The role of the mathematical practitioner seems to be related to that of engineer, particularly among the TM authors. Besson, notably, penned an entire monograph devoted to the Cosmolabe. Bachot also included designs for mathematical instruments. Ramelli included a single gunner's quadrant and also seems to have published a manuscript about his own mathematical instrument. In 1603, Levinus Hulsius--a dealer of mathematical instruments--was selling copies of Ramelli's book. Even Leupold was engaged in instrument building. His father--George Leupold--was cabinet-maker, turner, sculptor, and watch maker. Leupold became a math tutor and built his own mathematical instruments. In 1698 he even apprenticed as an instrument maker.
Nartov was particularly zealous about mathematical practitioners. He included a "Master of mathematical instruments" in the third rank of his proposed Russian Academy of Crafts. The first rank included masters of civil engineering, mills and locks, and visual arts like painting, drawing, sculpture, and engraving. The second rank included masters of iconography, woodcutting, inking, and special engraving (which is somehow different from just engraving). Other third rank masters included those for optical work, fountains, turning, medical instruments, and metalworking tool. Fourth rank masters represent trades such carpentry, joinery, printing, silversmith, etc.
There may even have been a lack of differentiation between what could be considered a true mathematical instrument like a compass or astrolabe and what we would consider a mechanical device. For example, in April of 1598, Edward Wright--Cambridge Master of Arts--was granted a patent for a "mathematical instrument" that was a water-draining device. It was conceived "by long and painful study of the mathematical sciences."
Mathematics in general became increasingly popular in the 16th century, largely due to mathematical practitioners that extolled the virtues of arithmetic and geometry for mechanical arts like astronomy, navigation, surveying, gunnery, architecture, and mensuration (Johnston). Turner notes:
"From the late sixteenth century on, it was becoming increasingly clear that if a gentleman were to maintain his position and engage in the activities traditional to his class, a measure of skill in mathematics was essential." (pg 51)
Indeed, all of these tasks seem to have been part of the working lives of practical engineers like Rogers, Aconcio, Ramelli, or Errard. Instruments had an important role in this increasing popularity. They simplified a variety of mathematical tasks and they also became a component of conspicuous consumption. The increasing importance of mathematical instruments as luxury items is made evident by Galileo's patronage tactics where he included telescopes with copies of his Siderus Nuncius.
Thomas Bedwell (1547-1595) presents a route for becoming a technical expert that differs from those of Besson or Ramelli. He graduated from Cambridge but instead of becoming a vicar he became a consultant at Dover harbour. He was involved with a variety of different stake holders, including Privy Councilors, gentlemen, craftsmen, and labourers. In 1585 he campaigned against the Spanish in the Netherlands, where he directed the efforts of a group of pioneers. On his return to England he became a sought after surveyor of military fortifications. In 1588 he turned to hydraulic engineering, working with Federico Genebelli on strengthening the defenses of the River Thames. In 1589 he became Keeper of the Ordnance Store for the Tower of London.
A key component of his success were the mathematical instruments he created--rules for carpenters and gunners--and the books he wrote to explain them. Notably, he lacked the typical training of engineers. He never spent time in the typical engineers centers of calculation/expertise: the construction sites of masons or the sieges of military professionals. In 1582 he petitioned Lord Treasurer Burghley for a position at Dover. He offered his instruments and a few others--a water clock for determining longitude, an instrument for measuring timber and stone, and a plan for ballistic investigations--as evidence of his competence. His Ordnance appointment eventually afforded him the opportunity to pursue these ballistic investigations.
Bedwell was a mathematical practitioner; he was not an engineer. As a result, he constantly strove to distinguish himself from mechanicians. In his books he pointed out the many errors of daily practice and suggested that only mathematics could provide a constant corrective action. This argument was a particularly effective approach for addressing the concerns of his patrons. He also had to deal with the practical problems of protecting his intellectual capital. He manufactured the instruments himself and only circulated his books in manuscript. In a tactic worthy of De Beers, he completely controlled distribution. This defensive posture effectively maintained both his reputation and the market value of the products.
Ash has also written extensively on the role of the mathematical practitioner. But his work requires additional attention at later date.
Johnston, Stephen. (1991). Mathematical practitioners and instruments in Elizabethan England. Annals of Science. 48. 319-344.
Turner, A.J. (1974). Mathematical instruments and the education of gentlemen. Annals of Science 30: 51-88.