Gerhard Mueller was born on 7 February 1835 at Darmstadt, in the grand duchy of Hesse, where his Danish father was a professor of mathematics at Darmstadt university. His ability was passed on to his son, for Gerhard soon showed promise of becoming an excellent mathematician, as well as an athlete and a linguist, even if his written English was, at times, delightfully erratic.
Gerhard Mueller began his wanderings when he was 18, as he did not wish to undertake compulsory military training. He spent five years travelling in England, America and Australia before coming to New Zealand, arriving first at Auckland and finally at Bluff in January 1859. He was naturalised as a British subject on 18 January 1860. For the next three years Mueller studied in Invercargill to qualify as a civil engineer and surveyor. He then entered into partnership in 1862 with Friederich Geissow. On 17 September 1862 he married Elizabeth Bannatyne McArthur at her parents' home in Invercargill. Friederich Geissow married Elizabeth's younger sister, Catherine, in the same ceremony.
Mueller practised as a surveyor and engineer in Southland until deteriorating economic conditions led him to leave for Dunedin in August 1865. John Turnbull Thomson advised him of the need for surveyors on the west coast of Canterbury province, and on 25 September he arrived at Hokitika. The first surveyor to be engaged on contract by the Canterbury Survey Department, Mueller was employed to define the boundaries of native reserves in South Westland. At the end of 1865 he laid out the town of Okarito.
Mueller was involved in surveying Greymouth in March 1866, and continued with his native reserves work until July. On 1 May 1866 he became assistant surveyor and mining assistant at Westland: he was in charge of the survey of South Westland, with his headquarters at Okarito. In April he had begun to design and build a cottage at Okarito to receive his wife and infant son, also named Gerhard. They disembarked at Hokitika on 18 August 1866. A daughter, Isabella, was born at Okarito.
Gerhard's characteristics, fortified by a happy marriage, carried him through an exceedingly difficult pioneering venture. He never complained. Indeed, with his whimsical humour he made light of privations, clearly evident in his letters to his wife, Bannie, during their 11 months' separation. He was continually shifting or setting up camp, was frequently wet through, and was sometimes reduced to near starvation. Yet his natural leadership kept his men loyal and hard-working. Mueller also had the services of a Maori guide, Kerei, who successfully saw them through many treacherous river journeys and crossings, frequently using dugout canoes made by themselves.
Westland was designated a separate land district in 1868, and on 12 January 1871 Mueller was appointed chief surveyor. The family then moved to Hokitika, where Mueller had his head office. Between January 1871 and February 1882 they had five more children, Frederick, Arthur, Mildred and Eric, and Bertha, who died in early childhood.
Mueller was still very active in leading field expeditions, one of which, in 1884, surveyed a road line from Jackson Bay to Martins Bay, and during which he located a line for a road from Jackson Bay to the Hollyford Valley. On 1 July 1885 he was appointed to the additional position of commissioner of Crown lands for Westland. He was also chairman of the education board for several years.
In 1891 all the family left the West Coast permanently when Mueller was transferred to Auckland as chief surveyor and commissioner of Crown lands. From 1 January 1902 until his retirement on a government pension at the end of July 1904, he held office as assistant surveyor general for New Zealand.
Mueller was a foundation member of the New Zealand Institute of Surveyors and a member of the Masonic order, holding various offices in Lodge St Andrew, Auckland. He also collected kiwi and other bird and insect specimens for scientists at Darmstadt university and other institutions, and corresponded with Julius von Haast over natural history matters. He was the first European to record the existence of the kotuku (white heron) nesting ground at Okarito. His letters to his wife from the West Coast in 1865 and 1866, give a lively account of his surveying activities and of life in the isolated goldmining settlements.
Gerhard Mueller died at his home in Auckland on 20 February 1918; Elizabeth Mueller had predeceased him in May 1915.