Amir Alexander, <i>Liberty’s Grid: A Founding Father, a Mathematical Dreamland, and the Shaping of America</i> Chicago: University of Chicago Press, 2024. Pp. 304. ISBN 978-0-226-82072-9. $30.00 (cloth).

On a wall of my home, I have a map of Iowa from 1854.The entire map is covered in a grid, with about a half-dozen different symbols indicating the status of each tiny square: surveyed, platted, field notes recorded and transcribed, under process of survey, and so on.In the upper north-west corner of the state, the grid is empty of symbols, yet the outcome feels inevitable: at some point in the near future, the map seems to declare, the furthest extent of Iowa will, too, be surveyed, platted and so on.Looking at a map of Iowa's county boundaries today, the grid is still there, although not nearly as perfect as the vision from 1854.Tama, Benton and Linn counties poke up higher than their neatly lined-up neighbours to the west.The line of counties starting with Woodbury and moving east have small notches in their north-east and north-west corners.Iowa's county boundaries were clearly governed by an imaginary grid, but today's map shows the concessions and deviations in the transition from idealized form to reality.Growing up in Iowa, I never had any reason to question why Iowa's counties were laid out in this way, although when quizzed on the names of all ninety-nine in school, I certainly wished that there were fewer of them.Amir Alexander's wonderful Liberty's Grid offers up the explanation for why Iowa -and, indeed, most of the United States -is laid out on this unrelenting pattern.The inevitability communicated on the 1854 map was, Alexander argues effectively, part of a larger ideological project, which by that point had been in place for decades.In tracing this project, Alexander skilfully and patiently directs the reader through a 'mathematical prologue', which lays out different conceptions of space from Aristotle to Descartes to Newton.Descartes gave us the coordinate system -with its familiar x and y axes -but Newton's concept of 'absolute geometrical space', separate from physical reality, meant that these axes could extend indefinitely.This Newtonian vision, Alexander argues, had a profound impact on Thomas Jefferson, who seized upon it as his concept for America: 'America was for him an empty, uniform, limitless, space -a blank slate on which enterprising settlers could employ their industry, write their own story, and create a new nation like no other' (p.85).Jefferson looked west and saw both promise and opportunity for independent yeomen farmers.This vision was aided by Jefferson's interest in mathematics, which had implications beyond geography; he successfully advocated for the United States to have the first decimal currency in the world, for example.Jefferson advocated for a plan to survey western lands, which was approved by Congress in modified form after he had departed to become the US envoy in Paris.But despite modifications, the 1785 plan included Jefferson's core idea of the grid, meaning that the West would be divided into 'precise, regular, and