The problem of fitting a polynomial to a set of observational data so that the sum of the squared residuals is a minimum has been frequently investigated. A.C. Aitken, in an appendix to his paper, "On the Graduation of Data by the Orthogonal Polynomials of Least Squares ", (Proc. Roy. Soc. Edin. Vol. LIII (1933) pp. 77 -78,) provides a list of the more important papers on this subject. Tchebychef and Gram were the first to expound, more than fifty years ago, the method of fitting my means of orthogonal polynomials. Their work has been followed up in more recent times by several writers, including Jordan, R.A. Fisher and A.C. Aitken. W.F. Sheppard and C.W.M. Sherriff develop the equivalent method of linear combination of data. However, in the latter method, the fitted value of the central observation only is considered in detail, and an odd number of data is therefore required. It has been shown by W.F. Sheppard, and recently, with much more conciseness by G.J. Lidstone that the methods of Least Square Fitting and Linear Combination of Minimal Reduction Co-efficient lead to identical results. It is proposed, in the following investigation, to express all the fitted values as linear combinations of the observed values. The data considered are either odd or even in number, equidistant, unweighted and uncorrelated. In Chapter I we shall investigate the form of the matrix and its properties, and shall give examples of its construction and practical use. The corresponding matrices of co-efficients, obtained in this way, of lower and lower order, are discussed in Chapter II, and appropriate examples given. The properties of these matrices are very similar to, and often identical with those of the matrix C Chapter III contains alternative methods of fitting, with simple checks on accuracy of working. Examples of each method are given. The appendix consists of tables of the numerical values of the matrices connecting observed and fitted values, for numbers of data equal to 4, 5, 6, 7 - - - 15, and for fitted polynomials of degree 0, 1, 2, 3, 4, 5, together with the corresponding matrices connecting the differences. There is also a short bibliography. Methods of fitting involving the matrices discussed here are particularly suitable for rapid calculation with a machine. Indeed the use of a machine is taken for granted. The same example is used throughout to simplify the comparison of the various methods.