In a previous post I identified the size of the pedestrian city as on the order of 50,000, let’s do this a bit more systematically.
Let’s illustrate with some assumptions:
- One-way Travel time budget (B) = 0.5 h
- Walking speed (S) = 5 km/h
- Walking network radius (Rn = S/B) = 2.5 km
- Network circuity (C) = 1.25
- Walking euclidean radius (Re = Rn/C) = 2 km
- Walking euclidean area (potential) (Ae=Pr*Re^2) = 12.56 km^2
- Population density (D) = 5,000 persons per km^2 [As a point of reference, the current population density of Manhattan is 27,485/km^2, which I would argue is only enabled by 19th century technologies like elevators and transit. Rome currently has a population density of 2,101/km^2]
- Population within TTB threshold (P=D*Ae) 62,800
Obviously you can construct a spreadsheet and play with population densities, which are highly disputed in ancient times. One sees claims that the City of Rome in ancient times had a population of 1 million people, but it is unclear over what area that was measured, and some estimates of those densities far exceed the densities of modern elevator cities (like Manhattan). I believe it is possible that high crowding occurred, but I think it unlikely that such crowding extended over large areas.
Also one can have a pedestrian city that exceeds the one-way walking travel time budget, but not a city one interacts with on a daily basis. This is more the equivalent of adjacent and overlapping cities, and likely have multiple cores.
David Levinson, Transportist 
While walking cities with populations ~50,000 almost definitely existed in the past, I don’t see a reason why population densities could not have been significantly higher.
The key question is whether population densities like those observed today in Manhattan were only enabled by technologies like the elevator.
Population densities > 5,000 km^2 are easily observed in many elevator lacking slums throughout the world where densities can reach 50k – 200k per km^2. Dharavi, in Mumbai, is the hyper-extreme case: some estimates peg densities at 300k per km^2.
Whether the densities observed in 20th and 21st slums correspond to densities of pre-19th century walking cities is somewhat of another matter but I mainly want to show that the technological explanation alone can’t explain a pop density of 27,485/km^2 in Manhattan today.
Come to think of it, the slums of today and ancient walking cities probably shared more than a few things in common: living within boundaries (defensive walls vs. slum boundaries), living at near subsistence levels in crowded quarters, and living within walking distance of each other/taking up very little land. So maybe the comparison isn’t totally off base.
Random aside: I was always struck by the passage in “The City in History” (p. 359) where Mumford claims, “…in the seventeenth century, these practices became universal: the systematic building of high tenements began –five or six stories high in old Geneva or in Paris, sometimes eight, ten, or more in Edinburgh.” If true, this would certainly have contributed to higher densities.
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This population limit seems surprisingly low to me, given that the Manhattan average density can, unless I am doing the math wrong, be easily achieved with four-unit apartment buildings with two residents apiece on 25-foot lots.
My understanding is that New York City didn’t have even horsecar transit until 1832, but according to the US Census it had an 1830 population of 202,589. Or is it too polycentric to count?
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Interesting take, and it reminds me of the most convincing solution to the Tragedy of the Commons I’ve seen, which is to organize ourselves into sufficiently small communities that the common good is easier to enforce (through the bonds of friends and family, namely, guilt and shame).
We can go back a very long way to see similar calculations of city size. Plato in his Laws determined the ideal city would have 5,040 heads of household (depending on the size of a household, probably not very far off your estimate of 50-60k). Of course his criteria were very different! He thought the size should accommodate a “moderate way of life,” allow a sufficient population to protect against invasion, be large enough to deliver aid to neighbors, and, being mathematically-inclined, divisible by a large number of divisors (5,040 is divisible 59 ways, including by 11 of the first 12 counting numbers). He stipulated that every household would have property in the dense center of town as well as on the periphery. Unfortunately he didn’t have much to say about transportation.
“Each of [the 5,040 lots] shall be divided into two, and every allotment shall be composed of two such sections; one of land near the city, the other of land which is at a distance. This arrangement shall be carried out in the following manner: The section which is near the city shall be added to that which is on borders, and form one lot, and the portion which is next nearest shall be added to the portion which is next farthest; and so of the rest… And [the legislator] shall distribute the twelve divisions of the city in the same way in which they divided the country; and every man shall have two habitations, one in the centre of the country, and the other at the extremity.”
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