With the arts gaining importance in the teaching of STEM subjects (now abbreviated to STEAM) it is prudent to remember that historically visual art worked alongside and complimented many scientific compositions. The teaching of the arts in STEM subjects has the benefit of nurturing creativity when looking at problem-solving. Visual arts can also help with the understanding of complex theories. Beautiful and elaborate illustrations that are integral to the scientific texts have become magnificent works of art in their own right.

One such publication in our rare books collection, Methodus Geometrica, combines the theories of geometry, mathematics and cartography with the beauty of pictorial art.

Methodus Geometrica is a 16th century early printed, hand coloured treatise on geometry and surveying by Paul Pfintzing. It was published in 1598 in Nuremburg, Germany. Its detail on surveying methods, instruments and cartographic techniques means that the Methodus Geometrica has an important place in the development of knowledge about surveying through geometry. In addition, it affords interesting insights into late 16th century printing, technology, costume and art.

Surveying during the Early Renaissance

The Renaissance in scientific knowledge and, in particular the revival of interest in Greek mathematics, provided the intellectual foundation for the development of modern surveying. This knowledge, including geometry, became more readily available as texts were translated from Latin and printing became more common.

Traditionally, the principle role of the surveyor was to determine the obligations and rights of the tenants of the estate and administer the recording of their agricultural production. During the Renaissance, however, this function became subservient to the role of the modern surveyor as one concerned with the location and measurement of plots of land and its features. The introduction, by Frisius in 1533, of the triangulation method using trigonometry and a greater understanding of geometry and proportion made surveying far more accurate.

Historically, the Egyptians, Babylonians, Greeks and then the Arabs and Muslims employed Pythagorean and later Euclidian geometry to divide and map plots of land. However, pursuit of these pure sciences fell away, in the West, with the spread of Christianity, until The Renaissance.

History of cartography

The earliest known maps, such as The Catal-Hyuk map and those discovered of Ancient Babylon and Egypt tended to be very localised and a reflection of religious rather than scientific beliefs about the form of the world. However, as the known world grew and societies developed, humans became more concerned with accurately measuring the size of, and distance between, places and natural features (surveying) and the accurate representation of this (cartography).
Consequently, these two disciplines, because of their concern with measurement, have played an important role in the development of mathematics.

Numerous ancient civilisations displayed a knowledge of geometry. The Egyptians and the Babylonians both applied Pythagorean relationships to construction and the Egyptians are known to have re-surveyed land following the annual flooding of the Nile.
The Greeks, created the first formal mathematics by organising the principles of geometry with rules of logic. It is Euclid’s, The Elements (circa. 350BC), a comprehensive treatise on geometry, proportions, and the theory of numbers that is the basis of all mathematics taught since.

The first major developments in cartography came from Ancient Greece where first, Pythagoras postulated the earth as spherical (6th century B.C) and then Aristotle proved it so (circa. 350 B.C). Fifty yeas later, a follower of Aristotle, Dicararchus, introduced the use of a grid to locate the position of places, a technique employed by the famous mapmaker Eratosthenes.
The Pillars of Hercules (east-west) and Rhodes (north-south) were chosen as the principal lines of his grid. Hipparchus, critical of Eratostheses arbitrary grid, proposed a grid based on astronomical observations whereby places on the same grid line would have the same length day.

The first attempt to mathematically position places was by Ptolemy (140 A.D). With the aim of producing maps that could be accurately copied, Ptolmey divided the world longitudinally at the equator and into 360° of latitude and then placed over 8000 places according to astronomical observations.
Unfortunately, despite Ptolemy’s correct mathematical theory, he started with an incorrect measurement of the circumference of the earth and he only had the necessary astronomical information for a few of the places.
This aside though, his use of the exact mathematics represented a major advance in cartographic technique and it was many centuries before more accurate maps were produced.

The rise of Christianity and the view of the Bible as the sole source of knowledge led to centuries of scientific and mathematic suppression across Europe. However, these techniques were continued and refined by Arabs and Muslims.
Following the translation of Ptolemy’s work into Arabic in the 9th century, various scholars expanded the known world and the accuracy of its representation.

Improvements were made in projecting a sphere onto two-dimensional maps. Around 1010, Al-Biruni, wrote a text on the solution of spherical triangulation and applied this to surveying, accurately measuring distances on the earth. He calculated the radius of the earth, a figure not known to the West until the 1700s.

A number of coincident factors bought about a renaissance of scientific knowledge during the 15th century that revolutionised surveying and cartography in Europe. The age of exploration had meant that newly discovered lands needed to be accurately surveyed. So, once more ancient Greek, Euclidian mathematics was embraced. The desire for accuracy in production was reflected in a renewed interest in absolute and provable truths.

Key scientific texts for the first time were translated from Latin and became more and more accessible following the invention of the printing press by Guttenberg during the mid 15th century. Similarly, printing revolutionised the means of producing maps and allowed for books of maps to be amended.

Methodus Geometrica-Linear Perspective

During the Renaissance, European artists began to paint with the goal of greater realism. Artists such as Brusnelleschi, Alberti, Piero Della Francesca and Leonardo da Vinci brought science to art, conducting investigations into the true nature of form. They learned to paint lifelike images and they became skilled at creating the illusion of depth and distance by using the techniques of linear perspective. Brunelleschi is credited with the first correct formulation of linear perspective in about 1413.

He understood that there should be a single vanishing point to which all parallel lines in a plane converge and that scale was dependant on an objects distance behind the plane of the canvas. It was Alberti though, who first described these rules.

The most mathematical of all the works written on perspective was Italian Renaissance artist, Piero della Francesca’s work, Trattato d’abaco (circa. 1450). It included a lengthy discussion on geometry and numerous illustrations drawn in perspective. In a later book Piero gives geometric theorems that relate to perspective along a plane.

The mathematician Luca Pacioli embraced the work of Piero in his own texts, which were illustrated by Leonardo da Vinci. Leonardo himself developed an acute interest in perspective and the two continued to learn from each other. Leonardo who, more so than anyone else, meshed the study of art and mathematics, studied the converse problem of perspective, eventually making the distinction between artificial and natural perspective.

These skills enabled surveyors and cartographers of the Renaissance to provide realistic depictions of construction and nature. In Methodus Geometrica, Pfintzing gives a detailed discussion of this newly understood science of perspective.

Fashion in the sixteenth century

The illustrations of surveyor’s reveal the fashion of the time. Elizabethan clothing for men emphasised the muscles in their legs. The hose and garter by the knee was popular for this purpose. Similarly, the waist would be highlighted by the wearing of breeches (short trousers) topped off with a fitted doublet (short tight jacket).

The wearing of the ruff, by both men and women, was popular across Europe during the Elizabethan period (1558-1603) as a symbol of high social class.

The large brimmed, feathered hat is also indicative of European fashion at the end of the sixteenth and start of the seventeenth centuries. The wearing of a hat also indicated social status- the taller the hat the more important one was.