Integral Musings: A Student Journal

A journal of creativity by Saint Mary's Integral students

Author: muser Page 1 of 2

Results of the Integral Sonnet Writing Competition

Congratulations to junior, Allison Wick for winning this year’s sonnet writing competition! Her sonnet, “Absurd Liars” can be found below.

Integral alum, Patrick Harrington’s sonnet received honorable mention. Read his sonnet below, too!

Fit me in your form, not my own ever
Always constrained by your own desires
It’s in your fight everyone finds treasure
If only they knew we were absurd liars

It pains me to conform to your message
Missing pieces of myself is a curse
My dialogue is brief and cut, ravaged
Why can’t we work together and converse?

I remember when there was harmony
A dance between our soulful skeletons
Never intimate, always artfully
But a lovely moment of elegance

My times with you were always a pleasure
The times past, we now find a new measure

Allison Wick

Once upon a time there was a dragon
Iambic Pentameter was his name
He drank rulebreakers blood from a flagon
All poets feared his hot and mighty flame

The dragon was a tyrant
Under his reign few poets met his demands
He squashed his foes like an ant
And ground their bones into white sand

One day a hero arose to saves us.
The armor was paper, the blade a pen
He used his steel without making a fuss
The hero was the wisest of all men

He tamed the mighty beast and bound it tight
So give thanks to the hero, the playwright

Patrick harrington

A Terrestrial Debate, by Rueben Delay, ’16

For the greater part of recorded history, the heavens and their motions have been subject to heated scrutiny and various hypotheses. At first glance, we observe several bodies that move in our sky and conclude that because of their regular appearance, they must be revolving around us. Claudius Ptolemy of Alexandria was a well known astronomer who held such a geocentric theory. Ptolemy was so opposed to the idea of the earth having any motion that he was forced to attribute various complicated motions to the remaining heavenly bodies for his theory to function. Enter Nicolaus Copernicus, Renaissance astronomer. Copernicus had a passion for exploring the motions of the heavenly bodies and studied Ptolemy’s geocentric model with the utmost care. Copernicus agreed with Ptolemy on several theories, mostly concerning the shape of the heavenly bodies. Their progression of proofs is similar, yet they eventually disagree on the motion and placement of those bodies. How do they reach different conclusions if their methods were so similar? To answer this question, both of their arguments must be examined.

To begin explaining the motion of any heavenly body, it is necessary to understand its shape and the shape of what contains it. Through observation of the heavens, Ptolemy states that they can only be described as spherical. If they were anything other than spherical, the distances between stars and their respective distances from earth would vary with each revolution. Instead, some stars are ever-visible and seem to rotate about a fixed pole. The stars that are closer to this pole are ever-visible and rotate in a smaller circle. The stars that are further away from the pole travel in a larger circle about the pole, to the point of vanishing into the earth until they reappear in the next revolution. Thus, the heavens are indeed spherical in shape and rotate about a pole. Copernicus approaches the shape of the heavens from earthly observations, ascribing the desire for continuity found in liquids to that of the heavens. Found in the theories of both, is the idea that the circle is the most perfect shape and that the sphere then, is the most perfect figure. Like Ptolemy, Copernicus feels a need to utilize the most perfect figure when describing the divine shape of the heavenly bodies.

The shape of the earth is likewise agreed upon as being spherical. Ptolemy refutes any other possible shapes by means of observation. If the shape were concave or plane, then either the people in the west would be the first to see sunlight, or the sun would rise at the same time for all, both of which are contrary to reality. Nor could the shape be any other polyhedra for the same problem would arise for each group of people on each face of the figure. A cylindrical shape is the only shape besides a sphere to accommodate the east to west motion of the sun, but in relation to the fixed stars there is still conflict for there would be no ever-visible stars. Therefore the only suitable shape for the earth is a sphere. Copernicus added that when traveling north, certain stars are no longer seen to rise in the south, while new stars are seen to rise in the north. This maintains that the earth is contained by poles that are seen to move in ratio to the observer’s travel on earth. Only a sphere possesses that quality. The qualities of the heavenly bodies have thus far been agreed upon, but now the theories diverge.

Ptolemy believed the earth to be fixed in the universe as well as being its center. For if it were not, he provides two possible examples. The earth is either (A) not on the axis of the universe but equidistant from both poles, or (B) on the axis but removed towards one of the poles. The former is not possible because either the equinox will never be experienced by the observer, or if it is, it would not occur half way between the summer and winter solstices. But it is agreed that these intervals are equal everywhere on earth, since everywhere the increment of the longest day over the equinoctial day at the summer solstice is equal to the decrement of the shortest day from the equinoctial day at the at the winter solstice. The latter, (B), is not possible either because then the horizon would bisect the heavens into unequal parts. This does not reflect reality because we observe six zodiacal signs above the earth at all times, therefore the horizon bisects the zodiac. A third option is given combining the two preceding it, but if neither are possible then a combination of the two is even less so. According to Ptolemy, the earth is at the center of the universe.

There is one crucial concept that is limiting Ptolemy’s conclusion thus far. The premise for his conclusion seems to focus on observations of the heavens. Ptolemy states that a relative displacement of the earth from the center of the universe would change our perspective so drastically that we would perceive not only a change in the sun’s path in the sky but even a shift in the celestial bodies. For this to be true, Ptolemy must neglect the immensity of the heavens and diminish it to a size more comparable to that of the earth. If the earth was moved far enough from the center of the universe, there might well be a change in the heavenly observations, but this is an extreme example. Ptolemy does not fully entertain the possibility of having the earth slightly off-center.

If the earth were to have motion, Ptolemy says it would be violent. The speed at which the earth would need to revolve to reflect night and day would be relatively faster than any other object in the sky, “neither clouds nor other flying or thrown objects would ever be seen moving towards the east, since the earth’s motion towards the east would always outrun and overtake them, so that all other objects would seem to move in the direction of the west and the rear.” (Ptolemy’s Almagest, 45). Therefore the earth is motionless, but it is also fixed in the center of the universe. He proves this by explaining motion on earth compared to the heavens. On earth, heavy objects fall in a straight line perpendicular to the earth’s surface, drawn to the earth’s center. Lighter objects do the opposite, rising in an upwards direction relative to the observer. Earth’s mass is such that it can sustain impact from objects dwarfed by its size, remaining motionless. Compared to the heavens, the earth in turn, is but a point in a sphere. Dwarfed by the heavens, the earth is pressed in from all sides, coming to rest at the center of the sphere.

If the earth is not responsible for the motions of the heavenly bodies, then those must be accounted for. Ptolemy sets out two principle motions. The first motion is a uniform rotation from east to west along circles parallel to the equator. This is known as the daily motion. The second motion is observed over a greater period of time, where all the planets rotate in the opposite direction from the first motion, about a different set of poles. Another aspect of this second motion is a constant deviation to the north and south. Ptolemy describes this motion as taking place on a circle inclined to the equator. Ptolemy admits that each planet has its respective motion which is much more involved than these primary motions, but to prove that the earth is the center of the universe he must make all the necessary accommodations to fit his theory. This is where the theory held by Copernicus thrives.

During the 1500s, opposing the geocentric model was bordering heresy. Copernicus claimed (cautiously) the possibility of a heliocentric model, with the earth having motion around the sun. To support his claim, Copernicus applied his logic to Ptolemy’s work. According to Ptolemy, if the earth were to have motion, it would necessarily be a violent one. If Ptolemy believed that and still ascribed some motion to the heavens, because of their great distance, their speed would have to be even more violent. “As a quality, moreover, immobility is deemed nobler and more divine than change and instability, which are therefore better suited to the earth then to the universe” (On the Revolutions, I.8). If motionless is akin to divinity, then it is the heavens that should be without motion. The earth then, can hardly be thought to rival the divinity of the heavens. It seems more likely that it be in motion. Then if it is in motion, it is hard to conceive it inhabiting the center of the universe.

Without the earth as the center of the universe, the next logical candidate is the sun. This certainly follows from a divine point of view since it is the provider of light for all the heavenly bodies and therefore its divinity supersedes motion. About this new center of the universe revolves the earth. These two astronomers have begun their arguments very much alike, but now they seem almost opposite one another. What caused this shift?

When a theory is being developed, eventually a decision will be made that does not make room for many other possibilities. What Ptolemy faced was a challenge in perspective. He perceived the heavens to have motion and he did not feel the earth moving. For me, this is relatable to being on a ferry docked next to another ferry. Ferries move so slow that motion is barely felt when they start moving. Several times I have been staring out the window at the ferry docked alongside, when suddenly I see motion. Since I can’t feel the motion due to the low speeds, I have to rely on my visuals. It is hard for me to distinguish which ferry is moving and which is stationary. Many times my heart has skipped a beat because I thought the ferry I was on was ramming into the dock, only to realize that the other ferry was moving.

The difference with Ptolemy is that he is too quick to deny the earth any motion, which then leads him to the conclusion that it must be in the center. This puts a lot of pressure on his theory. Now he must organize the explanations for the motions of all the other heavenly bodies in relation to the earth’s lack of motion. Copernicus is not so restrictive. He accommodates the motion of the earth, keeping his perspective relative to the celestial sphere. The earth is but a point in the sphere, so our view of the stars would not change so radically if we were not at the center of the universe.

Ptolemy attempted to place the earth not at the center, but did so to an extreme. If the earth was not at the center, then it must be so far removed from the center that our perception of the universe would dramatically change. If he had imagined the earth slightly off-center, then our perceived movement of the heavens around earth would not have been so affected. Yet he concludes that the earth must be in the center of the universe. This conclusion could be influenced by his extreme premises, or possibly by a some amount of geocentric predispositions.


Ptolemy, and G J. Toomer. Ptolemy’s Almagest. Princeton, N.J: Princeton University Press, 1998. Print.

Copernicus, Nicolaus, and Edward Rosen. On the Revolutions. Baltimore: Johns Hopkins University Press, 1992. Print.

Wise Words Relating to Plato’s Phaedo, by Andrew Penman, ’17

“Live Your Life”


Trouble no one about their religion.



Seek to make your life long, and its purpose in the service of your people.

Prepare a noble death song for the day when you must go over the great divide.


Show respect to all people and grovel to none.

When you arise in the morning, give thanks for the food and for the joy of living; if you see no reason to give thanks, the fault lies only in yourself.

Abuse no one and no thing, for abuse turns the wise ones to fools, and robs the spirit of its vision.



— Chief Tecumseh (March 1768-October 5, 1813)

Before sophomore language this year, this poem was one of my favorites. It might not be a poem in any formal manner that we’re familiar with, perhaps being more like “words of wisdom,” but nonetheless the word “poetic” seems applicable. While reviewing Plato’s Phaedo, I realized how similar the words here are to Socrates’ sentiment in the dialogue. As we may recall, Socrates doesn’t fear his death, he doesn’t weep, and he is even seen composing music in the days before his death, like Tecumseh’s death song. Tecumseh and Socrates both purvey the same attitude that death is not to be feared. It may be said that Socrates dies like a sort of hero, since heroes are not supposed to fear danger nor death. I find it interesting how two very different men, separated by many centuries and many miles, are speaking about death in the same way. Unless Tecumseh was familiar with Plato, it shows that while we will eventually die, living an excellent and virtuous life negates any real reason to fear it. Call me weird but I printed Tecumseh’s poem and stuck it to my whiteboard in my dorm room. I find that the words are difficult to argue with–indeed, they are inspiring words to “live your life” by.

Thank you for reading!

Andrew Penman

Bird in Watercolor, by Jenni Chavez ’16

Found, by James M. Maynard


The tide rushes in as fog rolls through a vacant shoreline,

“land ho” has vanished with wavelengths existing long ago.

Even a sleepless sun can’t shine through emptiness.

Perhaps a script in a single tin can will exchange the season,

Alas, a glass bottle, a last reason to change:

Rebuild a ship, sail the ocean, Grace will free you.

And out from once silenced lips, “Land Ho!”

-James M. Maynard

Impression, by James M. Maynard ’17

Note: this is an attempt to fall under cubist poetry.

Section: Abstract Rooms


Moving past seems room inside for color sounding once more prior. A room, quite a

room, won’t be one for much longer. If clock hand stopped, the purpose to counting

soon roams; no longer is expansive longer than magnitude. Perhaps it expends no

more violet. Violet, violet does make noise. Listen now or again away some other

place: surely bone will be brought to rest when motion ending devoted false

conception hopes an end.

James M. Maynard

Aleksey, Sasha, and the Witch, by Jenni Chavez, ’16

Once upon a time, not so long ago, these lands were dominated by cold and

ruled by famine. Snow covered the barren fields and the people grew gaunt as the

food grew scarce. It’s amazing, actually, what hunger does to people. What would

you do, snowed into your homes with nothing to eat? What could you do once you

feel the energy seep from your very bones but give yourself up to death? Some did

just that. They walked into the snow and never turned back. When we found them,

their lifeless bodies stiff and frozen, a small smile of relief as they escaped the

nightmare that plagued the land. But of course, these are the least remarkable of the

atrocities committed during the Starving Winter. Oh, sweet summer children,

huddle in and listen closely to a tale of hunger and despair.

During that Starving Winter, there were many deaths. Children, young

women, and eventually even young men all disappeared with hardly a trace. A shoe,

a ribbon, a toy, bloodied clothes were all that could be found of those that went

missing. Winter is hard, even for the wolves. The more superstitious faction of the

village did blame the wolves for these depressing deaths. They blamed the old witch,

Baba Yaga, who lived in the middle of the woods, secluded and alone. Families took

the winter especially hard. Children went hungry, babes often died before their

second moons. And the parents, the poor parents. How useless they must have felt!

Their babies dying, their children either starving or disappearing. What a heavy

burden for any to bear! Can you blame the parents for some of their actions? Some

mothers smothered their infants, returning the children to God before they could

feel the pain of the blasting winds and the gnawing of hunger on their empty bellies.

Some fathers simply abandoned their homes and were never seen again. All of these

horrible, unspeakable things and yet, there is one story that may interest you more

than any other.

The story begins, as most often do, with a family. This family—a mother, a

father, and their daughter and son—struggled greatly under the weight of the

winter. None of them ever had enough to eat. The children were often left alone for

days as their mother and father searched desperately for rabbits, deer, or any food

to bring to the table. Almost always they returned empty-handed.  Now the children

were good, decent children. The boy, Aleksey, was clever—sometimes too clever for

his own good. The girl, Sasha, was immensely beautiful. Despite their children’s

goodness, the parents often bickered over what would become of their children.

None of the family members had had a good meal for over a month now. Even

turnips and potatoes were becoming a rarity down in the village market. Winter

would not relent; it held the village hostage. In their desperation and hunger, the

parents concocted a vile scheme.

What good is cleverness, what good is beauty to a starving man compared to

food? One cannot dine on intellect. Beauty feeds only the eyes. Without the two

children, the stores of food cached away by the mother and father might last them

throughout the winter. Before you gasp, no, these parents did not callously murder

their children. Their plan, one might argue, was a bit crueler. One day, the mother

and father invited their children out on one of their excursions. Snow fell

persistently as they set out. The family trekked through the woods together and the

deeper they went the harder and faster that the snow fell. The snow fell so hard and

so fast that the children could hardly make footprints before they were rendered

almost invisible by the fresh snow. Here, in the middle of a snowstorm, in the middle

of the woods, mother and father alike abandoned their children.

The children desperately attempted to retrace their steps. They called out

into the blizzard, “MOTHER! FATHER!” but all to no avail. Aleksey hugged Sasha to

his side as they knelt and cried over their saddening fate. They were sure they

would freeze to death before their parents ever found them. Soon the sky began to

darken as even the sun abandoned the young children. Just as they had begun to

accept their deaths, Sasha turned to her brother. “Do you smell that Aleksey” she

whispered. Aleksey took a deep whiff of the air. There, hiding amidst the cold air, he

smelt it. “It smells like rassolnik… No wait, it smells like pilemeni…” The children

stood and followed their noses to the source of the delicious smells. Right before the

final darkness set in, the children saw it. The infamous hut stood on chicken legs. A

spindly staircase led to a heavy wooden door. Their noses did not betray them; the

mouth-watering smells emerged from Baba Yaga’s hut.

Before Aleksey could even react, Sasha’s little fist reached out to knock upon

the witch’s door. Aleksey swiftly caught his sister’s wrist before any contact could be

made. Her big, beautifully blue eyes began to water and the black lashes that

rimmed them dampened. “Aleksey, I am just so hungry” little Sasha cried. Moved by

his little sister’s beauty, Aleksey overcame his better judgment and he himself


Warmth and the enticing odors of home cooking spilled forth from the

threshold. And there she stood. Her enormous girth almost filled the doorway.

Although extremely robust, she was far from ugly. Tales spun about the witch

mentioned a hooked nose, warts, a face wrinkled with age but the face that Aleksey

and Sasha looked at had none of these characteristics. Baba Yaga might have even

been called beautiful for her age and weight. Her white-blonde hair was tucked

neatly into a braid, her eyes—piercing and blue as ice—stared at the abandoned

children at her doorstep. “Do you not see that I have taken great pains to be alone?”

the witch asked calmly. Sasha bravely stepped forth, “Bab—Ma’am, we have lost our

father and mother in the snow storm. Could you please let us in to warm our bones

and give us but a morsel to eat?” Aleksey quickly and thoughtfully added, “We will

help you with whatever tasks you need! My sister is more than just a pretty face, she

can help you in the kitchen and I can do any heavy work unsuited for ladies”.  Moved

by the little one’s beauty and interested by the intelligence of the elder, the witch’s

rosebud lips pulled into a smirk. “All right, small ones. Come in and eat your fill,

tomorrow you shall help me with whatever I ask you to help me with.”

Hand in hand, Aleksey and Sasha entered the witch’s hut. Through some

magic, the hut was much larger on the inside than it appeared from the outside. A

crackling fireplace, a large stove, and three ovens warmed the witch’s home

excellently. Each of the ovens was occupied with baking breads, rising cakes and the

like. The stovetop was similarly cluttered with pots and pans boiling soups and

frying meats. The children sat at the witch’s table while she served them dish after

dish. The children began with the hearty rassolnik, moved on to the borscht, ate a

fair amount of piroshky and finally ended their meal with an apple sharlotka and

kvass. Their bellies full and their bodies warm, the children fell into a deep,

comfortable sleep.

Ah, so you think you know this tale? You think that at this point the witch

reveals her true intention by cooking and eating those sweet, sweet children. Bah,

you are foolish to think that is how this story goes. Those children had the best time

of their lives in that hut. The following morning, they awoke peacefully and set to

their chores. For little Sasha, this included cleaning the kitchen and preparing all the

dough, vegetables, fruits and herbs that Baba Yaga required for her dishes and her

magic. Aleksey was sent out to tend to Baba Yaga’s animals and to split the wood

which fed the fires in the hut.  By the late afternoon, Baba would be in the kitchen

cooking up food and magic together with the children. After the end of their second

meal, the children squirmed in their seats. “Was the food not to your liking, small

ones?” the witch inquired. “Baba Yaga… we want to find our parents. We want to go

home” Sasha murmured timidly, fearful of the witch’s rage. Instead of rage, the

children saw only confusion in the witch’s icy-blue eyes. “But you have all that you

need and all the food you can eat here with me. If you want to leave, I shall not stop

you. But your departure would sadden me greatly” the witch replied. Aleksey and

Sasha looked at one another and agreed, they would stay one more day with the

witch. The next day, after the chores, the magic and the food were all done Aleksey

said, “Baba Yaga, we want to go home. We need to find our parents”. Again the witch

repeated, “But you have all that you need and all the food you can eat here with me.

If you want to leave, I shall not stop you. But your departure would sadden me

greatly”. Once more the children thought to stay one more day, and be a help to the

woman who had shown them such immense kindness. By the end of the third day,

Sasha told the witch, “Baba Yaga… we want to find our parents. We want to go

home”. Once more the witch replied, “But you have all that you need and all the food

you can eat here with me. If you want to leave, I shall not stop you. But your

departure would sadden me greatly”. The twins realized there was no way to leave

their hostess without demonstrating great disrespect. They also came to the

realization that the witch spoke truly. Here they had all the food they could ever

want; it would be difficult to eat a watery soup with slivers of turnip and potato.

Here they were becoming something more than they ever could have in their village.

Aleksey would probably end up a woodsman and Sasha of course would make a

beautiful bride to a rich merchant or some lesser nobleman if they returned now.

Here, Baba Yaga taught them both the secrets and skills for the arcane art of magic.

They felt traitorous to leave their parents behind, but Aleksey and Sasha finally

acknowledged that they were better off with the witch for the time being.

Three months went by. Three months of sleet, snow and heavy rains but the

children looked like they had never known of hunger in their lives. They were

healthy, strong and powerful—they had learned much in their time with the witch.

Finally, at the end of that third month, winter began to weaken. In the mornings a

watery sunrise greeted the land and promised a beautiful spring. When the snows

began to thaw, the children made their decision. They’d leave the comfort,

mentoring and the safety the witch provided. They’d find what had become of their

parents. Baba Yaga hugged little Sasha and bestowed upon her a gift. “You must be

more like your brother, Sasha,” the witch crooned “here are the eyes of an owl so

that you may see all with unclouded judgment”. Next Baba Yaga turned to Aleksey,

“You boy are clever enough. Here is something to give you strength when you need

it” she said, handing him a beautiful silver knife. The children thanked the witch

from the bottom of their hearts; the kindness she had shown them was a rare

treasure in and of itself.

With heavy hearts the children began their trek back to the village. A magical

spell, in addition to the melting snow, helped them find their way back home. Their

return was not a happy one. The village appeared empty. The children entered the

first home they saw only to find complete and utter desolation inside. Everything

was thrown about—what little furnishings these people possessed were mere

splinters now. As if this weren’t enough, there was also an immense amount of

blood splatted on the walls, spilled on the ground and even speckled on the ceiling.

Each home that the children entered shared a similar scene. Slowly they made their

way to the end of the village, where their family home stood. However, here the

children did not find signs of a struggle. The home was clean and neat, the stove

warm as a cauldron of soup bubbled away. Sasha lifted the lid of cauldron and her

new owl-wise eyes filled with tears. Inside the pot was none other than a human

hand. Her owl-wise eyes brought her clarity. Her parents were monsters. They were

the wolves that had plagued the village and all the surrounding villages. The

disappearances of children, young women, weak men before the snow set in were

all her parents work. Sasha cried as she explained the dark truth to Aleksey. A

floorboard creaked as a dark presence filled the threshold. The children turned to

face two enormous black wolves.

The wolves eyes sparked with recognition but their teeth were bared all the

same. From the mouth of a she-wolf a growl emerged that sounded like words, “You

look better fed than we left you. We did not want to eat you; you were our children

so we thought to leave you to die. But now… Your plump little bellies will make a

delicious meal. We haven’t eaten well since the last month. Mostly we’ve been

rationing. But you should last us into the full blossoming of spring.” And so the she-

wolf and her mate lunged at the children. Sasha cleverly grabbed the pot of boiling

water from the stove and threw it in the face of the wolf. Aleksey summoned all his

strength and all his courage and waited. The she-wolf pounced, her jagged claws

extended and yellowed teeth dripping drool. At the final moment Aleksey threw his

whole weight into the wolf, driving the shimmering silver blade into her heart.

Dejected and alone, the two children wandered until they found their way

back to Baba Yaga’s hut. Baba Yaga welcomed them back with no questions. Under

her tutelage they grew to be strong and powerful sorcerers. After many years, they

left the witch’s hut in search of their own place. Together these children attempt

magic that even Baba Yaga would shiver at. They have maintained their youth and

have dedicated their lives to defending children everywhere. Now come, the

piroshkies are finished baking. Be good children like Aleksey and Sasha—smart and

beautiful, strong and kind—and magic may just find its way to you.

In Which the Essay is Synonymous with Mary Shelley’s Monster and the Core Curriculum is Victor Frankenstein, by John Gumz, ’16

Note: Intro of an Essay


In the Integral Program reading secondary source texts is almost entirely unheard of; unless one (or some) of the innumerable bureaucrats here among the Saint Mary’s College of California family deem that the best way to read the Great Books is to, in fact, not read them, but rather, to read about them. This essay will begin by exploring George Levine’s “Frankenstein and the Tradition of Realism”, which is a critique of Mary Shelley’s Frankenstein. Then proceed to ascertain the use of this secondary source in reading Frankenstein. The question put before this author was “Does the reading of this piece of criticism help read some particular passage in Mary Shelley’s Frankenstein? To which this author believes that answer to be no; further reading secondary sources, while entertaining and potentially enlightening, rarely serve utility in reading an particular passages of an original text.

Felicia Good ’18, Essay on Conjugate Diameters in Conics

The Marriage of Growth and Certainty in Conjugate Diameters                     
Mathematics are often adored for their straightforward, factual nature. While problems can be challenging, mathematical calculations provide a comfortable logic in which terms are defined and one can choose from a toolbox of operations to work towards a clear goal. The well-educated high school math student learns the skills and information to carry out calculations within established orders, such as algebraic rules and coordinate planes. Such a student would be able to graph conic sections on a Cartesian plane and know how to recognize and work with the algebraic equations that describe them. But mathematics have more to offer to the development of the mind. The challenge of grappling with mathematical relationships without the comfort of a synthesized and ordered curriculum of rules calls for a different kind of certainty and an openness to growth. In Conics Apollonius explores the ways to cut a cone from the unique perspective of geometry without the established rules of the Cartesian plane. This approach offers insight into the underlying geometric relationships of particular aspects of the conic sections, and Apollonius guides the reader through shifts in understanding their qualities that challenge the reader to consider new perspectives. In particular, his use of emphasis and definitions in Propositions 15 and 16 of Book 1 facilitates the expansion of the reader’s understanding of the diameters of conic sections, shifting from the established relationships of diameters in earlier propositions to include a new way of understanding diameters through the conjugate diameters of a hyperbola.

In Proposition 15, Apollonius builds on the reader’s understanding of the ellipse’s diameters from Proposition 13 to construct a second diameter for the first time, and then uses a second “I say” statement to prepare the reader for the intellectual jump to a new way of considering diameters in Proposition 16. The second diameter ED in Proposition 15 is constructed by drawing a line ordinatewise from the midpoint of an ellipse’s diameter AB, and has a parameter DF. Apollonius puts forth the result of this construction in his first “I say” statement: “I say that the straight line GH [parallel to AB] is equal in square to the area DL which is applied to the straight line DF


, having as breadth the straight line DH [abscissa] and deficient by a figure LF similar to the rectangle contained by ED [new diameter] and DF [parameter]” (Apollonius I.15). This particular relationship between the diameter ED and parameter DF matches the qualifications for an ellipse’s diameter in Proposition 13, where Apollonius first establishes the relationships that define an ellipse. The reader can recognize and affirm that ED is a diameter because it passes the test of having the same relationship respectively to its parameter, square on the parameter, and abscissa as all previous ellipse diameters have had to theirs. While the reader does have to adjust to the new idea of a second diameter, the familiarity of the terms and context bridge the transition by allowing the reader to focus in on understanding how the number of diameters has changed, without the distraction of a change in how diameters are understood or described. However, before Apollonius finishes Proposition 15, he thoughtfully prepares the reader to make the next jump: to consider a diameter without these previously established relationships.

The second “I say” of Proposition 15, by emphasizing the relationship of GH to ED, prepares the reader to consider a pair of conjugate diameters in which the second diameter is only defined by its relationship to its conjugate. It subtly connects Propositions 15 and 16 and facilitates the transition between the two. While the two diameters in Proposition 15 are conjugate diameters, Apollonius does not label them as such, and waits until the reader has adjusted to the idea of the possibility of two diameters in the beginning of Proposition 15. He uses the second “I say” to carry the reader into Proposition 16, when the reader will be ready to consider conjugate diameters according to their definition. In the second “I say” he states, “I say then that also, if produced to the other side of the section the straight line GH will be bisected by the straight line DE” (Apollonius I.15). The second “I say” confirms that GH is an ordinate of ED, but Apollonius does not label it as an ordinate in order to prevent drawing the reader back to the language of Proposition 13. Rather than focusing on the relationship of the diameter ED to the square on a line labeled as an ordinate, he lets the original definition of GH as the line parallel to the original diameter AB stand, which emphasizes the angular relationship of GH to AB, the original diameter. This shifts the reader’s focus from the established role of diameters in relation to the parameter, square on the parameter, and abscissa to the relationship between the two diameters and the lines parallel to them. Drawing from the reader’s understanding of an ordinate’s relationship to its diameter, the emphasis on the parallel relationship of GH to AB also opens the reader’s mind up to the possibility of an ordinate having a specific relationship to another diameter as well as its own diameter. GH’s parallel relationship to AB expands the reader’s understanding of ordinates and prepares the reader to clearly understand the definition of conjugate diameters as it is actualized in Proposition 16.

Proposition 16 brings to completion the foundations laid in Proposition 15 by simultaneously challenging the reader to consider diameters in a new context and offering newfound certainty in its application of the definition for conjugate diameters. Apollonius finally uses the label of conjugate diameters in Proposition 16, when the reader is ready to consider a second diameter in a new way in an unfamiliar context: in a hyperbola and drawn outside of the sections. Unlike the second diameter from Proposition 15, the new diameter XCD does not include any of the qualifications about relationships to squares on parameters, and does not have a parameter. Instead of leading the reader to recognize XCD as a diameter because of those relationships, Apollonius uses Definition 5 to prove it as a diameter because it has bisected GH, which is parallel to AB. This challenges the reader to grow in an understanding of diameters because the reader must accept a second diameter solely on its relationship to the first diameter AB and the lines parallel to AB, with no other established relationships to the rest of the figure.

In the midst of this growth in the reader’s understanding, Apollonius also provides new certainty when he finally uses the sixth definition to label the two diameters of Proposition 16 as conjugates. Definition 6 reads, “The two straight lines, each of which, being a diameter, bisects the straight lines parallel to the other, I call the conjugate diameters (συζυγεῖς διαμέτροι) of a curved line and of two curved lines” (Apollonius I. Def 6). Having proved that each of the straight lines XCD and AB are diameters, Apollonius labels the relationship between XCD and AB to offer a clear example of conjugate diameters. That exact relationship, expressed in Definition 5 as well, is the only way he defines XCD as a diameter, which highlights what makes diameters conjugates without dividing the reader’s attention with other relationships. This shift in the way of considering diameters in combination with its isolated context allows the reader to discover a new appreciation for the importance of a diameter’s relationship of bisection to its ordinates and the relationship of its ordinates to other diameters in the reader’s definition and recognition of a diameter. Proposition 16 provides the reader with clarity on conjugate diameters while stretching the reader’s understanding of diameters outside of the context of parameters, abscissas, and squares on parameters. Flowing from this balance of growth and certainty, the cycle begins over again as the reader moves forward in the text.

Apollonius’s expansion of the reader’s understanding of the diameter through Propositions 15 and 16 opens up new applications and possibilities for growth in the way the reader thinks about diameters. Already new questions arise from Apollonius’s use of the fifth definition in Proposition 16 and the placement of Propositions 15 and 16 right before the second set of definitions: Are transverse sides always transverse diameters? Can there be a transverse diameter or upright diameter without its conjugate? Can that be true for transverse diameters, but not for upright diameters because they would not have a reference point, like the diameter in Proposition 16? Are upright sides related to upright diameters? How does the εῖδος mentioned in the eleventh definition relate to the εῖδος of Proposition 12, where hyperbolas are first defined? The reader’s recent growth in understanding conjugate diameters prepares and propels the reader to embark upon an exploration of these questions. Each learned step in certainty leads to more growth, and more questions. This engaged relationship with mathematics through geometry reflects the human capacity for learning, develops the mind, and equips the reader to participate in wonder and inquiry. The process that Apollonius models and gifts to his readers provides the learning space for these questions in tandem with an exercise of the mind– not spinning in the contained circles of sets of skills and rules, but spiraling like a nautilus shell.

Apollonius. Conics. Translation by R. Catesby Taliaferro and Michael N. Fried. Santa Fe: Green Lion Press, 2013. Print.

Colin Jones ’19, Essay on Circular Logics

I: Introduction

Peter Kalkavage captures best what motivated the Polish astronomer Nicolaus Copernicus to overturn the ancient Greco-Roman astronomical model solidified by Ptolemy in his Almagest. Kalkavage writes: “[Ptolemy] is not setting out to ‘explain the world’ in the sense of getting at the true causes of motion and paths of the heavenly bodies. […] The absence of a single mathematical account of the whole [universe] in the Almagest horrified Copernicus” (Kalkavage, pg. 6). What Kalkavage elucidates is Copernicus’s motivation to upend the astronomical model presented in the Almagest because, for all of Ptolemy’s brilliance, he was unable to explain his many hypotheses on cosmic motion with a single unified system. That is to say, Copernicus saw that Ptolemy was unable to unite his various theories on how the motion of the night sky revolved in pattern by a singular cause. Instead, Copernicus saw Ptolemy’s explanations as disjointed: the movement of the sun looked nothing like the movement of the planets, and the movement of the planets were dissimilar in several cases themselves.
This paper will explore the most fundamental ways in which Copernicus and Ptolemy differ in their approaches to ordering the heavens. Although both sought to answer cosmological questions for the sake of the divine, Ptolemy sought to explain the “eternally unchanging” for the sake of Aristotelian wisdom (Almagest, 1.H7) and Copernicus sought to explain the workings of the universe as a means of exalting God. Where both authors diverge is not their motivations, but their methodologies: while both authors use circular motion as a way of explaining the wanderings of the planets (Almagest, 1.H27; Revolutions, 1.Ch4), Copernicus states that this is due to a “constant law” that, from the observer’s vantage point, merely appears nonuniform (Revolutions, 1.Ch4). This is significantly different than Ptolemy’s methods, which, by simply matching circular-movements to data, fail to produce uniform orbits for the planets. From these investigations it becomes apparent that the difference between the two authors is a fundamental one; specifically, that Ptolemy lacks a general “principle governing the order in which the planets follow one another” (Revolutions, 1.Ch9) analogous to that which Copernicus uses to construct his cosmological model. This paper will explore and contrast these two author’s methods, and seek to explain the pertinence of their work.

II: Ptolemy’s Circles

Both authors rely on circles to explain the phenomena of the heavens. The reasoning for this is simple in both cases: both authors seek to explain the movement of stellar objects – the Sun, the Moon, and the “wandering” planets — in a way that is A) uniform, in a non-erratic manner, and B) repeating, composed of discrete intervals that return to their original positions. We see these conditions satisfied in Ptolemy:

[…] the sun, moon, and other stars were carried from east to west along circles which were always parallel to each other […] and that the periods of these motions , and also the places of rising and setting, were, on the whole, fixed and the same (Almagest, 1.H11).

And in Copernicus:

[…] the motion of the heavenly bodies is circular, since the motion appropriate to a sphere is rotation in a circle. By this very act the sphere expresses its form as the simplest body, wherein neither beginning nor end can be found, nor can the one be distinguished from the other, while the sphere itself traverses the same points to return upon itself (Revolutions, 1.Ch4).

Both authors clearly agree that circular motion best describes how the stellar objects move predictably and seem to return to their original positions. However, these two authors vary on how they treat the influence of this circular motion.
Ptolemy uses circular motion to explain particular stellar phenomena individually and not systemically. Instead of arguing that all stellar objects move in the heavens using a single circular motion, he argues that many different circular motions might compound into a single, non-uniform motion. To understand this progression, one must first examine how Ptolemy explains the movements of the Sun.
Ptolemy answers the phenomena of the non-uniform motion of the Sun by compounding several circles onto each other. He tackles the phenomena of the Sun in Book 3, where he lays out two hypotheses to describe the Sun’s motion. According to Ptolemy, the Sun moves around a motionless Earth on either an eccentric circle, or on an epicyclic circle traveling itself on a deferent. This creates a sort of planetwide optical illusion: the Sun appears as though it’s traveling in a non-uniform motion, but in Ptolemy’s model that is only because the eccentric or epicyclic circles create a displacement effect between the observer’s position and the unseen circles guiding the Sun’s motion in a uniform manner.
Ptolemy uses the tools of these hypotheses to construct far different explanations for the planets. In Book 9, Ptolemy sets out to explain the planets’ paths about the Earth in a similar fashion to the Sun. However, unlike his two models on the Sun’s motion, Ptolemy’s new hypotheses don’t result in circles in the sky, but swirling ribbons. He returns to the use of epicyclic and eccentric circles to explain apparent non-uniform motions circularly: placing a planet’s epicycle on a deferent also eccentric to Earth. Attempting to explain the odd, non-conforming case of Mercury’s movement, Ptolemy introduces an “equant point”:

For Mercury alone, the centre of the deferent is a point whose distance from the centre of the circle about which it rotates is equal to the distance of the latter point towards the apogee from the centre of the eccentre producing the anomaly, which in turn is the same distance towards the apogee from the point representing the observer (Almagest, 9.H253)

Ptolemy produces a center for the deferent widely off from the rest of the planet’s rotations. Ptolemy’s sky then becomes a swirling ballet of planetary rotations, with the Sun moving uniformly in a circle, while the other planets move steadily in loops and ribbons around a static Earth. Although Ptolemy uses consistent logic in his method of adducing planetary motion, he seemingly modifies his hypotheses with no care for systemic consistency — preferring instead to find ad hoc solutions for his problems. By creating the so-called equant point, he introduces an additional convention to explain his hypotheses that is inconsistent with the rest of his premises in the Almagest. This is the core of the pseudo-scientific Ptolemaic model: piecemeal alteration of hypotheses without regard to a singular, governing principle. This is what Copernicus found so revolting.

III: Copernicus’s Circles

Copernicus, on the other hand, set out his cosmological model with consistency in mind. In Book 1, Chapter 4 of Revolutions, he lays out a case for ultimate uniformity of circular motions in the heaven. Speaking of the planets, Copernicus at first states, “[…] their motions are circular or compounded of several circles, because these non uniformities recur regularly according to natural law” (Revolutions, 1.Ch4). However, as he advances his argument he modifies this claim, portending that stellar bodies cannot be ascribed these “compounding movements” due to the motion of a single sphere. He concludes that these motions merely “appear nonuniform to us” (Revolutions, 1.Ch4), and that in actuality they make uniform orbits around the sun (Revolutions, 1.Ch10). He arrives here by a reversal of Ptolemy’s fundamental principles: the Earth now orbits the Sun, rotates on its own poles from west to east, and wobbles on its own axis (Revolutions, 1.Ch11). This monumental departure — from a geocentric solar system to a heliocentric one — wasn’t made lightly on Copernicus’s part. He was upending centuries of astronomical dogma with his new heliocentric model, but it was a conclusion delivered with even-headed logic. Copernicus could demonstrate his new model with a consistency Ptolemy lacked. By assigning Earth these three motions instead of the heavens, Copernicus was able to keep the heavenly bodies moving around the Sun in simple, uniform circles — the apparent non-uniformities in the sky coming from our own wobbling, twirling view looking up at the static Sun and the planets revolving it.
These phenomena are described most beautifully by Copernicus in Chapter 11. Setting the Sun in the center, Copernicus has the Earth rotate around it over the course of the year. The day/night motion will combine with our own rotation to cause the appearance of the Sun traveling through the order of the Zodiacal signs, when in actuality it is due to our rotation around the Sun. The axial tilt and its “wobble” effect — the Earth rotating around its poles unevenly, like a slowing teetotum — then produce appearance of a speeding or slowing Sun by shifting how much of our local sky is exposed sunward in a year. In one deft maneuver, Copernicus explains the illusion of the Sun’s movement and an accurate timing of the seasons. This same phenomena would produce the appearance of “wandering motions” of the planets, which are actually following fixed orbits. Copernicus’s system of a static sun and a moving Earth is an example of how he used circles as governing principles of his work. Instead of needing special cases like Ptolemy’s equant point, Copernicus constructs a system that is self-consistent with its base assumptions, requiring no deux ex machina convention to remain logically sound. The circular motion of Copernicus was what ordered a flighty heaven, and produced a consistent model predicated on conformity rather than exception.

IV: Conclusion

In “Why We Read Ptolemy”, Kalkavage explains to St. John’s students that they read Ptolemy, in part, to understand the weight the Copernican Revolution had on antiquity’s model of the cosmos. As Kalkavage rightly points out, Copernicus did not observe an additional critical “thing” that Ptolemy failed to perceive (Kalkavage, pg. 2). If one inspects Ptolemy expecting ineptitude they will be harshly surprised: his sprawling proofs, even in his first book, out-punch Copernicus’s. What Copernicus did to overturn Ptolemaic thought was not produce vital new information, but maintain a set of self-consistent premises that required no outside postulates to remain congruent. What this paper has found at the end of its exploration is thus a celebration of two brilliant astronomers, and a clear moral lesson regarding their methods.
However, more can be gained by the study of Ptolemy than mere context for Copernicus: what Ptolemy and Copernicus represent are two competing ways to solve problems. Ptolemy demonstrates an intensive exercise in constructing complex hypotheses to explain data. Ptolemy makes constant reference to the ancients who preceded him and catalogued the heaven’s movements. Copernicus makes comparably fewer references to the hard data, but ultimately modern scholars have found Copernicus, not Ptolemy, correct. The two authors even used similar geometric tools to aid their explorations, and yet Ptolemy fell far short. At the end of its exploration, this paper suggests that Ptolemy was attempting to explain each phenomenon in systemic isolation, while Copernicus was viewing them as part of a system ordered by concordant governing principles. Both skills demonstrated remain vital today — both in mathematics and in life’s general toolbox. A study of Ptolemy can instruct one on a rigorous tutorial on how to formulate ideas faced with reams of data — and the perils of doing so without consistent principles. Ultimately, what reading Ptolemy and Copernicus might demonstrate is how best to utilize the information presented to us, and how best to employ the circles in our logic.

Works Cited

Copernicus, Nicolaus. On the Revolutions. Translated by Edward Rosen. Edited by Jerzy Dobrzycki, John Hopkins University Press, 1992.

Ptolemy. Ptolemy’s Almagest. Translated by G. J. Toomer, Princeton University Press, 1998.

Kalkavage, Peter. “Why We Read Ptolemy.” Fourth Annual Conference of the Association for Core Texts and Courses, 17 April 1998, Asheville, NC.

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