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Obeliscus Augusti, gnomon: an obelisk erected at Heliopolis in the seventh century B.C. by Psammetichus II, brought to Rome by Augustus in 10 B.C. and set up in campus Martius between the ara Pacis Augustae and the columna Antonini Pii (CIL VI.702; Amm. Marcell. XVII.4.12; Strabo XVII.805; Plin. NH XXXVI.71). It is of red granite, 21.79 metres high (cf. Plin. loc. cit.; Notit. Brev.: Jord. II.187), and covered with hieroglyphics (BC 1896, 273‑283 = Ob. Eg. 104‑114). It was standing in the eighth century (Eins. 2.5; 4.3), but was thrown down and broken at some unknown date (BC 1917, 23), and not discovered until 1512 (PBS II.3). It was excavated in 1748, but, in spite of various attempts (LS IV.151), it was not set up again in the Piazza di Montecitorio, its present site, until 1789 (BC 1914, 381). It was repaired with fragments from the columna Antonini.a
Augustus dedicated this obelisk to the Sun (CIL VI.702) and made it the gnomon, or needle, of a great meridian1 (horologium, solarium) p367formed by laying an extensive pavement of marble on the north side of the shaft, the lines indicating midday at the various seasons of the year (Lumisden, Remarks on the Antiquities of Rome, 262; JRS 1921, 265, 266, is wrong), being marked by strips of gilt metal inlaid in the marble (Plin. NH XXXVI.72; Richter 252‑253, fig. 26). Seventy years later the indications of the dial were incorrect, and it was supposed that the obelisk had been slightly displaced by an earthquakeb (Plin. NH XXXVI.73). About 1484, and at various times in the next century, portions of the pavement were found, with the gilt lines, and figures in mosaic around the edge representing the winds and different heavenly bodies, but they were covered up again and are not visible (LS I.83, 136, 169; HJ 611, n26, and literature there cited). The height of the obelisk would require a pavement extending about 110 metres east and west, and 60 north and southc (HJ 610‑612; LR 466‑468; CIL VI.29820).
1 The name 'ad Titan', applied to the neighbouring church of S. Lorenzo in Lucina in liturgies of the eighthtenth centuries, which originated perhaps as early as the fifth, may refer to it (RAP IV.261‑277).
a It was repaired with fragments from the columna Antonini: one of the interesting things about this is that the base of the Antonine Column, perfectly preserved and now in the Vatican Museums, depicts our obelisk and its placement in the campus Martius: see this page for some very nice photos and excellent text. The obelisk has since been restored again, in 1966.
I am indebted to Prof. François Hinard of the Sorbonne for the following more recent information:
The pavement of the Horologium was excavated by a German team, and the part under a caffé in the via del Campo Marzio may in fact be visited (apply to the German Institute). F. Coarelli published a large number of photographs of it in Roma sepolta, Rome, Armando Curcio, "Biblioteca di archeologia", 1984 (ISBN 88 7555 010 7, 191pp); and explanatory text in his recent Campo Marzio. See also E. Buchner, Die Sonnenuhr des Augustus, Mainz am Rhein, 1984 (112 pp., 25 illustrations).
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b the obelisk had been slightly displaced by an earthquake: Pliny's careful text, in fact, offers 5 different explanations: hydrological, seismic, structural and 2 astronomical.
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c 110 metres east and west, and 60 north and south: Very strictly speaking, this is a curious statement. South, the obelisk never cast any shadow at all; and east and west, it would have required an infinite pavement! Not surprisingly then, results differ: Bollettino Telematico #35 at the Università La Sapienza in Rome, for example, states that Prof. Rakob of the German Archaeological Institute calculated the required space at 160 by 75 meters.
Redoing the trigonometry here, let's start with the N shadow. The formula for the maximum length of the shadow northwards, i.e., at noon on the winter solstice, is
λ = the latitude of Rome, 41°54
ε = the obliquity of the ecliptic, subject to slight secular variation but roughly 23°30
The maximum space required "northsouth" is therefore 2.1842 times the height of the obelisk (21.79 m plus the globe at the top, specifically mentioned by Pliny, say roughly 23 m), and the maximum length of the shadow on the pavement must have been 50 meters.
To calculate the exact lengths of shadow, one would have to know the exact height of the capping globe.
We do have the globe placed by the Romans on another obelisk of similar size, that originally stood on the spina of the Circus of Nero: it is now in the Palazzo dei Conservatori. (If you know its exact dimensions, I would appreciate hearing from you. For an approximate idea, see this 16c engraving of the area around Old S. Peter's.)
If, mind you, the Romans constructed this sundial rationally — and there is no indication otherwise — we can approximate the minimum size of this globe. In order to cast a readable shadow, it would have had to cover the sun's disc completely: it would have had a minimum angular width of 32 minutes of arc as seen from the point where its shadow was cast. At noon on the winter solstice, the shadow and the globe would have been 55 m apart: the minimum diameter of the globe should thus have been
For the E‑W shadows, symmetrical, we are forced to be more vague. From a practical standpoint, the infinitely long shadows nominally cast at sunrise and sunset are of course irrelevant. The question becomes: at what angular distance from its noon height does the sun cease to cast a readable shadow? The answer of course will depend on the observer.
A good guideline is provided by a writer contemporaneous with the Horologium: Vitruvius. Although it is in Book IX of the de Architectura that he writes on the construction of sundials, it is in Book I that he recommends designing them based on shadow lengths measured 5 hours before and after noon: clearly to eliminate error due to diffraction and weak light (1.vi.6 and see my note there).
To follow the Roman architect's advice, we first need to convert 5 Roman civil hours (on the date of the winter solstice) to equinoctial hours: easy enough, that's 5/12 of the time between sunrise and sunset on that date. I cheated and used the U.S. Navy's online sunrise and sunset calculator (ignoring the year, of course, which merely causes a slight cyclical variation): at Rome, the shortest day of the year is 9h08m long, 5/12 of which is our hour angle H: 3.81 equinoctial hours, or 57°05.
From that, with only slightly more math than for noon, the altitude angle β is given by the general formula:
β = arcsin [(cos λ × cos δ × cos H) + (sin λ × sin δ)]
where δ is the Sun's declination.
At the winter solstice, the declination δ = ε, so that
β = arcsin [(cos 41°54 × cos ()23°30 × cos 57°05) + (sin 41°54 × sin ()23°30)]
The altitude angle β is thus arcsin (.37093  .26630 = .10463); β= 6°00 (the sun is 6°00 above the horizon), cot β = 9.49 and the shadow will be roughly 9.49 × 23 m = 218 m in each direction.
It appears then — so far — that the pavement of Augustus' sundial would have to be a semiellipse with an eastwest major axis 436 m long of which the obelisk would be the midpoint; and a semiminor axis (northwards only from the obelisk) 50 m long. This would be a huge piazza: by way of comparison, the main part of the piazza S. Pietro is an ellipse whose major axis measures 240 m; alternately, about 10 U.S. football fields.
Unfortunately, for the E‑W dimensions of the pavement, my little investigation made one greatly simplifying assumption: I viewed the obelisk as standing in solitary glory in a flat area with nothing else in sight. In fact, the area is a slight hollow surrounded by several natural hills at various distances; and, more importantly, was part of an environment of buildings, much nearer and some rather tall, which in the lingo of sundial aficionados, fall under "horizon pollution": at which point any estimate of what shadow might have been cast by our obelisk bogs down irretrievably in some very hypothetical reconstructions of the urban landscape to the SE and SW.
My best conclusion is that Platner was on target for the NS dimension (allowing a tenmeter margin) but that he almost certainly underestimates the E‑W dimension: at 110 meters, it would have restricted the usefulness of the timepiece to only about half the daylight hours.
Images with borders lead to more information.
The thicker the border, the more information. (Details here.) 

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