body, he is not inclined to overestimate his knowledge of “force” apart from the rate of change of momentum which accompanies it. Since students of ordinary endowment learn to observe things before words, we begin with some common examples. As for the term “momentum,” it is sufficient to begin by saying that