Final answer:
The information provided does not specify the company that first designed and installed the Type III hydrant system. The historical context given outlines the evolution of water supply systems, including William Mulholland's contributions to engineering in Los Angeles.
Step-by-step explanation:
The company first responsible for designing and installing what became known as the Type III hydrant system is not stated explicitly in the historical overviews provided. However, the development of water supply systems has been progressive, with several notable engineering projects contributing to their evolution.
From the construction of the "Great Conduit" in the thirteenth century to Sir Hugh Myddleton's creation of the New River in the early seventeenth century, many advancements have been made. Additionally, significant developments were made by William Mulholland, an engineer who notably planned and built the Los Angeles aqueduct, illustrating the progress in water engineering well into the beginning of the twentieth century.
Despite the information given, the specifics regarding the first company to design and install the Type III hydrant system requires further historical engineering research that is not covered by the available references.