The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.

The sequence 1, 3, 7, 13, 21, … arises in many ways, for example from the integers placed in a spiral on a square grid, the terms arising where the spiral intersects a diagonal on the grid, and as the central axis of a triangular array of numbers:

If p is a prime and ps is the greatest power of p that divides N!, then

The following discussion may be of interest on account of the way in which it leads to the axial and focal properties.

I. Let the locus of a point P such that A(CDEP) = B(CDEP), where A, B, C, D, E are fixed points, be called the conic (AB, CDE). Then it is clear that permuting A, B or C, D, E does not affect the conic, and that the conic passes through C, D, E, and also through A, B. (Give BP the four positions BC, BD, BE, BA and then give AP the position AB.)

The problem of finding a closed-form evaluation of baffled the pioneers of calculus such as Leibniz and James Bernoulli and, following the latter’s promulgation of the problem, it became known as the Basel problem after his home town (which was also Euler’s birthplace). Euler’s early sensational success in solving the Basel problem by identifying is extremely well-documented. In this paper, we give the full details of his subsequent derivation of the general formula

(1)

where (Bn) is a sequence of ‘strange constants’. Euler’s polished account of his discovery, in which he popularised the designation of the strange constants as ‘Bernoulli numbers’, appears in Chapter 5 of Volume 2 of his great textbook Institutiones calculi differentialis [1; E212]: see [2] for an online English translation. Here, we will focus on his initial step-by-step account which appeared in his paper with Eneström number E130, written c1739, carrying the rather nondescript title De seriebus quibusdam considerationes, ‘Considerations about certain series’. (For convenience, we will just use ‘Eneström numbers’ when referencing Euler’s work: all are readily available on-line at [1].) Euler’s proof is notable for its early, sophisticated and incisive use of generating functions and for his brilliant insight that the sequence (Bn) occurring in the coefficients of the general ζ(2n) formula (1) also occurs in the Euler-Maclaurin summation formula and in the Maclaurin expansion of . By retracing Euler’s original path, we shall not only be able to admire the master in full creative flow, but also appreciate the role played by recurrence relations such as

(2)

which, as our ample list of references (which will be reviewed later) suggests, have been rediscovered over and over again in the literature. Moreover, our historical approach makes it clear that, while deriving (2) is relatively straightforward (and may be used to calculate ζ(2n) recursively as a rational multiple of π2n), it is establishing the connection between ζ(2n) and the Bernoulli numbers that was for Euler the more difficult step. Even today, this step presents pedagogical challenges depending on one’s starting definition for the Bernoulli numbers and what identities satisfied by them one is prepared to assume or derive.

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.