Haworth's research of glucose
I cannot guarantee that the following explanation for the proof of the ring size of the principal cyclic form of glucose is due to Haworth. The link I provided in the comment above explains one method. In "Methylation of Pyranoses" methods are illustrated for the permethylation of the hemiacetal form of glucose and its subsequent hydrolysis to the tetramethoxy hemiacetal. Although the 6-membered ring is shown, a priori the ring size is unknown. This is the issue to be determined. continued below
In "Ring Size of Pyranosides" mild HNO3 oxidation of the tetramethoxy hemiacetal affords a keto carboxylic acid (upper left). Continued oxidation of either enol of the keto group leads to oxidative degradation to an  optically-active dimethoxysuccinic acid (red path) and a meso trimethoxyglutaric acid (green path). These two dicarboxylic acids are only derived from a 6-membered ring of glucose.

Source: http://ursula.chem.yale.edu/~chem220/chem220js/STUDYAIDS/carbohydrates44.pdf