328 11. POST-EUCLIDEAN GEOMETRY
FIGURE 5. Nipsus' method of computing the width of a river.
(on the left decumanus) and how many units it was ultra kardinem (on the far
kardo) or citra kardinem (on the near kardo).^7
A collection of Roman writings on surveying was collected, translated into Ger-
man, and published in Berlin in the middle of the nineteenth century. This two-
volume work bears the title Corpus Agrimensorum Romanorum, the word agrimen-
sor (field measurer) being the Latin name for a surveyor. Among the agrimensores
was one named M. Iunius Nipsus, a second-century surveyor, who, according to
Dilke (1985, p. 99), gives the following directions for measuring the width of a river
(Fig. 5).
You mark the point C on the opposite bank from Β (a part of the procedure
Nipsus neglects to mention until later), continue the straight line CB to some
convenient point A, lay down the crossroads sign at A, then move along the direction
perpendicular to AC until you reach a point G, where you erect a pole, then continue
on to D so that GD = AG. You then move away from D along the direction
perpendicular to AD until you see G and C in a straight line from the point H.
Since the triangles AGC and DGH are congruent (by angle-side-angle), it follows
that CB = CA - AB = HD - AB.
For this procedure to work in practice, it is necessary to have an accessible and
level piece of land covering the lines shown as AD and DH. If the river is large,
such a stretch of land may not exist, since the river banks are likely to be hilly. In
its neglect of similar triangles, this method seems a large step backward in applied
geometry.
3. Medieval geometry
Among the translations of Greek works into Latin mentioned above was a transla-
tion of Euclid's Elements written by Boethius (ca. 480-524). This work has been
lost, although references to it survive.^8 A "pseudo-Boethius" text of geometry,
written some centuries later, has survived. It may have been a standard text dur-
ing the Middle Ages. There were anonymous treatises on geometry during this time,
some attributed to Boethius, usually containing Latin translations of a few of the
(^7) In modern terms these would be First Avenue East, First Avenue West, North Main Street,
and South Main Street.
(^8) For example, in his Encyclopedia of Liberal and Literary Studies the early sixth-century writer
Magnus Aurelius Cassiodorus refers to the great Greek mathematicians "of whom Euclid was
given to us translated into the Latin language by the same great man Boethius."