HomeSite MapMap Fold-outsMap Projections - ContentsProjections for NavigatorsMap Projections

Premodern Projections

Introduction

Of several projections created before or at the Renaissance, most have fallen in disuse long ago, and only a few with some outstanding properties are remembered today. Much information about these projections is uncertain:

Projections mentioned here are of mainly historical significance; some where already presented when fitting one of the main projection groups. A strong Western bias is evident as I have little information on cartographic development in other cultures.

Equidistant Cylindrical and Trapezoidal Maps

The equidistant cylindrical projection was very common since before the Christian era, no doubt due to its simple and practical construction.
Trapezoidal map
Trapezoidal map in symmetrical variant, standard parallels 40° and 20° North and South

Maybe as old or older, the trapezoidal projection has similarly horizontal, straight, equally spaced parallels, but the meridians converge, not necessarily at a point. Thus at most two parallels (four in the broken meridian case) and only one meridian are standard lines. It is neither equivalent nor conformal.

The trapezoidal projection was credited to Hipparchus (in star maps), and then to Donnus Germanus, who used it to illustrate editions of Ptolemy's works. Its popular name, Donis, derives from a form of Donnus used by Nordenskiƶld.

Most maps for this primitive and obsolete pseudocylindrical projection show only part of the northern hemisphere; some include the whole Earth with symmetrical hemispheres, while others are truly trapezoidal with meridians unbroken at the Equator. The projection persisted well into the eighteenth century. Depending on the choice of standard parallels, a trapezoidal map may resemble the Collignon, Eckert's I and II projections. It is also a general case of the equidistant cylindrical, for standard parallels with different circumferences.

Azimuthal Projections

Equatorial azimuthal orthographic map Equatorial azimuthal stereographic map
Equatorial azimuthal orthographic (left) and stereographic (right) maps, central meridian 70°E
It is remarkable that some azimuthal projections, important even today, were known more than two thousand years ago by the Greeks and maybe by the Egyptians. The orthographic, stereographic and gnomonic projections are all based on solid principles of perspective and Euclidean geometry, while the azimuthal equidistant dates from the 15th century and is constructed arbitrarily.

The orthographic projection, one of the most realistic maps in a pictorial sense, provides an interesting contrast to the much later "globular" approaches.

Globular Projections

Apparently, much of the rich scientific inheritance from Antiquity was forgotten or ignored during most of the Western Middle Ages; however, despite the widespread modern myth, the idea of a spherical globe was never banned on religious terms and was accepted by most learned people. Christopher Columbus defended the idea not of a round, but smaller Earth; this mistaken claim made his proposed enterprise seem reasonably feasible.

The so-called globular maps were essentially simple pictorial devices for presenting general geographic features. Their main purpose was emphasizing the Earth's roundness; no globular projection is equal-area or conformal. Originally, all were restricted to one hemisphere bounded by a circle, with only equatorial aspects considered. Both the central meridian and Equator are straight, perpendicular lines. Basic geometrical constraints, summarized below, define all historic designs.

Reconstructed globular maps of the Eastern hemisphere; central meridian 70°E
Bacon map Apian 1 map Apian 2 map
Bacon First Apian Second Apian
Fournier 2 map Fournier 1 map Nicolosi map
Second Fournier First Fournier "Nicolosi"

In spite of superficial similarity to azimuthal projections, globular maps are not developed by proper perspective rules: the graticule is arbitrarily placed using easily drawn curves. As usual in cartography, no approach is perfect: although Georges Fournier's second work best matches the visual aspect of a three-dimensional globe, the "Nicolosi" map has possibly the least global shape distortion; latest to be widely published, the latter was almost certainly the very first to be recorded.

One of the oldest known arbitrary projections in a hemisphere was described by the great philosopher Roger Bacon (ca. 1265) and survived due to works by Monachus (ca. 1527) and d'Ailly. The design was modified in two proposals (1524) by Peter Apian (also known by the Latin name Petrus Apianus), one of which was used by Tramezzino (1554) and extended by Battista Agnese and Abraham Ortelius.

Fournier introduced two modifications to the globular style in 1643, the first using circular arcs instead of straight lines as parallels. A further change was popularized by Giovanni Nicolosi's projection of 1660 which, although also attributed to Philippe de La Hire in 1794, was probably first devised by the remarkable Islamic scholar al-Biruni (ca. 1000). Most modern mentions to "globular" maps refer to this "Nicolosi" design, which remained widely familiar even in the nineteenth century.

General features of classic globular projections
Projection Parallels Meridians
Common name Shape Equidistant at Shape Equidistant at
Bacon straight boundary meridians circular every parallel
Apian 1 straight central meridian circular every parallel
Apian 2 straight central meridian elliptical every parallel
Fournier 1 circular boundary and central meridians elliptical Equator
Fournier 2 straight boundary meridians elliptical every parallel
Nicolosi circular boundary and central meridians circular Equator

Oval and Extended Globular Maps

Extended Apian 1 map
Reconstruction of Ortelius's map, an approximate extension of Apian's first projection; central meridian 0°

Even though none of the classic globular projections were intended for showing more than one hemisphere at a time, all can be extended for spanning the whole world; the central circular hemisphere is identical to that of the original projection, except for the central meridian.

Oval maps have straight parallels and simple curves for meridians. The maps by Agnese (ca. 1540) and Ortelius (1570) are probably derived from Apian's first design, using semicircular arcs of fixed radius for the outer hemisphere. The result fits a 2:1 frame and looks like the modern projections III and IV by Eckert but it is not, of course, equal-area. Neither it is pseudocylindrical because the meridian spacing is not uniform in the inner and outer hemispheres.

Extended Apian 2 map
Second Apian map extended to whole world, central meridian 0°

The second design by Apian uses elliptical meridians and, as a modern curiosity, can be extended using arcs of inverse eccentricity. Despite being pseudocylindrical, its passing resemblance to Mollweide's elliptical map is only superficial.

A similar construction applied to Nicolosi's globular map produces an apple-shaped frame, an interesting contrast to Van der Grinten's IV projection.

Nicolosi map
"Nicolosi" projection extended to full world map, central meridian 0°

Octant Projections

Nicolosi map
South and North hemispheres of a possible reconstruction of Leonardo da Vinci's octant map; boundary meridians start from 45°. Gores were slightly spaced in the original map.

Resembling star projections, octant maps were briefly used contemporaneously to early oval maps. They divide the Earth surface in eight equal-shaped pieces, usually bounded by the Equator and four meridians. Each spherical triangle is separately projected into a roughly triangular octant; if its edges are circular arcs centered on the opposite vertex of an equilateral triangle, the octant's shape is called a Reuleaux triangle.

It is credited to Leonardo da Vinci a Reuleaux triangle-based octant world map (ca. 1514) with gores arranged in separate, shamrock-like hemispheres, but omitting the graticule; the projection method can only be speculated. Oronce Finé published a partial graticule (1551), again without specifying details. Quite possibly, parallels are nonconcentric circular arcs equally spaced along both meridian edges and central meridian of each octant; meridians are equally spaced along the Equator.



HomeSite MapMap Fold-outsMap Projections - ContentsProjections for Navigators  www.progonos.com/furuti    November  5, 2012
Copyright © 2001 Carlos A. Furuti