Medieval and renaissance mathematical arts and sciences
All the instruments included in the Epact database would have been described as ‘mathematical’ in the period when they were made. In the modern era, however, neither astronomy, nor surveying, nor gunnery, nor gnomonics (the making of sundials), nor most of the other disciplines represented in this collection of instruments, could be called a branch of mathematics in any straightforward or unqualified way, even though they all make some use of mathematical techniques. From this simple observation, we see already that the subject represented by these instruments is unfamiliar to us, and that mathematics was something different in the period before 1600. The collection cautions us to be careful about applying to the past the assumptions and categories that seem familiar and natural today.

This caution is reinforced when we seek to discover more about these instruments from accounts in books written in the period. Here we find that a number of mathematical disciplines are considered to be closely related, and that one of the characteristics that binds them together is their use of instruments. Astronomy is probably the oldest member of the group, though it may be that surveying could mount a rival claim to be the original branch of ‘geometry’, since the very term suggests an origin in the measuring of land. Gnomonics is based on the geometry of astronomy, while navigation begins to have a similar relationship to geometry and astronomy from the late 15th century. Cartography is linked to all the disciplines mentioned so far, while the renaissance subject ‘cosmography’ provided a convenient category that comprised aspects of astronomy, surveying, navigation, map-making and time-telling.

The range of application of practical mathematics expanded greatly as a coherent and sustained technical advance spread across Europe during the 15th and 16th centuries. The new branches of the mathematical arts included perspective, architecture, fortification, gunnery, and mechanics in the sense of the design and management of machines. Often such disciplinary developments exhibit a common pattern, as geometrical techniques are introduced with the aim of reforming more traditional practices, while instruments are offered as a convenient means for adopting such reforms.

While the main thrust of this movement occurred in the renaissance, it depended on both ancient sources and medieval developments. There were two routes for the transmission of ancient Greek learning that were important to the creation of European mathematical culture: the discovery through moorish Spain of Arabic texts, including translations of classical works such as Ptolemy’s astronomy, and the transfer of texts from Byzantium, such as those of Ptolemy’s geography, via renaissance Italy. There are also strong elements of continuity evident when medieval mathematical instruments are compared with those of succeeding centuries. Even though their number, the variety of their designs and the range of their applications are much smaller, they help to evaluate the originality of the renaissance mathematical programme and the claims it made regarding the reformation of established practice.

The development of mathematics was particularly marked throughout Europe in the 16th century, and its character was predominantly ‘practical’, rather than ‘theoretical’ or, better, ‘speculative’. This does not mean that there was a disjunction between speculative and practical interests, or that the ‘practical’ work was always particularly useful. But there was a strong emphasis on the solution of problems through geometry and by means of instruments, while designing instruments was accepted as an integral part of what it meant to be a mathematician. Further, these instruments solved problems, but they did not discover truths about the natural world. That is a later notion that does not apply to the period before 1600: these are ‘mathematical’ instruments in the sense of the time, not ‘scientific’ instruments.

There were three important social locations for mathematics which can for convenience be referred to as the university, the court and professional practice. Mathematics was a part of the university curriculum, though it was not always taught to a particularly high standard and was certainly considered inferior to such higher sciences as natural philosophy and divinity. Its practical character was probably better appreciated at court, ecclesiastical as well as civil, and mathematicians in princely employment had more freedom and greater esteem. Perhaps as a result, they tended to be more innovative and original, particularly in astronomy. Clerical mathematicians might be considered a special case of courtly ones, though the interests of the church tended to be focused more exclusively on astronomy and the calendar. Professional practice includes not only the surveyors and navigators, but also the instrument makers who supplied their needs. These categories are useful, but not inviolable, and there is much overlap in the careers of individuals. In particular, instruments could be designed or even made by mathematicians other than those in the professional category.

Given the character of mathematics in the period before 1600, the study of surviving instruments is essential to any attempt at historical characterisation. Too often the easy assumption of a congruence between the characters of the modern and ancient disciplines of mathematics has led to distortion and misunderstanding. The Epact database can be a step towards a better appreciation of the past on its own terms and in its own categories.

Yet, as with all historical evidence, we must avoid the dangerous assumption that the record is complete or even representative, and in this case we have every reason to suspect that it is profoundly biased. People collect and preserve what is special and valuable, while discarding everyday tools when better ones take their place. The selective survival of the beautiful, the ingenious and the exotic is a danger for historians, but a bonus for others browsing the instruments in Epact.

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