"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,

The Aqueducts of Rome (XXV.1, 4–5)Although once consul of Rome, Marcus Agrippa chose the lower office of

curule aedilein 33 BC so that, as magistrate of public buildings and works, he could assume responsibility for the maintenance of the city's aqueducts, fountains and sewers—all of which had been neglected during the civil war precipitated by Caesar's crossing of the Rubicon more than a decade and a half earlier. As the firstcuratoraquarumor superintendent of the water system, Agrippa was responsible for apportioning water to public buildings, basins and fountains, and private individuals (Frontinus, XCVIII.2).When Frontinus himself was appointed

curator aquarumby Nerva in 97 AD, he measured the capacity of Rome's water supply by thequinaria, 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). 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) 1825quinariae" (LXV.3). The area of the channel, in other words, was the aggregate capacity of two different pipe sizes: 2,240 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

quinariaor 5-pipe to be "a unit of capacity, for its size is most accurate and its standard best established" (XXVI.2). But how did diameter correspond to capacity? One way to make that determination was to install a pipe of a certain diameter in acastellumor 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

quinariacould be determined. The largest pipe (fistula) in regular use was thecentenaria, a 100-pipe, which was almost 23 cm in diameter. Assuming that such a pipe was completely submerged below water level (so as not to allow air to enter the line) and that thequinariawas set at a depth half the diameter of thecentenaria, 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 thequinaria(as later corrected) to be almost 0.48 liters per second (approximately 41.5 m^{3}/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

fistulaein 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 m^{3}/day. Too, if Di Fenizio's value for thequinariais 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 m^{3}/second, a figure that Taylor accepts in determining his own estimation for thequinaria: 32.8 m^{3}/day, which would be approximately 82,000 m^{3}/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

quinariacan 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 aquarumwas 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 thequinaria(CVI.1–2). An adjutage (calix) was to be fitted at thecastellum,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

calixwas set half-way up the wall of thecastellum, 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 thecalixwere 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 largercalixcould 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 tocalicesor, 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 14,018quinariaeofficially delivered by the nine aqueducts of Rome then in use, another 10,000quinariaewere 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,

On Architecture (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 thequinariaas defined by Frontinus.The

digitus(digit) is one-sixteenth of a Roman foot and theuncia(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 onedigitusto a circle with a diameter of onedigitus, Frontinus understoodpito be 3 1/7 (3.1429). Vitruvius, on the other hand, thoughtpito be 3 1/8 (3.1250) (X.9.1). Both are very close to the actual value ofpi, 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 Romaerepresents "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).

"When they are brought home to the walls of the city a reservoir (castellum) is built, with a triple cistern attached to it to receive the water. In the reservoir are three pipes of equal sizes, and so connected that when the water overflows at the extremities, it is discharged into the middle one, in which are placed pipes for the supply of the fountains, in the second those for the supply of the baths, thus affording a yearly revenue to the people; in the third, those for the supply of private houses. This is to be so managed that the water for public use may never be deficient, for that cannot be diverted if the mains from the heads are rightly constructed. I have made this division in order that the rent which is collected from private individuals who are supplied with water, may be applied by collectors to the maintenance of the aqueduct."

Vitruvius,

On Architecture (VI.1–2)Frontinus speaks of a similar distribution (LXXVIII–LXXXVI). Water was furnished in the name of Caesar (to supply the imperial palace and other official buildings), to private individuals as a favor of the emperor, and for public use, which included military camps, public works (such as baths, circuses, and markets), fountains, and open basins.

The striking marble bust is that of Marcus Agrippa and 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 RomanQuinariaMeasure" by Christer Bruun, inFrontinus: 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.