The History of Mathematics: A Brief Course

ROMAN GEOMETRY 327

of longitude as straight lines emanating from the common center, representing the

north pole. Thus, his plan is to map this portion of the Earth into the portion

of a sector of a disk bounded by two radii and two concentric circles. In terms

of Fig. 4, his first problem is to decide which radii and which circles are to form

these boundaries. Ptolemy recognized that it would be impossible in such a map

to place all the parallels of latitude at the correct distances from one another and

still get their lengths in proportion. He decided to keep his northernmost parallel,

through Thoule, in proportion to the parallel through the equator. That meant

these arcs should be in the proportion of about 9:20—to be precise, cos(63°) in

our terms. Since there would be 63 equal divisions between that parallel and the

equator, he needed the upper radius χ to satisfy χ : (χ + 63) :: 9 : 20. Solving this

proportion is not hard, and one finds that χ = 52, to the nearest integer. The next

task was to decide on the angular opening. For this principle he decided, like his

predecessor Marinus, to get the parallel of latitude through Rhodes in the correct

proportion. Since Rhodes is at 36° latitude, half of the parallel through it amounts

to about I of the 180° arc of a great circle, which is about 145°. Since the radius

of Rhodes must be 79 (27 great-circle degrees more than the radius of Thoule), he

needed the opening angle of the sectors è to satisfy è : 180° :: 146 : 79ð, so that

9 « 106°. After that, he inserted meridians of longitude every one-third of an hour

of longitude (5°) fanning out from the north pole to the equator.

Ptolemy recognized that continuing to draw the parallels of latitude in the

same way for points south of the equator would lead to serious distortion, since the

circles in the sector continue to increase as the distance south of the north pole

increases, while the actual parallels on the Earth begin to decrease at that point.

The simplest solution to that problem, he decided, was to let his southernmost

parallel at 16° 25' south have its actual length, then join the meridians through

that parallel by straight lines to the points where they intersect the equator. Once

that decision was made, he was ready to draw the map on a rectangular sheet of

paper. He gave instructions for how to do that: Begin with a rectangle that is

approximately twice as long as it is wide, draw the perpendicular bisector of the

horizontal (long) sides, and extend it above the upper edge so that the portion

above that edge and the whole bisector are in the ratio 34° : 131°, 25'. In that

way, the 106° arc through Thoule will begin and end just slightly above the upper

edge of the rectangle, while the lowest point of the map will be at the foot of the

bisector, being about 80 units below the lowest point on the parallel of Thoule, as

indicated by the dashed line in Fig. 4.

This way of mapping is not a conical projection, as it might appear to be, since

it preserves north-south distances. It does a tolerably good job of mapping the

parts of the world for which Ptolemy had reliable data. One can recognize Europe

and the Middle East in the map of Plate 8, constructed around the year 1300 CE

to accompany an edition of the Geography.

2.1. Roman civil engineering. Dilke (1985, pp. 88-90) describes the use of ge-

ometry in Roman civil engineering as follows. The center of a Roman village would

be at the intersection of two perpendicular roads, a (usually) north-south road

called the kardo maximus (literally, the main hinge) and an east-west road called

decumanus maximus, the main tenth. Lots were laid out in blocks (insula?) called

hundredths (centunae), each block being assigned a pair of numbers, telling how

many units it was dextra decumani (on the right decumanus) or sinistra decumani