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The Heresy of Friar Travolo

J.S Bangs lives somewhere in the American Midwest, where he lives with his family of four. His stories have appeared in Beneath Ceaseless Skies, Orson Scott Card's Intergalactic Medicine Show, Heroic Fantasy Quarterly, and other venues. You can see more at his site jsbangs.com and follow him on Twitter at @jsbpax.
Dearest Elizabeth, forgive me. The light is dim, and my hand trembles. The enemies of God have me under guard, but there is a maid who pities me. She brought me these instruments, and I pray she will take my letter to you when I have finished.
You have heard many monstrous things about me. None of them are true. I will tell you the truth, from the beginning, though the story will be long. Please, believe me.
I came to Rhoan at the Bishop's request, as innocently as a dove. We met for the first time in the dining room of the episcopal palace, where over a plate of veal and pears he claimed to have a heretic of the fifth dogma imprisoned.
"Ah, the fifth dogma," I said. "Always the favorite of madmen."
"And Travolo, the friar imprisoned, is madder than most." The Bishop took a tract from the fold of his velvet stole and presented it to me. I drew closer to the candlestick and began to read: An Explication of the Doctrines Arising From the Negation of the Fifth Dogma....
"The negation!" I said aloud.
The Bishop brushed aside my astonishment with a flick of his hand. "I assume that it will be a simple matter for you to expose the heresy."
The Bishop, please understand, has no deep grasp of dogmatics or exegesis. He is proficient in applied geometry and his political skill is formidable, which is why he holds his post to such general acclaim. But he was not equipped to confute the heretic himself. I attempted to explain. "The fifth dogma is the keystone of the geometric dogmas. Many have tried to do without it, but to deny it is a form of folly which has not gripped even the greatest madmen. Even Saint Serafano, in his Ab Initio--"
"I have no interest in the history of the dogma, Inquisitor," the Bishop said. "I simply want to know whether you can demonstrate that Travolo is a heretic."
"If the rest of the tract continues in the manner of its title, I expect it will be trivial."
"Good," the Bishop said. He fixed me with a stern glare. "Because without such a demonstration we have no chance of quelling the dissent of the Sussic princes."
At this point I became disinterested. I am a servant of God, not a politician. All I understood from the Bishop was that the Sussic princes were anxious to rid themselves of the Church's strictures, and had begun to search for an excuse to expel the Bishops and geometers. They were prevented by their dependence on the Church's universities to supply them with bursars, architects, cartographers, lawyers, and other masters of applied geometry. Should they simply expel the Church, they would lose the blessing of the guilds, and fall into penury. So when Travolo began promoting his heresy, they seized upon it as a pretext with which to break with the Church while retaining the loyalty of some portion of their clerics.
There were many more details which the Bishop conveyed to me, which I understood not at the time. If I had, perhaps this would have turned out differently.
Travolo was held in the keep belonging to the Comte of Rhoan, one of the dissident princes. This arrangement was a compromise, a way for the Sussics to protect their heretic without openly defying the Church. I showed the Comte and his men-at-arms the Bishop's emblems until they led me to a room in the tower of the keep. The men-at-arms opened the door with trepidation, as if fearing that the mad, mild monk within might devour them alive, then locked it behind me, instructing me to call out to them once I had finished.
And so I met the famed heretic: a small man, stooped at the neck, with mud-colored eyes and a thin, greasy beard. His chamber was floored with planks of wood and furnished only with a straw mattress and a burlap blanket. Light trickled in through the narrow windows. Without my lamp it would have been very dark, yet I could see that Travolo had fashioned a crude pencil from wood, and had begun to scratch the outlines of an exegesis into the stone walls.
I wore my ecclesial garb and carried the geometer's arc in my hand. He flinched from my shadow and drew up against the far wall. "You are Friar Travolo?" I said.
"Yes, yes," he said in a reedy voice. "Are you the Inquisitor?"
"I am here to examine your heresy."
He came up to me wringing his hands greedily. "Have you read my tract?"
"I have begun to," I said.
"Do you have it with you?"
I showed it to him. He seized the pages with immediate pleasure, as if greeting an old friend. "Would you like to work through it? I have yearned for the chance to speak to a true geometer."
I had been hoping for an immediate recantation, which many of the lesser heretics give the moment they see the shadow of an Inquisitor. Travolo was clearly not of this type. "Have you found no true geometers before now?" I said.
"Fah! These Bishops and princes? They couldn't exegete their nose out of their ass."
"Speak with respect, Brother Travolo! You are the one who is on trial for denying the fifth dogma."
"And I have succeeded in building a whole new geometry atop its negation."
I sighed, as I had heard similar claims from those who were both mad and vain. Nonetheless, I began my attack with reason. "Let us examine your exegesis."
Elizabeth, I pray you remember the fifth dogma as we have received it:
Given a line and a point not on that line, there is exactly one line which can be drawn through the point which does not meet the original line, which we name a parallel line.
Many have found the fifth dogma to be complicated and obscure given the divine clarity of the first four, and so have attempted to proceed without it or exegete it from the first four. Thus have arisen many heresies. But Travolo's heresy was of a different sort yet, as his tract began by asserting a different dogma.
Travolo's dogma. Given a line and a point not on that line, there are at least two other parallel lines which can be drawn through the point which do not meet the original line.
It is hard to express the outrage that I, a trained exegete and Inquisitor, felt upon encountering this. I confronted Travolo with the madness of this dogma, and he merely smiled at me. "Yes," he said, "now let us exegete the consequences."
For the rest of the day we engaged in exegesis, as he drew out doctrines from his insane dogma, his techniques growing gradually in complexity. At first I merely observed his heresies with an Inquisitor's eye, for his doctrines were in contradiction to those received, and were often absurd on the face of them. Yet I noted that his method was sound. He appeared to be working towards some conclusion whose shape was not immediately apparent to me, and I admit that I left his cell that evening in a state of mild curiosity.
In such a mind I supped with the Comte. His estate was spacious and well appointed, and he showed me the boundaries of his grounds before dinner, making sure I noted the livery stables, the barracks of his personal guard, and the county armory, which was contained within the estate. I was not at first especially impressed by this, but that evening, as we dined on grouse and raspberry sauce, the purpose of these demonstrations became clear.
"The nobles of this region are quite interested in the outcome of our investigations, Inquisitor," said the Comte at a lull in the conversation. He placed his wine glass on the table with deliberate care and folded his hands over his plate.
"I have heard," I said. "It's unusual for the civil authorities to take such interest in a simple monk. I understand he has incited no unrest or revolt."
"No, no," the Comte said. "Though some of us have an interest in dogmatics. Is it so unusual for a prince to study the doctrines of the church?"
"Not at all, if he does so under the guidance of a proper teacher. Untaught, few would ever cross the pons asinorum."
"Few of the asses. Yet Saint Evoclius formulated his doctrine untaught...."
"He is a saint."
"So anyone who discovers a novel doctrine is a saint? Perhaps I should bring Travolo down from the tower, so I should have the privilege of dining with a saint."
"Perhaps you should wait until I've given my verdict. Lest you fall into heresy."
"Naturally." He gave me a chilly smile, and fixed me with his stare. "Still, it would be a crime if I were to hinder the progress of a saint and give aid to his persecutors."
I very carefully pushed my plate away and looked at the Comte. "It would be worse to protect a heretic and hinder the truth of the Church."
"Yet should a saint appear, should I not protect him with every resource at my disposal?"
"If a saint should appear--"
"And where error appears, should I not raise every sword in my domain to rout it?"
"Undoubtedly."
"Well." He raised his wine glass to me as in a toast. "You have seen the armory of the county. Should the persecutors of truth appear, rest assured that my arms will not lie idle."
A chill passed over me. I paused to take a bite of grouse-flesh, and answered carefully. "The Church herself, of course, cannot be an instrument of error."
"Every flesh is fallible, as the Church herself teaches. Only geometry approaches God...."
"You quote Saint Tomasino to me?"
The Comte smiled at me. "And if the Church herself hinders the truth?"
"You draw near to heresy, dear Comte. Truth is the Church's foundation. She cannot fall into error any more than Evoclius's dogmas could be found false--"
"--and so we return to the matter of Travolo."
"You've read his tract?"
"As I said, I have an interest in dogmatics. But I leave it to you to find whether there is error in it." His eyes on me were sly and cold.
My food sat poorly in my stomach after that. I did not sleep well, and not merely because of the Comte's threats. Travolo's exegesis rumbled through my sleep, and in the morning I returned to the prisoner driven as much by curiosity as by my duties as Inquisitor.
When I appeared at the door of his cell, Travolo raised his head and rubbed his hands together happily. "Brother Inquisitor!" he said, extending his hands towards me as if greeting an old friend. "You have returned."
"Do not be eager," I said, disguising my own eagerness. "I have not completed my examination of your exegesis. We have much to do."
"We do!" Travolo cackled. He picked up the charcoaled stick with which we had worked yesterday. "Shall we begin?"
"First, eat," I said. I had hidden two buttered loaves of bread in my habit, out of pity for Travolo's miserable state.
He looked at the food with a mixture of desire and revulsion, glancing from the pencil to the bread, as if the passion of geometry might leave him if he stopped to dine. At last he took the loaves and stuffed them greedily into his mouth, letting crumbs and drops of butter fall to the ground, all the while turning his eyes to the dimly-lit exegesis drawn on the far wall. I pitied him; yet his devotion to geometry had a touch of the divine about it, and filled me with an uncanny fear.
When he had eaten, we resumed where we had left off, with an obscure exegesis of Sacciori, who had proven that the sum of the interior angles of a triangle cannot exceed 180 degrees. This may seem like a poor result to you, dear Elizabeth, since it has long been known that the sum of the interior angles of the triangle must be exactly 180 degrees, but Sacciori's exegesis is remarkable because it does not in any way require the fifth dogma. (Sacciori arrives at this result in his heretical De Quinto Dogmate, but this particular result was not condemned and has since been accepted by the Church.)
Yet with Travolo's dogma, it is trivial to show that there exists a triangle the sum of whose angles is less than 180 degrees. Consider for a few moments the consequences of the monk's dogma that there are multiple lines parallel to a baseline which all pass through the same point. You will see that given any "proper" 180-degree triangle, it is possible to construct another triangle with the same vertices which has more acute angles, the sum of whose angles must therefore be less than 180.
This result was unsettling, given that even an unlettered peasant has learned of the angles of a triangle from his parish dascal. Yet we continued from there to an even more impious conclusion: that given Travolo's dogma, there are no rectangles. The full proof I will not attempt to give here, but you may quickly see that any rectangle necessarily is composed of two right triangles. Following another result from Sacciori, it is proved that if there exists any right triangle, then all triangles have angles whose sum is equal to 180. Yet Travolo had already proved with his dogma there are triangles whose angles sum to less than 180, which presents us with a contradiction. And so we conclude that in Travolo's dogmatics there are no right triangles, and thus there are no rectangles.
Upon seeing this result, I laughed and looked at Travolo with disdain. "There are no rectangles? Not a one?" I pointed to the bricks in the wall of his cell. "Yet what is this? Are these not rectangles?"
"Perhaps," Travolo said. "Or perhaps you have not measured them with sufficient precision."
"And the great Court of the Geometers in Triema? Is it not a rectangle?"
Travolo spread his hands with indifference. "Such a grand structure is warped, surely, by the sphere of the earth. It is most certainly not a rectangle in the divine sense of Evoclius."
My blood boiled, though my mind was cold. Of course what Travolo said was true. No human structure is a true rectangle or a pure sphere, for we are bound to the imperfections of the material world. Thus only through dogmatic geometry do we approach the divine.
"Even so," Travolo added with a mild wave, "a small shape might approximate a rectangle. But we have no arc which measures the angles between the stars."
What Travolo meant to say, and which I quickly intuited, is that at small scales his madman's geometry approaches the metric of Evoclius. Perhaps, if we had tools to measure distances more vast than those found nearby, we might find that they bend to his crooked meter and not the straight rule of the saints. Perhaps. I began to shake with fury.
"You are damned," I said. "I have heard enough. I will see you tomorrow in the chapel where you will be judged."
I spoke harshly, but my heart burned with doubt.
I informed the Comte that I would not require his accommodation, but would spend the night at the bishopric. I sent a similar note to the Bishop. In truth I retired to a small chapel that lay in the burgh of Rhoan, just beyond the borders of the Comte's estate. In the morning I would pass between the wolf and the lion, and I sought solace.
The dascal was not present, but the doorkeeper recognized my monkish habit and opened the chapel at my request. The interior was simple, decorated with only a few icons, smelling of incense, oil, and charcoal. I lit a candle. Serpents writhed inside me and darkness boiled in my thoughts. I spent the night tormented by doubts, in prayer and fervent meditation.
Elizabeth, you know that I am not a saint. I have no pretensions to divine insight. Yet towards the morning my eyes alit upon the image of St. Evoclius, and I felt a certain kinship with him. For what have we other than the truth that God has vouchsafed us? And we must guard what truth we have, regardless of the power of its assailants, lest we fall into error and sin.
I arose at dawn from my place prostrate on the floor, feeling as though the angels carried me. And I walked out the door of the chapel and into the devil's mouth.
An armed guard waited at the door for me, and scarce could I cry out before they had seized me and struck me with many blows, and I do not know what followed.
I remember much pain, and darkness, and voices. My captors would not speak to me, except to order me to stand and to transfer me from a horse to a cart, and then to bind my eyes. When the cloth was loosed from my eyes, to my surprise I found myself before the Comte of Rhoan, in the chapel contained on his estate.
"What is the meaning of this?" I demanded.
"Brother Inquisitor," the Comte said with an icy smile. "I hope that my men-at-arms did not harm you. I merely wished to ensure that you arrived at this place on time to give your verdict."
"You molest a cleric of the Church--"
"I have molested no one. Now, if perhaps you were waylaid by ruffians after leaving the grounds of the estate and deceiving both myself and the Bishop, that would be a great crime. I would be sure to seek out and punish such evildoers."
"No one is deceived by this farce."
"Farce? I am entirely serious." He motioned to the captain of the guard standing behind him. "Bring in the monk. The Inquisition shall begin at once." Then he fixed me with a stare as bitter as iron. "I am here to get a verdict. A very particular verdict. It would be very unfortunate should I be disappointed."
"You think to buy a man of God with threats," I hissed.
The Comte turned without a word and marched to stand near the icon of St. Evoclius. Friar Travolo entered in chains, surrounded on either side by the Comte's men-at-arms, and was marched to the place of the accused before the ambo.
"You may begin, Inquisitor," the Comte said. "Do not take long."
I considered refusing, but my conviction steeled my heart. I would perform the Inquisition, and the prince's threats would not alter my verdict.
I clasped my hands together and prayed as I have done so many times before, Lucem scientiae, Domine Deus noster, da nobis, et salva nos ab ignorantia. All those gathered, including Travolo, said Amen; and as I continued with the words of the rite my peace returned. So it was with boldness of heart I turned to Travolo.
"Friar Travolo, you are charged with error and heresy. How do you plead?"
He grimaced, showing his teeth, and rubbed his head. "I have committed no error, Father Inquisitor."
I took up the tract which he had authored. "Did you write the words within this tract?"
"I did."
I opened the tract to the last page and read the conclusion: "Therefore the sum of the angles of every triangle are less than half of the degree of a circle. Quod erat demonstrandum." I cast the tract aside. "Rubbish. It is an elementary proof that the sum of the angles of the triangle are equal to half the degree of the circle. Is this not the case?"
Travolo did not appear troubled. He leaned forward and rubbed his hands together, then proclaimed, "It is, but if you look--"
"Quiet. I examine you here. I am quite aware of the reasoning behind your conclusions. In particular, I have considered in depth what you, in an act of singular hubris, have called 'Travolo's Dogma'--namely, a contradiction of the fifth dogma of Evoclius."
The Comte's reaction was not visible on his face, but his hand had formed a fist over the hilt of his sword. The sound of shouting from outside rattled through the windows of the chapel.
"Friar Travolo, have you contradicted the dogma of Evoclius?"
His face was still childish and eager, as if he did not understand the danger of his situation. "I have."
Something clattered against the wooden door of the chapel. The Comte directed his captain to see what the matter was with a silent gesture.
"Shall I wait?" I asked the Comte. "I would not want you to be interrupted in receiving your precious verdict."
"Delay will work in no one's favor," he said. "We will respect the formulas."
"Formula is nothing," I said. "Only divine truth is anything."
"You speak like a true son of the Church."
And in that moment the door of the chapel flew open and a score of episcopal militia charged through the door scattering the county guard. The purple-clad Bishop flew in at their heels boiling with righteous anger. My heart tripped in my chest.
"What is this blasphemy!" the Bishop shouted. The Comte's men-at-arms gathered around him in a defensive circle as the episcopal militia filled the chapel, cutting Travolo off from the noble.
The red-faced Bishop screamed at the Comte through the fence of swords around him. "You kidnapped my Inquisitor! I come to your estate and find armed guards and treachery--"
"Respect," the Comte hissed. "The Inquisition is underway."
"What?" The Bishop saw me then, holding my Inquisitor's staff, with Travolo in chains in front of me. "Has the rite begun?"
"Thus far all has been done according to canon," I said.
The Bishop smote the Comte with a spiteful glare, then faced me. He spoke in an oily voice.
"Well, Inquisitor. If you have begun, you may continue. You are no longer in their power, so you may speak the truth."
My tongue was suddenly dry. "Yes. I speak the truth."
My hands trembled on my geometer's arc. "I have examined Travolo's exegesis with the utmost care. I have spent two days in his company testing his mind and his reasoning. And I offer the following judgment, as an Inquisitor of the Church and a servant of God: there is no flaw in Travolo's exegesis. His proof is sound. His doctrines, though novel, are unassailable."
The chapel was silent. All watched me with silent fixation.
"The only question that remains is whether a geometry founded on dogmas other than those handed down may be considered divine. And here I beg us remember a rule of the Church, that the Holy Spirit gives illumination, but exegesis gives proof. The saints are known not because they claim to have received divine insight, for a madman may make this claim, but because their insights are proven by the principles of exegesis. And Travolo's doctrines demonstrate that his insights truly represent some geometry--but it is not the geometry of Evoclius."
"The geometry of demons--" hissed the Bishop.
"Or another geometry in the mind of God. Or did you imagine"--and here I fixed the Bishop with a stare--"that God in his vastness knows only five dogmas?"
The Bishop stared back at me in silent, venomous shock. Even the Comte's face was riven with surprise.
I went on. "Let us consider this the true insight of Friar Travolo. For Travolo's dogma does not describe a false geometry, but a different one, one whose features are as vast and subtle as the depths which the saints have explored for centuries. And do the saints reject this novelty? They do not. Evoclius is not put to shame by the introduction of a new geometry, but expanded. The depths of God are not plumbed by a single page of dogma, but they go on forever, vast, unknown, and incomprehensible. There is room in Him for both the geometry of Evoclius and the geometry of Travolo, and for a thousand geometries besides. Glory be to God--"
And I turned to Travolo. "As an Inquisitor of the Church and a servant of God, I find you innocent of heresy. Go in peace."
I managed to bless him with my geometer's arc before the militia seized me.
You may imagine what followed. Having come to the chapel by force, the Bishop took me, Travolo, and the Comte under arrest, and what has happened to the other two I do not know. I have been confined to a single chamber within the episcopal palace, though not otherwise mistreated. I can only surmise what has transpired in the broader world.
Dearest Elizabeth, I hope that this letter may comfort you in the days to come. Do not pity me. I understand now why Travolo was so blissful even when he had every expectation of being condemned. Who can be afraid who has glimpsed a fragment of the mind of God? I rejoice that I was found worthy to see with him, if only for a little while. I regret nothing.
With eternal love, your brother--
The End
This story was first published on Friday, July 11th, 2014


This story started with my ongoing lay interest in hyperbolic and other non-Euclidean geometries. When I first learned about non-Euclidean geometry in high school it seemed too weird and counterintuitive to be true, and when I later learned a little about the history of non-Euclidean geometries, including how they were almost (but not quite) discovered during the Middle Ages, it just got weirder. Eventually I was moved to write a story which attempted to give an outline of these ideas in an accessible frame, and a religious disputation between mathematicians seemed like the perfect setup.

For the curious, you can find a much more rigorous proof of the theorem in Travolo's tract, along with many other fundamentals of hyperbolic geometry, here. This introduction should be accessible to anyone comfortable with high-school geometry. Travolo himself does not correspond to any one mathematician, nor does the Inquisitor, but Evoclius is Euclid, and Sacciori is in fact the 17th-century Jesuit mathematician Giovanni Saccheri.

- J.S. Bangs

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