Final answer:
The specific discovery by Mathews & Vine in 1959 is not provided, but it may be related to genetic or molecular biology research based on the context of genetic discoveries mentioned.
Step-by-step explanation:
In 1959, Mathews & Vine presumably made a significant discovery, although the specific discovery is not mentioned in the background provided. Given the context of genetic research milestones, one may conjecture this could be related to genetics or molecular biology. Significant milestones in the history of genetics include the experiments by George Beadle and Edward Tatum with Neurospora crassa in the 1940s which led to the "one gene-one enzyme" hypothesis, Frederick Griffith's 1928 discovery regarding the transforming principle of bacteria, Avery, McLeod, and McCarty's experiments in 1944 which identified DNA as this transforming principle, and the Hershey-Chase experiment of 1952 which established DNA's role in genetic information transmission.
Further research and detailed information are required to conclusively identify what Mathews & Vine discovered in 1959, as this is not disclosed within the material presented.