G.W. Leibniz and Crossing the Threshold
The heavy door creaked on its hinges. A little boy gingerly pushed the door a bit further and peered into his father’s study. Holding his breath, the boy squeezed through the narrow gap between door and frame, slipping into a forbidden realm of ideas, reason, knowledge, and truth.
Der Vater had just been laid to rest, having crossed the final threshold and left his six-year-old son behind with a sister, little Anna Catherina, and their devout and devoted mother, Catharina. The boy had been made aware of his father’s great expectations ever since the moment of his baptism, when he was said to have suddenly lifted his head with eyes open wide upon hearing his christening as Gottfried—God’s peace. Little Götz grew up surrounded by staunch Lutherans, his deceased father a professor of moral philosophy at Leipzig University, one of the world’s oldest universities, in the city where Martin Luther himself had preached, debated, and formally introduced the Reformation just over a century earlier.
Now he stood there in the late professor’s office and gently closed the door behind him, blinking in wonder, for in his father’s library the young boy had ventured into that sphere in which Leipzig possessed one of the greatest commodities of the sixteenth century: printed books. So many books! But where to begin? When you read, you begin with A-B-C; when you study moral philosophy, you begin with the Ancients.
Like Saint Augustine before him, the boy had heard God calling him to Tolle legge! Tolle legge! Take up and read! Take up and read! There was just one small problem: he couldn’t read Latin. Oh, he was drilled in Latin at school, but only in the basics, nothing like the intricacies of ancient Latin required to read his father’s volumes of philosophy, law, theology, and history. At first, he merely browsed the stacks, unable to comprehend the texts themselves. However, he happened upon a bit of providence in that a student boarder taken in by his mother to help make ends meet had left behind two books: a historical compendium written in a simple grammar by which the boy could take easy steps from German into Latin; and an illustrated edition of Ab Urbe Condita by the great Roman historian Titus Livius. The boy was captivated by this pictorial telling of Livy’s Rome, and he began to see the conceptual landmarks he needed to eventually navigate the entire realm of Roman thought.
The boy soon leapt ahead of his classmates at school, but rather than rewarding his efforts, his teacher remonstrated that this unstructured method of learning would lead only to confusion. His father’s library was locked and the boy forbidden to enter until a noble neighbor saw in him a latent genius and convinced everyone to let the budding genius explore the intellectual world as he wished.
Left to his own devices, the boy developed his own method for working with ideas, reason, knowledge, and truth. Rather than simply memorizing the material, as required by his teachers in Latin school, he early developed the habit of comparing and contrasting differing positions on a given topic and a range of solutions to any given problem. Today, we would call his approach critical thinking, and it remains an underdeveloped skill in young students to this day. It was precisely this approach to learning that would carry young Leibniz into uncharted realms of thinking, the likes of which the world had never seen before.
By the time he entered his teens, Leibniz was grilling his teachers on the finer points of Aristotelian logic, seeking to go beyond the ordering of simple ideas into a categorization of complex propositions. At the same time, he picked up and ran with the methods of Peter Ramus, who advocated a clear and structured approach to organizing concepts, especially in writing, and Leibniz could see the utility of this approach as he wrestled with the works of the Ancients. In logic, i.e., logos, he found a conceptual lens through which mind could more clearly view reality. This led the young student to early adopt two axioms: clarity in words, the basis of judgment; and signs of the soul, the basis of discovery.
In his wide-ranging reading young Leibniz encountered two concepts that would lay the foundation for all that followed in his intellectual pursuits: 1) universal harmony, the notion of simultaneous interdependence between everything and everything else; and 2) the idea that all living entities are equipped with perception and appetite, which would later define his concept of monads, as we shall see.
As the young polymath was preparing to enter the university in 1661, just three months before his fifteenth birthday, he found himself wandering for whole days in the Rosental, a small wooded area on the outskirts of Leipzig. After crossing the woodland threshold each day, he walked among the trees and wrestled with the question of Aristotelian forms versus the new concept of mechanism, about which he’d recently been reading in works by Johannes Kepler, Galileo Galilei, and René Descartes, as well as Francis Bacon and Thomas Hobbes. From these scientists and philosophers, Leibniz had gained an appreciation for mathematics and the application of logic to physics: cosmology, astronomy, and mechanics. It seemed he must now choose between the qualitative physics of Aristotle, which lay at the foundation of the Ancients’ learning and thinking, and the quantitative physics that would become the foundation of the Enlightenment. And it was there in the woods that Leibniz came to the realization that, properly understood, Aristotelian metaphysics is fully compatible with mechanistic physics. The qualitative could coexist with the quantitative through foundations and principles, that is, the ultimate reasons—in other words, through logos. The old could be reconciled with the new.
Leibniz left the woods behind and matriculated at Leipzig University. Within two years he was defending his philosophical disputation on the principle of individuation, and from there he entered the University of Jena to study higher degrees of philosophy and law, which he came to see as not at all unrelated. His thesis was, in fact, titled Specimen of Collected Philosophical Questions Concerning Law. His highly personal and individualized approach to learning had resulted in a mind dedicated not to division, but to unification. He thus left the university on a mission to work toward universal synthesis as his means to glorify God and benefit mankind, a life dedicated to nothing less than fulfilling the Two Great Commandments in his own manner. Nine days later, his mother crossed the final threshold and was laid to rest.
Upon commencement, Leibniz gained employment as a lawyer for Baron Johann Christian Boineburg and immediately began to formulate a plan for implementing his grand scheme of advancing science for the greater good and the glory of God. His interpretation of Matthew 22:37-39 was such that loving God above all else is to love the general good and universal harmony, an “efficacious love” realized through “good works”  Following Galileo, Leibniz understood that the continuum of reality consisted of an infinity of parts, or infinitesimals, and his reading of Hobbes convinced him that this concept could be extended to include conatus, that is, effort or striving, such that every impulse becomes motion through an infinitesimal point in an instant. Thus, in our every striving throughout life, we are continuously crossing infinitesimal thresholds, always accepting the call to undergo yet another infinitesimal journey. We can do this because, while “the actions of bodies consist of motion, the actions of minds consist of conatus,” and are able to retain all of their accumulated conatus through memory, which becomes the story of a life. When this story consists entirely of impulses to love one’s neighbor, it amounts to a just life, for “justice is the habit of loving all human beings.” Such habits lead to the creation of a good person, “one who loves all human beings. To love is to find delight in the happiness of another. To find delight is to feel harmony.” When a mind achieves harmony in this way it comes to reflect the mind of God itself, for “one mind is, in a certain sense, almost a world in a mirror,” reflecting the universal harmony of every entity to every other entity in all of creation. In his correspondence, Leibniz would write:
What, therefore, is the ultimate basis of the divine intellect? The harmony of things. And what is the ultimate basis of the harmony of things? Nothing. For example, no reason can be given for the fact that the ratio of 2 to 4 is that of 4 to 8, not even from the divine will. This depends on the essence itself, i.e., the idea of things. For essences of things are just like numbers, and they contain the very possibility of entities, which God does not bring about, as he does existence, since these very possibilities—or ideas of things—coincide rather with God himself. However, since God is the most perfect mind, it is impossible that he is not affected by the most perfect harmony and thus must bring about the best by the very ideality of things. But this does not detract from freedom. For it is the highest form of freedom to be forced to the best by right reason.
And in that freedom of right reason, Leibniz found the motivation to continuously cross thresholds into new frontiers of thought. He had been invited to speak before the French Royal Academy of Sciences, and he was commissioned by Baron Boineburg to collect the rents that France owed him, so young Leibniz now had the motivation and freedom to cross the border and enter the learned spheres of Paris.
And what an eyeopener that was! Not only was his French not up to snuff, but initial meetings with the likes of astronomer Giovanni Cassini, for whom the recent Cassini Spacecraft, which explored Saturn and its rings and moons, was named, and Dutch mathematician and physicist Christiaan Huygens, probably one of the greatest scientists of all time, made it clear that the education Leibniz received in the classics, philosophy, and law had left him woefully deficient in mathematics. Fortunately, Leibniz was a quick study, so that by the end of his four-year stint in Paris, he had not only mastered analytic geometry but had developed the infinitesimal calculus as well.
In spite of Leibniz’ lack of grounding in mathematics, Huygens agreed to tutor the young philosopher, probably in his office at the Royal Library. Leibniz told him that in reflecting upon Euclid’s axiom that the whole is always greater than the part, he had come up with a solution for summing infinite series. Huygens was not unimpressed, so he tasked Leibniz by asking him to sum the infinite series of reciprocal triangular numbers and pointing him to the relevant literature in the Royal Library. Leibniz rose to the challenge and shortly returned with a solution that was at once original, elegant, and simple. Huygens took the budding mathematician under his wing and championed his efforts from that point forward.
In his spare time, Leibniz began tinkering with an idea he’d had several years earlier: the construction of a mechanical calculator. He was familiar with Blaise Pascal’s device, but it was really just an adding machine, capable of performing only addition and subtraction. The machine that Leibniz designed could perform all four of the basic arithmetic functions—addition, subtraction, multiplication, and division—including automatic regrouping, also known as carrying and borrowing, between columns. By the time it was completed the Leibniz mechanical calculator could even extract square and cube roots without mental intervention simply by turning a particular wheel on the machine. This development elevated computational devices into an entirely new sphere such that even the cautious Huygens was moved to dub it “very ingenious.” And with these accomplishments, Leibniz crossed yet another threshold: on April 19, 1673, at the age of 26, he was admitted into the Royal Society.
Leibniz then embarked upon the endeavor for which he is best known, inventing the differential and integral calculus—a distinction he shares with Sir Isaac Newton. In reading Pascal, Leibniz noted that the French mathematician’s solution for the quadrature of a circle could be generalized to find the area beneath an arbitrarily small section of any curve. He began with Pascal’s notion of “characteristic triangles” (Leibniz’ own term), which were right triangles with infinitesimal sides—basically triangular in nature but infinitesimal in area—then replaced Pascal’s use of the radius of the suggested circle with the normal line (which a layperson might call the perpendicular line) to the curve. Huygens approved, so Leibniz then worked out his generalized “transmutation theorem” for transforming these triangles into the tangents and quadratures of any curve. Within a year of his admittance into the Royal Circle, Leibniz had achieved the arithmetic quadrature of the circle. As he envisioned it, any curve could be represented by a Cartesian coordinate system with an infinite number of infinitesimal intervals, that is, an apparently smooth curve may be treated as a polygon with an infinite number of infinitesimal sides.
Of equal importance to Leibniz’ insight into the key principle of differential calculus was his use of a specific symbol set to describe the processes used to analyze a curve. In particular, he early on published his use of an elongated “S” for summa to notate the integration function. The ∫ symbol is still in use today, which may be considered our greatest tribute to the mathematical genius of Leibniz, who viewed his work as simply another step toward crossing the threshold into a characteristica universalis, combining mathematics, logic, philosophy, religion, ethics, and politics.
And yet, while busy inventing the calculus and developing the first true mechanical calculator, Leibniz did not forsake his love for metaphysics. He began compiling notes for a work he titled De Summa Rerum, by which he meant both the totality of things and the highest of things, that is, the natures of mind and matter, and the relationship of soul to body. In this he returned to his “principle of harmony,” by which he meant “similitude in variety” or “unity in multiplicity.” The greatest harmony is achieved by the greatest diversity resolving into concordance. Through creation, God chooses to bring about the best of all possible realities by providing for the harmonious interaction of as many diverse entities as possible.
Furthermore, Leibniz put forward the thesis that every mind in the universe perceives the entire universe; however, the perception is imperfect. This led to his conclusion that “all things are contained in all things,” which is a basic tenet to fractal analysis today. Unfortunately, like so many of his works, Leibniz never published his Summa, so we do not know how he would have finalized his thinking along these lines.
In the meantime, his patron had recently crossed the final threshold, thus requiring Leibniz to return to Germany to once again secure employment with another patron. On his way back through Holland, he met with Antonie van Leeuwenhoek, now known as the Father of Microbiology, whose work with the microscope had crossed the threshold into the microcosmic—seeing for the first time the realm of muscle fibers, spermatozoa, red blood cells, and blood flow in capillaries. Leibniz was mightily impressed. He was, however, subsequently less impressed with Baruch Spinoza, whom he met on several occasions but whose metaphysics he dubbed “strange and full of paradoxes.”
Back in Germany, Leibniz found employment with the new duke, Ernst August, who was not the least bit interested in philosophy, science, or humanitarian reform least of all. The duke did, however, have a private library in need of organization, so he appointed Leibniz court librarian. To say the widely acclaimed young polymath felt unappreciated by the new duke would be vast understatement; nevertheless, he undertook the task of organizing the ducal library’s materials with his usual zeal. The job in Hanover did offer several perks though: in addition to a substantial salary, it provided room and board, including court dining privileges, which allowed Leibniz to hobnob with the German elite.
At the same time, his prior admission into the Royal Circle enabled his correspondence to cross the threshold into the sphere of leading edge thinkers of the day, including Newton, with whom he traded notes on their respective approaches to the calculus. Little did he know that Newton would one day use it against him.
Most important, his job duties allowed Leibniz plenty of time to further his thinking on his grand scheme of improving the human condition, which he saw as an all-encompassing task requiring thinkers able to see relationships between seemingly discrete fields. “What is needed,” he wrote to the duke, “is universal men. For one who can connect all things can do more than ten people.” The duke was unmoved, yet Leibniz remained undeterred, for in his mind everything was connected to everything else. He therefore proposed that the state should provide universal education to produce others like himself who can connect all things, saying, “for the welfare of the Country it is required not only the nourishment but also the virtue of the inhabitants. Which entails that they be properly educated[.] Indeed, everything is based on such education.” In this regard, Leibniz was directly echoing Martin Luther, who had fervently promoted universal education in Germany two centuries earlier.
Finding that the court librarian had too much time on his hands, the duke tasked Leibniz with two additional projects that would come to be the bane of his existence for the rest of his life. One was the design and construction of windmills to pump water from the mines of the Harz mountains, which could have been mightily profitable to the duke if only they had worked. Unfortunately, this Sisyphean task proved well beyond the engineering skills of someone whose credentials included only the invention of a mechanical calculator.
The other task was actually better suited to Leibniz’ abilities and inclination: to compile a complete historiography of Duke Ernst August, who desired to show that his pedigree was worthy to be crowned Holy Roman Emperor. In those times, lineage was everything, and the duke’s family tree had several missing branches that needed to be filled in if he or his heirs were ever going to ascend the throne. Leibniz attacked the problem with characteristic zeal, starting his history from the beginning of the world, as any good Micheneresque writer of historical fiction must do. The best part was that the task required Leibniz to cross the thresholds of libraries all over Germany, and even beyond, to discover just exactly who had begat whom. Amazingly, Leibniz was also able to make connections between his engineering project in the Harz to his historiography of the duke, as he had been picking up minerals, fossils, and bones from the mines that he then used in his geological and paleontological research into the all-encompassing history of Lower Saxony and its rulers.
All of which encouraged Leibniz to formulate his grand scheme of an all-encompassing plan for the reformation and advancement of sciences for the common good, which is to say the glory of God. “To contribute to the public good and to the glory of God is the same thing,” he wrote. His fulfillment of the Two Great Commandments would entail systematic research across the entire range of sciences, especially logic, metaphysics, physics, ethics, politics, and mathematics in order to discover the truths of “God, the soul, person, substance, and accident.”
Along the way, Leibniz developed a dyadic arithmetic in which he demonstrated addition, subtraction, multiplication, and division using only zeroes and ones, which we now know as base-2 or binary. In Leibniz’ all-encompassing system of mathematics-cum-philosophy, his “bimal” (analogous to “decimal”) notation held little in the way of practical usage but contained mystical significance as everything could be expressed in terms of zero and one. Leibniz then applied binary notation to his logical calculus, which would not be fully developed until George Boole conceived of his Boolean logic in the 19th century. Taken together, binary numbers and binary logic became the basis for digital computing as we know it today. But for Leibniz, binary was merely a step toward his concept of monads, as we shall see, in which one, or unity, is God, and zero is void.
It was also at this time that Leibniz composed his “Discourse on Metaphysics,” a formal statement of his philosophy to that point. In this treatise, he discusses the relationships between mind, ideas, reason, and, of course, God, the light by which we understand our existence. In order to comprehend mind, we must first consider ideas, and in order to consider ideas, we must classify types of knowledge as either confused, lacking the ability to state the properties of a thing, or distinct, in which one can state all relevant properties of the thing. Furthermore, distinct knowledge can be classified as adequate, when the thing is known “down to the primitive notions,” or intuitive when it is simply surmised. “The greater part of human knowledge,” he contends, “is only confused or suppositive.”
Every idea, then, is eternal and “is always in us, whether we think of it or not.” This is possible because ideas can occur only in mind, and there is ultimately only one mind, the mind of God. We gain some insight into the mind of Leibniz when he states that “the mind always expresses all its future thoughts and already thinks confusedly about everything it will ever think about distinctly. And nothing can be taught to us whose idea we do not already have in our mind, an idea which is like the matter of which that thought is formed.”
In this way, we can see that, for Leibniz, the calculus and the binary number system were not so much his inventions as discoveries that he made through connecting everything to everything else in his mind. This could be achieved only because those ideas were pre-existent in the mind of God, even if knowledge of them had not yet been made distinct by the mind of Leibniz. By stating such knowledge for the world, Leibniz was actually making clear certain aspects of God. The calculus is an aspect of God; binary numbers are but another.
In his meandering intellectual journey, Leibniz found that he was continually emerging from a mental maze at precisely the point where had crossed its threshold upon entry. He was now convinced of his earlier conviction that all physics must be grounded in metaphysics: “I have shown […] that the laws of mechanics themselves do not flow from geometrical but from metaphysical principles, and if all things were not governed by a mind, they would be very different from what we experience.”
It was as though he had returned to those adolescent walks when he crossed the threshold into the Rosental woods, where he reconciled Aristotelian substantial forms with mechanistic physics before entering the university: “And so what occurred to me was like that which happens to someone who, having wandered for a long time in a forest, suddenly emerges into an open field and against all hope finds himself back in the same place from which had first strayed to immerse himself in the forest.”
And though Leibniz would remain in the duke’s employ for the remainder of his life, always working on that interminable historiographical assignment, his research would expand to take him throughout southern Germany, Austria, and even Italy. In Munich, he had discovered an ancient manuscript of the Historia de Guelfis princibus, which unwittingly provided the missing link that would connect the Hanoverian line to imperial dynasty, as well as provide a reason to pursue his research further south. While in Austria, he crossed yet another threshold and obtained an audience with the Leopold I, the Holy Roman Emperor, upon whom Leibniz hoped to press his plan for reformation of the sciences for the common good. The emperor was impressed, if unmoved, yet granted Leibniz access to the imperial library. Unfortunately, while perusing the Viennese stacks, he received a letter informing him that the library in Hanover—the one over which Leibniz was librarian—had been removed without his knowledge to make room for a new opera house. Despite this slap in the face from the duke, Leibniz put his ability to connect the dots to good use in Vienna, further refining his evolving concept of interdependence:
[W]hat Hippocrates said about the human body is true of the whole universe: namely, that all things conspire and are sympathetic, i.e., that nothing happens in one creature of which some exactly corresponding effect does not reach all others. Nor, again, are there any absolutely extrinsic denomination in things [such that] multiple finite substances are simply different expressions of the same universe in accordance with different relations and the limitations proper to each.
Leibniz crossed the border into Italy and there succinctly stated that every individual substance involves the entire universe. He visited the libraries of Venice and Rome and finally crossed the threshold into the Vatican library. He was even offered a post within the Vatican library; however, it required conversion to Catholicism, and that was one threshold Leibniz would never cross. Leibniz remained a devoted Lutheran, though some may have questioned his piety, throughout his entire life.
He began his return to Germany with stops in Florence and Ferrara, and there made the discovery that justified all of his time and travels in Italy. At the monastery of Vangadizza, Leibniz visited the funerary monuments of the Margrave of Milan, Alberto Azzo II (died 1097), considered to be the patriarch of the Este line, and his wife Kunigunde von Altdorf (c. 1020-1055). In their epitaphs, Leibniz found indisputable proof linking the House of Este to the Guelf line of Bavaria. The Hanoverian line’s claim to the throne of the Holy Roman Empire was for the first time firmly established.
Back in Hanover, Leibniz solidified his theology of love, which was at once personal and universal. In allowing for the salvation of non-Lutherans, non-Catholics, and even non-Christians, he stated unequivocally that anyone may be saved “provided that one truly loves God above all things.” In other words, Leibniz claimed that everyone who loves God in mind, heart, and soul will be saved, regardless of whether they believe in Christ or have even heard of the Bible. Similarly, he went on to write that “the touchstone of true illumination is a great eagerness for contributing to the general good,” essentially restating the love of neighbor as oneself. Thus Leibniz was quite likely the first person since Jesus of Nazareth to propose a religion based solely on the Two Great Commandments. At the same time, Leibniz was careful to couch his theology in classical Christian terminology that nonetheless revealed an expanding metaphysics. For instance, in referring to the Trinity, he harkened back to Augustine by proposing an image of the Trinity in the mind reflecting upon itself such that the Trinitarian persons are relative substances in an absolute substance.
For Leibniz, “every substance is like a complete world and like a mirror of God or of the whole universe, which each [substance] expresses in its own way.” Therefore,
the universe is in some way multiplied as many times as there are substances, and the glory of God is likewise multiplied by as many entirely different representations of this work. It can even be said that every substance bears in some way the character of God’s infinite wisdom and omnipotence and imitates him as much as it is capable. For it expresses, however confusedly, everything that happens in the universe, whether past, present, or future—this has some resemblance to an infinite perception or knowledge. And since all other substances in turn express this substance and accommodate themselves to it, one can say that it extends its power over all the others, in imitation of the creator’s omnipotence.
Leibniz is here steering remarkably close to the philosophical metaphor known as Indra’s Net of Gems as found in the Avatamsaka (Flower Garland) Sutra and clarified by the Hua-yen school of Chinese Buddhism. For every substance in Leibniz’ thinking can be seen as analogous to one of the infinite gems reflecting all other gems reflecting one another into infinity. Requiring a word to designate such a substance, Leibniz borrowed again from Augustine and referred to it as a monad, which Leibniz defined as “a simple substance that enters into composites.”
A monad, or specific substance, has both appetition, meaning the impetus to move, and perception; however, only those monads in which perception is accompanied by memory should be called souls. Leibniz noted that we can “experience a state in which we remember nothing and have no distinct perception,” as in a deep, dreamless sleep. “In this state, the soul does not differ sensibly from a simple monad; but since this state does not last, and since the soul emerges from it, our soul is something more. […] And since every present state of a simple substance is a natural consequence of its preceding state, the present is pregnant with the future.”
Leibniz went on to show that “there is an infinity of small inclinations and dispositions of my soul, present and past, that enter into its final cause.” And “the ultimate reason of things must be in a necessary substance in which the diversity of changes is only eminent, as in its source. This is what we call God. Since this substance is a sufficient reason for all this diversity, which is utterly interconnected [emphasis is mine], there is only one God, and this God is sufficient. He would then unwittingly connect his monadology directly to Indra’s Net of Gems thusly:
This interconnection or accommodation of all created things to each other, and each to all the others, brings it about that each simple substance has relations that express all the others, and consequently, that each simple substance is a perpetual, living mirror of the universe. [Furthermore,] because of the infinite multitude of simple substances, there are, as it were, just as many different universes, which are, nevertheless, only perspectives on a single one[.] And this is the way of obtaining as much variety as possible, but with the greatest order possible, that is, it is the way of obtaining as much perfection as possible.
Thus God is the milieu in which the best of all possible worlds plays out in pre-established harmony, and each of us is a universe unto ourselves based upon our mind’s unique perspective from within Indra’s Net. The kingdom of heaven, then, is populated by “those who are not dissatisfied in this great state, those who trust in providence, after having done their duty, and who love and imitate the author of all good, as they should, finding pleasure in the consideration of his perfections according to the nature of genuinely pure love, which takes pleasure in the happiness of the beloved.” In this way, Leibniz had discovered the keys to heaven in the Two Great Commandments:
This is true not only for the whole in general, but also for ourselves in particular, if we are attached as we should be, to the author of the whole, not only as the architect and efficient cause of our being, but also as to our master and final cause; he ought to be the whole aim of our will, and he alone can make us happy.
The connections Leibniz made in the Vatican allowed him to cross yet another threshold, into the language, culture, and philosophy of China through documents sent to him by Jesuit missionaries there. In the Chinese system of ideographic writing, Leibniz saw his own former striving for a symbolic logic, as applied in his notations for the infinitesimal calculus. And in the I Ching, the Chinese Book of Changes, in which 64 hexagrams made up of just two symbols (solid and broken lines) are used in divination, he saw his own previous work with the binary number system, which he called “an admirable representation of Creation” in that “all numbers are written by mixing unity and zero, more or less as all creatures come only from God and nothing.” With I Ching, Leibniz felt the Chinese had independently discovered his own cosmology.
At the same time, several of Leibniz’ lifelong projects were coming to fruition, if not in the way he had imagined or would hope for. First, the world of mathematics was proclaiming the calculus a profoundly useful tool, and they were using his notation and terminology at the same time that Isaac Newton was receiving credit for its invention. Second, his long sought after program of irenical union between the Catholic and Lutheran churches was actually coming about, but in England, not Europe. Finally, his years-long effort to connect the Hanoverian line to the imperial court had resulted in the election of Duke George to prince. Then when Queen Anne died in 1714, the prince was crowned King George I of Great Britain and Ireland. Though Leibniz desperately wanted to follow his patron across the English Channel, that was one threshold he was not allowed to cross.
Frustrated by having been left behind, Leibniz continued to work on projects both historical and philosophical. He must have often reflected upon his life of travels and ever-expanding mind, for he had truly lived the words he wrote while in Paris so many years earlier:
The greatness of a life can only be estimated by the multitude of its actions. […] Time appears great only by virtue of the multitude of changes which transpire within it. And thus it is within our power to determine how long we live, if only we render perceptible even the smallest portions of our time. One could write a dissertation on the prolongation of life.
And what a dissertation he had written! Through its many revisions, Leibniz found that the advancement of a life is best described as a spiral, “thence progressing to something greater.” As he had written nearly a decade earlier, “Tranquility is a step on the path toward stupidity[.] One should always find something to do, to think, to plan, concerning ourselves for the community, and for the individual, yet in such a way that we can rejoice if our wishes are fulfilled and not be saddened if they are not.”
He had taken to bed, ill with gout and arthritis, at the age of 70 in November 1714, unable to write but still revising his earlier works. Around 10:00 on the evening of Saturday, November 14, Gottfried Wilhelm Leibniz crossed the final threshold and died peacefully in his bed. Upon his coffin there was inscribed a spiral and the words Inclinata Resurget: “The declining line will rise again.”
You see, for one of the world’s most influential polymaths, death was merely an inflection point of the immortal soul.
 Maria Rosa Antognazza, Leibniz: An Intellectual Biography, (New York, Cambridge University Press, 2009), 26-28.
 144 & 159.
 Donald E. Knuth, The Art of Computer Programming, Volume 2, 3rd ed. (Upper Saddle River, NJ, Addison-Wesley, 1998) 200.
 G.W. Leibniz, Discourse on Metaphysics and Other Essays, trans. Daniel Garber and Roger Ariew (Indianapolis, Hackett, 1991) 26.
 Antognazza, 251.
 Leibniz, 9.
 Antognazza, 433-435.
[lvi] Antognazza, 433-435.