1) The author gives a short resume of the principal breeding methods in maize. Since mass selection cannot be considered as a method for improving corn, two groups of methods remain: the hybrid corn method and the method of the synthetics. Reasons are given why it seems important that the latter should be applied, after a further improvement of the breeding technique and its theoreticsl basis. The method may still be subdivided into the method of simple synthetics and of balanced synthetics. In the preparation of the former, only the following two points have to be considered: selection for combining ability before the constitution of the synthetic, and mass selection aganst weak descendants of consanguineous matings after the establishment of the synthetic. In the case of the balanced synthetics, a third element is added: selection against or rather previous elination of all hybrids which give too strong mendelian segregation in a synthetic. 2) The first proposal to use synthetics has been made by Hayes and Gardner in 1919. Positive results were obtained howewer only much later, since before 1940 the importance of selection for combining ability was not recognized. Hayes, Rinke and Tsiang (1944) obtained a synthetic which equalled the double hybrid Minhybrid 403.Lonnquist (1949) obtained a synthetic which eaqulled the double hybrid US13. Roberts, Wellhausen, Palacios and Cuevas (1949), Roberts and Wellhausen (1948) and Wellhausen (1950) reported on satisfatory results from Mexico. Brieger (1944 and later) produced balanced synthetics of subtropical sweet corn. 3) The author demostrates that the formulae of Sewall Wright (1922) and of P. C. Magelsdorf (1939) cannot be used satisfactorily to explain the composition of synthetics. Both these formulae start from certain assumptions which are not allways satisfeid, and disregard the principles of genetics in populations under selection. 4) In oder to avoid confusion in discussions on the theory of synthetics, a basic scheme is proposed for the identficiation of subsequent generations. The first generation of plants, planted out in mixture and left to free pollinisation should be called So. and it should be composed of the offspring of individual selected plants which had suffered at least one selfing or more. The first generation after free pollinisation, or generation Sy1, consists thus mainly of simple hybrids except for some individuals resulting from consanguineous matings between sister plants. If there were no selection, the next generation Sy2 should be composed of individuals from matings between individuals which may have none,, one, two, three or even four lines in their ancestry in common. Howewer if artificial selection against weaker and less productive plants is carried out in the generations Syl and Sy2, than we may assume that only individuals remain from matings wich had from none up to two ancestral lines in common. 5) Using this classification we can say that the generation Syl in Lonnquist's experiment corresponds to the generation Syl of the basic scheme, Syl of Hayes, Rinke and Tsiang correspond to Sy2 of the basic scheme and Syl of the Mexican authors corresponds to a generation of about the order Sy3. 6) A correct theory of synthetics should take fully into consideration the principals of population genetics, taking furthermore into consideration modern theories on the genetic basis of heterosis in maize.