Methodus Geometrica 1598 by Paul Pfintzing

Methodus Geometrica is a printed hand coloured treatise on geometry and surveying published in 1598 in Nuremburg, Germany. It is held in the SP Thompson Library, part of the Rare Book Collection at the IET Archives.

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 in the development of mathematics.

Methodus Geometrica- cartographers at work

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.

Methodus Geometrica- cartographers at work

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.

Methodus Geometrica- cartographers at work

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.

Methodus Geometrica- cartographers at work

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 1700's.

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.

Find out how the Methodus Geometrica gave advice on the study of Perspective and Geometry on the next page.