was the instrument from which the Roman gromatici
or land-surveyors took their name. The
word is generally said to be a Latin version of γνώμονα
(acc. of γνώμων
though the two instruments are in no respect alike, but more recently a
connexion has been suggested with norma
that may be, groma
is certainly a very ancient name, for which later writers
sometimes used machinula
and perhaps tetrans,
this name belongs rather to cross lines drawn on the ground (cf. Nonius, p.
63, for gruma
in this sense).
The groma is represented (fig. 1) on the gravestone of a gromaticus
found some years ago at lvrea. (Cavedoni in
Bull. Arch. Nap., N. S.,
Anno I.; Cantor, Vorles.
zur Gesch. der Math.
1.456; Rossi, Groma e
1877, p. 43.) The design is not in perspective, but, if
allowances be made for the inexperience of the artist, it explains fairly
well the nature of the instrument. Two small planks crossing one another at
right angles are supported on a column or post (ferramentum
). Plummets (probably four, though there are only
two in the monument), are suspended from the
Groma, fig. 1. (From a gravestone.)
planks to guide the operator in securing a vertical position of
the column, and a horizontal for the cross-pieces. The small circles at the
point of section in the drawing may represent a hole in the continuation of
the column for the operator to look through, or a large hole in the
cross-pieces to allow of their being tipped up to a certain angle if
necessary. The latter is the more likely, for in that case the continuation
of the column would serve as a support to prevent the cross from falling. In
any case it obstructs the view along the planks.
The use of the instrument is obvious. It is intended to guide a surveyor in
drawing real or imaginary lines at right angles to one another, more
especially in fixing the cardo
and S. line) and decumanus
(or E. and W. line) essential to the orientation of any templum
or to the [p. 1.924]
division of a
Roman camp [CASTRA
]. A very
large T square was sometimes used for the same purpose (see below).
Another very ancient method of drawing long lines at right angles was to
stretch a rope round three pegs fixed in the ground, at such distances from
one another that the sides of the triangle formed by the rope were in the
ratio 3: 4: 5. The angle contained between the shorter sides is a right
angle (Eucl. 1.48). This method was very early used by the Egyptians, whose
“rope-stretchers” were considered by Democritus the masters of
geometry in his day (Clemens Al. Strom.
1.357). It was
employed also at the building of the temple of Edfu, as the inscriptions
thereon declare, and was used by Heron himself on occasion (περὶ διόπτρας,
Props. 21, 22; cf. Cantor,
pp. 55, 56, 324, 325). The knowledge of the ratio
3: 4: 5 for the sides of a right-angled triangle was introduced into Greece
by Pythagoras, and was for a long time one of the most interesting subjects
of Greek geometry (cf. Proclus, Comm. in Eucl. i.
Friedlein, p. 426).
But the groma
and all other instruments of the
same kind were ultimately superseded, for all purposes requiring any nicety
of measurement, by the dioptra,
which very closely resembles the modern theodolite. It seems to have been
the outcome of many gradual improvements on the groma.
Heron, who describes it, does not claim it as his own
invention, though he is probably responsible for many details of the
instrument which he describes. Biton, who seems to be earlier than Heron,
says that he had, in his Optics, described a διοπίον
to be used in ascertaining the heights of walls,
&c. (Vett. Mathematici,
ed. Thevenot and La Hire,
1693, p. 108), and Heron himself says that the διόπτραι
of previous writers were many and various. The
described by Heron (Not.
et Extraits des MSS. de la Bibl. Impér.
pt. 2, ed. Vincent, p. 157 ff., with a plate) is constructed as follows. A
flat brass rod, about eight feet long, with two small sights at each end, is
supported on a pillar, which may be adjusted to the perpendicular by
comparison with a plummet hanging by its side, the rod being at the same
time adjusted to the horizontal by comparison with a water-level attached to
it in a groove. The rod is not immediately attached to the pillar, but is
supported on a frame which allows it to swing horizontally, and to a less
degree vertically. Two screws (κοχλίδια
turning cog-wheels (ὀδοντωτὰ τυμπάνια
provided in an ingenious manner to regulate the two movements of the rod. It
seems also that a button fixed in the plate of the horizontal cog-wheel
stopped the movement when the rod had turned exactly through a right angle
in the horizontal plane. The vertical cog-wheel allowed only for a swing of
about 45° in that plane. With the dioptra
were used also two poles of precisely the same kind as
those used by modern surveyors. The poles were 20 feet long, marked with
measures and provided with an internal slide (χελωνάριον
), also nearly 20 feet long. The slide was
surmounted with a circular disc, coloured half black and half white, which
of course was always visible at the top of the pole. The slide could be
pulled up by a string until the disc was elevated 40 feet high. . It is
evident from Vitruvius (8.6
) and other writers
that the dioptra
was adopted for nice
calculations by the Roman surveyors, and the terms used by the gromatici
ed. Lachmann, &c.,
Berlin, 1848), e.g. hortogonium, hypotenusa, embadum,
&c., show how largely their art was derived from Greek sources,
especially the works of Heron (Cantor, Vorles.
i. p. 470).
The treatise on the dioptra
by Heron, above
mentioned, contains a number of propositions illustrative of the use of the
instrument. A few examples will suffice: the general aim of all the problems
being to find the difference of level between two given points, or to draw a
straight line from a point to an unseen point. Thus Prop.
is to cut a straight tunnel through a hill from one given
point to another: 7 is to find the height of an inaccessible point: 14 and
15 are to sink a shaft which shall meet a horizontal tunnel: 19 and 20 are
to find two points at a given distance from one another. The examples given
by Heron are seldom purely theoretical. He prefers, when a distance is
given, to state it in actual measurement, and thus makes continual use of
arithmetic and even of algebraical operations. His geometry is confined
almost entirely to the first and the sixth books of Euclid. The measurement
of angles by degrees, &c., was never used by the Greeks or Romans
for any but astronomical purposes, and even in that field is not found
before Ptolemy, though he derived the practice from Hipparchus (vid.
ed. Halma, i. p.
The most common problem in the writings of ancient surveyors which have come
down to us is that of finding the breadth of a river without crossing it,
the Latin fluminis va<*>atio.
The solution of this is given in the treatise on the dioptra
(pr. 4) in the 21st chap. of the Κεστοὶ
of Sex. Julius Africanus. (in the
pp. 295-6), and by Nipsus in the
(1.285-6). In Africanus it is put in
strategical form, the inaccessible shore of the river being supposed to be
in the occupation of the enemy. His solution is as follows:--εε
is the accessible shore of the river, α
a point on the opposite side. The dioptra
is set up at a point ι,
obviously further from the bank than the breadth of the
river, and the sights are directed so that the line ια
shall cut the river at right angles in φ.
is then turned
Groma, fig. 2.
through a right angle and the point υ
is taken on ιυ
at right angles
to at. ιυ
is bisected at κ,
drawn parallel to αι,
meeting in θ
the line joining αυ.
drawn parallel to ιυ,
A preceding proposition (Cantor, ϝορλες.
p. 372) of the simplest character has
shown that the sides αι, αυ
of the triangle
α ι υ
are bisected in ρ
and it remains only to measure ρι
and to deduct the
measure [p. 1.925]
that of ρι.
The remainder is the measure of
Another mode of solving the same problem is given by the same writer in the
same chapter of the Κεστοί.
only a very large solid right angle or Τ
Groma, fig. 3.
figure 3 the thickened lines εδ
represent such a right angle. Then,
by similar triangles, αβ
Of these three distances the last three are
known. In the same figure, if αβ
height of a tower, the triangles are drawn in a vertical plane: εδ
is the pillar supporting a tilted dioptra,
the line from the top of the tower passing through the sights of the
and produced to meet the ground.
The superior advantages of the dioptra
are obvious from this last example.
It should perhaps be added that the Gromatici
Frontinus, Hyginus, &c., above mentioned, were, so far as the extant
fragments enable us to judge, judicial officers, engaged chiefly in taking
areas, &c., for the purposes of taxation, division, &c.
Hence there are not, in the works attributed to them, many of the more nice
and difficult problems which are discussed with so much acuteness by Heron
and Julius Africanus. Enough, however, is said on such subjects to show that
the Roman surveyors were fully learned in the works of their Greek
predecessors. (Cf. Rudorff on Gromatische Institut.
ii. p. 230 sqq.
Leipzig, 1875; and AGRIMENSORES