Let the curve be given by x=f(t), y = ɸ(t)

Asymptotes parallel to the coodinate axes.

If as t→tl, y = ɸ(t)→∞ while x=f(t)→a, where a is finite, then x = a is an asymptote.

If as t→t2, x=f(t)→∞while y = ɸ(t)→b, where b is finite, then y = b is an asymptote.

This article offers suggestions, not for formal lessons on astronomy, but for occasional “raids” into astronomical territory—excursions intended to show how even an elementary knowledge of mathematics may be applied fruitfully to problems of real importance and abiding interest. The mathematical knowledge assumed is acquaintance with the three main trigonometrical ratios and their reciprocals, familiarity with the “natural” trigonometrical tables, and a little practice in changing the subject of a straightforward formula containing the ratios. Values taken to three places generally suffice, and skill in using logarithms, though desirable, is not demanded.

We live in days when more than ordinary care is taken to extract from jam-pots, gardens, dust-bins and ourselves the maximum of productivity; and in such times why should mathematical tables be exempt from the general squeeze? To all of us, I suppose, there come moments when we say “For this calculation I wish I had tables giving more figures”: such cases occur especially when anything is found as the difference of two much larger quantities, in which a small percentage-error may appear greatly magnified in their difference (weighing the ship’s cat by the displacements before and after she fell overboard). I shall outline here and attempt to justify the least laborious that I know of methods by which the inevitable approximation-errors can be cut down to a much smaller mean than is involved in using the crude values as printed.

Whether it is because of the utility aspect or whether it is because of some other reason, boys are invariably interested in logarithms. Whenever I take up the subject with a form, at least one boy will ask: “Please, Sir, how were logarithm tables constructed?” When a master is confronted with this question, he usually thinks of logarithmic expansions, and he is obliged to tell the boy that the explanation is beyond his understanding.

Several difficulties arise in the definition of the radius of curvature ρ of a plane curve. In the first place, it does not seem to have been previously noticed that the convention adopted in many textbooks for the sign of ρ leads to ridiculous consequences. Consider the very simple case of a circle of unit radius, with centre at the origin. Use A, B, C, D to denote the points (1, 0), (0, 1), ( −1, 0), (0, −1) respectively. Then the upper semicircle ABC is convex upwards and the lower semicircle CDA is concave upwards. G. A. Gibson, in his Elementary Treatise on the Calculus (p. 355), defines the sign of ρ at any point P as being that of d2y/dx2 at that point, or, what is equivalent, as being positive or negative according as the curve is concave upwards or convex upwards at P. From this definition it follows that ρ = −1 on the upper semicircle, but ρ=+1 on the lower semicircle! At A and C d2y\dx2 does not exist, so presumably ρ does not exist at these points! We cannot obtain a value by postulating continuity, since ρ is certainly discontinuous.