This section is from the book "The American Woman's Cook Book", by Ruth Berolzheimer. Also available from Amazon: The Domestic Arts Edition of the American Woman's Cook Book.
When Little or No Fat is Used - Heat three-fourths of the liquid. Stir the remainder of the liquid gradually into the thickening agent. If sugar is used it may be mixed with the thickening agent before the liquid is stirred in or added to the sauce after the thickening is completed. Stir into the thickening agent at first only enough of the cold liquid to make a thick batter. Beat this batter until smooth and free from lumps, then add the rest of the cold liquid. The mixture should be about as thick as medium cream. Beat this gradually into the hot liquid and cook, stirring constantly, until the mixture is thickened. If fat is used, it may be added at this time. After thickening, the sauce may be cooked in a covered double boiler with occasional stirring.
When Amount of Fat Equals or Exceeds Amount of Thickening Agent - Melt the fat, add the flour or corn-starch and cook, stirring constantly, until thoroughly blended. This is called a roux. Stir in the liquid, a little at first, then immediately enough to thin the roux perceptibly and finally the remainder. Cook, stirring constantly, until thick. Complete cooking in a double boiler, stirring occasionally.
Heat the liquid; cream together the fat and thickening agent; add this modification of roux to the hot liquid and stir constantly while the fat melts and the particles of flour or cornstarch are being spread through the liquid and cooked. Complete cooking in a double boiler, stirring occasionally.