On the reverse side of the Moon, just south of the Sea of Moscow, is a crater named Tsu Ch’ung-Chih. Who is Tsu Ch’ung-Chih? What did he do to be thus admitted to the honoured company of Maxwell and Hertz, of Mendele’ev and the Curies, of Lomonosov and Tsiolkovsky?

Tsu Ch'ung-Chih was a Chinese mathematician-astronomer who nourished in the fifth century A.D. He made notable contributions to the calendar calculation and determined several constants with remarkable accuracy. For example, he gave a value of 27·21223 days for the length of the nodical month, the modern value being 27·21222 days. As another example, he found that the planet Jupiter completes seven and one-twelfth circuits of the heavens in every seven cycles of 12 years; this corresponds to a sidereal period of Jupiter of 11-859 years, which differs from the modern value by only 1 part in 4000.

It is part of the folklore of symbolic logic, and more specifically of the statement calculus, that there is an arithmetic method for determining the truth values of statements of that calculus. A brief survey of texts in logic reveals that the matter seems to be dealt with rarely if at all, and usually only as an exercise ([1], p. 76). It is the purpose of this article to draw the attention of the teachers of mathematics to this rather amusing sidelight of symbolic logic.

Statisticians are commonly confronted with the problem of expressing the relationships among three variables and often wish to illustrate these relationships. Various ways have been tried with more or less success. For some types of data isometric block diagrams or graphs have been found satisfactory. But unless there is a fairly steady surface trend this solution is not practicable, and the whole picture can be seen only in a solid model. Where the data are a number of points the information can be represented as small balls or other symbols standing on stalks . The co-ordinates of each point are given by the height of its stalk and the position of the base of the stalk on the horizontal plane. But the problem becomes acute with scatter diagrams where the purpose of the exercise is to see the relationships among the points and to identify clusters, particularly when the variables themselves are principal components and of little interest or meaning. Gyllenberg and Rauramaahave attempted a solution by plotting the points from a component analysis with respect to two perpendicular axes and representing the distance in the third dimension, i.e. height above the plane of the paper, by symbols. Although the height of each individual point can be roughly inferred, it is impossible to see the overall distribution of the points. Other similar techniques are commonly employed, but the impressions they give are usually rather confused.

It is well known that any line through the “mid-point” of a parallelogram bisects the area. What can be discovered about lines which bisect the area of a triangle?

Obviously any median bisects the area. It is also bisected by eachline which is parallel to one side and divides the others in the ratio

Shiu has shown [1] that there are infinitely many triangular numbers that are also square. He remarked that while his theorem did give a list of such numbers, it did not give all of them. It seemed that there might be a reiterative method, working in integers, that gave more of these triangular square numbers, and which included all the known smaller numbers.