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 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, mapmaking and timetelling.
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.

