tained with only a few dimensions.
 The following cases illustrate how the given number of dimensions affects the procedure's ability to arrive at a configuration of points that adequately reflects the supplied distances. Using classical MDS to analyse the distance matrix for Australian cities with only two dimensions produces this result:
 The resulting map has an unconventional orientation, which is to be expected because the underlying mathematical analysis does not include any constraints concerning what should be regarded as up, down, left, or right. Consequently, it is quite appropriate to rotate or reflect such a map in order to place it in a preferred orientation. Even though it has an unconventional orientation, this map accurately reflects the relative positions of the cities included in the distance matrix. A figure called the “proportion of variance” indicates how well the procedure has managed to fit the extracted configuration of points into the available number of dimensions. In this case, the proportion of variance is 1.00 which indicates that two dimensions are sufficient to achieve a perfect fit between the original distances and those derived from the extracted coordinates. This is because the original distance matrix was derived from a flat (i.e. two-dimensional) map.
 Subjecting the distance matrix for the apexes of a tetrahedron to classical MDS while allowing only two dimensions produces a proportion of variance figure of 0.67, indicating that the resulting map conveys two-thirds of the analysed distance information. Allowing three dimensions produces a perfect score of 1.00. Three dimensions is sufficient to produce a perfect configuration for any four points. Two dimensions would be sufficient if all of the points were in a plane, and only one dimension would be required if they were all in a line.
 The distance matrix derived from the UBS4 apparatus data for Hebrews can be analysed by classical MDS as well. This matrix includes 54 objects, meaning that up to 53 dimensions would be required to perfectly convey the distance information. However, as it is hard for the human mind to cope with more than three spatial dimensions at once, it is necessary to settle for a three-dimensional representation:
 The proportion of variance figure for this analysis is 0.42, meaning that three dimensions explain 42% of the distance information contained in the matrix. This map is optimal in the sense that it is the best representation of the distance information that can be achieved with only three dimensions.
 Chatfield and Collins describe cluster analysis as follows:
The basic aim of cluster analysis is to find the “natural groupings,” if any, of a set of individuals (or objects, or points, or units, or whatever)... More formally, cluster analysis aims to allocate a set of individuals to a set of mutually exclusive, exhaustive, groups such that individuals within a group are similar to one another while individuals in different groups are dissimilar. This set of groups is usually called a partition.
 This mode of analysis needs to be used with discretion. Even if the data set does not have a natural group structure, clustering techniques will still divide its members into groups. As Chatfield and Collins say, “The ability of clustering methods to detect non-existent clusters is well established.” There are numerous clustering techniques, each liable to produce its own set of clusters when applied to the same data. Subjecting the distance matrix for Australian cities to a mode called “divisive clustering” produces the following result:
 According to this chart, called a “dendrogram,” Australian cities can be partitioned in a number of ways. In order to define a partition, the tree is “chopped” at some vertical level to produce a series of clusters. For example, cutting across at a height of about 3500 km produces two clusters: one comprised of Adelaide, Hobart, Melbourne, Brisbane, and Sydney; the other containing Alice Springs, Darwin, and Perth. Other numbers of clusters can be produced by chopping the tree a