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The Quinaria and Frontinus

"Later, a pipe called the 5-pipe (quinaria) came into use in the City to the exclusion of all former sizes. Its origin was based neither on the inch nor on either of the two kinds of digit. Some think that Agrippa was responsible for its introduction, others that this was done by the lead-workers under the influence of the architect Vitruvius....Most probable is the explanation that the name of the 5-pipe came from its diameter of five quarter-digits, according to a system which remains consistent in pipes of increasing size up as far as the 20-pipe: the diameter of each increases in size by the addition of one quarter-digit."

Frontinus, De Aquaeductu Urbis Romae (XXV.1, 4-5)

Even though Marcus Agrippa already had held higher office, he chose to be elected aedile in 33 BC so he could assume responsibility for the maintenance of Rome's aqueducts, fountains, and sewers, all of which had been neglected during the civil wars precipitated by Caesar's crossing of the Rubicon more than a decade and a half earlier. As the first curator of the city's water system, Agrippa apportioned water to public buildings, basins and fountains, and private individuals (Frontinus, XCVIII.2), allocations no doubt made easier by having been quantified.

When Frontinus was appointed superintendent of aqueducts by Nerva in 97 AD, he measured the capacity of Rome's water supply by the quinaria, a standard unit which was both the name of the smallest size of lead pipe and its capacity. It measured five quarter-digits in diameter (0.91 inches or 2.3 centimeters) and had a circumference of 3 89/96 digits (0.65 sq in or 4.15 sq cm) (XXXIX). Here  Frontinus describes the capacity of the Aqua Appia, the oldest of Rome's aqueducts and the first to have been restored by Agrippa: "I found the water had a depth of 5 feet and a width of 1 3/4 feet. This gives an area of 8 3/4 square feet, the equivalent of 22 100-pipes plus one 40-pipe or (expressed in terms of the 5-pipe) 1825 quinariae" (LXV.3). The area of the channel, in other words, was the aggregate capacity of two different pipe sizes: 2240 square digits, or the equivalent of 22 x 100 (the cross-section of a 100-pipe) plus 40 (the cross-section of a 40-pipe). The problem, of course, is what is meant by this unit, which is simply the diameter of the pipe (and its cross-sectional area), and how it was used to express a standard volume.

To be sure, Frontinus understands the quinaria or 5-pipe to be "a unit of capacity, for its size is most accurate and its standard best established" (XXVI.2). But how does diameter correspond to capacity? One way to make that determination is to install a pipe of a certain diameter in a castellum or distribution tank where the aqueduct terminated and then measure its flow. Frontinus himself remarks that his own measurements are "all the more reliable" because they were taken at a settling tank (LXXII.3). Set below the surface of the water under the same static pressure, a pipe discharges the same volume of water at a uniform rate of flow.

In 1916, Di Fenizio reasoned that, if a steady head of pressure could be established, a value for the quinaria could be determined. The largest pipe (fistula) in regular use was the centenaria, a 100-pipe, which was almost 23 cm in diameter. Assuming that such a pipe was completely submerged below the water level of the castellum (so as not to allow air to enter the line) and that the quinaria was set at a depth half the diameter of the centenaria, Di Fenizio posited that the minimum head would have to be at least 12 cm. Using a standard formula, he then calculated the capacity of the quinaria (as later corrected) to be almost 0.48 liters per second (approximately 41.5 cubic meters/day).

Such an estimation is a minimum figure, however, and depends upon the value attributed to the head, which may not have been uniform. Nor need it have been determined by the diameter of the largest pipe. Most fistulae in Rome averaged only 15 cm, and even a 19 cm pipe, which would have a head of 10 cm, would reduce the flow to 0.44 l/sec or 38.0 m3/day. Too, if Di Fenizio's value for the quinaria is used, Blackman found that some channels in Rome's four largest aqueducts were not deep enough to convey the amount of water reported by Frontinus. He calculates their total volume to be approximately 7 m3/second, a figure that Taylor accepts in determining his own estimation for the quinaria—32.8 m3/day, which would be approximately 82,000 m3/day for the Aqua Virgo.

As long as a fixed head is established and the rate of flow is uniform, as it would be under the same head, the diameter of the quinaria can serve as a standard unit for capacity, quantity being the capacity of the pipe to deliver it. When Frontinus states that "the rather rapid current of the water, taken from a broad and swift-flowing river, increases the quantity by its very velocity" (LXXIII.6) or complains that a measurement at the source of the Aqua Virgo cannot be taken because the current is "too gentle" (LXX.2), it is apparent that he was aware of the effect of velocity on capacity. But he was not able to measure speed of flow nor was it likely even  relevant for him.

In 11 BC, the year after Agrippa's death, the cura aquarum was established by Augustus to continue the work of his friend (XCIX.2). Water pipes for private use were required to be connected at the distribution tank and not directly to the aqueduct. Nor could they be larger than the quinaria (CVI.1-2). An adjutage (calix) was to be fitted at the castellum, to which a pipe of the same size was attached. Made of bronze to ensure that its diameter was uniform and could not be enlarged, this collar or nozzle controlled how much water was discharged (XXXVI.3-5; CV.4-5).

Correctly inserted, the calix was set half-way up the wall of the castellum, perpendicular and horizontal to it. But if it were placed lower in the tank or angled downward (whether deliberately or accidentally), there would be more pressure and more water, just as there would be if the calix were directed into the flow (XXXVI.2, CXIII.1-2). Other ways to circumvent the official distribution of water were enumerated by Frontinus (CXII-CXV). A larger calix could be placed in the tank or a larger pipe fixed to it. Or pipes could be placed at different levels below the surface of the water. Some were not even fitted to calices or, if a new pipe was installed, the old one was left in place to draw water, which then was sold. Lead pipes also could be punctured. The result was that, of the 14,018 quinariae officially delivered by the nine aqueducts of Rome then in use, another 10,000 quinariae were diverted illegally (LXIV.2, LXXIV.4).

"If the water is to be brought in leaden pipes, a reservoir is first made near the spring, from whence to the reservoir in the city, pipes are laid proportioned to the quantity of water. The pipes must be made in lengths of not less than ten feet: hence if they be one hundred digits wide (centenarić), each length will weigh twelve hundred pounds....if five digits (quinarić), sixty pounds. It is to be observed that the pipes take the names of their sizes from the quantity of digits in width of the sheets, before they are bent round: thus, if the sheet be fifty digits wide, before bending into a pipe, it is called a fifty-digit pipe; and so of the rest."

Vitruvius, De Architectura (VIII.6.4)

Writing more a century and a quarter after Vitruvius, Frontinus regarded the width of the lead sheet as an inaccurate description of the quinaria, "because in forming a cylindrical shape the inner surface is contracted while the outer surface is extended" (XXV.3), i.e., the exterior circumference of the pipe would be larger than its interior circumference. A rolled sheet of lead five digits wide would have a larger diameter than the quinaria as defined by Frontinus.

The digitus (digit) is one-sixteenth of a Roman foot and the uncia (inch), one-twelfth (approximately three-quarters and one inch, respectively). And "Just as there is a distinction between the inch and the digit, there are also two kinds of digits. One is called square, the other round. The square digit is larger than the round by three-fourteenths of its own size; the round digit is smaller than the square by three-elevenths of its size (because, of course, the corners are taken away)" (XXIV.3-5). In expressing this relationship of a square with a side of one digitus to a circle with a diameter of one digitus, Frontinus reveals that he understood pi to be 3 1/7 (= 3.1429). Vitruvius, on the other hand, thought it to be 3 1/8 (= 3.1250) (X.9.1). Both are very close to the actual value of pi, which is 3.1416.

Maher and Makowski contend that Frontinus, in this discussion of the relative area of the square and the circle, is using a general fraction (one in which the numerator and denominator are both whole numbers) for the first time in Roman literature and that De Aquaeductu Urbis Romae represents "the peak of extant Roman accomplishments in arithmetic."

"Pressure" and "head" are two terms that can mean the same thing, although from a different perspective. They are related by the mathematical constant 2.31, which converts head into pressure (and vice versa). A head in feet of water divided by this constant equals pressure in pounds per square inch. Pressure in psi times the constant equals feet of head. The figure 2.31, itself, derives from the fact that one pound of water fills a one-inch square column to a height of 2.31 feet. Water, for example, from an aqueduct 160 feet high (that of the Pont du Gard, the highest Roman aqueduct bridge) would provide water pressure at ground level of about 69 psi (160 ÷ 2.31).

The striking marble bust above is that of Marcus Agrippa and is a copy of what may have been a bronze original, the most famous of which was the statue of Agrippa in the Pantheon, which he completed in 25 BC. Originally part of the art collection of Camillo Borghese, it was acquired in 1807 by Napoleon, whose sister was married to the prince, and now is in the Louvre.

References: "Copia Aquarum: Frontinus' Measurements and the Perspective of Capacity" (1986) by R. H. Rodgers, Transactions of the American Philological Association, 116, 353-360; "The Impossibility of Reaching an Exact Value for the Roman Quinaria Measure" by Christer Bruun, in Frontinus: De Aquaeductu Urbis Romae (2004) edited and translated by R. H. Rogers; "How Did Frontinus Measure the Quinaria?" (1984) by A. Trevor Hodge, American Journal of Archaeology, 88(2), 205-216; Public Needs and Private Pleasures: Water Distribution, the Tiber River and the Urban Development of Ancient Rome (2000) by Rabun Taylor; "Literary Evidence for Roman Arithmetic with Fractions" (2001) by David W. Maher and John F. Makowski, Classical Philology, 96(4), 376-399; Frontinus' Legacy: Essays on Frontinus' de aquis urbis Romae (2001) by Deane R. Blackman and A. Trevor Hodge; The Two Books on the Water Supply of the City of Rome of Sextus Julius Frontinus (1899) translated by Clemens Herschel; "Torrent or Trickle? The Aqua Alsietina, the Naumachia Augusti, and the Transtiberim" (1997) by Rabun Taylor, American Journal of Archaeology, 101(3), 465-492.


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