Learned Failure and the Untutored Mind

Mathematics, in several of its Renaissance guises, lies in the background throughout Atalanta fugiens. In the preface, optics and music are among the “innumerable arts and sciences” the reader needs to search out God’s hidden secrets.¹ Explaining Emblem 1, “Wind carried him in his belly,” Maier presents three examples of the mysterious “him”: in arithmetic it is the cubed root; in music, the perfect fifth; and in astronomy it is the center of the planets Saturn, Mars, and Jupiter.² Emblem 8 uses the common technique of a perspective tunnel to focus the image. As a whole, the book is held together by music, as Eric Bianchi shows. Any Renaissance reader would have found these allusions to perspective, astronomy, and music instantly recognizable as “applied” or mixed sciences, rooted in the theoretical knowledge of arithmetic and geometry.

In Emblem 21 (fig. 1), mathematics moves into the foreground: “Make of the man and the woman a circle, of that a quadrangle, of this a triangle, of the same a circle, and you will have the Philosoph. Stone.”³ The emblem itself comes from the influential Rosarium philosophorum (Rose garden of the philosophers; 1550), in the chapter on conception by the alchemical king and queen.⁴ In the Rosarium philosophorum, the emblem concludes a discussion of how substances mix and putrefy and their seeds begin to grow and transform: “Aristotle” presents the squaring of the circle to sum up the process known as conception. The dichotomy of male and female is a familiar piece in the metaphorical repertory of early modern alchemy.⁵ Indeed, Emblem 21 here evokes themes that permeate Atalanta fugiens — the first two emblems play upon the male and female elements of the philosophers’ stone, its conception and generation, supplying the book with one of its governing metaphors. Nevertheless, Emblem 21 does not dwell on that rich dichotomy. Instead it spends most of its time on the ancient problem of squaring the circle. The epigram below the image underscores the explanatory power of mathematics, adding that “[i]f this [matter of conception] too high and too abstruse you find, / Geometry will soon inform your mind.”⁶

Squaring the circle

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

Why should squaring the circle help understand nature’s secrets, even the philosophers’ stone? This essay takes up Maier’s injunction to consider how geometrical learning might support the alchemist, as part of Maier’s aim to present chymistry as serviceable to the state. By considering how Emblem 21 redeploys commonplaces about the famous mathematical problem, I wish to suggest the ambivalence of mathematics as an archetypic form of innate knowledge and how this ambivalence could fuel a growing, fertile tension between learned and artisanal forms of knowledge.

Emblem 21 uses mathematics to make a point about the possibility of transformation in knowledge. The accompanying discourse makes clear that the emblem is not intended to give advice on the actual process of refining the philosophers’ stone, as some of the other emblems are, such as those on the philosophers’ bath (e.g., Emblem 28). This holds true even near the end of the discourse, where Maier is most explicit about the emblem’s alchemical meaning. There he compares the transformation of a square into a circle to the conversion of inferior matter into the philosophers’ stone. Matter is a square, like the four elements, which metamorphoses through the three alchemical colors of earthy black, silvery white, and golden yellow, before ending up the final “invariable redness” (rubedo invariabilis) that characterizes the completed work. But what interests Maier is the metaphor of fertile conversion, as we see in an arithmetical metaphor for the same process: the resolution of the perfect number six into its divisors, ultimately into even and odd, until resolved into the unchanging divine One or Monad. The single sentence hints at the standard Pythagorean table of opposites, where even numbers are female, odd ones are male, and the One generates all numbers from within itself. As Maier forefronts the generation and transformation of these mathematical objects, the operational details fade. In the context of the alchemical work’s stage of conception, mathematics is a metaphor for fertility.⁷

This is not the fertility of nature’s secret workings. Instead, Maier uses mathematics to consider the human mind’s power and its sources. Mathematics helps Maier think through a fundamental problem for any defender of a new science: how is human knowledge generated and transformed? Are human minds simply vessels for ideas they were born with, or do they generate and nurture new knowledge? Maier begins with a classic place in Plato, whose Meno used mathematics to supply the paradigmatic argument for all knowledge actually being remembered:

Mathematics supplied the strongest basis for the claim that minds do not change — they simply remember.

On this account, knowledge is a gift, “whether . . . first taught by heaven, or by the Gods of the heathen.”⁹ To explore this idea, Maier uses the metaphor of fire: sparks and glowing coals evoke the flash of inspiration. This was a credible association for Renaissance thinkers, who might follow Marsilio Ficino in setting human intellectual power upon a Platonic footing, so that exceptional inventors of the arts and sciences might get their knowledge from ecstatic insight. But even though Maier accepts the image, he shifts the emphasis:

Everyone has these “small sparks,” presumably given by God. But they must be cultivated. Nurture transforms such sparks into effective, useful knowledge.

Such an emphasis on human “industry, art, and care” opens up space for knowledge as the product of labor. Maier sets up a dichotomy between those faculties that Renaissance thinkers widely distrusted for being unreliable (imagination and phantasy; “imaginatio et phantasia”) and those that might supply certain knowledge (reason and experience; “ratio et experientia”). Moreover, Maier links the former to Plato and the latter to Aristotle. Imagination was known to concoct unreal links, being therefore an untrustworthy guide to reality; every practitioner of alchemy, in contrast, insisted that his craft was difficult and laborious, led by the twin guides of reason and experience.¹¹

For someone like Maier, mathematics posed a peculiar challenge to reason and experience. Mathematics was associated with the opposite pole of remembered or inspired knowledge, not only because of Plato’s story of the untutored boy but also because of the special powers of music and mathematics to move the soul. Heinrich Khunrath’s famous image of the learned alchemist nods to such associations (fig. 2). The adept kneels in his tabernacle, praying before a mise en abyme: an open book just like Khunrath’s own, opened to the page that visually correlates the divine and material world in the shape of a triangle inscribed within a square within a circle.¹² Above, the top-most beam reminds our devout alchemist that “without divine inspiration, no man was ever great.”¹³ The tools of the musical arts — penknife, inkwell, notebook lined with musical staves, dividers, measuring scales, harp, violin, and lute — lie thrust toward the reader, with the most central text of the print advising on the special power of music: “With holy music, flee sadness and malignant spirits, for the Spirit of Yahweh freely sings in the heart drenched in pious joy.”¹⁴

The image builds on the biblical story of young David’s harp and song driving out the wicked spirits of a melancholic King Saul and confirms commonplace tales about Pythagoras curing the Dionysiac madness of a youth with music. Such tales were recounted in handbooks on the liberal arts that introduced students to the rudiments of mathematical music theory.¹⁵ This iconography also deepened the link between mathematics and a knowledge sourced in ecstatic imagination — after all, imagination manipulates images, shapes. It can be no accident that Khunrath places the table of tools a short distance away from the forge, which is supported by the two prominent pillars of ratio and experientia. Arithmetic and geometry were widely reputed to be certain, yet they appear to support an account of knowledge depending on inspiration, or at least imagination.

In the discourse to Emblem 21, Maier solves this problematic association with inspiration by taking quadrature as a question of history. Squaring the circle is here an argument against Plato and for Aristotle. Plato said mathematical knowledge was remembered. Plato’s own student Aristotle, however, reported that the solution to the quadrature of the circle was not yet known — even though it was in principle knowable. “Whereupon it might be demanded, why Plato wrote upon the door of his school, that he that was ignorant of Geometry should not be admitted into it, he also having affirmed that children actually knew it?”¹⁶ In other words, if such knowledge is innate, surely Plato and his students would have known about the squaring of the circle.

To buttress his argument against Plato, Maier turns to the animal world, comparing a moth and a child. Most animals have instinct, he points out, and from birth they naturally know to avoid dangers. But humans are different. “An infant knows not or shuns such things, except hurt, or his finger being a little burnt at the flame of a candle like a Fire-fly [pyrausta], which burns its wings and falls down.”¹⁷ Here Maier deploys a proverb from the ancient Greek collection of Zenobius, popularized by Erasmus in Adage 851, on “the moth’s demise” (pyraustae interitus), a saying that applies to those who seek their own destruction just as a moth flies to the flame.¹⁸ In emblem books of the sixteenth century, the proverbial moth was a well-known caution against dangerous innate tendencies — against the dangerous potential of one’s given character.¹⁹ Gabrieli Symeoni deployed the proverb in his own book of emblems: “just as the moth out of its nature [naturel], loving the brilliance of the fire in which it unwisely and knowingly is consumed, so he who too imprudently loves anything, gains only dishonor and misfortune.”²⁰ The inborn nature — the ingenium — of the moth compels it to self-destruction. To emphasize the tragic nature of the moth, Maier compares it with other insects. “Why do not the young bee, the fly and gnat fly directly into fire, they not knowing by experience the danger from thence arising?”²¹ These other insects have self-preserving instincts, unlike the moth and unlike the helpless newborn human.

The proverbial fact of the moth’s tragic ingenium allows Maier to say something about human knowledge, pushing back against the Platonist assumption that mathematics is innate. In fact, the quadrature of the circle is an example of knowledge that no one had in Plato’s time. “If Geometry be so natural and easy to children, how comes it to pass that the Square of a circle was not known to Plato himself, So that Aristotle the Scholar of Plato affirmed it possible to be known, but not as yet known?”²² The immediate implication is clear. The arts and sciences are not eternal verities, to be simply intuited or remembered. Even mathematics — the discipline most susceptible to that description — changes and develops with time. Therefore, space must be made for new forms of knowledge. Squaring the circle becomes an example of how unknown disciplines can legitimately be developed through ratio et experientia.

The transformation of knowledge does not happen overnight. Epigram 21 also alludes to the time it may take for a new discipline to emerge, when translated most literally: “If such a great matter does not quickly come to mind, if you take up the teaching of geometry, you will know all.”²³ Patience is required. Maier confidently assumes his reader will agree that geometry — and specifically the geometry of circle squaring — will repay patience. To understand this confidence, we need to consider the status of quadrature of the circle in early modern mathematics.

Like the moth and the flame, “squaring the circle” has become a modern proverb for quixotic attempts of human knowledge that are doomed to fail. We have known since the 1890s that a solution is formally impossible. Earlier in the nineteenth century, Augustus de Morgan already cast the circle-squarer as the archetype of thinkers lured to foolish, wasted study on unsolvable paradoxes such as perpetual motion machines.²⁴ Yet de Morgan satirized circle-squarers even while producing a ream of notes on the history of the problem — a history that revealed squaring the circle as a legitimate problem that had exercised the best mathematical minds since antiquity, along with problems such as duplicating the cube or trisecting an angle. The challenge was not to find a close approximation between a circle and a square but rather to prove an exact equivalence, using only Euclidean principles, a compass, and a straightedge.

In Maier’s time, squaring the circle was not a species of learned folly but precisely the opposite: it was the archetype of knowledge that was possible but not yet possessed. Every university student knew this from Aristotle’s Categories, which had been used since antiquity to introduce students to the basics of logic. To argue that an object of knowledge exists even when no one knows of it (a tree falls in the forest even if nobody hears), Aristotle used quadrature as his evidence. “Take the squaring of the circle, for instance, if that can be called such object. Although it exists as an object, the knowledge does not yet exist.”²⁵ Aristotle set the quadrature as a central problem for Renaissance mathematicians. He also gave them examples of how his own contemporaries had failed. In his Physics, he sketched three attempts, one by Bryson, another by Antiphon, and the fullest by Hippocrates of Chios.²⁶

In fact, Aristotle’s statements about quadrature were widely familiar to members of the republic of learning. When Nicholas of Cusa described his own attempt, he described it as a topic for learned conversation, in Aristotle’s language of knowability.²⁷ Oronce Fine opened his attempt at a solution with a discussion of Aristotle’s statements on the topic; having given his subject an august history, from Aristotle and Archimedes to Cusanus, he demurred that he would hardly be worthy of his office if, as royal professor of mathematics, he left quadrature aside. A high-water mark was reached by Fine’s student Jean Borrel (Buteo), whose De quadratura circuli (On squaring the circle; 1559) based its critical reading of the problem on a step-by-step reconstruction of Bryson, Antiphon, and Hippocrates, out of Aristotle, before moving on to an analysis of the modern failures of Cusanus, Fine, and others.²⁸ Borrel’s book served as a manual for would-be circle-squarers, as well as a cautionary tale.

As a whole, the history of quadrature is one of failure, of ingenious solutions followed by ever sharper analysis and critique. Some of the more spectacular failures involved the tallest intellectual giants of the day. Maier and indeed the whole learned world heard about Julius Caesar Scaliger, who deigned to turn his princely expertise in philology to expurgate the errors of mere mathematicians. But his gorgeously printed Cyclometria (Measurement of the circle) revealed his own fundamental errors of mathematical judgment, and mathematicians all over Europe darted at the great man’s exposed flank. Christoph Clavius penned the most widely read refutation, though rumors circulated also that formidable mathematicians such as Henry Savile found Scaliger’s quadrature nonsense.²⁹ Over half a century later, Thomas Hobbes would play from the same script, convinced like Scaliger that his towering intellect would finally realize the possibility Aristotle had mentioned — only to be disabused of his mathematical prowess by the Oxford Savilian professor John Wallis.³⁰

Yet spectacular failures did not eliminate hope that the circle might be squared. The very hubris of Scaliger and Hobbes in taking on quadrature indicates their high regard for the problem as the gold standard for the possibility of new knowledge.

Precisely because it was the reputable example of knowledge that was possible but not yet possessed, the squaring of the circle could legitimate the kind of new knowledge Maier proposed. Recall the tenor of his argument: that “reason and experience” are the way forward in developing new knowledge; and that, contrary to what the Platonists might say, mathematics is evidence for the productivity of reason and experience. Near the end of the discourse, however, Maier’s argument moves from the lofty eminences of philosophy, Plato and Aristotle, to practical mathematics.

Money and politics were certainly on Maier’s mind. He had lost a powerful patron in 1612 when Rudolf II died, and afterward he began traveling Europe and writing in search of a new one. After time in Rotterdam and London, he settled in Frankfurt am Main. In the two years leading up to the publication of Atalanta fugiens early in 1618, Maier had written several “chemical tracts” while he lay ill, including a De circulo physico quadrato (On the natural [or: physical] circle squared; 1616), dedicated to Moritz of Hessen-Kassel, a prince widely known for his alchemical interests.³¹ In April 1618, the ink on Atalanta fugiens barely dry, Maier sent it along with ten of his other books to the prince and was promptly appointed Medicus und Chymicus von Hauß aus (court physician and chymist).³² The books had achieved their main goal of catching the count’s eye. Maier himself later admitted these books were meant to gain attention — indeed to sell, as they were “inspired more by the small payment which I received for them rather than by the improvement and perfection of the works themselves.”³³ The contents of De circulo physico quadrato in no way attempted to contribute to the mathematical scholarship on quadrature. It developed a reflection on the “circular” motions of the natural four elements, which tend ultimately to circular perfection, in the heavens, on earth, and also in human hearts. The book culminated in reflections on the prince as the heart of his people, the center who unified disparate members within the political ambit.³⁴

Maier opened De circulo physico quadrato with a familiar hook in the first sentence of the dedication:

Thankfully, Maier was available to ensure that the prince would have the chemical, medical, and indeed moral knowledge he needed to care for his subjects. Squaring the circle — or at least the “physical circle” — was a matter of state utility.

Throughout the discourse attached to Emblem 21 of Atalanta fugiens, Maier is concerned with practical knowledge; the “sparks of inspiration” resolve in a cooking analogy, where the flame serves digestion and the cook’s tasks. Near the end of the discourse he assures the reader that he intends something much more useful than the mathematicians can give: “Now that this quadration is Physical and agreeable to nature, every man understands; by which far more utility accrues to a Commonwealth as also more illustration to the mind of man, than by that Mathematical and merely theoretical or from an abstracted matter.”³⁶

Then Maier takes this argument one step further, tying successful quadrature to practical utility. Not only is squaring the circle possible, but it has in fact been done by those with practical wit. After promising all manner of utility to the commonweal, he says, “To learn that perfectly, a Geometrician acting about solid bodies must inquire what depth of Solid figures, for example, of Sphere and Cube can be known, and transferred to manual use or practice.”³⁷ What follows is a garbled example of practical mathematics, using a rule of thumb to approximate the size of a cube of the same area as a given sphere. By the time Maier has moved on to say that chymical philosophers’ work does the same thing, the point has been made: in fact, people do cube spheres, which shows the power of practice. Just as mathematical quadrature has actually been found by artisans who measure columns, dig wells, and gauge barrels, so too actually letting alchemists get on with their practice will result in new, useful knowledge.

Maier here makes a point every circle-squarer knew — and struggled mightily to avoid. Craftsmen had always used rules of thumb for the ratio of a circle’s circumference to its radius (π = approx. 22/7) to convert circles to squares. Some theorists, such as Campanus, Nicholas of Cusa, and Charles de Bovelles, tried to formalize these techniques.³⁸ Theorists often ridiculed such practical efforts as formally fallacious. Nevertheless, they could hardly forget their existence. It was common knowledge that an approximation can get arbitrarily close to success; the challenge is to make it perfect, within the self-imposed framework of Euclidean demonstration. It is perfectly possible to square the circle in a way that “satisfies the senses,” as Jakob Christmann put it.³⁹ Because such solutions, even though off limits in theory, are rather well suited to experience, the problem taunts. Common intuition is precisely what makes the problem alluring.

Pragmatic accounts of the untutored mind’s success seem to have inflated the value of quadrature among the likes of Maier. For Maier, with his rhetoric of utility, must be seen as part of a larger group of projectors who stoked the desires of funders with wish lists, accounts of desiderata in knowledge that promised outstanding utility to princes and governors.⁴⁰ The genre’s founder, the Paduan law professor Guido Pancirolli, included the quadrature in his Nova reperta sive Rerum memorabilium, recens inventarum, et veteribus plane incognitarum (Two books of things lost and newly found) — he offered a solution using mechanicians’ rules of thumb.⁴¹ His student Heinrich Salmuth set Pancirolli’s solution within the long history of mathematical solutions, from Antiphon and Brysson to Cusanus and Borrel. But his last two authorities were particularly telling: Peter Ramus and Jakob Christmann.⁴² Both had reflected upon how artisans had been able to satisfy the senses “honestly” (ingenue).⁴³

Quadrature was prominent on the lists of seventeenth-century utopian thinkers, who hoped to regain for mankind the knowledge lost after the Fall into sin.⁴⁴ In his Mythologiae Christianae (Christian mythology), Johann Valentin Andreae, the Lutheran theologian who like Maier contributed to the early swell of Rosicrucian tracts, surveyed a fabulous marketplace filled with desirable knowledge from the art of memory to the quintessence. Andreae’s survey of the marketplace began with the quadrature of the circle.⁴⁵ Jakob Bornitz, a politician who was connected to Maier’s circles through the Rudolphine court, likewise included quadrature on his list of things to be rediscovered and investigated, next to perpetual fire and techniques for navigating against the wind and under water.⁴⁶ Bornitz’s list was influential: it was excerpted by the well-connected Samuel Hartlib into his own Ephemerides, as well as by the pedagogical reformer Jan Amos Comenius.⁴⁷ These projects used squaring the circle as an intellectual desideratum in ways similar to Maier’s. Bornitz himself cited Scaliger to support the worthiness of quadrature as a pursuit, but he also referenced two less-learned examples. The Leiden mathematical teacher Ludolph van Ceulen, teacher to Willebrord Snell, wrote a study of quadrature which refined the practical method for approximating quadrature to the limit of the senses.⁴⁸ Bornitz also cited an even more telling work from his and Maier’s own circles: the painter Philipp Uffenbach, who wrote a work on quadrature that celebrated the use of instruments beyond the compass and straightedge to offer a “new, short, extremely useful, and effortless mechanical treatise and account.”⁴⁹ Such works contravened theoretical rules but vindicated Maier’s promises at two levels. They played on learned assumptions about possible knowledge; they also fostered a growing assumption that letting artisans get on with it would bring such knowledge about.

This was polemic, of course, defending the associations of chymia with grubby labor. Maier had used quadrature to show the balance required between innate talent and cunning industry. Certainly, witless labor was no answer. At the end of De circulo physico quadrato, in some Cantilena Acreontica (Anacreontic songs), Maier excoriated rivals for being entirely without inspired talent or acquired skill. The Ancreontic genre offered a chance to play with the themes of inspiration, wine, and sensory pleasure.⁵⁰ These “stumbling verses” (claudicanta camaena) deride those who only think of vital powers lying in “smelly filth, and the limbs of animals” and not “in the metals dug from the deep hollows of the earth”; such people are “led by ignorance and madness of the mind.”⁵¹ Ultimately, Maier associates such people with a certain naïveté:

Despite his praise for practice, Maier intended to eliminate neither class nor talent. One needed at least a spark of inborn wit to go with industry.

Maier offers us a way to reconsider the usual dichotomies we use to understand early modern knowledge. The usual dichotomy is drawn between Descartes and Bacon: between rationalism and experimentalism, between reason and experience. But Maier — and the network of mathematicians, alchemists, and projectors who made up his world — drew the dichotomy differently. He was eager to claim ratio et experientia together as a pleonasm for chymistry; their contrast was innate knowledge, inspiration. An option was to put mathematics on the innate side of the ledger, aligning reason, imagination, and memory with the certainty of mathematical proof. Instead, Maier pushed for a vision of the human mind that admitted the origins of knowledge somehow in sparks of divinity — but to be useful, productive, these must be developed by reason and experience. On his retelling, mathematics became evidence that instinct was not enough but must be nurtured through reason and experience.

Once we see this formulation in Maier, we can begin to see its significance elsewhere. Around the time that Maier was writing De circulo physico quadrato and Atalanta fugiens, René Descartes was beginning his own study of mathematical learning, but with very different effects. For a brief while, Descartes was intrigued by Rosicrucian themes. In 1619, during his time in Germany, he wrote his old teacher Isaac Beeckman, asking after the art of Lull.⁵³ When he returned to Paris in 1623, he found himself under suspicion of being a member of the Rosicrucian brotherhood — a suspicion that reemerged later in the century.⁵⁴ What he shared with Maier and others linked to the Rosicrucian circles was a conviction in the need to develop one’s inborn talent. A famous passage in Descartes’s early Regulae ad directionem ingenium (Rules for the direction of the mind) reflects on the origins of knowledge in terms akin to those Maier used in Emblem 21:

The goal of the Regulae would have been quite familiar to Maier: to provide the reason and experience needed to develop the inborn light, seeds, or mind (lumen, semina, ingenium). But the two differed in their evaluation of whether mathematical reasoning might itself be reason and experience. Descartes took the old bedrock disciplines, geometry and arithmetic, and transformed their exercise into a method for all the arts — what he called a mathesis universalis.⁵⁶ Maier instead kept mathematics fixed within the cycle of arts, as a rhetorical and historical metaphor of transformation — leaving reason and experience to work without mathematics at the furnace.

Acknowledgment: The writing of this article has been supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 617391.