Two recent problems appearing in die September 1997 and September 1999 issues of the NCTM publication Mathematics Teacher asked the following:

A. An integer is called decreasing if each digit is less than the one to its left. Determine the number of decreasing integers between 2000 and 7000.

B. A 5-digit integer is a peak number if its digits strictly increase to the middle one then strictly descend. Determine the number of 5-digit peak numbers that are greater than 70,000.

A new and general proof of theorem [1] is presented, which accommodates directed angles and covers all possible configurations of line and conic. A sign convention for the use of directed angles in this setting is developed, and examples are given showing how this fundamental theorem can be used to establish, much more directly and succinctly, the many tangent/focus properties of conies.

The natural numbers are the elements of the set N = {1,2,3,…}, the integers are the elements of the set Z = {0,± 1,±2,…} and the rational numbers are the elements of the set Q = {p/q:pεZ, qε6 N}. A real number is called algebraic if it is the root of a polynomial equation

P(x) = anxn + an−1xn−1 +…+ a1x + a0 = 0

1. Professor C. E. Weatherburn’s paper in the Gazette, 1936, p. 320, gives an interesting proof by vector-analysis of the condition for a twisted curve C to have as binormals the principal normals of another curve C1.

It may be of some interest to give a few bibliographical notes on the subject and to mention new or not generally known properties of certain of these pairs of curves.

It is tempting to state the problems of the foundations of mathematics in general logical. terms, with words like “truth”, “proposition”, “necessary conclusions” In this way a sometimes useful, and always at any rate verbal, connection is made with old problems of logic; and for this reason also mathematical investigations into the foundations of mathematics have found a good number of popular exponents. But it seems that at times these expositions suffer considerably from rather ill-chosen slogans.