Author: Some Random Dude

Review of “The Italian Renaissance of Machines”

Paolo Galluzzi is the director of the Museo Galileo in Florence, Italy.  A historian of science, he’s published extensively on the Galileo, Leonardo, and Italian Renaissance aspects of the scientific revolution.  He is a member of the Royal Swedish Academy of Sciences and the American Philosophical Society, and has teaching appointments at the Universities of Siena and Florence.

In “Machines”, Galluzzi profiles the emergence of a particular historical archetype familiar to us even today on the TED Talk stage — the artist-engineer.  Before Da Vinci, engineers were mostly anonymous, but the Renaissance established draftsmanship as a high art in and of itself, and cultivated rockstar reputations for the many engineers who emerged in the tradition of Leonardo.  None of the works of these engineers were ever printed, but their manuscripts were circulated and widely read.  Galluzzi argues that these manuscripts represent the first systematic recourse to illustrations in engineering text, and prose largely in dialogue with them.  The sciences of architecture, anatomy, hydraulics, geology, and military engineering all required skilled draftsmanship, and these documents introduced a new concept of learning — characterized by observation and visual expression — which was distinct from traditional knowledge, which was based on eloquence and rhetoric.  Later generations of the movement built on this foundation with increasingly abstract geometry, incorporating the emerging science of mechanics, and culminating in Galileo’s bright line between those practicing mechanics with and without mathematical skill.

Galluzzi compiles source drawings from national libraries in Florence, Madrid, the British Museum, and the Vatican.  Chapter one profiles Taccola (1382-1453), dubbed the “Sienese Archimedes”, who was the first effective promoter of a movement for the cultural recognition of technical knowledge and practice.  Taccola’s last compilation was found in Paris in the early 19th century, having been preserved at the Ottoman court until 1687.  It contains over two thousand drawings, and while they were perhaps intended for eminent patrons, it appears that none of them ever left his study.  These were neither presentational works, nor workshop notebooks, instead constituting a new literary genre.  They contained Taccola’s interview with Brunelleschi, with illustrated methods for laying bridge foundations underwater, the earliest documented visual representation of the three-speed hoist which was critical to Brunelleschi’s construction of the dome at Il Duomo (the Florence cathedral).  

Francesco Di Giorgio, also Sienese, adapted many of Taccola’s designs, including machines for moving and lifting obelisks and columns, raising water, and wagons with complex transmissions — worm screws, metal racks, and reduction gears.  Galluzzi explains that it is unclear if any of Giorgio’s designs were ever built; they may constitute a kind of experimental archaeology interpreting ancient Roman sources.  Giorgio’s work contains the first-ever known depiction of a human descending under a parachute, military subjects combining traditional war machines (ie. siege engines) with depictions of firearms and artillery, as well as instructions for targeting them.  Methods for moving and filtering water were omnipresent, given Siena’s reliance upon complex hydraulic culture.  Landscape and animals appear in a manner more realistic than the machines, since mathematical perspective is not yet present.  

In chapter two, Galluzzi surveys Leonardo’s machine-oriented output, and describes how it lays a conceptual foundation for his later anatomical work.  Leonardo’s earliest drawings of machines date from the 1470s.  These are now in mathematical perspective, a significant advance over the work of Di Giorgio, but retain occasional deliberate distortions which appear to be intentional, so as to better expose the inner workings of various mechanisms.  Galluzzi characterizes this as Leonardo’s “anatomical approach”, which constitutes a major shift from the traditional analysis of machines as indivisible entities to one more focused on a limited number of basic mechanisms, which can be combined to yield a variety of devices.  He explains that there is significant disagreement among scholars as to whether or not Leonardo, in these machine studies, was in fact grappling with principles of mechanics in a manner that prefigured and anticipated Galileo.

Galluzzi presents ample evidence of Leonardo’s virtuosity here — this is the first documented evidence of the use of plan and elevation perspectives, exploded views, and lettered/numbered components.  Leonardo further uses many sequences of drawings to depict kinematic relationships between components, in the style of modern animation.  Shadows are employed to depict spatial relationships, and he uses hatching with curved lines to demonstrate patterns of wear.  Galluzzi contends that this is evidence of Leonardo attempting to gain insight into the sources and directions of the forces being applied inside the machines.  All of this “density of information” is almost impossible to communicate with text, and from 1490 onward, drawings occupy more space in Leonardo’s written work than text.  

Next Galluzzi draws a causal line between the machine studies and Leonardo’s anatomical illustrations after 1500.  Galluzzi believes that Leonardo perceived the human body as a series of “motor contrivances”, pointing to vivid examples like depictions of the action of the intercostal muscles in breathing, the reduction of jaw movements to the laws of levers, and the replacement of muscle with lines of force, representing them as wires.  He describes Leonardo’s plan for a “visual anatomical encyclopedia” of the body, at various structural levels (musculature, circulation, membranes, skeletal, etc.).  Indeed, Leonardo boasted that drawings were superior to direct observation during dissection.  Here he claimed illustration as capable of visualizing a level of reality screened from the distorting effects of sensory input. 

In chapter three, Galluzzi surveys attempts by Giocondo (1511, a frequent collaborator of di Giorgio), Cesariano (1521, a member of Bramante’s circle), and Barbaro (1567) to render Escheresque illustrated editions of the rediscovered works of the Roman architect Vitruvius.  In many cases, these engineers could not figure out how Vitruvius’ catapults, ship speedometers, and water pumps worked, but they published their interpretations, hoping that others would build on their efforts.  Cesariano was notable for publishing the first printed edition of Vitruvius in Italian (rather than Latin), and a short manifesto praising the function of a “science of representation — painting and drawing”, as a prerequisite for any form of knowledge.  Barbaro’s aim was not to offer a realistic depiction of any of Vitruvian devices, but rather to speculate on the principles governing their operation.

Galluzzi closes the book by asserting the influence of these engineer-draftsmen on Galileo himself.  After encountering a somewhat hostile reception to the abstract geometric renderings in “Two New Sciences”, his treatise on mechanics, Galileo replaced the illustrations at the beginning of the book with engravings set at a construction site, the Venice Arsenal.  These engravings have concrete realism; an irregular stone hangs from an iron hook at the end of a wooden cantilevered beam anchored in a ruined stone wall.  Having thus anchored the reader in reality, Galileo returns to his geometrical diagrams, but the continuity with the tradition Galluzzi identifies is clear.

“The Italian Renaissance of Machines” is a monumental curation effort.  In contrast to works compiling the complete notebooks of Leonardo, the depiction of a coherent, focused slice of Leonardo’s work in the context of his progenitors and successors is refreshing.  Galluzzi’ s work reveals how art slowly came into the service of science and engineering throughout the 15th century.  

Review of “Reckoning with Matter:  Calculating Machines, Innovation, and Thinking about Thinking from Pascal to Babbage”

Matthew L. Jones is the James R. Barker Professor of Contemporary Civilization in the Department of History at Columbia University.  He obtained his BA and PhD from Harvard, and his masters from Cambridge.  His focus is on the history of science and technology in early modern Europe, as well as the development of information technology.  

In Reckoning, Jones traces the development of mechanical devices capable of automating basic arithmetic operations, beginning in the mid-17th century.  He structures the book as a chronologically ordered series of profiles of the key “inventors” in the field, and the skilled artisanal collaborators they worked with over centuries of attempts.  Between each profile, he delivers an episodic saga of Charles Babbage’s attempts to realize his Difference Engine.  He closes the book with a meditation on how human perceptions of the limits to machine “creativity” have evolved over the last four centuries.   

In chapters one and two, Jones describes the essential problems requiring solutions in the construction of such a device.  First, it must “mechanize carry” — it must somehow apply sufficient force so as to propagate the carrying of successive digits across the various columns in addition or subtraction.  Second, it must “keep things digital” — in other words, it must track and express whole numbers, despite being constructed from analog mechanical components which possess continuous, rather than discrete, state.  Third, it must match the excellence of manual instruments in simplicity, compactness, and ease of use.  To these ends, device designs relied upon discs and gears, precise division of those discs and gears, and an internal mechanism to drive them.  These components the skills of craftsmen experienced in striking coinage, precision engraving, and clockmaking.  As a pioneer in 1646, Pascal found himself in the position of having to make the case for the utility of such a device.  Pascal pointed to the error-prone and boring process of compiling a long series of sums, common in accounting and bookkeeping.  At that time, numeracy was concentrated in the commercial centers, and an increasing sophistication and density of commercial transactions was creating plausible demand for automation.  Meanwhile, “computation” was seen as a mechanical activity of lower intellectual status, distinct from logic and reasoning (then perceived as not reducible to arithmetic operations).  

Pascal’s machine attempted to automate the cascade of carries, and the necessary force propagation, by means of progressively heavier weights in each column of numbers.  As a result, it was not reversible (incapable of subtraction), and required a level surface for operation.  Jones’ ability to explain intricate and arcane engineering design principles shines through; in an analogical example typical of his explanations throughout the book, he explicates what happens inside a modern mechanical odometer when it turns over one million miles!  Pascal made more than fifty attempts to ruggedize his device, but it never caught on.  Meanwhile, across the Channel in 1669, Morland produced a less ambitious, more portable device that relied upon manual carries.  It automated John Napier’s “bones” method for multiplication.  In a feat of sourcing typical of his extensive primary bibliography, Jones details an itemized invoice for a prototype to Lennox, Morland’s patron, for a sum equivalent to a month’s wages for the entire Lennox household.  Here Jones also introduces an ongoing exploration of the interplay between inventors and artisans, praising Morland’s “knowledge of his limits and taste in collaborators”.  Morland perceived two “classes” — inventors and copiers; Pascal thought artisans lacked theory and weren’t inventors, but acknowledged a symbiotic relationship, where his need for artisan skill was matched by their inability to execute without his theories.  Furthermore, Pascal thought that theory-guided exploration was more efficient than trial-and-error.

Chapter two also relies upon the extensive correspondence between Leibniz and his collaborator Ollivier concerning the four-decade quest to construct Leibniz’s design.  In 1672, Leibniz presented a model of a device capable of multiplication via repeated addition to the Royal Society in London.  After a favorable reception, he embarked on a search in Paris for someone whose fascination with machines exceeded his self-interest, and found Ollivier.  Leibniz was impressed by Colbert’s policy of attracting artisans and then stealing their secrets.  Leibniz awarded Ollivier a three-year fixed price contract for a project that ultimately consumed forty years of effort.  Here Jones emphasizes that the custom of presenting a model enabled an inventor to claim priority, while allowing others to assist the acknowledged inventor.  Since Leibniz was based in Hanover, and Ollivier remained in Paris, almost the entirety of their collaboration is preserved in letters, which are a fascinating dialogue between vague qualitative descriptions by Leibniz, requiring Ollivier’s skilled and discretionary interpretation, and Ollivier’s frustrations with the limits of written communications:  “I cannot explain, the machine would need to be present for you to understand.”  This depiction strengthens Jones’ argument that “technology transfer” in the 17th century was chiefly a matter of moving artisans between geographic locales.  Jones uncovers an unexecuted contract dated 1679 which lays bear Leibniz’s frustrations with his engineer.  It specifies an almost absurd degree of micromanagement that belies the considerable autonomy conveyed in Leibniz’s letters.  Here Jones also admits the limitations imposed by his sources — eventually the Ollivier correspondence falls silent, and Jones is unable to prove that this was due to a move by Ollivier to Hanover.  This seems unlikely, given that Ollivier’s frustrations had grown to a point where he threatened to “melt down everything” on account of Leibniz’s failure to recognize him as more than just a journeyman technician.

Chapter three examines nascent concepts of intellectual property, beginning with an exploration of the 17th century concept of “privilege” as distinct from the modern conception of a “patent”.  Privilege protected an already extant manufacturing process and required “reduction to practice”.  It other words, privilege was seldom awarded to a mere idea on paper.  In this regard, Pascal received special treatment:  a perpetual monopoly for what was then largely a thought experiment.  The reasons for this appear to be a unique confluence of the Pascal family’s relationships in the French court, an ongoing Colbertist process of state primacy exertion over the cities and regions, and the fact that privilege was more a symbolic gift than any actual mechanism for enhanced commercial renumeration.  Across the Channel, Morland ran aground on a general suspicion of patents following the Glorious Revolution; patents were seen as examples of royal overreach and corruption in the pre-Cromwell era.  Leibniz also failed to procure protection, since the typical “window” for successful reduction to practice was only six months.

Chapter four’s title is the splendid “Productivity of Ignorance”.  Leibniz’s ultimate failure was legendary, and this “ignorance” and struggle inspired others to attempt and persist.  After all, if the great Leibniz couldn’t achieve a solution, the problem must be truly worthy.  Jones explores the prevalence of imitation as a virtue — “challenge alone can force the improvement of a talent”, whereas mere copying lacks the same rigor.  This burst of innovation by many competitors produced the first well-working machine, a cylindrical approach which bore an inscription assigning nearly equal credit:  “Braun invented it, Vayringe made it”.

Chapter five chronicles the effort of the Earl of Stanhope, beginning in 1777.  Stanhope was brought up in the middle of the Geneva clockmaking ferment and served a technical apprenticeship in the construction of marine chronometers.  He was a habitual sketcher, tacking between “paper and metal”, but he was ultimately unable to recreate the constellation of talent he’d encountered in Geneva at his Chevening workshop, with the exception of his engineer Varley, to whom he bequeathed its entire contents.  Stanhope and Varley did not publish or patent their designs.  However, it seems that Stanhope’s prototypes exerted great influence on Charles Babbage, and Babbage was scrupulous about dividing credit for his ideas, meeting with Stanhope’s son and heir and procuring Stanhope’s prototypes when the Royal Society requested a “literature review” of sorts, detailing prior attempts and why they had failed.  Jones again acknowledges the limitations of his sources, noting that it is impossible to determine if any of Stanhope’s innovations made their way into Babbage’s designs.

Jones’ interwoven description of Babbage’s quest to build his Difference Engine is a clever structural conceit, but at times the reader struggles to hold the plot thread across the span of the entire book — not unlike the mental persistence required to perform arithmetic.  Babbage conceived of the Difference Engine as a means of automating the production of numerical tables by the method of adding differences, and then automatic typesetting and printing of the resulting value series.  He constructed his first model in 1822, and abandoned the pursuit in 1833 with only a conceptual demonstration of the calculating component.  Jones details Babbage’s iterative relationship with Clement, a “draftsman of the highest order to economize the labor of my [Babbage’s] own head, and a workman to execute the experimental machinery to which I am obliged constantly to have recourse”, as well as his constant struggle to extract sufficient research and development funding from the British government.  Periodically, British officials requested that the Royal Society effectively audit Babbage’s project, a position of considerable discomfort for the Royal Society, given that it had no mandate for the endorsement or rejection of national scientific investments.  Eventually Peel became Prime Minister, and as a longtime opponent of the project, ended Babbage’s patronage.  During Babbage’s meeting with Peel, he cited the opportunity cost of the Difference Engine and its exclusion of other efforts, as well as the government’s irresponsibility in “stringing him along”, leading him to believe that there was a firm commitment to the realization of a functioning device.  In reports on the project, the Royal Society cited the spinoffs from Babbage’s efforts — tools for large-scale draftsmanship, as well as precision toolmaking, which eventually paid dividends for British industry.  In fact, such were the quality of Clement’s drawings that the British Science Museum was able to construct a functioning prototype of the Difference Engine in 1991.  During that project, the museum uncovered a flaw that Babbage would have doubtless encountered and corrected, if he’d had the resources to build it.  Eventually, a Swedish father/son duo, the Scheutzes, build a working machine in 1853 with Babbage’s blessing, and dedicated the first printing of their numerical tables to him.

Chapter six considers the position of arithmetic in the early modern conception of reasoning.  In 1843 Ada Lovelace, Babbage’s longtime collaborator and the daughter of Lord Byron, cautioned that their engines could “only do whatever we know how to order them to perform”.  Calculating machines were seen as simply the wrong sort of matter to think with, since arithmetic was perceived as a small slice of human reasoning.  However, Stanhope had mused that reasoning is merely the “arrangement of signs”, which is reducible to mechanism, implying that reasoning itself is reducible to mechanism.  Leibniz’ view was that the human mind can “transplant itself, such that it gives matter the power of doing more than it could by itself”.  Babbage, grappling with the problem of runtime efficiency, endowed his design with a prediction capability of sorts — “teaching the machine to foresee and then act on that foresight — performing carries before the additions that cause them”.  

This anticipatory calculation aspect is highly analogous to branch prediction in modern microprocessors, and shows that the earliest hardware and software designer was operating at an extremely high level of conceptual sophistication by modern standards.  Indeed, Jones points out that, even today, the performance characteristics of a state-of-the-art chip are not known until the chip is physically produced.  Almost four hundred years later, “rational design” remains a goal rather than a reality, and that is the central message of Jones’ book:  a design on paper seldom survives contact with the physical world, and his work restores a long-neglected balance between the marquee “idea guys” and their engineering collaborators, who actually brought their designs into the physical world.  Jones also shows that the current Information Age, far from a modern phenomenon, is the result of a slow-burn process that began when data was becoming large, if not yet “big”. 

Margaret Schotte received her BA in history from Harvard, an MA from the University of Toronto, and her PhD from Princeton.  “Sailing School” is her first book, and it builds upon the topical foundation of her publication history, which concerns the systemization of nautical logbook-keeping processes, as well as a study of data-gathering practices by the Dutch East India Company (the “VOC”).  Overall, her research focuses on processes of formal maritime knowledge formation during the Age of Discovery – a period when growing theoretical knowledge from mathematics and astronomy integrated steadily with centuries of “small navigation” seamanship grounded in tacit knowledge.  The unprecedented demands of “large navigation” — wayfinding across vast distances on the open ocean, without the benefit of landmarks — drove inexorable systematization and professionalization of navigational techniques by all the major European naval powers over a period of three centuries.  In Sailing School, Schotte chronicles the emergence of a dialogue between pure and applied science that persists to this day.

Schotte structures her book around five “schools” of maritime navigation, arranged in chronological order and framed as case studies.  The prologue describes the formation of the Casa de la Contratacion, the “House of Trade”, in Seville, under the patronage of the Spanish crown, headed by a chair of cosmography.  This school served as a state-sponsored template for its later northern European counterparts, and Schotte points out that the variations between these northern European players, in their respective implementations of the Spanish model, reveal the subtle influences on national-level navigational cultures.  For example, she cites the importance of maritime trade and business arithmetic in England as a key factor in the eventual English emphasis on arithmetic over astronomy, and a Mediterranean safely traversable largely without recourse to theory as an explanation for the paucity of Italian contributions to the large-navigational literature.  This revealed pattern of cultural evolution under competitive selection pressure is typical of Schotte’s ability to synthesize patterns across a truly vast trove of primary sources, drawn almost exclusively from maritime archives across Europe.  The book features over fifty pages of endnotes and a forty-page bibliography, and this extraordinary compilation reflects both the author’s exhaustive efforts and maritime history’s remarkable legacy of documentation and preservation.

Chapter one chronicles a thriving commercial market for maritime reference material in Amsterdam, owing to high rates of literacy in the population, the involvement of a large cross-section of Dutch society in the growth of Dutch sea power, and the relatively high social status of sailors.  Chapter two examines Dieppe, on the French channel coast, and the efforts of Guillaume Denys to introduce the powerful mathematical tools of trigonometry and tables of logarithms to standard navigational practice.  Chapter three details three distinct class-oriented paths to navigational certification in Greenwich, England: a high-aptitude theoretical track for promising math students at the Royal Mathematical School, seasoned “small-navigators” who wished to upgrade their skills for larger vessels (not unlike modern airline pilots qualifying for multiple engines and instrument ratings), and sons of the nobility seeking commissions in the Royal Navy.  Here Schotte cultivates a “you are there” atmosphere as she puts readers in the shoes of examinees, particularly those subject to curricula influenced heavily by mathematical and astronomical luminaries like Isaac Newton and Edmund Halley.  She also makes great use of the legendarily prolific output of Samuel Pepys, the administrator of the Royal Navy.  The fourth chapter returns to the Netherlands and depicts the emergence of the modern test preparation industry — a network of private educators selling practice tests and cramming regimens for the exacting VOC navigational certification process, almost 250 years before SATs, ACTs, LSATs, GREs, GMATs, and MCATs.  

The book culminates with chapter five’s harrowing tale of Lieutenant Edward Riou’s experience in the crucible of an iceberg collision, causing a hull breach and a broken rudder, after rounding the Cape of Good Hope en route to New South Wales.  Riou guided his ship Guardian and much of the crew to safety at Table Bay in South Africa through two months of dogged navigation and extreme privation across the vast Southern Ocean.  Early in his career, Riou had served in Captain James Cook’s fleet on Cook’s third and final voyage of discovery.  The story of the Guardian brings together all the techniques Schotte describes, as Riou draws upon every ounce of his theoretical training and practical experience to accomplish the impossible.

Sailing School’s most penetrating insights are twofold and of profound import for modern economic and policy debates regarding the nature of scientific and technological development, as well as concerns about human skill growth and maintenance in a world of increasing automation and complexity.  First, that there was no oversimplified narrative of one-way navigational progress from theory to application.  Schotte writes, “no consensus will emerge; the pendulum of opinion will continue to swing between shipboard practice and classroom study. . .[there is] no straightforward trajectory from traditional to scientific sailing or from memory to mathematics.”  Second, that new theories and technologies, far from outright superseding human capability, require ever-more-sophisticated modes of human-machine collaboration.  For parents and teachers concerned that their children and students cannot navigate without Google Maps or remember once-hallowed facts now available on-demand from Wikipedia or ChatGPT, Schotte offers some comfort: “We find this is not a teleological story of inexorable mathematization, where traditional techniques were sloughed aside to make room for rational modernity,  One of the most traditional hallmarks of the professional navigator maintained its paramount importance:  his memory.  There was simply a sea change in the nature of what he was expected to memorize.  Instead of geographic and calendrical details, which were increasingly easy to look up in reference works (especially as tables proliferated), the eighteenth-century navigator had to master mathematical calculations and remember formulas — and comprehend them as well.  Yet, even when offered a panoply of labor-saving inventions, he still continued to use a range of older techniques”.  

Reading Sailing School, one is reminded of the primary finding in the loss of an Air France flight over the south Atlantic between Buenos Aires and Paris: when foul weather caused automation systems to fail, the pilots proved incapable of snapping out of their state of decreased vigilance and simply flying the airplane, as they were once capable of doing.  Arguably, the lessons of Edward Riou’s nuts-and-bolts training and experience had been forgotten.  Schotte’s work provides a fascinating window into the earliest efforts to extend human cognitive and sensory capabilities with technology.

Robyn Arianrhod is an Australian historian of science.  She was trained in mathematics at Monash University in Melbourne, Victoria, and remains an adjunct researcher there.  Her most notable prior work, Einstein’s Heroes:  Imagining the World through the Language of Mathematics (2005), focused on James Clerk Maxwell’s derivation of the laws of electromagnetism, and the concept of mathematics as the language of modern physics.  The full extent of her powers of conceptual explication in mathematics, as well as her ability to convey the process and sensation of “discovering new math” to a lay audience, are on full display in Thomas Harriot:  A Life in Science.

Arianrhod’s prologue frames her story as an archaeological dig in the British Library, where she undertakes the monumental task of organizing and reverse-engineering Harriot’s papers, attempting to gain insight into his creative process.  In chapters one and two, she relates Harriot’s common birth, his attendance at Oxford via the surprisingly modern practice of need-based tuition, and his initial encounters with and patronage by Sir Walter Raleigh, against the backdrop of the Elizabethan Golden Age, the competition for naval power between the existing superpower Spain and the rising power of Britain, and the British desire to gain a colonial foothold in a New World thereto monopolized by Spain and Portugal.  This motivates her discussion in chapters three and four of the practicalities of ocean navigation in the age of sail, and Harriot’s employment as Raleigh’s expert tutor in navigational techniques.  In these chapters, she describes the development and transmission of millennia of astronomical knowledge underpinning the science of navigation, and their state of synthesis at the end of the 16th century.  This magisterial technique, deployed time and again in the book, establishes the intellectual foundations upon which Harriot stood, and enables the reader to attempt an understanding of the practical problems Harriot needed to solve in the context of the knowledge that was available to him at the time — including logical, repeatable procedures amenable to reliable execution by non-experts.

Chapters five through eight chronicle the ill-fated Roanoke expedition and colony, while establishing Harriot’s status as a bona fide polymath capable of learning and documenting the Algonquin language and culture.  In chapters nine and ten, she relates the defeat of the Spanish Armada, Raleigh’s shifting fortunes, the development of Harriot’s relationship with Northumberland, and, in a particularly compelling section, an illustration of Harriot’s interest in probability theory, likely spurred by Northumberland’s gambling habits.  This vignette is an example of Arianrhod’s technique of motivating a theoretical tangent with a mundane scenario — providing insight into Harriot’s talent for abstraction and generalization.  

In chapter eleven, Harriot and his patron Raleigh endure accusations of atheism, and Harriot survives Raleigh’s self-inflicted fall from grace at court via Northumberland’s gift of a life endowment, moving his residence from Raleigh’s estate in Devon to Northumberland’s, closer to London at Syon.  In chapter twelve, Harriot painstakingly derives the law of refraction via experiment (1602), establishing his precedence over Snell (1621) by almost twenty years, and in chapter thirteen, he wrestles with the problem of projecting three-dimensional curves onto two-dimensional surfaces while measuring their length — of obvious import for the utility of flat maps in a time when globes were expensive and rare.

Chapter fourteen sees Raleigh’s fall from grace nearly complete, as the transition of power to James I in the wake of Elizabeth’s death leaves Raleigh imprisoned in the Tower of London for life.  Chapter fifteen relates Harriot’s development of efficient algebraic notation and his brush with accusations of treason, as he is briefly detained in the wake of Guy Fawkes’ Gunpowder Plot, having done nothing more than attended a dinner party which happened to host one of the conspirators.  After having been released, Harriot’s work on the physics of rainbows is repeatedly distracted and derailed by the continued capricious detention of Northumberland.  This story illustrates the extreme precarity of court status and position during this tumultuous period in English history, and the resulting insecurity it generated for those of lesser status whose fortunes ebbed and flowed with their patron.  

Chapter seventeen is a fascinating interlude detailing Harriot’s correspondence with Kepler, Kepler’s knowledge of Harriot’s domestic troubles, and their mutual frustration with their inability to theorize without fear of retribution.  Of note, the role of the merchant Jan Erikson in shuttling letters between Kepler and Harriot illustrates the criticality of of early modern businessmen, transacting with wealthy patrons, in transmitting knowledge between scholars across Europe — especially prior to the development of reliable national postal services and a fully-fledged so-called “Republic of Letters”.  Chapter 18 continues this dialogue concerning early notions of atomic theory, arising from Harriott and Kepler’s quest to understand why refraction occurs.  While Kepler rejected the notion of atoms because he refused to acknowledge the concept of a pure vacuum, Harriot at points appeared to emphasize the role of empty space between particles of matter, between which light would “slip through”.  While he did not fully grasp the profound truth just out of reach, his conception of empty space as playing a key role predated Ernest Rutherford’s discovery that atoms (and thus matter) are composed mainly of nothing — by nearly three centuries.

Arianrhod’s final chapters detail Harriot’s telescopic observations and theorizations on ballistic curves, roughly contemporaneous with Galileo’s work on similar problems.  Again her strengths are clear — explaining the errors in reasoning committed by Harriot and Galileo, while making it clear that Newton could not have derived the concept of inertia without the half-correct notions hammered out by Galileo.  She closes the book with an account of Raleigh’s ultimate fate on the scaffold and Harriot’s death from cancer at the hands of the pipe-smoking habit he’d acquired among the Algonquin.  Her epilogue reads like a detective story, chronicling Harriot’s fall into obscurity, and in one particularly riveting sequence, historian Johannes Lohne’s chance encounter with a book owned by Harriot and containing Harriot’s marginalia, which spurred further investigation by Lohne and the modern era of Harriot scholarship.

Arianrhod’s dual themes are the case for Harriot’s sheer intellectual prowess, as well as an apologia of sorts for his failures to publish his work with sufficient frequency.  After all, life in the shadow of the Elizabethan court was a stressful affair.  The tragedy of Harriot’s life and obscurity throws the modern imperative for publication into stark relief — publish or perish, yes — but worse yet, publish or be guaranteed oblivion to posterity.  This reader found only one major fault with Arianrhod’s work.  Her endnotes contain exhaustively created and compiled diagrams illustrating the key mathematical concepts, and her mathematical prose would have been far easier to digest had it been interlaced with these diagrams.  In their current position at the end of the book, they have far less impact than they otherwise could have had, and give the reader no respite from a textual onslaught of facts and logic that made her book a challenging slog at times, albeit a rewarding one.

Ulinka Rublack is a professor of Early Modern European History at the University of Cambridge and a Fellow of the British Academy.  While her primary expertise is in the history of gender roles in early modern Germany, she’s also published a history of the Protestant Reformations.  Her undergraduate studies were at the University of Hamburg, but she was born in Tübingen, whose university was Johannes Kepler’s alma mater.  Thus one might say she was destined to tell the story of the Kepler family and its milieu, culminating in Kepler’s half-decade struggle to save his mother Katharina from being burned as a witch.  “The Astronomer & the Witch” mines the Leonberg and other local archives, as well as Kepler’s own papers, to render an immersive portrait of Kepler’s world, and the precarity of middle-class academic existence amid the religious and political upheavals that culminated in the Thirty Years’ War.

Rublack begins the book with a timeline of Kepler’s life, laying out his major life events, publications, and geographic moves.  This is an effective and necessary technique against the subsequent structure of her narrative, which groups events and biographical details thematically, often entailing flashbacks and large time jumps, making it easy for the reader to become chronologically confused.  This reader referred back to the timeline regularly.  After setting the scene with a description of the witch craze in Germany, Kepler’s philosophy, and intellectual ferment among the ruling classes in the Holy Roman Empire, she describes the Kepler family’s history, with a focus on Katharina’s journey and the practical challenges of survival, and launching productive children, as a middle-class widowed single mother. 

In the second chapter, Rublack shifts to the nearly opposite end of the social hierarchy, with a portrait of the Duchess Sibylle of Württemburg, also a widow, who moved her court from Stuttgart to Leonberg (Katharina’s home) after Frederick’s death.  It paints a picture of courtly intellectual life in the German principalities, and makes it clear that a phenomenon similar to Renaissance Italian city-state competition was playing out north of the Alps.  The aim of Rublack’s digression into herbalism as the domain of women unable to attend university here was unclear, except perhaps to propose that the alchemical activities of highborn women demonstrate an inescapable class element in witchcraft accusations.

Chapter three depicts three compounding sources of growing political and social instability at regional and local levels.  First, the diplomatic efforts of Lutheran princes to secure alliances with Protestant England, pinned as they were between the Catholic Austrian and Spanish Hapsburgs.  Second, worsening food insecurity amid failed harvests and Little Ice Age winters; and third, a ducal governor in Leonberg newly emboldened by the death of Duchess Sibylle.  Rublack provides the impression that Governor Einhorn saw witchhunts as a pressure relieve valve for a population under increasing stress.  The chapter culminates in accusations against Katharina Kepler. 

Chapter four relates the development of Johannes Kepler’s skills as a logician, rhetorician, marketer, and early “crowd-sourcer” of experimental data.  It makes a powerful case that, were these traits uncoupled from his brilliance as a mathematician, Kepler might not have been able to survive as a professional mathematician and astronomer in Holy Roman Imperial society, particularly in a confessional environment where disagreements among Protestant factions over esoteric doctrines like transubstantiation could render one ineligible for a university teaching position (as it did for Kepler).  It sets the stage for chapter five, which describes the Kepler family’s reaction to the accusations against Katharina, and Johannes’ rapid appraisal of the situation.  As Rublack puts it, “a towering intellectual, he [Kepler] had identified how the German persecution of witches worked”. 

Chapter six posits astrology as an early form of psychoanalysis, utilizing horoscopes he compiled on himself and his mother to document Kepler’s exhaustive introspection and insecurities throughout his career.  These tendencies culminated in efforts to protect his academic reputation from his mother’s case — even as he labored to defend her in correspondence with the authorities —  via quasi-disavowal of her in one section of his “Harmonies of the Worlds”, wherein he lays out his third law of planetary motion.  This chapter is utterly fascinating, insofar as it provides some insight into Kepler’s creative process.  He attempts a synthesis of astrology and free will, much as his elliptical orbits are a synthesis of dynamism and Ptolemaic stability/perfection.

Chapters seven, eight, and nine lay out details of the witchcraft allegations, the credibility of the accusers, the dire situations of similarly accused defendants under the jurisdiction of Leonberg’s governor Einhorn, Katharina’s imprisonment and public interrogation, and the petition of Johannes’ younger brother Christoph for a change of venue from Leonberg, so as to limit collateral damage to Christoph’s pewtering business. 

Chapters ten, eleven, and twelve detail Kepler’s decision to put his life on hold, uproot his family from Linz, and move to Stuttgart full-time in order to mount a more effective defense of his mother.  They also explain Kepler’s ability to transfer techniques honed in scholarly argument – meticulous analysis and refutation of details and inconsistencies in testimony – to his mother’s case.  The chapter closes with Tübingen law professors deciding that, while likely not guilty, Katharina must be shown the instruments of torture, in an effort to once more induce a confession. 

In chapter thirteen, the ordeal ends as Katharina resists coercive methods, but dies (likely of exhaustion) six months after the trial ends.  It follows Kepler to the end of his life as he settles her affairs, navigates the thicket of increasing intolerance among potential patrons in a markedly less pluralistic empire descending into religious war, struggles to self-publish important scientific work, and arguably attempts to process the trauma of his mother’s trial by publishing a bizarre science fiction story he penned in his youth containing uncanny parallels to later events (such was the foreshadowing that Kepler became convinced his own story was the seed of later accusations).

Rublack’s epilogue attempts to shift the focus back onto Katharina and various historiographical perspectives on the witchhunt craze.  While her intent was likely to illuminate a less luminous subject (Kepler’s mother, rather than Kepler himself, in the current fashion of social histories of under-studied demographics), the predominant effect of the book on the reader is to humanize Johannes on a profound level, and make his accomplishments all the more astonishing in the context of his stressors.  Modern scholars, some of them representing the first generation in their family to attend university, vying for grant money under the weight of student loans, while remaining alert to the risk of professional immolation at the hands of Twitter mobs, can identify with Kepler’s constant need to hustle, bow, and scrape for patronage while paying off his education, finely parse religious/ideological sensitivities, tutor children of the nobility with little aptitude or interest in mathematics, and subsidize transformative theoretical work with mundane, boring applied work (in Kepler’s case, horoscopes and land surveys).  Such was his grit that he managed to formulate his third law of motion while sparing his mother a painful and humiliating death.