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Proof!: How the World Became Geometrical
An eye-opening narrative of how geometric principles fundamentally shaped our world
On a cloudy day in 1413, a balding young man stood at the entrance to the Cathedral of Florence, facing the ancient Baptistery across the piazza. As puzzled passers-by looked on, he raised a small painting to his face, then held a mirror in front of the painting. Few at the time understood what he was up to; even he barely had an inkling of what was at stake. But on that day, the master craftsman and engineer Filippo Brunelleschi would prove that the world and everything within it was governed by the ancient science of geometry.
In Proof!, the award-winning historian Amir Alexander traces the path of the geometrical vision of the world as it coursed its way from the Renaissance to the present, shaping our societies, our politics, and our ideals. Geometry came to stand for a fixed and unchallengeable universal order, and kings, empire-builders, and even republican revolutionaries would rush to cast their rule as the apex of the geometrical universe. For who could doubt the right of a ruler or the legitimacy of a government that drew its power from the immutable principles of Euclidean geometry?
From the elegant terraces of Versailles to the broad avenues of Washington, DC and on to the boulevards of New Delhi and Manila, the geometrical vision was carved into the landscape of modernity. Euclid, Alexander shows, made the world as we know it possible.
320 pages, Kindle Edition
First published September 10, 2019
41 people are currently reading
479 people want to read
About the author
Amir Alexander
8 books34 followersAmir Alexander teaches history at UCLA. He is the author of Geometrical Landscapes and Duel at Dawn. His work has been featured in Nature, the Guardian, among others. He lives in Los Angeles, California.
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Displaying 1 - 30 of 32 reviews
September 28, 2019
The hypothesis proposed by Alexander is intriguing, and it may in fact be true, but ironically, it seems his own 'proof' of his central claim is only weakly supported.
In this treatment of the relations between geometry and politics in the 17th Century, it is striking that the only support for this proposition are a handful of metaphorical connections between the absolute sovereign and geometric concepts. Are these metaphors really sufficient to support that claim that the 'world became geometrical'? In particular, this is true in the handful of pages on Thomas Hobbes, who I note was an avid enthusiast of Euclidean Geometry and published a number of attempts to square the circle. Alexander here focuses on Hobbes use of a geometric method in his political philosophy. But it should be noted, that the use of a geometric method in philosophy is not exactly to claim that 'the world' is structured geometrically. In fact Alexander's treatment here seems so awfully superficial, and relies on dubious readings of the key texts discussed.
Disappointing.
In this treatment of the relations between geometry and politics in the 17th Century, it is striking that the only support for this proposition are a handful of metaphorical connections between the absolute sovereign and geometric concepts. Are these metaphors really sufficient to support that claim that the 'world became geometrical'? In particular, this is true in the handful of pages on Thomas Hobbes, who I note was an avid enthusiast of Euclidean Geometry and published a number of attempts to square the circle. Alexander here focuses on Hobbes use of a geometric method in his political philosophy. But it should be noted, that the use of a geometric method in philosophy is not exactly to claim that 'the world' is structured geometrically. In fact Alexander's treatment here seems so awfully superficial, and relies on dubious readings of the key texts discussed.
Disappointing.
October 22, 2023
This was really a dull read - I tried twice to get through it, and though I am a total recreational math nerd, I could not convince myself this was worth my time. It is not at all heavy on math - more a history book on royal gardens, as far as I could tell. The author keeps claiming this is where geometry really entered the mainstream, but he never seems to actually say anything about geometry. Highly recommend skipping this one.
February 8, 2025
I went into Proof! expecting a dive into a wide arrange of topics related to geometry and geometry applications: perhaps how geometry influences and dictates architecture, or how geometry can accentuate beautiful art, or how it appears in nature in immutable ways. Instead, I practically only got a lengthy lesson on the history of French palaces and gardens. It was a decent starting point: many of those 18th-19th century gardens *are* geometrical, so sure, I’ll play along. I expected that to be an introductory topic, but when the author continued into the next chapters talking about how jealous Frenchmen kept constructing different and more elaborate gardens, I knew we’d be stuck here for a while.
It was still interesting at times, but almost entirely more of a history lesson rather than an exploration of geometry. That French influence seeped into America in the 18th century, but the topic of the book never changed: it remained a history lesson with the presence or relevance of an architect or urban designer being involved serving as a thin veil of “geometry!” just to be able to still relate to the purported topic of the book. In the introduction, the author brings up ancient mathematicians of legend, discuses how they proved things, what they discovered, but then they are never mentioned again once we get to the topic of palace gardens. Even Chapter 1 still had me, with its explanations of how 15th-century artists “discovered” and incorporated perspective in their paintings. From Chapter 2 onward, though, it was not the book I thought I was signing up for. Just interesting enough in a few areas to keep me moving on, I ultimately ended up finishing the book disappointed. I wouldn’t say it’s a bad book, but it’s poorly explained what you’ll be getting into. That subversion, interventional or not, irked me. If you’re interested in this book, I would just be sure you know what you’re getting into first.
1.75/5 ⛲️
It was still interesting at times, but almost entirely more of a history lesson rather than an exploration of geometry. That French influence seeped into America in the 18th century, but the topic of the book never changed: it remained a history lesson with the presence or relevance of an architect or urban designer being involved serving as a thin veil of “geometry!” just to be able to still relate to the purported topic of the book. In the introduction, the author brings up ancient mathematicians of legend, discuses how they proved things, what they discovered, but then they are never mentioned again once we get to the topic of palace gardens. Even Chapter 1 still had me, with its explanations of how 15th-century artists “discovered” and incorporated perspective in their paintings. From Chapter 2 onward, though, it was not the book I thought I was signing up for. Just interesting enough in a few areas to keep me moving on, I ultimately ended up finishing the book disappointed. I wouldn’t say it’s a bad book, but it’s poorly explained what you’ll be getting into. That subversion, interventional or not, irked me. If you’re interested in this book, I would just be sure you know what you’re getting into first.
1.75/5 ⛲️
April 30, 2020
The book turned out to be a very good book for the wrong reasons. Everything on the book cover to the description on the inside cover of the book led me to believe this was going to be a global exploration in how geometry was used in a variety of contexts.
It turned out to be a primarily French historical perspective.
I rated it three stars because the book delves into geometry sparsely. Instead, there are long tangents providing historical context for his points.
Those tangents were really fascinating and I would have enjoyed the book more had I expected a history exploration.
In the end, it wasn’t what I signed up for, so I gave it three stars.
Still fascinating though and am glad I read it.
It turned out to be a primarily French historical perspective.
I rated it three stars because the book delves into geometry sparsely. Instead, there are long tangents providing historical context for his points.
Those tangents were really fascinating and I would have enjoyed the book more had I expected a history exploration.
In the end, it wasn’t what I signed up for, so I gave it three stars.
Still fascinating though and am glad I read it.
January 20, 2020
Interesting idea, point made a bit too repetitively and perhaps stretched to fit the examples.
Applies Euclidean geometry to city planning and architecture and wants us to accept that applying it to landscape design expressed the message that the corresponding social structure was immutable.
The examples, starting with high Renaissance perspective and proceeding from Italian to French palaces and gardens, to the designs of Paris and Washington, DC, with several Asian colonial capitals thrown in, were fascinating.
Applies Euclidean geometry to city planning and architecture and wants us to accept that applying it to landscape design expressed the message that the corresponding social structure was immutable.
The examples, starting with high Renaissance perspective and proceeding from Italian to French palaces and gardens, to the designs of Paris and Washington, DC, with several Asian colonial capitals thrown in, were fascinating.
August 6, 2023
Possibly more mathematics would have been better...and less French history!
June 25, 2024
I came for the math but it was mostly history. I guess it's good if you like that?
July 25, 2020
In this latest offering to issue from his pen, the science writer Amir Alexander reveals his true colors to be more those of a historian of culture than one of mathematics, per se, although the latter remains the inspiration from which he jumps off. Put briefly, his thesis in Proof! How the World Became Geometrical is that the discipline of geometry, which comes to us from the ancient world where it occupied the leisure time of a tiny elite and had little to no discernible impact on the course of great political events, was enlisted during the early modern period by political regimes interested in centralizing and consolidating power. For one of the great problems of governance all during the medieval and Renaissance periods was that, as a result of feudalism, local nobility possessed considerable rights and effective power vis-à-vis the often weak central authority of the king or emperor (as we see, for instance, with the signing of the Magna Carta in England in 1215); this led to interminable strife and warfare. The kings had a vested interest in promoting political order, not merely to magnify their own power—this too, of course!—but also on behalf of the domestic tranquility of their subjects.
What, pray tell, does this have to do with geometry? To be sure, geometry plays a crucial role in the intellectual history of the West, which more perhaps than any other great civilization has been founded on a scientific mindset, as opposed to a religious, philosophical or humanistic one (for a good study of what this meant during the middle ages, see Edward Grant (Foundations of Modern Science in the Middle Ages), who documents the extent to which even theological speculation was in surprising measure pursued as a pretext for doing what we would call science, or a precursor thereof). Yet this is not Alexander’s line. Rather, after a somewhat cursory review of the history of mathematics ever since the Greeks in the fifth century BC (he doesn’t concern himself very much with the Egyptians or Babylonians), he commences with the discovery of the laws of perspective during the Renaissance in early fifteenth-century Italy—which, as everyone knows, exercised a major influence on art and architecture: Brunelleschi, Alberti, Masaccio. At a time when aesthetic theory was, in the main, ruled by traditional ideas of mimesis, people were very impressed by the verisimilitude that the great painters could achieve by incorporating perspective into their work. Alexander sees in this development a profound change from the medieval concept of order. Now, everything must be subject to the rational laws of Euclid. For Alexander, this points the way to modern science, with its penchant for seeking out a deep mathematical pattern in nature behind the seemingly chaotic surface level of appearances.
Unfortunately, Alexander does not attempt to document this claim in the texts of the scientific revolutionaries, as one might expect him to. Perhaps he judges it obvious. At any rate, he changes course and switches to the political scene. Here, he unfolds a very circumstantial narrative on political and military conditions as they evolved during the early modern period, beginning with Charles VIII of France (1470-1498). The particular angle Alexander takes is to view things from the aspect of the royal interest in gardens, which he dates to the retreat Charles VIII took at the villa Poggio Reale near Naples, upon the conclusion of one of his military campaigns in the Italian peninsula. Charles must have undergone something like an epiphany there, for he retained Mercogliano to design a new garden for him at the château of Amoise in the Loire valley. Why? In Alexander’s interpretation, the sight of the exquisitely ordered garden at Poggio Reale must have inspired in Charles a vision of what royal rule over a country should be, everything in its place, obedient. The quiet garden represented in miniature the state of an ideal realm.
For a hundred pages, Alexander pursues his analogy and goes into great detail on the vicissitudes encountered by the kings in France as they sought to assert control over their turbulent subjects, culminating in the reign of Louis XIV. Along the way, he catalogues the kings’ sponsorship of royal gardens at their palaces, There can be no doubt that Louis XIV saw in the perfect Euclidean order of his garden at Versailles and the elaborate ceremony that attended court life there a model of his rule over his kingdom, and that he chose to locate his court there in order to remove the nobles from their ancestral lands in the provinces, which, in previous generations, had supported independence and even armed resistance to the crown.
In the concluding chapters, Alexander ranges beyond France and examines how the French ideal of the rational state, as expressed most perfectly at Versailles, influenced other countries. We can see this reflected in architecture at Berlin in Prussia, St. Petersburg in Russia, the new capital for the United States founded at Washington, DC (after debate between Jefferson and L’Enfant, in which Washington’s support for the latter ultimately prevailed), and in European colonial cities throughout the world.
Yet, what we have here is just a piling up of illustrations of his well-worn theme. Alexander declines to enter into any number of questions that might occur to any curious reader. First, the problem of causality; gardens are, after all, only an emblematic symptom of, not the driving force behind the rationalization of the political order; there is no political theory to speak of here. Thus, while the reader can certainly agree that French gardens do represent an ideal of rational order, he will be dissatisfied if he looks for a genealogical account of why absolutism arose and became so prominent during the early modern period. Second, what about France after the French revolution, or England for that matter? As to France during the nineteenth century, wasn’t its political life rather turbulent? What happened to the ideal of royal order? True, under emperor Napoleon III, Georges-Eugène Haussmann oversaw a radical transformation of the city of Paris, involving the demolition of entire neighborhoods, which put it into the form we know today, but doesn’t the resulting linear order stand in contradiction to the disorderly political scene of the times? As for England, doesn’t it have a rather different constitutional order? Isn’t there a pretty great contrast between English common law and its unwritten constitution versus continental European law as promulgated by Napoleon, on the model of the Justinian code? Alexander does discuss the plans of John Evelyn and Christopher Wren to rebuild London after the great fire of 1666, which would have made it into something like modern Paris, but fails to note the significance of the fact that the plans were shelved. Doesn’t this, partly at least, contradict his thesis of a correlation between rational political order and a rational built environment?
What about us today, moreover? Haven’t our aesthetic tastes evolved? With the incipient Romantic movement, didn’t people begin to find the strict linear order of the French garden too rigid, wooden, and to prefer the greater naturalism of the English garden? Alexander doesn’t discuss English gardens or the aesthetic implications to be drawn from them at all; he merely suggests that, despite everything (multiculturalism and all), the geometrical vision of political order persists in our day in the checks and balances of the American constitution and in the rise of the modern bureaucratic state, the smooth functioning of which depends on the ability of those at the top to enforce their directives all the way down to the local level.
A final point: the observant reader could object that there is very little actual mathematics in the book, apart from a short epilogue on non-Euclidean geometry which doesn’t say much and seems to be tacked on as an afterthought. One wishes that Alexander could have drawn a tighter connection between Euclid’s methodology and early modern political theory. For this one would surely want a close examination of, inter alia, Hobbes, Locke, Spinoza, Montesquieu, the American founding fathers etc., but there is nothing of the sort here. Alexander’s stylistic tendency to repetitiveness seems, at times, to get the better of him; a good editor could have helped make his argument much more concise. Hence, the reader can expect from this book Alexander’s hallmark easy-going prose style and competent historical narrative, embellished with incidental detail, but scarcely anything in the way of hard philosophical analysis. To be read through quickly, leaves one wanting for more (not details, plenty of those, but conceptual apparatus and prognosis for us today).
What, pray tell, does this have to do with geometry? To be sure, geometry plays a crucial role in the intellectual history of the West, which more perhaps than any other great civilization has been founded on a scientific mindset, as opposed to a religious, philosophical or humanistic one (for a good study of what this meant during the middle ages, see Edward Grant (Foundations of Modern Science in the Middle Ages), who documents the extent to which even theological speculation was in surprising measure pursued as a pretext for doing what we would call science, or a precursor thereof). Yet this is not Alexander’s line. Rather, after a somewhat cursory review of the history of mathematics ever since the Greeks in the fifth century BC (he doesn’t concern himself very much with the Egyptians or Babylonians), he commences with the discovery of the laws of perspective during the Renaissance in early fifteenth-century Italy—which, as everyone knows, exercised a major influence on art and architecture: Brunelleschi, Alberti, Masaccio. At a time when aesthetic theory was, in the main, ruled by traditional ideas of mimesis, people were very impressed by the verisimilitude that the great painters could achieve by incorporating perspective into their work. Alexander sees in this development a profound change from the medieval concept of order. Now, everything must be subject to the rational laws of Euclid. For Alexander, this points the way to modern science, with its penchant for seeking out a deep mathematical pattern in nature behind the seemingly chaotic surface level of appearances.
Unfortunately, Alexander does not attempt to document this claim in the texts of the scientific revolutionaries, as one might expect him to. Perhaps he judges it obvious. At any rate, he changes course and switches to the political scene. Here, he unfolds a very circumstantial narrative on political and military conditions as they evolved during the early modern period, beginning with Charles VIII of France (1470-1498). The particular angle Alexander takes is to view things from the aspect of the royal interest in gardens, which he dates to the retreat Charles VIII took at the villa Poggio Reale near Naples, upon the conclusion of one of his military campaigns in the Italian peninsula. Charles must have undergone something like an epiphany there, for he retained Mercogliano to design a new garden for him at the château of Amoise in the Loire valley. Why? In Alexander’s interpretation, the sight of the exquisitely ordered garden at Poggio Reale must have inspired in Charles a vision of what royal rule over a country should be, everything in its place, obedient. The quiet garden represented in miniature the state of an ideal realm.
For a hundred pages, Alexander pursues his analogy and goes into great detail on the vicissitudes encountered by the kings in France as they sought to assert control over their turbulent subjects, culminating in the reign of Louis XIV. Along the way, he catalogues the kings’ sponsorship of royal gardens at their palaces, There can be no doubt that Louis XIV saw in the perfect Euclidean order of his garden at Versailles and the elaborate ceremony that attended court life there a model of his rule over his kingdom, and that he chose to locate his court there in order to remove the nobles from their ancestral lands in the provinces, which, in previous generations, had supported independence and even armed resistance to the crown.
In the concluding chapters, Alexander ranges beyond France and examines how the French ideal of the rational state, as expressed most perfectly at Versailles, influenced other countries. We can see this reflected in architecture at Berlin in Prussia, St. Petersburg in Russia, the new capital for the United States founded at Washington, DC (after debate between Jefferson and L’Enfant, in which Washington’s support for the latter ultimately prevailed), and in European colonial cities throughout the world.
Yet, what we have here is just a piling up of illustrations of his well-worn theme. Alexander declines to enter into any number of questions that might occur to any curious reader. First, the problem of causality; gardens are, after all, only an emblematic symptom of, not the driving force behind the rationalization of the political order; there is no political theory to speak of here. Thus, while the reader can certainly agree that French gardens do represent an ideal of rational order, he will be dissatisfied if he looks for a genealogical account of why absolutism arose and became so prominent during the early modern period. Second, what about France after the French revolution, or England for that matter? As to France during the nineteenth century, wasn’t its political life rather turbulent? What happened to the ideal of royal order? True, under emperor Napoleon III, Georges-Eugène Haussmann oversaw a radical transformation of the city of Paris, involving the demolition of entire neighborhoods, which put it into the form we know today, but doesn’t the resulting linear order stand in contradiction to the disorderly political scene of the times? As for England, doesn’t it have a rather different constitutional order? Isn’t there a pretty great contrast between English common law and its unwritten constitution versus continental European law as promulgated by Napoleon, on the model of the Justinian code? Alexander does discuss the plans of John Evelyn and Christopher Wren to rebuild London after the great fire of 1666, which would have made it into something like modern Paris, but fails to note the significance of the fact that the plans were shelved. Doesn’t this, partly at least, contradict his thesis of a correlation between rational political order and a rational built environment?
What about us today, moreover? Haven’t our aesthetic tastes evolved? With the incipient Romantic movement, didn’t people begin to find the strict linear order of the French garden too rigid, wooden, and to prefer the greater naturalism of the English garden? Alexander doesn’t discuss English gardens or the aesthetic implications to be drawn from them at all; he merely suggests that, despite everything (multiculturalism and all), the geometrical vision of political order persists in our day in the checks and balances of the American constitution and in the rise of the modern bureaucratic state, the smooth functioning of which depends on the ability of those at the top to enforce their directives all the way down to the local level.
A final point: the observant reader could object that there is very little actual mathematics in the book, apart from a short epilogue on non-Euclidean geometry which doesn’t say much and seems to be tacked on as an afterthought. One wishes that Alexander could have drawn a tighter connection between Euclid’s methodology and early modern political theory. For this one would surely want a close examination of, inter alia, Hobbes, Locke, Spinoza, Montesquieu, the American founding fathers etc., but there is nothing of the sort here. Alexander’s stylistic tendency to repetitiveness seems, at times, to get the better of him; a good editor could have helped make his argument much more concise. Hence, the reader can expect from this book Alexander’s hallmark easy-going prose style and competent historical narrative, embellished with incidental detail, but scarcely anything in the way of hard philosophical analysis. To be read through quickly, leaves one wanting for more (not details, plenty of those, but conceptual apparatus and prognosis for us today).
August 5, 2020
There is something deeply ironic in this book. The irony exists on several levels. Some of the book's ironies are intentional, such as the way the author spends most of his time talking about the way that geometry of a precise and planar form informed the artistic and gardening and architectural worldview of an absolutist European kind that also serves at the basis of Washington DC's own design, demonstrating its imperialist ambitions in spatial organization, with a surprise ending of sorts that discusses non-Euclidian geometry as destroying the supposed perfection of planar geometry and its assumptions. Not all of the book's ironies are intentional, for all of the author's desire to show himself superior to the supposed imperialism of the Euro-American tradition as enshrined in artistic perspective, formal gardens, and city design, the more notable irony that is not recognized is that the Europeans were by no means the developers of these connections, for among the first ever appearances of geometry comes from India, where geometry was viewed as a pivotal part of the art of making religious altars properly, and from India a great deal of mathematical knowledge spread to Europe through the Middle East, a connection that the author barely acknowledges in a perfunctory way.
This particular book is a bit more than 250 pages long and it is divided into three parts and seven long chapters. The book begins with an introduction. The first part of the book then explores the author's view on how and when it was that the world became geometrical in the sense that the author focuses on, namely the Italian renaissance (I), with chapters on the importance of the mirror image (1), and the mathematical code that was related to the art that followed the revolutionary discovery of the focal point (2). After that the author discusses Euclid's Kingdom (II) and pays attention to the attempts of the late Valois kings to utilize royal geometries developed in Italy (3) that were found when these French kings repeatedly invaded Italy (4) to build their own fancy royal gardens in various palazzi in France upon their return, culminating in Versailles (5). Finally, the author closes with a discussion of the enormous influence of Louis XIV's design of Versailles (III), looking first beyond Versailles to the gardens and civic architecture of other European empires (6), and then looking to Washington DC and its importance as a Euclidian and imperial Republic (7), after which the book ends with a conclusion about non-Euclidian geometry as well as notes, acknowledgements, and an index.
In a sense, it is not that the world suddenly became geometrical during the renaissance and early modern periods, only to become more chaotic during the postmodern period as different geometries were discovered. It is that the world itself has always been geometrical and mathematical, and different aspects of this have been chosen for different reasons at different times. Specifically, the development of artistic perspective was easy to exploit by absolutist early modern royals because they wanted to be viewed and to view themselves as the focus of the state and the larger society, with everyone revolving around them. Once such a technique was discovered its obvious political importance could not fail to be utilized by rulers. And it should not surprise us that these insecure monarchs (like the French rulers the author emphasizes in his study) should be so intensely aware of the symbolism of orderly gardens and focusing, and how it is that such designs can endure in cities like Washington DC and New Dehli long after the original designers of those places are dead and gone, giving a symbolic meaning that casts a heavy weight in the world.
This particular book is a bit more than 250 pages long and it is divided into three parts and seven long chapters. The book begins with an introduction. The first part of the book then explores the author's view on how and when it was that the world became geometrical in the sense that the author focuses on, namely the Italian renaissance (I), with chapters on the importance of the mirror image (1), and the mathematical code that was related to the art that followed the revolutionary discovery of the focal point (2). After that the author discusses Euclid's Kingdom (II) and pays attention to the attempts of the late Valois kings to utilize royal geometries developed in Italy (3) that were found when these French kings repeatedly invaded Italy (4) to build their own fancy royal gardens in various palazzi in France upon their return, culminating in Versailles (5). Finally, the author closes with a discussion of the enormous influence of Louis XIV's design of Versailles (III), looking first beyond Versailles to the gardens and civic architecture of other European empires (6), and then looking to Washington DC and its importance as a Euclidian and imperial Republic (7), after which the book ends with a conclusion about non-Euclidian geometry as well as notes, acknowledgements, and an index.
In a sense, it is not that the world suddenly became geometrical during the renaissance and early modern periods, only to become more chaotic during the postmodern period as different geometries were discovered. It is that the world itself has always been geometrical and mathematical, and different aspects of this have been chosen for different reasons at different times. Specifically, the development of artistic perspective was easy to exploit by absolutist early modern royals because they wanted to be viewed and to view themselves as the focus of the state and the larger society, with everyone revolving around them. Once such a technique was discovered its obvious political importance could not fail to be utilized by rulers. And it should not surprise us that these insecure monarchs (like the French rulers the author emphasizes in his study) should be so intensely aware of the symbolism of orderly gardens and focusing, and how it is that such designs can endure in cities like Washington DC and New Dehli long after the original designers of those places are dead and gone, giving a symbolic meaning that casts a heavy weight in the world.
June 13, 2023
The author's discussions of Renaissance art and architecture, Versailles, Second Empire Paris, New Delhi, Vienna, St. Petersburg, and DC are fascinating. However, they only add up to about a hundred pages. And this isn't "the world" becoming geometrical; it's almost entirely construction of gardens and new cities, with some reconstruction of devastated areas.
The book is padded with mind-numbing repetition and blow-by-blow accounts of the French Wars of Religion, L' Enfant's rocky relationship with Thomas Jefferson, and other digressions.
One such digression is on the Burnham Plan for Manila. He doesn't notice that (a) the plan *was never implemented* (so much for the colonial power quickly imposing its will on city plans) and (b) the its proposed avenues focused on the legislature, not the executive (Versailles) or both legislature and executive (DC).
As a result, as far as its topic was concerned, about two-thirds of the book was wasted. He could at least have cited other interesting examples, like Canberra, (where avenues radiate from parliament, not Government House), and Victoria, BC, (with a long vista ending in the legislative building), expressing the parliamentary form of government.
By "Euclidean geometry," he seems to mean logically linked tricks you can do with a compass and straightedge, that fascinated artists, French garden planners, and emperors.
But Euclidean geometry in the more important sense includes all geometry based on Euclid's five postulates, including the parallel postulate. It includes analytic geometry using a grid of Cartesian coordinates, used in blueprints and civil engineering, even where the designs are curved. It also includes the descriptive geometry used to represent 3D objects in engineering and architecture. These are pervasive. Compared to them, 18th century planners' infatuation with one corner of Euclidean geometry is an interesting footnote.
He discusses non-Euclidean geometries that aren't based in the parallel postulate as if they superseded Euclidean geometry. They're important for large scale mapping (Earth isn't flat) and relativistic astrophysics (space is curved). But for design and engineering purposes the difference from Euclidean geometry is negligible. Differences that are invisible on the human scale have no psychological impact. French Enlightenment designs for gardens and streets went out of style mainly because they were generally too expensive for anyone but an absolute monarch to design, build, and maintain, especially in cities. Non-Euclidean geometries had nothing to do with it. After they were discovered in the 19th century, Euclidean geometry in the broad sense became more pervasive than ever.
In short, this book omits the most important aspects of its alleged subject. Recommended with 3 stars mainly because of the chapters on Renaissance art and French gardens, but also to those who are curious about the French Wars of Religion, and to anyone who enjoys learning by rote repetition of the same points in the same words.
The book is padded with mind-numbing repetition and blow-by-blow accounts of the French Wars of Religion, L' Enfant's rocky relationship with Thomas Jefferson, and other digressions.
One such digression is on the Burnham Plan for Manila. He doesn't notice that (a) the plan *was never implemented* (so much for the colonial power quickly imposing its will on city plans) and (b) the its proposed avenues focused on the legislature, not the executive (Versailles) or both legislature and executive (DC).
As a result, as far as its topic was concerned, about two-thirds of the book was wasted. He could at least have cited other interesting examples, like Canberra, (where avenues radiate from parliament, not Government House), and Victoria, BC, (with a long vista ending in the legislative building), expressing the parliamentary form of government.
By "Euclidean geometry," he seems to mean logically linked tricks you can do with a compass and straightedge, that fascinated artists, French garden planners, and emperors.
But Euclidean geometry in the more important sense includes all geometry based on Euclid's five postulates, including the parallel postulate. It includes analytic geometry using a grid of Cartesian coordinates, used in blueprints and civil engineering, even where the designs are curved. It also includes the descriptive geometry used to represent 3D objects in engineering and architecture. These are pervasive. Compared to them, 18th century planners' infatuation with one corner of Euclidean geometry is an interesting footnote.
He discusses non-Euclidean geometries that aren't based in the parallel postulate as if they superseded Euclidean geometry. They're important for large scale mapping (Earth isn't flat) and relativistic astrophysics (space is curved). But for design and engineering purposes the difference from Euclidean geometry is negligible. Differences that are invisible on the human scale have no psychological impact. French Enlightenment designs for gardens and streets went out of style mainly because they were generally too expensive for anyone but an absolute monarch to design, build, and maintain, especially in cities. Non-Euclidean geometries had nothing to do with it. After they were discovered in the 19th century, Euclidean geometry in the broad sense became more pervasive than ever.
In short, this book omits the most important aspects of its alleged subject. Recommended with 3 stars mainly because of the chapters on Renaissance art and French gardens, but also to those who are curious about the French Wars of Religion, and to anyone who enjoys learning by rote repetition of the same points in the same words.
October 4, 2019
I wasn't entirely sure what to expect from this book. The title sounded interesting, and so I decided to try it. What I learned was that gardens, architecture, and painting were considered a part of geometric unity. That is, starting from the Renaissance, people began to assume that geometry and its inescapable proofs could be used for other endeavors. Painting (leading to linear perspective), architecture, and even the structure of gardens could be used to reinforce a view of eternal hierarchy based on objective rules. I liked that Alexander explains how to view this, and while he emphasizes that these were often conscious decisions, that there was a strong current of these things being subconsciously absorbed. People were so used to thinking about things in terms of things being all about an absolutist monarch that a garden that subtly and not so subtly emphasizes this simply reinforces the thought.
Alexander goes through the Renaissance to WWII, with much of the book focused on the times before and during Louis XIV's reign. He then talks about the making of the city of Washington DC and how L'Enfant had a design that he wouldn't compromise on because of its eternal geometric design.
Alexander argues well, and I think he doesn't push this point too far. He looks at the context of the time, and simply explains how the proofs of Euclidean geometry were an influence on people's thinking in other realms. The passage of non-Euclidean geometry into humanity's knowledge is commented on, which is short but interesting.
Overall, an interesting book that I think people interested in the historical importance of hierarchy in people's thought will find fascinating. There is not as much math as I would have expected, but it was still fun to read. The prose is well put together and it looks at history well. If you want more mathematics than history, though, this is not the book for you in my opinion. If you're fine with a mix, more strongly on the history side, this is a good option.
Alexander goes through the Renaissance to WWII, with much of the book focused on the times before and during Louis XIV's reign. He then talks about the making of the city of Washington DC and how L'Enfant had a design that he wouldn't compromise on because of its eternal geometric design.
Alexander argues well, and I think he doesn't push this point too far. He looks at the context of the time, and simply explains how the proofs of Euclidean geometry were an influence on people's thinking in other realms. The passage of non-Euclidean geometry into humanity's knowledge is commented on, which is short but interesting.
Overall, an interesting book that I think people interested in the historical importance of hierarchy in people's thought will find fascinating. There is not as much math as I would have expected, but it was still fun to read. The prose is well put together and it looks at history well. If you want more mathematics than history, though, this is not the book for you in my opinion. If you're fine with a mix, more strongly on the history side, this is a good option.
May 24, 2024
In Proof! Alexander takes on European history, focusing on France, from the later Middle Ages through to the fall of the monarchies, and finally settling in the USA. He begins with a bow to Euclid and demonstrates how Euclidean geometry was understood as the structure of the world, including social, political, and religious (Christian) relationships. Thus, the book treats many aspects of history as organized around a general geometrical theme. In an intriguing development, Alexander presents the transition of the garden from an area for growing food to an expression of the societal and political order in parallel with the development of monarchy as a moveable court to a stationary one, finally to the democracy seen in the US in its foundation. These insights elucidate the organization of these societies and their shifts, especially by emphasizing the symbolic or model nature of the gardens, landscape, and cities in their organization and geometrical nature -- something which is probably lost on current US citizens. In conclusion, Alexander brings in non-Euclidean geometries and what these mean for a continuation of this historical story.
The book has an Introduction, 3 Parts divided into chapters with each chapter having subsections (set off with headings), a Conclusion, a section of Notes, Acknowledgements, and an Index. There are numerous black and white illustrations (figures) and photographs.
The title is a bit misleading since there is little actual geometry and I did find the book a bit tedious and repetitive but enjoyable and insightful just the same. Alexander does a good job of pointing to encompassing themes in history by spinning the smaller portion he focuses on in this book. I found the final chapter on the planning of Washington, D.C. and the Conclusion presenting non-Euclidean geometries of particular interest. I recommend the book for those interested in how societies are organized and operate in a more complete sense and how these patterns shift in historical time.
The book has an Introduction, 3 Parts divided into chapters with each chapter having subsections (set off with headings), a Conclusion, a section of Notes, Acknowledgements, and an Index. There are numerous black and white illustrations (figures) and photographs.
The title is a bit misleading since there is little actual geometry and I did find the book a bit tedious and repetitive but enjoyable and insightful just the same. Alexander does a good job of pointing to encompassing themes in history by spinning the smaller portion he focuses on in this book. I found the final chapter on the planning of Washington, D.C. and the Conclusion presenting non-Euclidean geometries of particular interest. I recommend the book for those interested in how societies are organized and operate in a more complete sense and how these patterns shift in historical time.
September 25, 2019
Another tour de force from the brilliant and engaging historian of science and mathematics Amir Alexander. Like Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World before it, Alexander's latest book is a fascinating page-turner; here he takes us on a journey from Euclid's Elements and the irrefutable, absolute truths of Euclid's geometric proofs to 15th century Florence and the discovery of perspective, and then on to the grand, perfectly ordered, geometric gardens of Versailles and their critical significance as a tool of political power. Alexander leads us to other geometric destinations as well, covering wide terrain–– spatially and temporally–– and geometry's necessary role in art, architecture, formal garden design, city planning, political power and more. He's a wonderful storyteller; there's enough drama and conflict in Proof! that it's tough to put down.
As a Washington DC native, I particularly enjoyed his investigation into the how America's capital city became geometric, designed by a Frenchman, Pierre L'Enfant, who applied the geometric principles of the gardens of Versailles not to buttress a monarchy, but rather to instantiate a Republic. As Alexander tells it: "Taken as a whole, L'Enfant's plan was nothing less than the Constitutional power structure of the United States set in stone, pavement, trees, and shrubs.... Where Louis XIV's capital pointed to a single glorious seat of authority, the American capital featured two coequal seats of power, as well as a network of lesser ones spread throughout the city." It was a "geometrical construct," physical and "visible proof that in the American capital -- just as at Versailles-- the political order is at its core a geometrical order." Thanks to Alexander's deep dive into the layers of meaning contained in the geometry of D.C., I have a newfound appreciation for the city I call home, and a deep hope that the balance between the coequal seats of power will soon be righted to geometrical perfection.
As a Washington DC native, I particularly enjoyed his investigation into the how America's capital city became geometric, designed by a Frenchman, Pierre L'Enfant, who applied the geometric principles of the gardens of Versailles not to buttress a monarchy, but rather to instantiate a Republic. As Alexander tells it: "Taken as a whole, L'Enfant's plan was nothing less than the Constitutional power structure of the United States set in stone, pavement, trees, and shrubs.... Where Louis XIV's capital pointed to a single glorious seat of authority, the American capital featured two coequal seats of power, as well as a network of lesser ones spread throughout the city." It was a "geometrical construct," physical and "visible proof that in the American capital -- just as at Versailles-- the political order is at its core a geometrical order." Thanks to Alexander's deep dive into the layers of meaning contained in the geometry of D.C., I have a newfound appreciation for the city I call home, and a deep hope that the balance between the coequal seats of power will soon be righted to geometrical perfection.
July 17, 2022
A spectacularly ironic title. Sadly, it does not seem intentional.
The good:
- As someone who enjoys gardens, art, and European history, I found the content diverting.
- The author is a beautiful writer. One might even say he “brings history to life,” and it’s a quick and easy read, because of that. This would make a delightful beach read, and for that I've given it 3 stars instead of 2.
The bad:
- I think it’s possible that I’ve never before in my whole life read an author more taken to hyperbole. And in what is ultimately an attempt at history, that’s not great. There are so many ridiculous claims that it’s difficult to trust what might be more accurate (and interesting!) parts of the book.
- As other reviewers have noted, despite the book’s title, the author provides shockingly little evidence to support his many hyperbolic statements. A work of academic rigor, this ain’t.
- In the year of our lord 2019 (when this book was published), it is no longer acceptable to call Europe and North America “the world”.
- This is not about geometry.
The good:
- As someone who enjoys gardens, art, and European history, I found the content diverting.
- The author is a beautiful writer. One might even say he “brings history to life,” and it’s a quick and easy read, because of that. This would make a delightful beach read, and for that I've given it 3 stars instead of 2.
The bad:
- I think it’s possible that I’ve never before in my whole life read an author more taken to hyperbole. And in what is ultimately an attempt at history, that’s not great. There are so many ridiculous claims that it’s difficult to trust what might be more accurate (and interesting!) parts of the book.
- As other reviewers have noted, despite the book’s title, the author provides shockingly little evidence to support his many hyperbolic statements. A work of academic rigor, this ain’t.
- In the year of our lord 2019 (when this book was published), it is no longer acceptable to call Europe and North America “the world”.
- This is not about geometry.
October 29, 2019
I was hoping for something that pointed out geometrical truths (proof) around the world and through history. Instead, I got a history of European (mostly French) kings and stories about their lineage and gardens. I kept reading, hoping for some insight.
It did kind of come together near the end. There was discussion about the designs proposed for Washington D.C. and how they represent the hope for a new nation versus the traditions of Versailles, and the "universal truths" they contain.
It did kind of come together near the end. There was discussion about the designs proposed for Washington D.C. and how they represent the hope for a new nation versus the traditions of Versailles, and the "universal truths" they contain.
February 13, 2020
By far, the best parts were the introduction and the conclusion. Those seemed to belong to an entirely different book. Everything in the middle was a history lesson on France. I wanted geometry and I got Louis the XIV , wars, politics and only the occasional triangle. No proofs except the title. Lots of happenings around Versailles, yet only a single mention of Marie Antoinette. The one chapter on the streets of Washington DC was interesting, but also more of a history lesson.
October 7, 2022
Nowhere near as good as his earlier book Infinitesimal, despite emerging from the same historical theme that powerful interests in Renaissance and Enlightenment Europe harnessed the methods of Euclid's geometry to validate and express theories of cultural supremacy. But where Infinitesimal easily sustained its argument, Proof! is repetitive and hyperbolic. A snappy read, however, and Amir Alexander is a lovely writer, so worth a dip.
December 25, 2019
I was severely disappointed with this book. It consistently tries to make grand claims concerning the relationship between geometry and royal power but fails to provide enough evidence. It also often tries to attribute certain motives to historical figures when there is not enough evidence to do so. Finally, its descriptions of geometrical gardens are extremely dry and boring.
April 8, 2020
I wish this book was more about geometry. The sections that focus on the impact on geometry on buildings, art, and gardens are really good. The problem is that much of this is general history of France and which is unnecessary to the larger narrative. I understand that the author was trying to bridge two time periods but he did not need fifty pages of history to do so.
February 6, 2023
This was not what I thought it was going to be. At all. It was more a work of French history than a deep look at how geometry impacted world history. I did, however, stick with it. It was an incredibly interesting read but the thesis of the book was just too forced.
The writer is incredibly gifted. The prose is absolutely beautiful. For that, it was worth the time and I learned a ton as well. Taking the glass half full approach, I would say I enjoyed this work.
The writer is incredibly gifted. The prose is absolutely beautiful. For that, it was worth the time and I learned a ton as well. Taking the glass half full approach, I would say I enjoyed this work.
December 20, 2024
Tedious Descriptions of Gardens
Title suggests the book describes the rise of mathematical proofs. A surprising number of pages is devoted to the elaborate gardens of European kings. It seems well researched, and I imagine it's an interesting book for the right reader. I found it a chore.
Title suggests the book describes the rise of mathematical proofs. A surprising number of pages is devoted to the elaborate gardens of European kings. It seems well researched, and I imagine it's an interesting book for the right reader. I found it a chore.
November 8, 2019
WAY more than I ever needed to know about the structure and "philosophy" of the gardens at Versailles. The chapter on Washington, D.C. was interested; made me want to visit and look at the city through the geometer's lens.
April 30, 2024
This book is a 2-star book about geometry and 3-star book about French history (all of which is encased inside a 5-star cover). Using the powers of geometry which rationally govern the order of the universe, I have found that the 3.33 average of these values constitutes this work's ideal rating.
November 12, 2019
Nothing was proven in this book. There was a lot of surmising and leaps of logic. Worse, it was dull. The descriptions of the gardens were interminable, and seemingly to no purpose.
Read
March 2, 2020Really only glanced through it, enjoyed what i read.
February 22, 2022
This book was kind of silly, but fun if you like history
April 19, 2022
This book was fun. i enjoyed reading about geometry. It was really fun.
June 29, 2022
Title is misleading
February 27, 2023
When this book stops giving you a basic level intro to random moments in history it randomly throws in "the overarching nature of geometry" and calls it a day.
Read
December 27, 2024Sometimes got bogged down in repeating that "geometry is the final and most high law", but fed my part-time obsession with the history of city planning very nicely, and the coda was very nice!
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