#### GROMA

GROMA 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 for (g However that may be, groma or gruma is certainly a very ancient name, for which later writers sometimes used machinula or and perhaps tetrans, though 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 Squadro, 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 (or N. 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 ἁρπεδονάπταιor “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 ( Props. 21, 22; cf. Cantor, Vorles. 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. 47, ed. 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 an instrument which very closely resembles the modern theodolite. It seems to have been the outcome of many gradual improvements on the 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 dioptra described by Heron (Not. et Extraits des MSS. de la Bibl. Impér. vol. xix. 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 ( are 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 (Grom. Vett. or Römische Feldmesser, ed. Lachmann, &c., Berlin, 1848), e.g. hortogonium, hypotenusa, embadum, podismus, praecisura (ἀποτομή), &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. 13 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. Theon's Comm. ed. Halma, i. p. 110).

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 Vett. Mathematici, pp. 295-6), and by Nipsus in the Gromat. Vett. (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. The dioptra 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, and κθ is drawn parallel to αι, meeting in θ the line joining αυ. From θ again θρ drawn parallel to ιυ, meeting αι in ρ. A preceding proposition (Cantor, p. 372) of the simplest character has shown that the sides αι, αυ of the triangle are bisected in ρ and θ. Therefore αρ==ρι, and it remains only to measure ρι and φρ, and to deduct the measure [p. 1.925]of φρ from that of ρι. The remainder is the measure of αφ.

Another mode of solving the same problem is given by the same writer in the same chapter of the This requires only a very large solid right angle or Τ square (γνώμων). In

 Groma, fig. 3.

figure 3 the thickened lines εδ and δγ represent such a right angle. Then, by similar triangles, αβ: εδ:: βγ: δγ. Of these three distances the last three are known. In the same figure, if αβ be the height of a tower, the triangles are drawn in a vertical plane: εδ is the pillar supporting a tilted and αεγ is the line from the top of the tower passing through the sights of the dioptra and produced to meet the ground. The superior advantages of the dioptra over the groma 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. in the Grom. Vett. ii. p. 230 sqq.; Cantor's Agrimensoren, Leipzig, 1875; and AGRIMENSORES

[J.G]

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