- ROMAN GEOMETRY 325
the area rotated and the distance traversed by its center of gravity (which is 2ð
times the length of the line from the center of gravity to the axis of rotation). In
the modern form this theorem was first stated in 1609 by the Swiss astronomer-
mathematician Paul Guldin (1577-1643), a Jesuit priest, and published between
1635 and 1640 in the second volume of his four-volume work Centrobaryca seu
de centra gravitatis trium specierum quantitatis continuae (The Barycenter, or on
the Center of Gravity of the Three Kinds of Continuous Magnitude). Guldin had
apparently not read Pappus and made the discovery independently. He did not
prove the result, and the first proof is due to Bonaventura Cavalicri (1598-1647).
2. Roman geometry
In the Roman Empire geometry found applications in mapmaking. The way back
from the abstractness of Euclidean geometry was led by Heron, Ptolemy, and
other geometers who lived during the early Empire. We have already mentioned
Ptolemy's Almagest, which was an elegant arithmetization of some basic Euclidean
geometry applied to astronomy. In it, concrete computations using the table of
chords are combined with rigorous geometric demonstration of the relations in-
volved. But Ptolemy studied the Earth as well as the sky, and his contribution to
geography is also a large one, and also very geometric.
Ptolemy was one of the first scholars to look at the problem of representing
large portions of the Earth's surface on a flat map. His data, understandably very
inaccurate from the modern point of view, came from his predecessors, including the
astronomers Eratosthenes (276-194) and Hipparchus (190-120) and the geographers
Strabo (ca. 64 BCE-24 CE) and Marinus of Tyre (70-130), whom he followed
in using the now-familiar lines of latitude and longitude. These lines have the
advantage of being perpendicular to one another, but the disadvantage that the
parallels of latitude are of different sizes. Hence a degree of longitude stands for
different distances at different latitudes.
Ptolemy assigned latitudes to the inhabited spots that he knew about by com-
puting the length of sunlight on the longest day of the year. This computational
procedure is described in Book 2, Chapter 6 of the Almagest, where Ptolemy de-
scribes the latitudes at which the longest day lasts 12j hours, 12| hours, and so
on up to 18 hours, then at half-hour intervals up to 20 hours, and finally at 1-hour
intervals up to 24. Although he knew theoretically what the Arctic Circle is, he
didn't know of anyone living north of it, and took the northernmost location on the
maps in his Geography to be Thoule, described by the historian Polybius around
150 BCE as an island six days sail north of Britain that had been discovered by the
merchant-explorer Pytheas (380-310) of Masillia (Marseille) some two centuries
earlier.^4 It has been suggested that Thoule is the Shetland Islands (part of Scot-
land since 1471), located between 60° and 61° north; that is just a few degrees south
of the Arctic Circle, which is at 66° 30'. It is also sometimes said to be Iceland,
which is on the Arctic Circle, but west of Britain rather than north. Whatever
it was, Ptolemy assigned it a latitude of 63°, although he said in the Almagest
that some "Scythians" (Scandinavians and Slavs) lived still farther north at 645°.
Ptolemy did know of people living south of the equator and took account of places
as far south as Agisymba (Ethiopia) and the promontory of Prasum (perhaps Cabo
Delgado in Mozambique, which is 14° south). Ptolemy placed it 12° 30' south of
(^4) The Latin idiom uiiiraa Thule means roughly the last extremity.