“Aleksey, Sasha and the Witch” by Jenni Chavez, 2016

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.

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In Which The Essay Is Synonymous with Mary Shelley’s Monster And The Core Curriculum Is Victor Frankenstein by Jon Gumz, 2016

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.

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Impression by James M. Maynard, 2017

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

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Found by James M. Maynard, 2017


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

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Bird in Watercolor by Jenni Chavez, 2016

Jenni's Bird

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Wise Words Relating to Plato’s Phaedo by Andrew Penman, 2017

“Live Your Life”

So live your life that the fear of death can never enter your heart.

Trouble no one about their religion.

Respect others in their view, and demand that they respect yours.

Love your life, perfect your life, beautify all things in your life.

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.

Always give a word or sign of salute when meeting or passing a friend, even a stranger, when in a lonely place.

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.

When it comes your time to die, be not like those whose hearts are filled with the fear of death, so that when their time comes they weep and pray for a little more time to live their lives over again in a different way.

Sing your death song, and die like a hero going home.

— 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

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A Terrestial Debate by Reuben Delay, 2016

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.

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Defining the Numberless Magnitudes: A Euclid Essay by Joshua Spooner, 2018

Euclid’s strange world of geometry, despite its peculiarities, progresses smoothly through four books from angles, to triangles on to squares, from circles to polygons and even combinations of all of these geometric objects. Book V of Euclid’s Elements presents its reader with a strikingly different and complex realm of magnitudes and ratios which cannot be represented visually nor simply defined. In fact, the place of definitions in Book V is radically different than those in books I through IV, for they no longer describe a geometric object. Instead, the Book V definitions are merely descriptions of characteristics and patterns found in the phenomena of ratios. Not only are these definitions incapable of providing a clear framework for Book V, many cannot be used in proofs until they are themselves proven, changing the nature of Euclid’s definitions.

The most obvious of the changes found in Book V is the object of the definitions, or the lack thereof. Prior to this book, Euclid uses definitions to describe objects he uses for geometric proofs. For example, Definition I. 2, “A line is breadthless length” shows a clear image of what a line is in order for Euclid to represent lines in his propositions. These definitions may also present an image of interactions between these objects, such as Definition III. 3 which explains what is happening when circles touch one another. These definitions, while varying in type and depth, all commonly define something in a way that it can be theoretically drawn, or at least imagined, so that readers of Euclid may have a clear perception of what the object is and how it may relate to other objects and geometry. The definitions of Book V, on the other hand, are the complete opposite. The very first definition, for example, reads “A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.” This definition has three concepts, “magnitude”, “greater”, and “lesser” which cannot be perceived as images. Even Euclid’s ‘drawings’ of magnitudes as straight lines, with unspecified lengths, is only a fabricated representation and does not clearly show a magnitude, only lines as substitutes. This change continues as Euclid defines in Definition V. 3, “A ratio is a sort of relation in respect to size between two magnitudes of the same kind.” Although this definition is similar to III. 3, which defines an interaction between other defined objects, it does not provide a perceivable example of the interaction, for since a magnitude is not an object, one cannot picture relating magnitudes. Euclid evidently means these definitions to be simply understood by logic rather than perception. This is further complicated by Euclid defining methods of using ratios, such as separation of a ratio in Definition V. 15. Even the editor’s note in Heath’s translation of Euclid, which uses letters to represent magnitudes, can only create a logical framework to these definitions, in no way actually representing actual magnitudes being manipulated in these ratios. This launches The Elements into a new phase dominated only by logic and devoid of construction.

Not only does the nature of the definitions change in Book V, but the definitions’ relation with Euclid’s proofs also shifts. In the first four books, the definitions logically are used to augment the process of proving Euclid’s theorems. For example, Proposition I. 1 uses the definition of a circle in order to project its image, then proceeding to use overlapping circles to construct an equilateral triangle through construction. Definitions are likewise used for demonstration proofs in the early books, such as I. 27, which postulates a straight line falling on two others and creating alternate angles. After showing that the two former lines do not meet, he uses the definition of parallel lines to prove that these parallel lines, when interacting with another straight line, will create alternate angles. The complete opposite of these types of methods are found in Book V. For example, Proposition V. 16 is a proof of a definition. It shows that Definition V. 12, “Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent,” can be applied for all pairs of four proportional magnitudes. It builds from earlier proofs which uses the definitions describing concepts of magnitude, ratio, and proportion to demonstrate this. In this case, the definition is a method, not a concept. In the case of V. 16 and others like it, the proof does not use the definition to demonstrate a concept, it uses previously learned concepts to make a definition applicable so the method, in this example taking ratios alternately, can be used in the world of magnitudes.

Not only does Euclid reverse his propositions to use theorems to prove definitions applicable, he even manages to prove definitions through other definitions. This is shown in Propositions V. 17 and V. 18, when Euclid proves that magnitudes proportional componendo, Definition V. 14 “the antecedent together with the consequent as one in relation to the consequent by itself” are also proportional separando, Definition V. 15 “taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself”, and vice versa. Euclid achieves this by proposing magnitudes that fit both the definition of componendo and that of separando, and then proceeding to demonstrate using equimultiples of these magnitudes that componendo can be applied to separando and vice versa. Through this major shift in the method of proofs, Euclid’s definitions in Book V change from being perceived to being comprehended. For if one creates the image of a line, a circle, or any other object found in Books I through IV, the image itself can serve as a wordless definition, sufficiently providing the description of an object on its own. The new definitions such as Ex aequali cannot be understood simply with an image, since none exists. These definitions can only be grasped by comprehending the nature of ratios, proportions, and the relationships between them. The Book V definitions also change from being applied to proofs to being proved applicable. The definitions no longer describe geometric objects, they evolve to describe ways of working with concepts.

In Book V, the scope and use of Euclid’s definitions change radically, and for some, but not all, their overall role is changed. The exceptions in this are Definitions V. 1 through V. 8, those which define magnitudes, ratios, and proportions. Although these describe only concepts instead of perceivable objects, these definitions are still used in the conventional way found in earlier books to prove theorems. However, Definitions V. 9 through V.18, which discuss methods applied to ratios such as inverses and separando, are the result, not the groundwork of the propositions. For these special definitions, a new role is introduced in Book V. These definitions are defining what can be, or what must be, proven in order to understand magnitudes and ratios, the basis of Book V. Without these definitions to show what methods of geometry need to be validated, Book V would be much smaller, and lose its depth of insight into Euclid’s world of magnitudes.

Works Cited

Euclid. Euclid’s Elements. Translation by Thomas L. Heath. Santa Fe: Green Lion Press, 2007.


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Playing Euclid: Alternative Proof by Julia Stanislav, 2018

Freshman Julia Stanislav has taken up the Euclidean challenge and given her own account of an alternate proof to Euclid Book V Propostion 22. Here is what Julia had to offer:

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.

The alternate proof avoids taking equimultiples; instead it takes the original
given ratio of magnitudes,
A:B::D:E and B:C::E:F
and by Proposition V.16, takes the alternates of these ratios:
A:D::B:E and B:E::C:F.
Next, by Proposition V.11, which states that “ratios which are the same with the same ratio are
also the same with one another”(V.11), the alternate proof shows that
Next, the alternate proof proves the enunciation by Proposition V.16, which again allows the
alternate to be taken, so
an ex aequali relationship as set out in Definition V.17.
If you wish to read more about Julia’s process and how she thought through the Euclid check out her essay here!
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The Point of a Proof: to Prove the Enunciation or go Further? (2018)

     Euclid has shown through Books 1-4 that he has a specific method of organizing his work; each definition seems to be carefully placed in relation to each proposition, each Book put together thoughtfully. Euclid seems to abandon his previous mode of organization and coordination between books, however, when he begins Book 5. Suddenly, none of the propositions from previous Books are used to prove the propositions in Book 5, and Euclid begins working with magnitudes rather than geometric shapes. While Book 5 does seem to stand alone from Euclid’s first four Books, it maintains a specific organization within itself. I have discovered an alternate proof for proposition 22, and while it proves the same enunciation as Euclid’s version, it does so differently, changing the the way in which the proposition contributes to Book 5 overall. A Euclidean proof must be understandable while also contributing to the Book as a whole; while my alternate proof makes more logical sense to me, Euclid’s proof of Proposition V.22 does a better job of contributing to Book 5 as a whole. The question becomes, what is the point of a proof? to contribute to the book as a whole or to make logical sense?

Proposition 22’s enunciation, that is, Euclid’s original and my alternate, states that “If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali”(V.22). Just as the original and alternate propositions share the same enunciation, they also share a given,“any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F”(V.22), and what needs to be proved, “that they will also be in the same ratio ex aequali”(V.22). Euclid’s proof goes on to take equimultiples of the given magnitudes, G of A, H of D, K of B, L of E, M of C, and N of F, and prove, based on Proposition V.4, that since the original first magnitude has the same ratio to the second as the third has to the fourth the equimultiples of the original first and third magnitudes will have the same ratio to any equimultiples of the original second and fourth magnitudes. In other words, with regard to Proposition 22, Proposition 4 proves that G:K::H:L and K:M::L:N. Euclid then goes on to prove by Proposition V.20 that since “there [are] three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than sixth; if equal, equal; and if less, less”(V.20), or, that since there are six multiples of the original six magnitudes and they are in the same ratio when taken two and two together, by ex aequali, if G is greater than, equal to, or less than M, the same relationship holds true for M and H. Finally, Euclid proves his enunciation. By Definition V.5, which states that magnitudes are in the same ratio if equimultiples taken of the original first and third and the original second and fourth are greater, equal, or less than their equimultiples respectively, Euclid proves that the original magnitudes are in ratio ex aequali.

The alternate proof, however, avoids taking equimultiples; instead it takes the original given ratio of magnitudes,

A:B::D:E and B:C::E:F
and by Proposition V.16, takes the alternates of these ratios:

A:D::B:E and B:E::C:F.
Next, by Proposition V.11, which states that “ratios which are the same with the same ratio are also the same with one another”(V.11), the alternate proof shows that

Next, the alternate proof proves the enunciation by Proposition V.16, which again allows the alternate to be taken, so
an ex aequali relationship as set out in Definition V.17.

While both Euclid’s Proposition 22 and the alternate of Proposition 22 prove the same enunciation, they do so differently. Euclid’s version uses Propositions V.4 and V.20 in conjunction with Definition V.5 while the alternate version of Proposition 22 uses Propositions V.11 and V.16 in conjunction with Definition V.17. The obvious loss in the alternate proof is the use of multiples, though the extensions of Propositions V.4 and V.20 are also lost in the alternate version. Euclid’s Proposition 22 extends Proposition V.4, which states that “if a first magnitude have to a second the same ratio as a third to a fourth, and equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order” by showing that the same relationship that exists in Proposition V.4 between four given magnitudes and their four equimultiples also exists ex aequali between four of six given magnitudes and four of their six equimultiples. While Euclid’s Proposition 22 seems like a more universal Proposition V.20, the alternate proof eliminates the elaboration on Proposition V.20. Proposition V.20 states that “if there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; if less, less” while Proposition 22 states nearly the same thing, but, instead of specifically three magnitudes and three other magnitudes equal to them in multitude, Proposition 22 allows for “any number of magnitudes whatever, and others equal to them in multitude” to be in the same ratio ex aequali if they are in the same ratio when taken two and two together.

     Another difference between the enunciations of Propositions V.20 and V.22 is that the former provides the ex aeqauli relationship as a given which is what has to be proved in Proposition 22. This brings up the question, then, of why Euclid decided to use three magnitudes in his proof of Proposition 22 if Proposition V.20 also uses three magnitudes. Since Proposition 22 extends Proposition V.20 to hold true for any number of magnitudes it would seem that it might be more useful for Euclid to use a different number of magnitudes than are used in the Proposition it is built off of, but Euclid has a point. By using the same number of magnitudes in each proposition but by taking vertical multiples, that is, multiples of A and D, B and E, and C and F, Euclid shows a relationship between the original magnitudes that may not have been evident from the outset of the proposition. By relating the original magnitudes(A,B,C) to the others equal to them in multitude(D,E,F) through multiples which he can manipulate, Euclid is illustrating Proposition V.16, that alternate relationships exist, without explicitly stating so. Therefore, though the reasoning may seem convoluted, multiples do indeed have a purpose. Not only do Euclid’s vertical multiples create a visualization of alternate ratios, without taking multiples in Proposition 22, Definition 5 would have no place in the Proposition and it would contribute less to the Book.

     The fact that both versions of the Proposition prove the enunciation does not render them equally valuable. While the alternate proof may be logically sound and easier to understand, it is ultimately not as useful since it does not contribute as much to Book 5. Not only does Euclid’s version of Proposition 22 use the cited Propositions V.4 and V.20, it also uses Proposition V.16 without stating it while the alternate proof of Proposition 22 only uses Propositions V.11 and V. 16. While both versions of the Proposition share in their use Definition 5, ultimately Euclid’s version connects back to the most information from Book 5, therefore contributing more to the book as a whole. Euclid’s organization is purposeful and intricate, as can be seen through the comparison of these two different methods of proving Proposition 22. This comparison convincingly shows also that the point of a proof is not simply to prove the enunciation, but to go further and contribute to the overall Book, which Euclid’s Proposition 22 does while the alternate proof falls short.

Euclid. Elements. Translated by Thomas L. Heath. Santa Fey: Green Lion Press: 2002. Print

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