Arthur Beverly was born on 22 March 1822 at Alford, Aberdeenshire, Scotland. His father was George Beverly, a farmer; his mother's identity is unknown. Arthur Beverly was educated at home. Occasionally he attended a parish school, and evening classes conducted by James Taylor, a shoemaker, from whom he learnt Euclid, trigonometry, astronomy, surveying and navigation. At about the age of 14 he was apprenticed to Aberdeen watchmaker and optician Baillie Berry. He soon gained a considerable reputation as a lensmaker, and made a set of microscope lenses for Dr George Dickie, professor of botany at Marischal College, University of Aberdeen. Dickie was sufficiently pleased with these to recommend Beverly to other scientists.
After eight years as a journeyman in Aberdeen, Beverly sailed for Melbourne, Australia, in 1852. He tried his luck in the Victorian goldfields, but quickly realised that he could earn more money as a watchmaker in Melbourne. There he became friends with William Strachan, who worked for the firm of Patterson and Strachan, provision merchants. In 1857 Strachan emigrated to Dunedin, New Zealand, and at his encouragement Beverly followed in January 1858. In order to set up a new watchmaking business, he brought with him a stock of jewellery, clocks and watches. He also brought materials for daguerreotypy, as he was interested in photography.
Beverly set about acquiring property in Dunedin, and by 1861 owned the buildings which housed his shop in Princes Street, and a large block of residential land. On part of his land there was a natural spring. He offered this section as a brewery site to his friend William Strachan, who was then a brewer at Port Chalmers. Strachan paid Beverly £2,000 for land which had cost Beverly only £34 10s. a few years previously.
Beverly was briefly involved in local politics, as a member of the Dunedin Town Board from 1859 to 1861. When news of Gabriel Read's gold strike reached Dunedin, Beverly was asked to report on the find, and on the strength of his opinion many Dunedin men joined the large numbers bound for the diggings. In 1862 he was a member of the Otago provincial expedition to the West Coast, during which he made a collection of plants between Preservation Inlet and Breaksea Sound.
The widespread financial crisis of 1864 caused Beverly to sell his business. He devoted the second half of his life to his scientific interests and to gardening. Most of his known writings, including notebooks which are now in the Hocken Library, Dunedin, date from this period. Beverly's ability to apply his mathematical skills to practical problems caused him to be consulted by mining engineers and surveyors working on the Otago goldfields. His notebooks contain examples of water race plans and other design calculations relevant to mining. When the public of Oamaru doubted their engineer's opinion that no motive power could be derived from the flow of water in the race which supplied the town, Beverly was consulted. He confidently demolished the engineer's theory, and gave advice on how maximum energy extraction could be achieved.
When Dunedin hosted the New Zealand Exhibition in 1865, Beverly was a member of the panel of judges and also an exhibitor. He won praise for two of his exhibits: a side-board clock, with compensation balance and apparatus to keep it wound up by the variations in temperature of the atmosphere, and a planimeter for the measurement of irregular areas on maps and plans. On 27 November 1865 the Royal Scottish Society of Arts awarded Beverly the Makdougall-Brisbane Prize for the specimen planimeter. The medal which was part of the prize, and the planimeter, have since disappeared. However, the clock is in the possession of the University of Otago Physics Department. Later, Beverly believed his planimeter had been copied and improved by Jakob Amsler, but it is now clear that Amsler was first to invent the device. Beverly's planimeter differed from Amsler's by using a rectangular, rather than a circular, reference frame.
Astronomy was another of Beverly's interests. Several of his notebooks are devoted to astronomical calculations which derive the orbits of comets observed by him and predict the circumstances of lunar occultations. Beverly's observations were made with the three inch refracting telescope constructed by him and now in the possession of the Beverly–Begg Observatory, Dunedin. For many years his predictions of astronomical phenomena were published as 'Rough astronomical notes' in Dunedin's Evening Star. He also regularly contributed to 'Notes and queries', a column in the Otago Witness which answered readers' questions.
In the early years of the settlement Beverly made transit observations to establish local time for the purposes of correcting ships' chronometers. When C. W. Adams, then chief surveyor at Dunedin, wrote a paper on the measurement of distances with long steel tapes, it was Beverly who devised the series, later used world wide, to account for the sag of the tapes. Beverly insisted on measuring the elasticity of the tapes by actual experiment, but scorned the idea of further experimentation to verify the underlying mathematics.
Beverly was intrigued by the theories of Professor C. Piazzi Smyth, which were described in Life and work at the Great Pyramid (1867). As an agnostic he was sceptical of Piazzi Smyth's religious ideas, but devoted much effort to testing and extending the mathematical base which they suggested.
Beverly was described as a small, inconspicuous man. His appearance matched his temperament to a degree: he was always reluctant to publish his work and did so only at the strongest urgings of his friends. He died a bachelor on 25 October 1907 at Dunedin. He left to the University of Otago an estate valued at £57,000. The will contained such detailed instructions regarding the management of the bequest that the university sought clarification from the Supreme Court. A judgement freed the university from obligation to comply in detail. Income from the Beverly bequest now funds prizes and fellowships in the departments of physics and mathematics.