28.4: Fetal Development
By the end of this section, you will be able to:
- Differentiate between the embryonic period and the fetal period
- Briefly describe the process of sexual differentiation
- Describe the fetal circulatory system and explain the role of the shunts
- Trace the development of a fetus from the end of the embryonic period to birth
As you will recall, a developing human is called a fetus from the ninth week of gestation until birth. This 30-week period of development is marked by continued cell growth and differentiation, which fully develop the structures and functions of the immature organ systems formed during the embryonic period. The completion of fetal development results in a newborn who, although still immature in many ways, is capable of survival outside the womb.
Sexual Differentiation
Sexual differentiation does not begin until the fetal period, during weeks 9–12. Embryonic males and females, though genetically distinguishable, are morphologically identical (Figure \(\PageIndex{1}\)). Bipotential gonads, or gonads that can develop into male or female sexual organs, are connected to a central cavity called the cloaca via Müllerian ducts and Wolffian ducts. (The cloaca is an extension of the primitive gut.) Several events lead to sexual differentiation during this period.
During male fetal development, the bipotential gonads become the testes and associated epididymis. The Müllerian ducts degenerate. The Wolffian ducts become the vas deferens, and the cloaca becomes the urethra and rectum.
During female fetal development, the bipotential gonads develop into ovaries. The Wolffian ducts degenerate. The Müllerian ducts become the uterine tubes and uterus, and the cloaca divides and develops into a vagina, a urethra, and a rectum.
The Fetal Circulatory System
During prenatal development, the fetal circulatory system is integrated with the placenta via the umbilical cord so that the fetus receives both oxygen and nutrients from the placenta. However, after childbirth, the umbilical cord is severed, and the newborn’s circulatory system must be reconfigured. When the heart first forms in the embryo, it exists as two parallel tubes derived from mesoderm and lined with endothelium, which then fuse together. As the embryo develops into a fetus, the tube-shaped heart folds and further differentiates into the four chambers present in a mature heart. Unlike a mature cardiovascular system, however, the fetal cardiovascular system also includes circulatory shortcuts, or shunts. A shunt is an anatomical (or sometimes surgical) diversion that allows blood flow to bypass immature organs such as the lungs and liver until childbirth.
The placenta provides the fetus with necessary oxygen and nutrients via the umbilical vein. (Remember that veins carry blood toward the heart. In this case, the blood flowing to the fetal heart is oxygenated because it comes from the placenta. The respiratory system is immature and cannot yet oxygenate blood on its own.) From the umbilical vein, the oxygenated blood flows toward the inferior vena cava, all but bypassing the immature liver, via the ductus venosus shunt (Figure \(\PageIndex{2}\)). The liver receives just a trickle of blood, which is all that it needs in its immature, semifunctional state. Blood flows from the inferior vena cava to the right atrium, mixing with fetal venous blood along the way.
Although the fetal liver is semifunctional, the fetal lungs are nonfunctional. The fetal circulation therefore bypasses the lungs by shifting some of the blood through the foramen ovale , a shunt that directly connects the right and left atria and avoids the pulmonary trunk altogether. Most of the rest of the blood is pumped to the right ventricle, and from there, into the pulmonary trunk, which splits into pulmonary arteries. However, a shunt within the pulmonary artery, the ductus arteriosus , diverts a portion of this blood into the aorta. This ensures that only a small volume of oxygenated blood passes through the immature pulmonary circuit, which has only minor metabolic requirements. Blood vessels of uninflated lungs have high resistance to flow, a condition that encourages blood to flow to the aorta, which presents much lower resistance. The oxygenated blood moves through the foramen ovale into the left atrium, where it mixes with the now deoxygenated blood returning from the pulmonary circuit. This blood then moves into the left ventricle, where it is pumped into the aorta. Some of this blood moves through the coronary arteries into the myocardium, and some moves through the carotid arteries to the brain.
The descending aorta carries partially oxygenated and partially deoxygenated blood into the lower regions of the body. It eventually passes into the umbilical arteries through branches of the internal iliac arteries. The deoxygenated blood collects waste as it circulates through the fetal body and returns to the umbilical cord. Thus, the two umbilical arteries carry blood low in oxygen and high in carbon dioxide and fetal wastes. This blood is filtered through the placenta, where wastes diffuse into the maternal circulation. Oxygen and nutrients from the pregnant person diffuse into the placenta and from there into the fetal blood, and the process repeats.
Other Organ Systems
During weeks 9–12 of fetal development, the brain continues to expand, the body elongates, and ossification continues. Fetal movements are frequent during this period, but are jerky and not well-controlled. The bone marrow begins to take over the process of erythrocyte production—a task that the liver performed during the embryonic period. The liver now secretes bile. The fetus circulates amniotic fluid by swallowing it and producing urine. The eyes are well-developed by this stage, but the eyelids are fused shut. The fingers and toes begin to develop nails. By the end of week 12, the fetus measures approximately 9 cm (3.5 in) from crown to rump.
Weeks 13–16 are marked by sensory organ development. The eyes move closer together; blinking motions begin, although the eyes remain sealed shut.  The ears move upward and lie flatter against the head. The scalp begins to grow hair. The excretory system is also developing: the kidneys are well-formed, and meconium , or fetal feces, begins to accumulate in the intestines. Meconium consists of ingested amniotic fluid, cellular debris, mucus, and bile.
During approximately weeks 16–20, as the fetus grows and limb movements become more powerful, the pregnant person may begin to feel quickening , or fetal movements. However, space restrictions limit these movements and typically force the growing fetus into the “fetal position,” with the arms crossed and the legs bent at the knees. Sebaceous glands coat the skin with a waxy, protective substance called vernix caseosa that protects and moisturizes the skin and may provide lubrication during childbirth. A silky hair called lanugo also covers the skin during weeks 17–20, but it is shed as the fetus continues to grow. Extremely premature infants sometimes exhibit residual lanugo.
Developmental weeks 21–30 are characterized by rapid weight gain, which is important for maintaining a stable body temperature after birth. The bone marrow completely takes over erythrocyte synthesis, and the axons of the spinal cord begin to be myelinated, or coated in the electrically insulating glial cell sheaths that are necessary for efficient nervous system functioning. (The process of myelination is not completed until adolescence.) During this period, the fetus grows eyelashes. The eyelids are no longer fused and can be opened and closed. The lungs begin producing surfactant, a substance that reduces surface tension in the lungs and assists proper lung expansion after birth. Inadequate surfactant production in premature newborns may result in respiratory distress syndrome, and as a result, the newborn may require surfactant replacement therapy, supplemental oxygen, or maintenance in a continuous positive airway pressure (CPAP) chamber during their first days or weeks of life. In male fetuses, the testes descend into the scrotum near the end of this period. The fetus at 30 weeks measures 28 cm (11 in) from crown to rump and exhibits the approximate body proportions of a full-term newborn, but still is much leaner.
Interactive Link
Visit this site for a summary of the stages of pregnancy, as experienced by the pregnant person, and view the stages of development of the fetus throughout gestation. At what point in fetal development can a regular heartbeat be detected?
The fetus continues to lay down subcutaneous fat from week 31 until birth. The added fat fills out the hypodermis, and the skin transitions from red and wrinkled to soft and pink. Lanugo is shed, and the nails grow to the tips of the fingers and toes. Immediately before birth, the average crown-to-rump length is 35.5–40.5 cm (14–16 in), and the fetus weighs approximately 2.5–4 kg (5.5–8.8 lbs). Once born, the newborn is no longer confined to the fetal position, so subsequent measurements are made from head-to-toe instead of from crown-to-rump. At birth, the average length is approximately 51 cm (20 in).
Disorders of the...
Developing Fetus
Throughout the second half of gestation, the fetal intestines accumulate a tarry, greenish black meconium. The newborn’s first stools consist almost entirely of meconium; they later transition to seedy yellow stools or slightly formed tan stools as meconium is cleared and replaced with digested breast milk or formula, respectively. Unlike these later stools, meconium is sterile; it is devoid of bacteria because the fetus is in a sterile environment and has not consumed any breast milk or formula. Typically, an infant does not pass meconium until after birth. However, in 5–20 percent of births, the fetus has a bowel movement in utero, which can cause major complications in the newborn.
The passage of meconium in the uterus signals fetal distress, particularly fetal hypoxia (i.e., oxygen deprivation). This may be caused by maternal drug abuse (especially tobacco or cocaine), maternal hypertension, depletion of amniotic fluid, long labor or difficult birth, or a defect in the placenta that prevents it from delivering adequate oxygen to the fetus. Meconium passage is typically a complication of full-term or post-term newborns because it is rarely passed before 34 weeks of gestation, when the gastrointestinal system has matured and is appropriately controlled by nervous system stimuli. Fetal distress can stimulate the vagus nerve to trigger gastrointestinal peristalsis and relaxation of the anal sphincter. Notably, fetal hypoxic stress also induces a gasping reflex, increasing the likelihood that meconium will be inhaled into the fetal lungs.
Although meconium is a sterile substance, it interferes with the antibiotic properties of the amniotic fluid and makes the newborn and pregnant person more vulnerable to bacterial infections at birth and during the perinatal period. Specifically, inflammation of the fetal membranes, inflammation of the uterine lining, or neonatal sepsis (infection in the newborn) may occur. Meconium also irritates delicate fetal skin and can cause a rash.
The first sign that a fetus has passed meconium usually does not come until childbirth, when the amniotic sac ruptures. Normal amniotic fluid is clear and watery, but amniotic fluid in which meconium has been passed is stained greenish or yellowish.
Aspiration of meconium with the first breath can result in labored breathing, a barrel-shaped chest, or a low Apgar score. An obstetrician can identify meconium aspiration by listening to the lungs with a stethoscope for a coarse rattling sound. Blood gas tests and chest X-rays of the infant can confirm meconium aspiration. Inhaled meconium after birth could obstruct a newborn’s airways leading to alveolar collapse, interfere with surfactant function by stripping it from the lungs, or cause pulmonary inflammation or hypertension. Any of these complications will make the newborn much more vulnerable to pulmonary infection, including pneumonia.


1.4.2: Communication and Culture
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- Media Hack Team
- University of Otago and others via Creative Commons
What is communication, why do we communication, how do we communicate, and to what end, are all questions we ask in the study of communication. At its most basic, communication is the exchange of information and meaning. We are constantly communicating, in a wide range of different contexts, such as with each other (interpersonal communication), with different cultural groups or subgroups (intercultural communication), or to large audiences (mass communication), just to name a few. However, to understand communication, we need to understand the place of communication in culture.
Culture as a term is widely used in academic as well as in daily speech and discourse, referring to different concepts and understandings. While the term originally stems from ancient Greek and Roman cultures (Latin: cultura ) it has various dimensions today built from the different needs and uses of each field, be it anthropology, sociology or communication studies. For communication studies, we might start by defining culture as a set of learned behaviours shared by a group of people through interaction.
Cultures are not fixed, monolithic entities, but are fluid, always changing and responding to pressures and influences, such as the changing experiences of its members, or interaction with other cultures. However, to its members, the artefacts and even the existence of cultural behaviours and schemas may seem invisible or unremarkable. A culture may even have within it certain subcultures which exist within the main cultural framework of a society, but share within it specific peculiarities or modalities that also set it apart from the mainstream. These subcultures may continue to exist for many years or only a short period of time. They may die out, or may become incorporated into the mainstream as part of this ongoing evolution of culture.
While there are specific differences to each culture, generally speaking, cultures share a number of traits, such as a shared language or linguistic marker, definition of proper and improper behaviour, a notion of kinship and social relationship (i.e.: mother, friend, etc), ornamentation and art, and a notion of leadership or decision making process.
Culture and society, though similar, are different things. Cultures are defined by these learned behaviours and schemas. Societies at their simplest can be defined as groups of interacting individuals. However, it is through this interaction that individuals develop and communicate the markers of culture, and so in human societies, it is very difficult to separate out ‘culture’ and ‘society.’
And thus we come back to the role of communication within culture. The idea of culture as something that is shared means that it is vital to understand culture and communication in relation to one another. The relationship between culture and communication, in all its forms, is tightly interwoven and interlinked. We can see that communication enables the spread and reiteration of culture. Both communications and the media propagate the values and schemas of a culture through the repeated interaction and exchange enabled by the communications process.
Notice the emphasis on repeated there: it is not in single instances of communication that culture is made, but rather in the repeated exchange of information and the reinforcement of the ideals and values it embodies, all conveyed within a particular moment. One way we can think about this complex interplay is by looking at du Gay, et al (1997) notion of the circuit of culture.
The circuit of culture is a way of exploring a product of a culture as a complex object that is affected by and has an impact on a number of different aspects of that culture.
Image (c) du Gay et al, 1997
It’s worth briefly going through each of these five variables on the diagram now, and then applying it to a familiar form of communication in culture. This text will return in later sections to deal more deeply with many of the idea’s that this diagram introduces.
Representation – how is the meaning conveyed to the audience, user, or co-communicator? What signs, modes and discourses help convey the meaning – not only the ‘factual’ or informational meaning, but also the social meaning. For example, what does the colour pink represent in your cultural context?
Identity – refers to how meaning is internalized by the receiver or audience. Our identity is shaped by our culture, which creates a range of viable and non-viable identity options that are presented, refined and renegotiated through our communication and exchange of cultural objects. By consuming and displaying certain communicative texts and strategies, we are both claiming certain identity positions, and simultaneously rejecting others.
Production — here refers to the production of meaning. Meaning can be produced and reproduced in a number of ways. An individual may produce meaning about themselves in the way they dress or wear their hat. Apple™ produces meaning about itself in the way they design and build the iPhone™. A terrorist organization may produce meaning about itself by making videos they put on Youtube. This act of meaning production may be unproblematic within mainstream culture, and help maintain the hegemony, the dominance of a particular set of schema or values. Alternatively, this production may challenge dominant beliefs or values in some. A pop culture example of this might have been early Lady Gaga, whose mode of dress was confrontational because it deviated from existing cultural schema about appropriate dress for someone of her class, race, gender and occupation.
Consumption – The flip side of production is consumption. Consumption of texts, whether they be an outfit, a conversation, or a pop-song, reflects cultural values and expectations – conforming to values and expectations leads to unproblematic consumption – it’s what is expected, it fits our internalized schema. Texts that do not fit this schema are confronting, challenging, even shocking. To continue to use Lady Gaga as an example, when she released a nine-minute long music video centered around a narrative of female violence, it was shocking both in terms of its format (which wasn’t standard MTV fare) and its narrative structure.
Regulation – finally, regulation refers to the forces which constrain the production, distribution, and consumption of texts. These forces may be explicit, such as the television broadcasters code of conduct, or they may be implicit, such as the blogger litmus test of ‘would you say this in front of your mother?’
Finally, linking together these areas are these arrows – the arrows are very important, because as du Gay says, none of these variables can really be considered in isolation. It would be like looking only at the tyres to try and figure out why your car isn’t running. So it is important, when considering communication within a cultural context, to remember that there are multiple factors influencing the production of text and meaning. These factors may support the text, reinforce a cultural position, or alternatively, they may challenge or confront a cultural schema.
du Gay’s model is regularly applied to analysing the interplay between communication and culture within one cultural situation. However, with the globalisation of the media and communication landscape, it is becoming increasingly important to think about the specificities of intercultural communication.
Discussion
- How would you explain what culture is in your own words?
- Choose a form of communication or a media text, and see if you can apply the five variables du Gay suggests to understand that example.
References
du Gay et al. (1997) Doing Cultural Studies: The story of the Sony Walkman Milton Keynes: Open University; Thousand Oaks, CA: Sage.


3.1: Beowulf Prelude
LO, praise of the prowess of people-kings
of spear-armed Danes, in days long sped,
we have heard, and what honor the athelings won!
Oft Scyld the Scefing from squadroned foes,
from many a tribe, the mead-bench tore,
awing the earls. Since erst he lay
friendless, a foundling, fate repaid him:
for he waxed under welkin, in wealth he throve,
till before him the folk, both far and near,
who house by the whale-path, heard his mandate,
gave him gifts: a good king he!
To him an heir was afterward born,
a son in his halls, whom heaven sent
to favor the folk, feeling their woe
that erst they had lacked an earl for leader
so long a while; the Lord endowed him,
the Wielder of Wonder, with world’s renown.
Famed was this Beowulf
[1]
far flew the boast of him,
son of Scyld, in the Scandian lands.
So becomes it a youth to quit him well
with his father’s friends, by fee and gift,
that to aid him, aged, in after days,
come warriors willing, should war draw nigh,
liegemen loyal: by lauded deeds
shall an earl have honor in every clan.
Forth he fared at the fated moment,
sturdy Scyld to the shelter of God.
Then they bore him over to ocean’s billow,
loving clansmen, as late he charged them,
while wielded words the winsome Scyld,
the leader beloved who long had ruled….
In the roadstead rocked a ring-dight vessel,
ice-flecked, outbound, atheling’s barge:
there laid they down their darling lord
on the breast of the boat, the breaker-of-rings,
[2]
by the mast the mighty one. Many a treasure
fetched from far was freighted with him.
No ship have I known so nobly dight
with weapons of war and weeds of battle,
with breastplate and blade: on his bosom lay
a heaped hoard that hence should go
far o’er the flood with him floating away.
No less these loaded the lordly gifts,
thanes’ huge treasure, than those had done
who in former time forth had sent him
sole on the seas, a suckling child.
High o’er his head they hoist the standard,
a gold-wove banner; let billows take him,
gave him to ocean. Grave were their spirits,
mournful their mood. No man is able
to say in sooth, no son of the halls,
no hero ’neath heaven, — who harbored that freight!
- Beowulf. Authored by : Trans. Gummere. Provided by : Project Gutenberg. Located at : http://www.gutenberg.org/files/981/981-h/981-h.htm#linkfootnote2b . License : Public Domain: No Known Copyright


7.2: Add and Subtract Polynomials
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Learning Objectives
By the end of this section, you will be able to:
- Determine the degree of polynomials
- Add and subtract polynomials
- Evaluate a polynomial function for a given value
- Add and subtract polynomial functions
Determine the Degree of Polynomials
We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form \(ax^m\), where \(a\) is a constant and \(m\) is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and \(−4a^2b^3c^2.\)
Definition: MONOMIAL
A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form \(ax^m\), where \(a\) is a constant and \(m\) is a whole number.
A monomial, or two or more monomials combined by addition or subtraction, is a polynomial . Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.
Definition: POLYNOMIALS
- polynomial —A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
- monomial —A polynomial with exactly one term is called a monomial.
- binomial —A polynomial with exactly two terms is called a binomial.
- trinomial —A polynomial with exactly three terms is called a trinomial.
Here are some examples of polynomials.
| Polynomial | \(y+1\) | \(4a^2−7ab+2b^2\) | \(4x^4+x^3+8x^2−9x+1\) | |
|---|---|---|---|---|
| Monomial | \(14\) | \(8y^2\) | \(−9x^3y^5\) | \(−13a^3b^2c\) |
| Binomial | \(a+7ba+7b\) | \(4x^2−y^2\) | \(y^2−16\) | \(3p^3q−9p^2q\) |
| Trinomial | \(x^2−7x+12\) | \(9m^2+2mn−8n^2\) | \(6k^4−k^3+8k\) | \(z^4+3z^2−1\) |
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial , binomial , and trinomial when referring to these special polynomials and just call all the rest polynomials .
The degree of a polynomial and the degree of its terms are determined by the exponents of the variable. A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.
Definition: DEGREE OF A POLYNOMIAL
- The degree of a term is the sum of the exponents of its variables.
- The degree of a constant is 0.
- The degree of a polynomial is the highest degree of all its terms.
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms. Let's start by looking at a monomial. The monomial \(8ab^2\) has two variables \(a\) and \(b\). To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of \(b\) is 2. The sum of the exponents, 1+2,1+2, is 3 so the degree is 3.
Here are some additional examples.
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial . Get in the habit of writing the term with the highest degree first.
Example \(\PageIndex{1}\)
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
- \(7y2−5y+3\)
- \(−2a^4b^2\)
- \(3x5−4x3−6x2+x−8\)
- \(2y−8xy^3\)
- \(15\)
- Answer
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Polynomial Number of terms Type Degree of terms Degree of polynomial ⓐ \(7y^2−5y+3\) 3 Trinomial 2, 1, 0 2 ⓑ \(−2a^4b^2−2a^4b^2\) 1 Monomial 4, 2 6 ⓒ \(3x5−4x3−6x2+x−8\) 5 Polynomial 5, 3, 2, 1, 0 5 ⓓ \(2y−8xy^3\) 2 Binomial 1, 4 4 ⓔ \(15\) 1 Monomial 0 0
Example \(\PageIndex{2}\)
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
- \(−5\)
- \(8y^3−7y^2−y−3\)
- \(−3x^2y−5xy+9xy^3\)
- \(81m^2−4n^2\)
- \(−3x^6y^3z\)
- Answer a
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monomial, 0
- Answer b
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polynomial, 3
- Answer c
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trinomial, 3
- Answer d
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binomial, 2
- Answer b
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monomial, 10
Example \(\PageIndex{3}\)
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.
- \(64k^3−8\)
- \(9m^3+4m^2−2\)
- \(56\)
- \(8a^4−7a^3b−6a^2b^2−4ab^3+7b^4\)
- \(-p^4q^3\)
- Answer
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ⓐbinomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7
Add and Subtract Polynomials
We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.
Example \(\PageIndex{4}\)
Add or subtract:
- \(25y^2+15y^2\)
- \(16pq^3−(−7pq^3)\).
- Answer a
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\( \begin{array} {ll} {} &{25y^2+15y^2} \\ {\text{Combine like terms.}} &{40y^2} \\ \end{array} \nonumber \)
- Answer b
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\( \begin{array} {ll} {} &{16pq^3−(−7pq^3)} \\ {\text{Combine like terms.}} &{23pq^3} \\ \end{array} \nonumber \)
Example \(\PageIndex{5}\)
Add or subtract:
- \(12q^2+9q^2\)
- \(8mn^3−(−5mn^3)\).
- Answer
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ⓐ \(21q^2\) ⓑ \(13mn^3\)
Example \(\PageIndex{6}\)
Add or subtract:
- \(−15c^2+8c^2\)
- \(−15y^2z^3−(−5y^2z^3)\)
- Answer
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ⓐ \(−7c^2\) ⓑ \(−10y^2z^3\)
Remember that like terms must have the same variables with the same exponents.
Example \(\PageIndex{7}\)
Simplify:
- \(a^2+7b^2−6a^2\)
- \(u^2v+5u^2−3v^2\)
- Answer
-
ⓐ Combine like terms.
\(a^2+7b^2−6a^2 \;=\; −5a^2+7b^2\)ⓑ There are no like terms to combine. In this case, the polynomial is unchanged.
\(u^2v+5u^2−3v^2\)
Example \(\PageIndex{8}\)
Add:
- \(8y^2+3z^2−3y^2\)
- \(m^2n^2−8m^2+4n^2\)
- Answer
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ⓐ \(5y^2+3z^2\)
ⓑ \(m^2n^2−8m^2+4n^2\)
Example \(\PageIndex{9}\)
Add:
- \(3m^2+n^2−7m^2\)
- \(pq^2−6p−5q^2\)
- Answer
-
ⓐ \(−4m^2+n^2\)
ⓑ \(pq^2−6p−5q^2\)
We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.
Example \(\PageIndex{10}\)
Find the sum: \((7y^2−2y+9)\;+\;(4y^2−8y−7)\).
- Answer
-
\(\begin{align*} &\text{Identify like terms.} & & (\underline{\underline{7y^2}}−\underline{2y}+9)+(\underline{\underline{4y^2}}−\underline{8y}−7) \\[6pt]
&\text{Rewrite without the parentheses,} \\
&\text{rearranging to get the like terms together.} & & \underline{\underline{7y^2+4y^2}}−\underline{2y−8y}+9−7\\[6pt]
&\text{Combine like terms.} & & 11y^2−10y+2 \end{align*} \)
Example \(\PageIndex{11}\)
Find the sum: \( (7x^2−4x+5)\;+\;(x^2−7x+3)\)
- Answer
-
\(8x^2−11x+8\)
Example \(\PageIndex{12}\)
Find the sum: \((14y^2+6y−4)\;+\;(3y^2+8y+5)\)
- Answer
-
\(17y^2+14y+1\)
Be careful with the signs as you distribute while subtracting the polynomials in the next example.
Example \(\PageIndex{13}\)
Find the difference: \((9w^2−7w+5)\;−\;(2w^2−4)\)
- Answer
-
\(\begin{align*} & & & (9w^2−7w+5)\;−\;(2w^2−4) \\[6pt]
&\text{Distribute and identify like terms.} & & \underline{\underline{9w^2}}−\underline{7w}+5-\underline{\underline{2w^2}}+4 \\[6pt]
&\text{Rearrange the terms.} & & \underline{\underline{9w^2-2w^2}}−\underline{7w}+5+4\\[6pt]
&\text{Combine like terms.} & & 7w^2−7w+9 \end{align*} \)
Example \(\PageIndex{14}\)
Find the difference: \((8x^2+3x−19)\;−\;(7x^2−14)\)
- Answer
-
\(x^2+3x−5\)
Example \(\PageIndex{15}\)
Find the difference: \((9b^2−5b−4)\;−\;(3b^2−5b−7)\)
- Answer
-
\(6b^2+3\)
Example \(\PageIndex{16}\)
Subtract \((p^2+10pq−2q^2)\) from \((p^2+q^2)\).
- Answer
-
\(\begin{align*} & & & (p^2+q^2)\;−\;(p^2+10pq−2q^2) \\[6pt]
&\text{Distribute and identify like terms.} & & \underline{\underline{p^2}}+\underline{q^2}-\underline{\underline{p^2}}-10pq + \underline{2q^2} \\[6pt]
&\text{Rearrange the terms, putting like terms together.} & & \underline{\underline{p^2-p^2}}−10pq +\underline{q^2 + 2q^2}\\[6pt]
&\text{Combine like terms.} & & −10pq+3q^2 \end{align*} \)
Example \(\PageIndex{17}\)
Subtract \((a^2+5ab−6b^2)\) from \((a^2+b^2)\)
- Answer
-
\(−5ab+7b^2\)
Example \(\PageIndex{18}\)
Subtract \((m^2−7mn−3n^2)\) from \((m^2+n^2)\).
- Answer
-
7mn+4n^2
Example \(\PageIndex{19}\)
Find the sum: \((u^2−6uv+5v^2)\;+\;(3u^2+2uv)\)
- Answer
-
\(\begin{align*} & & & (u^2−6uv+5v^2)\;+\;(3u^2+2uv) \\[6pt]
&\text{Distribute and identify like terms.} & & \underline{\underline{u^2}}-\underline{6uv}+5v^2+\underline{\underline{3u^2}}+ \underline{2uv} \\[6pt]
&\text{Rearrange the terms to put like terms together.} & & \underline{\underline{u^2}}+\underline{\underline{3u^2}}- \underline{6uv}+ \underline{2uv}+5v^2\\[6pt]
&\text{Combine like terms.} & & 4u^2−4uv+5v^2 \end{align*} \)
Example \(\PageIndex{20}\)
Find the sum: \((3x^2−4xy+5y^2)\;+\;(2x^2−xy)\)
- Answer
-
\(5x^2−5xy+5y^2\)
Example \(\PageIndex{21}\)
Find the sum: \((2x^2−3xy−2y^2)\;+\;(5x^2−3xy)\)
- Answer
-
\(7x^2−6xy−2y^2\)
When we add and subtract more than two polynomials, the process is the same.
Example \(\PageIndex{22}\)
Simplify: \((a^3−a^2b)\;−\;(ab^2+b^3)\;+\;(a^2b+ab^2)\)
- Answer
-
\(\begin{align*} & & & (a^3−a^2b)\;−\;(ab^2+b^3)\;+\;(a^2b+ab^2) \\[6pt]
&\text{Distribute} & & a^3−a^2b − ab^2 - b^3 + a^2b+ab^2\\[6pt]
&\text{Rearrange the terms to put like terms together.} & & a^3−a^2b + a^2b− ab^2 + ab^2 - b^3 \\[6pt]
&\text{Combine like terms.} & & a^3−b^3 \end{align*} \)
Example \(\PageIndex{23}\)
Simplify: \((x^3−x^2y)\;−\;(xy^2+y^3)\;+\;(x^2y+xy^2)\)
- Answer
-
\(x^3+y^3\)
Example \(\PageIndex{24}\)
Simplify: \((p^3−p^2q)\;+\;(pq^2+q^3)\;−\;(p^2q+pq^2)\)
- Answer
-
\(p^3−3p^2q+q^3\)
Evaluate a Polynomial Function for a Given Value
A polynomial function is a function defined by a polynomial. For example, \(f(x)=x^2+5x+6\) and \(g(x)=3x−4\) are polynomial functions, because \(x^2+5x+6\) and \(3x−4\) are polynomials.
Definition: POLYNOMIAL FUNCTION
A polynomial function is a function whose range values are defined by a polynomial.
In Graphs and Functions , where we first introduced functions, we learned that evaluating a function means to find the value of \(f(x)\) for a given value of \(x\). To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.
Example \(\PageIndex{25}\)
For the function \(f(x)=5x^2−8x+4\) find:
- \(f(4)\)
- \(f(−2)\)
- \(f(0)\).
- Answer
-
ⓐ
Simplify the exponents. Multiply. Simplify. ⓑ
Simplify the exponents. Multiply. Simplify. ⓒ
Simplify the exponents. Multiply.
Example \(\PageIndex{26}\)
For the function \(f(x)=3x^2+2x−15\), find
- \(f(3)\)
- \(f(−5)\)
- \(f(0)\).
- Answer
-
ⓐ 18 ⓑ 50 ⓒ \(−15\)
Example \(\PageIndex{27}\)
For the function \(g(x)=5x^2−x−4\), find
- \(g(−2)\)
- \(g(−1)\)
- \(g(0)\).
- Answer
-
ⓐ 20 ⓑ 2 ⓒ \(−4\)
The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.
Example \(\PageIndex{28}\)
The polynomial function \(h(t)=−16t^2+250\) gives the height of a ball t seconds after it is dropped from a 250-foot tall building. Find the height after \(t=2\) seconds.
- Answer
-
\( \begin{array} {ll} {} &{h(t)=−16t^2+250} \\ {} &{} \\ {\text{To find }h(2)\text{, substitute }t=2.} &{h(2)=−16(2)^2+250} \\ {\text{Simplify.}} &{h(2)=−16·4+250} \\ {} &{}\\ {\text{Simplify.}} &{h(2)=−64+250} \\ {} &{} \\ {\text{Simplify.}} &{h(2)=186} \\ {} &{\text{After 2 seconds the height of the ball is 186 feet.}} \\ \end{array} \nonumber \)
Example \(\PageIndex{29}\)
The polynomial function \(h(t)=−16t^2+150\) gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after \(t=0\) seconds (the initial height of the object).
- Answer
-
The height is \(150\) feet.
Example \(\PageIndex{30}\)
The polynomial function \(h(t)=−16t^2+175\) gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after \(t=3\) seconds.
- Answer
-
The height is \(31\) feet.
Add and Subtract Polynomial Functions
Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.
Definition: ADDITION AND SUBTRACTION OF POLYNOMIAL FUNCTIONS
For functions \(f(x)\) and \(g(x)\),
\[(f+g)(x)=f(x)+g(x)\]
\[(f−g)(x)=f(x)−g(x)\]
Example \(\PageIndex{31}\)
For functions \(f(x)=3x^2−5x+7\) and \(g(x)=x^2−4x−3\), find:
- \((f+g)(x)\)
- \((f+g)(3)\)
- \((f−g)(x)\)
- \((f−g)(−2)\).
- Answer
-
ⓐ
Rewrite without the parentheses. Put like terms together. Combine like terms. ⓑ In part (a) we found \((f+g)(x)\) and now are asked to find \((f+g)(3)\).
\( \begin{array} {ll} {} &{(f+g)(x)=4x^2−9x+4} \\ {} &{} \\ {\text{To find }(f+g)\space(3),\text{ substitute }x=3.} &{(f+g)(3)=4(3)^2−9·3+4} \\ {} &{} \\ {} &{(f+g)(3)=4·9−9·3+4} \\ {} &{} \\ {} &{(f+g)(3)=36−27+4} \\ \end{array} \nonumber \)
Notice that we could have found \((f+g)(3)\) by first finding the values of \(f(3)\) and \(g(3)\) separately and then adding the results.
Find \(f(3)\). Find \(g(3)\). Find \((f+g)(3)\). ⓒ
Rewrite without the parentheses. Put like terms together. Combine like terms. ⓓ
Example \(\PageIndex{32}\)
For functions \(f(x)=2x^2−4x+3\) and \(g(x)=x^2−2x−6\), find: ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f−g)(x)\) ⓓ \((f−g)(−2)\).
- Answer
-
ⓐ \((f+g)(x)=3x^2−6x−3\)
ⓑ \((f+g)(3)=6\)
ⓒ \((f−g)(x)=x^2−2x+9\)
ⓓ \((f−g)(−2)=17\)
Example \(\PageIndex{33}\)
For functions \(f(x)=5x^2−4x−1\) and \(g(x)=x^2+3x+8\), find ⓐ \((f+g)(x)\) ⓑ \((f+g)(3)\) ⓒ \((f−g)(x)\) ⓓ \((f−g)(−2)\).
- Answer
-
ⓐ \((f+g)(x)=6x^2−x+7\)
ⓑ \((f+g)(3)=58\)
ⓒ \((f−g)(x)=4x^2−7x−9\)
ⓓ \((f−g)(−2)=21\)
Access this online resource for additional instruction and practice with adding and subtracting polynomials.
- Adding and Subtracting Polynomials
Key Concepts
-
Monomial
- A monomial is an algebraic expression with one term.
- A monomial in one variable is a term of the form axm,axm, where a is a constant and m is a whole number.
-
Polynomials
- Polynomial —A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.
- monomial —A polynomial with exactly one term is called a monomial.
- binomial — A polynomial with exactly two terms is called a binomial.
- trinomial —A polynomial with exactly three terms is called a trinomial.
-
Degree of a Polynomial
- The degree of a term is the sum of the exponents of its variables.
- The degree of a constant is 0.
- The degree of a polynomial is the highest degree of all its terms.
Glossary
- binomial
- A binomial is a polynomial with exactly two terms.
- degree of a constant
- The degree of any constant is 0.
- degree of a polynomial
- The degree of a polynomial is the highest degree of all its terms.
- degree of a term
- The degree of a term is the sum of the exponents of its variables.
- monomial
- A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form axm,axm, where a is a constant and m is a whole number.
- polynomial
- A monomial or two or more monomials combined by addition or subtraction is a polynomial.
- standard form of a polynomial
- A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.
- trinomial
- A trinomial is a polynomial with exactly three terms.
- polynomial function
- A polynomial function is a function whose range values are defined by a polynomial.


1.11: Ancient Aesthetics
If one were being pedantic, one would say that there was no “ancient aesthetics,” certainly in the ways that aesthetics emerged as one part of philosophy in the 18 th century (Mason 2016, 3). Later moderns’ exclusive focus on aesthêsis , on how art and beauty “makes us feel,” is foreign to the Greeks and Romans. Beauty for them was firstly what we would call an objective thing. If one were being more liberal-minded, we could say that all of Greek existence was meaningfully “aesthetic,” characterised by an overarching sense of the beautiful, to kalon . The arts were bound up from their inception with religious ritual and worship, and the classical Greeks could think of no better designation for the ethically excellent man than to call him kalos k’agathos , the beautiful and good man. Indeed, the very term kalon , designating beauty, could be used to describe nobility of action or character, as well as physical beauty (Mason 2016, 64).
This is not to say that the Greeks and Romans did not produce arts in great abundance. In the renaissance and enlightenment, their architecture, painting, sculpture and literature would be held up as timeless standards by artists and theoreticians of the arts. From the classical period (5 th –4 th century BCE), Greek artists joined the philosophers in theorising concerning beauty and the arts, and in the attempt to lay down “canons” for the production of music, architecture, sculpture, painting, rhetoric, and poetics (Tatarkiewicz 1970, 24–25, 49–63).
To give an opening generalisation, we can say that four different areas of concern emerge within “ancient aesthetics,” if we take the latter term to describe ancient authors’ attempts to theoretically comprehend beauty and the arts:
- The attempt to understand beauty ( to kalon ) as an “objective” quality in the world that characterises some objects, people, and nature;
- The attempt to understand what we would call the “subjective” dimension involved in human responsiveness to beauty and the arts: the way that beautiful things please or move us, and the way that their effect upon us can be edifying, purifying us from negative beliefs or emotions ( katharsis ), or morally elevating us to be better citizens or human beings (in paideia );
- Attempts to understand how artistic objects, from poems to sculptures, are produced, whether through madness or inspiration, or by following codifiable technical norms, and with what ends;
- As it were in between (i) and (ii), attempts to theorise the ethical and political significance of the arts, given their capacities to powerfully affect and transform individuals or groups. So, in both Plato and Aristotle, the arts are addressed very largely in their political dialogues, like Plato’s Republic and Aristotle’s Politics , in ways we would not today associate with political theory. [1]
With these four concerns in view as a preliminary rubric, we will proceed in what follows in a more or less chronological fashion. We begin with the aesthetic practices and reflections of the preclassical artists and poets, and end with the Stoic philosophers’ views on art and beauty. As we will see, in different authors and periods, different considerations become predominant and pass out of focus. This again reflects the absence of a codified (sub)discipline of philosophical thought of aesthetics, like our own, in the classical world.
The prephilosophical aesthetics of the Greeks
As Wladyslaw Tatarkiewicz has underscored, we must be careful before assuming that the prephilosophical Greeks’ experiences and assumptions concerning the arts and beauty are identical to our own (Tatarkiewicz 1970, 25–30, 166–167). Firstly, the extensions of key terms, like mousikê, poeisis , and to kalon , differ from those of our own seeming equivalents—“music,” “poetry,” and “beauty” (Mason 2016, 64). Moreover, what the Greeks deemed worthy of expressing or producing in the arts, the nature, number, and classification of those arts, and their understandings of what an artist was engaged in when they created an artwork all differ greatly from modern views.
One framing consideration to approach ancient aesthetics must be that each of the arts, from architecture and sculpture to music, poetry, and drama, emerged from the ancients’ polytheistic cults and worship. Homer and Hesiod each sang pre-eminently of the gods and semi-divine heroes. The preeminent achievements in archaic Greek architecture were temples; early sculptors carved gods, mythological reliefs, and documented religious events on the pediments of temples, or else shaped archetypal male and female nudes ( koroi , korai ) without distinguishing individuality (Durant 1939, 221–226). In music, an art associated mythologically with the lyre-playing god Orpheus, the paean was developed as a medium to praise Apollo, the dithyramb to hymn Dionysus, and prosodies as accompaniments for religious processions (Tatarkiewicz 1970, 18; Durant 1939, 228–230). Dance, too, emerged from religious ceremony, as did poetry, which was from the beginning closely associated with musical performance. The Greek word choreuein , from which the word for the “chorus” of tragedies would come, originally meant group dancing and singing. Flute ( aulus ) music was closely associated with the Dionysean cult, and lyre-playing with sacrificial and other sacred rituals (Tatarkiewicz 1970, 18–19). Consensus has now been established that Greek tragedies emerged from Dionysean rituals surrounding goat sacrifice, with the heroes’ diegetic destruction coming to stand in for the sacrificial victim (Burkert 1970). Comedies, Aristotle tells us, hailed from the kômos, religious processions in which a company of males sang and danced around the likeness of a ceremonial phallus ( Poetics 1449a, cf. 1448a; [2] Durant 1939, 230–233).
Again, unlike today, artists were not valued highly as a specific cultural type. Sculptors and architects would leave no individuating marks on their works. Whilst Terpander of Lesbos and Thaletas the Cretan are known to us for establishing “norms” in music (Tatarkiewicz 1970, 19), the actors in Greek theatres wore stylised masks. When the great tragedian Aeschylus died, it was as someone who had fought at Marathon that he wished to be remembered in his epitaph (Mason 2016, 7). There are several converging reasons behind these (for us) strange phenomena. The first is that the arts in general were considered species of technê or craft; they were not therefore essentially distant from the servile pursuits of cobblers, tanners, smiths, and the like. To the extent that the creation of art required knowledge, it was considered noble; to the extent to which it required manual work, for instance in shaping the stone, it was considered unfree (Tatarkiewicz 1970, 29).
The second consideration here is that the Greeks and Romans did not place any great value on novelty, creativity, or the imagination, three of the key features of the postromantic ideology of “genius” (Tatarkiewicz 1970, 24, 29). Over time, giants of proverbial stature emerged in architecture, like Iktinos, or sculpture, like Phidias or Praxiteles, or painting, like Apelles or Zeuxis (Mason 2016, 13–14). But their greatness was not exactly a matter of individual innovation. For the Greeks and Romans all art’s greatness could only come from its beauty. But all beauty could only come from its approximation to, or idealisation of, the larger order that the artistic object expressed or represented. The closest approximation to the modern understandings comes in the ancient sense that poets, unlike other artists, were divinely inspired: they were more like soothsayers or diviners than craftspeople, a position most famously expressed in Plato’s Phaedrus (244e–245a).
Music and dance were conceived by the Greeks as pre-eminently expressive arts, and poetry was only slowly differentiated from the former (Mason 2016, 14; Tatarkiewicz 1970, 18–19). Music, as the Pythagoreans would develop, was held to be able to reproduce or express the inner harmonies of the soul (Mason 2016, 3, 14). The very term mimêsis , which we usually translate as “imitation,” was first used to describe the expression of dancers’ inner feelings in dance, before being developed to encompass the representation of things in words, objects, and images (Tatarkiewicz 1970, 16–17, cf. 81–82). Alongside these expressive arts came the constructive arts of architecture and sculpture, although it is notable that neither of these arts had one of the nine mythological muses associated with it (Tatarkiewicz 1970, 28). Our category of fine arts also has no exact ancient equivalent, although the Greeks and Romans always divided the arts requiring manual work from other, “free” arts.
In all ancient arts, what stands out is the sense that what is beautiful and pleasing for human beings—for what is beautiful always pleases—is above all what has inner order, harmonia or symmetria : between the sounds, between the lengths of lines or columns, between shapes and colours, between the parts of a body (Tatarkiewicz 1970, 25–26; Mason 2016, 3, 66–67, 125). In Pythagorean philosophy, music was accorded the highest significance, insofar as the Pythagoreans were the first to discover the connection between mathematical ratios and musical intervals, like thirds, fifths, and octaves (Anderson 1983; Mason 2016, 14, 66–67; Tatarkiewicz 1970, 81–82). In the Greek temples, every part was crafted according to mathematical ratios, based on calculations building upon a unit module, usually half the width of the base of a column. Thus, the Athenian Temple of Hephaestus is a six-column temple with 27 modules. The relation of the column to the middle aisles is 5:8, and the triglyphs are each one module wide, with their widths relative to the metopes’ again forming a ratio of 5:8 (Tatarkiewicz 1970, 51).
Similar mathematical ratios were determined by Greek sculptors between the different parts of the human body, and even the three parts of the human face: forehead, nose, and lips to chin (Tatarkiewicz 1970, 57–59).
It is this sense of the mathematical ordering of reality, so that natural forms have their own intrinsic, governing rationality that underlies the emergence of “canons”: works codifying the principles of beautiful architecture, sculpture, music, even vase pottery (Tatarkiewicz 1970, 24–25, 49–63). It is also the sense that underlies the Greeks’ deep association of beauty per se with the natural human form, so evident in the great proliferation of sculptures of anthropomorphic Olympian deities, athletes, and later, of other human figures and busts (Durant 1939, 217–218). As the great Russian aesthetician, Alexei Losev comments,
the beautiful in the antiquity presents itself in those circumstances when physical elements harmonise with each other in a perfect human body, when the principle of the unified bodily life, which the Greeks called “soul,” fully subsumes all bodily elements. A body formed in accordance with this principle is the ideal in question. The phenomenon of beauty transpires as the ideal manifests itself in physical elements. (Losev [1963] 2000, I, 87; cf. Grube 1927, 629)
Let us close this section by nevertheless remarking that the beauty of non-human nature was not wholly lost on the Greeks or Romans (P. Hadot 2010a). Indeed, a sense of cosmic order and beauty is present within all of the ancient philosophies, perhaps excluding only the Sceptics, but including even the followers of Epicurus (341–271 BCE), who conceived of the world as the product of atoms, void, and motion. At different periods throughout antiquity, we find poets and philosophers raising paeans to the beauty of the natural world, and the surpassing excellence of rustic life close to nature. Many Greek temples are located in the most sublime natural locations, like the Temple of Apollo the Healer at Bassae and the Temple to Poseidon on Cape Sounion (P. Hadot 2010b).
Many of the Roman residences unearthed in Herculaneum and Pompei in the 18 th century are covered with delicately stylised landscape paintings (P. Hadot 2010a). If anything, an increasing sense of the beauty of the natural world can be traced as antiquity proceeds, as we will duly see.
From Pythagoreanism to Plato
Pythagoras (c. 570–490 BCE) is said to have coined the term “philosophy.” And it is in his school that the first developed set of theoretical reflections on “aesthetic” phenomena emerged. The qualification implied in the inverted commas is needed. As we have indicated, the Greeks owed to the Pythagoreans the developed sense that the principles of mathematics “were the principles of all things,” based pre-eminently on their researches in acoustics (Tatarkiewicz 1970, 80–81; Mason 2016, 3, 14, 30, 66–67). But for the Pythagoreans, it was above all the cosmos itself, a perfect sphere containing all things, which was superlatively beautiful. Excepting music, they showed little interest in the other arts. The objective beauty at issue here (i) was characterised by harmonia , “a Unity of many elements and an agreement between disagreeing elements,” like a musical harmony (Tatarkiewicz 1970, 80; Mason 2016, 30). Ultimately, it was held to characterise the orderly, spherical circuits of the planets, creating a harmony of the spheres which we cannot hear, since it is sounding all the time.
There is, however, a second decisive contribution the Pythagoreans made to ancient aesthetics. It concerns the subjective dimension of our experience of art (ii). Musical harmonies, the Pythagoreans maintained, had the power to both evoke and express feelings, due to their mimesis (imitation, reproduction) of the inner constitution of the psyche (psychology or mind; Anderson 1983). So different types of music, even different musical scales, could be used to affect audiences’ souls in different ways. Music, that is, could be used as a means of psychagogia , the guidance or direction of souls, leading hearers towards “good” or “bad” forms of ethos or character. Indeed, adapting Orphic beliefs, the Pythagoreans believed that music could be used therapeutically, in order to purify people of negative affects: the process of katharsis . As Aristoxenus tells us, “the Pythagoreans employed medicine to purge the body, and music to purge the soul” (Tatarkiewicz 1970, 82). According to the fifth century Pythagorean, Damon of Athens, the singing and playing of music can form the young to courage, moderation, justice, and spiritual order or eunomia (Lord 1978) . All of these Pythagorean claims were to abidingly affect classical aesthetics, including Plato’s and Aristotle’s political reflections on art (iv).
On the subject of beauty and the arts, as on many subjects, Plato (428/7–348/7 BCE) says different things in different places. Modern commentators have hence divided as to how we should read his corpus: some dividing earlier “Socratic” from “later” dialogues, others arguing that Plato aims to address different levels of readers in different texts (Strauss 1964), others again returning to the ancient idea that the dialogues are to be read in a single pedagogical order, tied to a project of shaping the ideal student (Altman 2013). Beauty and the arts are addressed in the Ion, Gorgias, Hippias Major, Republic, Phaedrus , Symposium , Philebus, Statesman, Laws , and Timaeus.
In both the Gorgias and Hippias Major , Plato’s Socrates entertains and queries common Greek opinions on beauty, including views that the Xenophontic Socrates variously entertains in the Memorabilia (Grube 1927, 271–273; Sider 2012; Tatarkiewicz 1970, 100–102) . To kalon would name here the “appropriate” or “fitting” ( to prepon ); whatever thing (from the sublime to the mundane) is well shaped to particular human ends and circumstances. Or else beauty is whatever gives pleasure. Or it is simply what is useful ( Hippias Major 293e, 295c–296e).
Socrates expresses hesitation as to whether whatever pleases will be beautiful. For this seems to make aesthetic phenomena wholly subjectively relative, as we say. As for whether the beautiful is just whatever is fitting or useful, Socrates notes that many fitting and useful things can serve bad ends, like a beautiful sword in the hands of a murderer. Yet the beautiful is for him axiomatically always also good. Moreover, there are many things which the Greeks admire as kalon which are not simply useful. These include phenomena as different as ornamental statues or acts of self-sacrificing bravery (Tatarkiewicz 1970, 115–116).
In other dialogues, Plato has Socrates presenting different views, closer in orientation to the objective Pythagorean view (i). So, in the Philebus (64e) we are told that “measure ( metriotes ) and proportion ( symmetria ) are … beauty and virtue”; or again, in the Timaeus , we read that “all that is good is beautiful, and what is good cannot lack proportion” (87c). Plato’s great innovation, here as elsewhere, turns upon how he develops this Pythagorean orientation into a fully-fledged metaphysical conception of Beauty as what he terms an “Idea” or “Form” ( eidos ).
In Hippias Major , the relative beauty of particular women was measured against that of the goddesses (289a). Just so, in the Republic, Philebus , and Symposium , Plato will have his Socrates argue that all of the particular things we find beautiful in the physical world are all only relatively beautiful. And if we ask “relative to what?”, the answer comes: to the very Idea of Beauty Itself ( Philebus 51b–c). As the climactic passage in the famous “ladder of love” in Diotima’s speech in the Symposium makes clear (210d–211e), this Idea alone is absolutely Beautiful, neither coming into being nor passing away, not beautiful in some parts but ugly in others, neither beautiful for some only, not for others, but existing alone, the very measure by which all other beauties are adjudged. Indeed, all these other beauties are only beautiful at all to the extent that they “participate” in the Idea.
It is easy for moderns, in the wake of thinkers like Friedrich Nietzsche, to think that nothing could be more “life-denying” and abstract than such a metaphysical vision of Beauty. The sometimes-overlooked subjective side to Plato’s doctrine (ii), very much carried forward into later Neoplatonists like Plotinus, responds to the sense that the Idea is what is most, eternally Real, the very source of order and life (cf. Mason 2016, 142–145). Indeed, the Symposium introduces a desiderative component into the ancient aesthetic, tying to kalon very closely to human Eros or desire (cf. Plotinus, Enneads VI, 7, 22). Beauty is what moves human beings to desire, Plato’s Diotima argues in the Symposium , in a thought to which Socrates’ palinode in the Phaedrus will give mythicopoetic form. Indeed, she tells us, “to love is to desire to give birth in beauty ( en tô kalô) ” ( Symposium 206e): whether to other human beings through sexual union with a beautiful other, or to elevated speeches through the intellectual love of the Idea of Beauty characterising the true philosopher (210d–211e). Beauty intimates immortality to we hapless mortals, on Plato’s view. It awakens the “wings of desire” of our souls, which have not wholly forgotten their other-worldly origins (so Plato seemingly believes), and still long to gaze directly on the generative, other-worldly Ideas ( Phaedrus 248c–251c).
One might well imagine that such an elevated conception of the metaphysical place of Beauty would have led Plato into the deepest appreciation of the arts. Famously, however, in his political writings (iv), Plato delivers a very qualified assessment of poetry, positioning the poets as involved in an “ancient quarrel” with the philosophers ( Republic 607b). One ground of this view, which we see in the Ion (533e) and Phaedrus (244d–245a), is the claim that poets “do not know what they do,” but must be moved by divine madness to produce great verse. The philosopher, by contrast, desires knowledge above all things, and that knowledge should shape human speech and action. Another ground, evident in books II–IV of the Republic , is Plato’s deep appreciation of the power of poetic representations of gods and men, as well as different forms of music, to move the young to emulation ( Republic 377a–d, 397d; cf. Mason 2016, 31; Tatarkiewicz 1970, 126). “Homer and Hesiod, and the rest of the poets” were “the great myth-makers ( mythopoious )” of the Greeks, Plato sees ( Republic 377b). It is surely not stretching the interpretive bow too far to surmise that Plato wished for philosophy to take on this august culture-forming place, and that this was the real stake of his “ancient quarrel” ( Republic 607b). Indeed, the Stranger in the Laws declares as much to the poets:
Best of strangers, we will say to them, we also according to our ability are tragic poets, and our tragedy is the best and noblest; for our whole polis is an imitation of the best and noblest life, which we affirm to be indeed the very truth of tragedy. You are poets and we are poets, both makers of the same strains; rivals and antagonists in the noblest of dramas, which true law can alone perfect. ( Laws 817a–b)
Plato in the political texts approaches a deeply moralising, censorious approach to the arts. This would have them wholly serve the needs of government and education. His Socrates even famously proposes to expel the poets from the ideal city ( Republic 607a–b). Republic X adds to this political dimension a criticism of the representative or mimetic (as against expressive) arts, principally painting ( Republic 596b–598a; Mason 2016, 30, 34–36). Although he elsewhere recognises that artists aim exactly to produce idealised images of people, actions and things— eikónes (images) seemingly close therefore to his own Ideas—Socrates argues here that what “mimetic” artists produce are mere copies of the physical objects we see. These are as such “at three removes ( tritou )” from the metaphysical Ideas that shape physical reality ( Republic 597e; Phaedrus 248e; Mason 2016, 35–36). Small wonder that in the Statesman , we can be told that all imitative arts none of them is practised for any serious purpose, but all of them merely for play” ( Statesman 288c).
Aristotle on art, beauty and poetics
Outside of his Poetics , a treatise devoted to considering how a certain art is shaped and produced (iii), Aristotle (384–322 BCE) also gives most space to art in his political writings (iv). Here as elsewhere, however, he challenges his great teacher, Plato. The highest goal of government, Aristotle maintains, is to enable citizens to achieve “the actualisation and complete practice of virtue” ( Politics 1332a9). Healthy peoples make war for the sake of peace. Yet the goal of all peaceable activities, including work and politics, will be the cultivation and enjoyment of activities which are their own ends in themselves. The goal of education in turn should accordingly be to teach citizens “to be capable of being at leisure ( scholazein ) in a kalos fashion” ( Politics 1337b30–32; cf. 1329a1–2; 1334a36–39). Such leisure is not mere idleness, passing the time. Rather, it should be filled with the arts, whose pleasurable forms have misled many people (and perhaps Plato is intended) into considering them as mere games. Citing Homer, Aristotle protests that “Odysseus says that this is the best pastime, when human beings are enjoying good cheer and ‘the banqueters seated in order throughout the hall listen to the singer’” ( Politics 1338a27–29).
In Aristotle’s more liberal purview, the arts “should be practised not for the sake of a single benefit but for the sake of several” ( Politics 1341b35–37). Likewise, contra his teacher, they can be judged according to at least five different concerns, including the plausibility, consistency, and reasonableness of their subjects, their (non)adherence to specifically artistic norms, as well as moral considerations (Tatarkiewicz 1970, 149). A key part of the goal of political life in “pastime and phronêsis (practical wisdom),” the arts led by music also stand as a principal means to educate citizens towards a kalos leisure: “It is rather to be supposed that music contributes something to virtue, the assumption being that, just as gymnastics makes the body of a certain quality, so also is music capable of making the character of a certain quality by habituating it to be capable of enjoying in the correct fashion” ( Politics 1339a21–25).
Significantly, it is in the context of reflecting in a Pythagorean vein upon the ability of music, and the different scales (Lydian, mixed Lydian, Phrygian, etc.) to shape character that Aristotle’s longest passage on the famous thesis concerning art and katharsis (ii), also central to the Poetics , is found (Mason 2016, 89–94). Implicitly contesting Plato’s Socrates’ and his friends’ exiling of certain musical modes from the best city in the Republic (398d–399a), Aristotle comments,
It is evident that all the harmonies are to be used, but that all are not to be used in the same manner, but with a view to paideia [education, training, cultivation] those most relating to character. … For there are certain persons who are possessed by the passion of enthusiasm, but as a result of the sacred tunes … we see them calming down as if obtaining a cure and catharsis. This same thing, then, must necessarily be experienced also by the pitying and fearful as well as the generally passionate, and by others insofar as each individual has a share in these things, and there must occur for all a certain purification and a feeling of relief accompanied by pleasure. ( Politics 1342a2–14)
Notably, when Aristotle considers beauty (i), he does so under the heading of theoretical philosophy, treating it accordingly in the fifth book of his Metaphysics (Marshall 1953, 228–229) . Given his famous criticisms of Plato’s post-Pythagorean idealism, it is also notable that one main thrust of his claims here is nevertheless to assert the mathematical dimensions of to kalon (Tatarkiewicz 1970, 151):
Now since the good and the beautiful are different (for the former always implies conduct as its subject, while the beautiful is found also in motionless things), those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them. … The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. (Aristotle, Metaphysics 1078a, 32 ff.)
In the Topics , we are told that the beautiful is to prepon (102a, 6; Marshall 1953, 229) . Nevertheless, there is no sense in which the beautiful in Aristotle would be reducible to what well fits human ends. Nature herself, for Aristotle, is the master craftsman of order and symmetry (cf. Marshall 1953, 229–230). His criticisms of Plato’s Ideas aside, Aristotle thus affords a very high place in his aesthetic regard to the perfect, orderly movements of the heavenly bodies. It is an order and perfection to which we sublunar creatures can scarcely aspire, and of which we have no direct experience on earth. Above even the encircling heavens, however, comes the God of the philosopher, “who in might is most powerful, in beauty most fair, in life immortal, in virtue supreme; for, though he is invisible to all mortal nature, yet he is seen in his very works. For all that happens in the air, on the earth, and in the water, may truly be said to be the work of God” (Aristotle, De Mundi 399a 19–21).
Aristotle’s most famous contribution to aesthetics is, however, his Poetics , which has survived in part, and contains an extensive discussion of the literary form of tragedy, and how it might be produced (iii). Tragedy, Aristotle famously declares, is “an imitation of an action that is serious, complete, and of a certain magnitude; in language embellished with each kind of artistic ornament, the several kinds being found in separate parts of the play; in the form of action, not of narrative; through pity and fear effecting the proper purgation of these emotions” ( Poetics 1449b).
Poetry is generally afforded a much higher place, once again, in Aristotle’s thought than in Plato’s, reflecting his more favourably conception of mimesis . Poetry is more philosophical than history as Aristotle proverbially says ( Poetics 1451a; Mason 2016, 84). For it deals with individuals who are also representative of entire types (heroes, villains, kings, priests, etc.), rather than being bound to report on actual events and people. What the poet “mimes,” then, when he imitates reality is not historical fact. It is some idealised (or demonised) figure and action, like the fall of an iconic hero like Oedipus or a tragic heroine like Antigone. The tragedian should render people and actions “as they were or are … as they are said or thought to be or … as they ought to be” ( Poetics 1460b). Aristotle hence praises Sophocles for rendering his characters as ideals, and Zeuxis for painting men as more beautiful than they were: “The ideal type must surpass the reality” (Tatarkiewicz 1970, 142).
How then does tragedy’s idealised presentation of noble, fallen characters’ actions and reversals purge audiences of pity and fear, and what does Aristotle contribute to the Pythagorean conception of art as a means of katharsis (ii)? The aim of art’s idealisations, Aristotle says, is exactly to provoke emotions: “to make things more moving” (Tatarkiewicz 1970, 147). Through what we call identification with the tragic protagonist, as his fate unfolds, pity for his plight is awakened in us, as well as fear that something like this could happen to us:
Fear and pity may be aroused by spectacular means; but they may also result from the inner structure of the piece, which is the better way, and indicates a superior poet. For the plot ought to be so constructed that, even without the aid of the eye, he who hears the tale told will thrill with horror and melt to pity at what takes place. This is the impression we should receive from hearing the story of the Oedipus. (Aristotle, Poetics 1453b)
The emotions of fear and pity, Aristotle maintains, can become debilitating, or even politically troublesome. However, through witnessing the virtual reality of what transpires on stage, audiences are able to experience and “let them out” in a controlled environment, without direct threat or consequence to themselves or the polis (Mason 2016, 91–94). Katharsis for Aristotle is thus akin to purging in Greek medicine (Jones 1962, 39–40; Tatarkiewicz 1970, 146; Mason 2016, 72–80). However, as we have seen, the thought clearly looks back to Pythagorean musical theory and beyond it, to Orphic ritual (see the first section of this chapter, above).
The Hellenistic and Roman periods
The classical period of the Greek “golden age” is generally assigned no more than two centuries preceding Alexander’s conquest of Greece in 338 BCE. The Hellenistic period, until the Roman conquest (146 BCE), and then the Roman epoch (generally dated until 476 CE) spans eight hundred years. A 19 th century convention which is still widespread sees the Hellenistic period as one of decline, even as Grecian and Greek-inspired thought and arts were gradually spread throughout the Mediterranean world, conquering Rome culturally at the same time as Greece lost its political independence. It is the arts of this period that inspired the classicism and philhellenism of Winckelmann, Lessing, and Goethe. And even if we see the Hellenistic arts as in decline, despite great works like the Lacoön and his Sons or the Venus de Milo , the Hellenistic and Roman periods have bequeathed us invaluable theoretical works advancing the study of painting and sculpture (by Pliny), music (by Theophrastus, Aristoxenus), architecture (by Vitruvius), and literary theory (pre-eminently On the Sublime , long attributed to Cassius Longinus, and Horace’s Ars Poetica ) (Tatarkiewicz 1970, 173, 216–224, 235–237). Hellenistic and then Roman sculpture, turning ever-farther away from the austere, near-Platonic stylisation of the classical period, began more and more to model human actions and distinct individuals (Durant 1939, 616–626). The rise of Rome saw golden and silver ages of Latin literature, led by giants such as Virgil, Horace, Cicero, Lucretius, Seneca, and Juvenal, and the mastery of new genres, led by satire. The Roman textbooks in rhetoric, such as the Rhetorica ad Herennium and Cicero’s De Oratore , continue to shape rhetorical studies and reward rereading. The Romans made literally monumental advances in architecture, enabled by their mastery of brick, concrete, and the arch, and achieving engineering and aesthetic wonders like the dome of the Pantheon (Durant 1944, 357–362).
The period is also not without innovations in philosophical considerations of art and beauty, albeit reframed in the terms of the larger Hellenistic interest in philosophical ethics and self-transformation through practicing regimens of spiritual exercises (P. Hadot 1995; 1998; I. Hadot 2014). The lasting classical designations of beauty as characterised by order, symmetry, harmony of parts in relation to the whole, and appropriateness remain canonical. Despite Epicurus’ advice that his students should flee paideia and the liberal arts, the greatest extant Epicurean work is the great Latin poem De Rerum Natura by Lucretius, in which the hard truths of Epicurean philosophy are sweetened with the “wormwood” of vigorous verse (Book 4, Proem). Despite the Sceptics’ attacks on music, musical, and literary theory (Tatarkiewicz 1970, 181–185), the eclectic Platonist Cicero (106–43 BCE) introduces the notion of an innate, distinctly human aesthetic sense: “And it is no mean manifestation of Nature and Reason that man is the only creature that has a feeling for order, for decorum, for moderation in word and deed” ( De Officiis I, 4, 14). In his Orator , we find the first sketch of an adaption of Plato’s conception of the metaphysical Ideas to explain the creative vision of the artist:
For example, in the case of the statues of Phidias, the most perfect of their kind that we have ever seen, and in the case of the paintings I have mentioned, we can, in spite of their beauty, imagine something more beautiful. Surely, that great sculptor, while making the image of Jupiter or Minerva, did not look at any person whom he was using as a model, but in his own mind there dwelt a surpassing image of beauty; at this he gazed and, all intent on this, he guided his artist’s hand to produce the likeness of a god. ( Orator 2, 8; cf. Plato, Timaeus 28a)
In Cicero, too, as in the Stoics, we find in the wonderful vision of human nature of Laws I the specification that humans have been uniquely formed to contemplate the beauties of the heavens: “while [nature] has debased the forms of other animals, who live to eat rather than eat to live, she has bestowed on man an erect stature, and an open countenance, and thus prompted him to the contemplation of heaven, the ancient home of his kindred immortals” (Cicero, Laws I, 26–27).
Amongst the Stoics themselves, the arts tend to be assessed pre-eminently in an ethical, if not political purview (iv). They are aids or hindrances for shaping people to the virtues. Like Plato ( Republic 606b), some Stoics worry that the tragic poets’ staging of extreme passions and suffering is a potential source of ethical corruption. The physical beauty of men and women so adored by wider ancient culture they deem as something “indifferent,” being capable of harming as well as helping their possessors, if not guided by wisdom (Tatarkiewicz 1970, 187–188).
Nevertheless, the Stoics’ larger conception of the kosmos depicted the world as the product and embodiment of the “poetic” fire of the form-giving Logos immanent in all things. The kosmos for the Stoics was hence the supremely beautiful thing, as for the Pythagoreans. As the middle Stoic Posidonius wrote, “the world is beautiful. This is clear from its shape, colour, and rich array of stars” (at Tatarkiewicz 1970, 188). So, in Seneca ( De Otio 5) as in Epictetus (c. 50–135 CE), there are passages assigning to the contemplation of the world and living things a much higher place than the enjoyment of human-made objects, even the most beautiful:
God has introduced man, as a spectator of Himself and of his works; and not only as a spectator, but an interpreter of them … [people should] end where nature itself has fixed our end; and that is, in contemplation and understanding, and in a scheme of life conformable to nature.  You take a journey to Olympia to behold the work of Phidias, and each of you thinks it a misfortune to die without a knowledge of such things; and will you have no inclination to see and understand those works for which there is no need to take a journey, but which are ready and at hand even to those who bestow no pains! Will you never perceive what you are, or for what you were born, or for what purpose you are admitted to behold this spectacle? (Epictetus, Discourses I, 8)
For the Stoics, since the Logos shapes all things, the fully enlightened “sage” will indeed be able to see the value and beauty in even the smallest things, in their relation to the whole, as in this striking passage from Marcus Aurelius’ (121–180 CE) Meditations :
We must also observe closely points of this kind, that even the secondary effects of Nature’s processes possess a sort of grace and attraction. To take one instance, bread when it is being baked breaks open at some places; now even these cracks, which in one way contradict the promise of the baker’s art, somehow catch the eye and stimulate in a special way our appetite for the food. … Ears of corn too when they bend downwards, the lion’s wrinkled brow, the foam flowing from the boar’s mouth, and many other characteristics that are far from beautiful if we look at them in isolation, do nevertheless because they follow from Nature’s processes lend those a further ornament and a fascination. ( Meditations III, 2; P. Hadot 1998, 168–69).
In order to achieve such a vision, however, the philosopher must retrain his vision (ii) to “a thorough knowledge of the workings of the universe” (Marcus Aurelius, Meditations III, 2). This will involve cultivating the ability to see natural things as existing independently of and indifferent to the habitual purposes with which human hopes and fears usually clothe them (P. Hadot 1972). For such reasons as these, the philologist-philosopher Pierre Hadot has intriguingly argued that, if we were to seek the closest analogy to the modern concept of aesthetic perception, we should look at it in ancient philosophical discourses concerning the figure of the sage (1995, 251–263). Only a figure who has fully conquered their fears, prejudices, and desires could fully “see” and savour the world, and every one of their experiences within it, in the ways most of us only experience in moments of absorption in beautiful works of art (cf. Sharpe 2018). The question of whether such a figure, one with deep parallels in near and far Eastern conceptions of the wise person, is possible today is a question worth deep reflection.
Concluding remarks
We can only hope here to have introduced readers to some of the key features, figures, ideas, and debates in the vast field of the ancients’ reflections on art and to kalon . We have seen their overarching sense of beauty as order, symmetry, and harmony (i). We have approached the ways that this was at once shaped by as it shaped their understandings of music, architecture, sculpture, and human beauty. We have considered how, whilst the Greeks did not pre-eminently focus on the subjective experience of art and beauty, they nevertheless understood how the arts such as music and drama can powerfully affect us (ii). The Pythagoreans and Aristotle developed theories of art’s psychagogic and psychotherapeutic, cathartic capacities. Plato and the Platonists stressed beauty’s power to animate our desire. In Cicero and the Stoics, the notion of an innate human sense of beauty are spelled out (Mason 2016, 124–126). Throughout antiquity, artistic canons were produced codifying technical standards of beauty and excellence in the arts, a tradition in which Aristotle’s Poetics may be placed (iii).
Plato and Aristotle, in these lights, centrally considered the arts in their political writings, each of them moved by a sense of the centrality which art either does, regrettably, or should, ideally, play in the education of good citizens and human beings (iv). Plato and the Stoics, in particular, were anxious that the powers of art could be used for evil as well as good ends. We saw that Aristotle nevertheless thinks that the leisured cultivation and enjoyment of the arts should be the key part of the goal or excellence of the ideal city. We closed by seeing how in the Stoics in particular—but examples from other philosophies could be given—there is also a sense that the fully enlightened person or sage would perceive the world with the same absorbed but impartial, captivated but contemplative mode of perception that modern theorists have assigned to aesthetic experience.
The classical achievements in the plastic, constructive, and literary arts, as well as in poetics and reflections upon art, continue to exert a determinative effect on Western and now global cultures. Their larger sense of order and beauty will attract admirers and may even be surmised to be something the modern world urgently needs to rediscover as great, destabilising ecological and political crises again beckon. As Albert Camus (1913–1960), a great 20 th century philhellene, wrote in 1948,
the Mediterranean has a solar tragedy that has nothing to do with mists. There are evenings, at the foot of mountains by the sea, when night falls on the perfect curve of a little bay and an anguished fullness rises from the silent waters. Such moments make one realize that if the Greeks knew despair, they experienced it always through beauty and its oppressive quality. In this golden sadness, tragedy reaches its highest point. But the despair of our world—quite the opposite—has fed on ugliness and upheavals. That is why Europe would be ignoble if suffering ever could be. We have exiled beauty; the Greeks took arms for it. A basic difference—but one that has a long history. (Camus 1970, 148)
References
Altman, William H. F. 2013. Plato the Teacher: The Crisis of the “Republic.” Lanham, MD: Lexington.
Anderson, Gene H. 1983. “Pythagoras and the Origin of Music Theory.” Indiana Theory Review 6, no. 3: 35-61.
Burkert, Walter. 1966. “Greek Tragedy and Sacrificial Ritual.” GRBS 7: 87–121.
Camus, Albert. 1970. “Helen’s Exile.” In Albert Camus: Lyrical and Critical Essays , edited by Phillip Thody, translated by Ellen Conroy Kennedy, 148–153. New York: Vintage.
Durant, Will. 1939. The Life of Greece. New York: Simon & Schuster.
Durant, Will. 1944. Caesar and Christ. New York: Simon & Schuster.
Grube, G. M. 1927. “Plato’s Theory of Beauty.” The Monist 37, no. 2 (April): 269–288.
Hadot, Ilsetraut. 2014. Sénèque: Direction spirituelle et pratique de la philosophie . Paris: Vrin.
Hadot, Pierre. 1972. “La physique comme exercise spirituel ou pessimisme et optimisme chez Marc Aurèle.” Revue de théologie et de philosophie 22: 225–239.
Hadot, Pierre. 1995. Philosophy as a Way of Life. Trans. Michael Chase. London: Wiley-Blackwell.
Hadot, Pierre. 1998. The Inner Citadel : The Meditations of Marcus Aurelius. Trans. Michael Chase. Harvard: Belknap Press.
Hadot, Pierre. 2010a. “L’Homme antique et la nature.” In Études de philosophie ancienne , 307–318. Paris: Belles Lettres.
Hadot, Pierre. 2010b. “La Génie du lieu dans la Grèce antique.” In Études de philosophie ancienne, 319-326. Paris: Belles Lettres.
Jones, John. 1962. On Aristotle and Greek Tragedy. Stanford: Stanford University Press.
Lord, Carnes. 1978. “On Damon and Music Education.” Hermes 106: 32–43.
Lord, Carnes. 1982. Education and Culture in the Political Thought of Aristotle. Ithaca, NY: Cornell University Press.
Losev, Alexei. (1963) 2000. History of Ancient Aesthetics, Early Classics, Vol. 1 . Moscow: AST.
Marshall, John. 1953. “Art and Aesthetic in Aristotle.” The Journal of Aesthetics and Art Criticism 12, no. 2 (December): 228–231.
Mason, Andrew. 2016. Ancient Aesthetics. London: Taylor & Francis.
Sharpe, Matthew. 2018. “Towards a Phenomenology of Sagesse : On the Unique Philosophical Problematic of Pierre Hadot.” Angelaki 234, no. 2: 125–138.
Sider, David. 2012. “Plato’s Early Aesthetics: The Hippias Major.” In Plato on Art and Beauty , edited by A.E. Denham, 75–83. London: Palgrave-Macmillan.
Strauss, Leo. 1964. The City and Man . Chicago: University of Chicago Press.
Tatarkiewicz, Wladyslaw. 1970. Ancient Aesthetics. Mouton, the Hague: Polish Scientific Publishers.
Translations of Classical Texts
Aristotle. 1984. De Mundi. In The Complete Works of Aristotle; The Revised Oxford Translation , Vol. 1 , edited by Jonathan Barnes. Princeton, NJ: Princeton University Press.
Aristotle. 1980. Metaphysics. Translated by H. Treddenick. Loeb Classical Library 271. Cambridge, MA: Harvard University Press.
Aristotle. 1902. Poetics . Edited and translated by S.H. Butcher. 3rd ed. London: Macmillan & Co.
Aristotle. 1944. Politics. Translated by H. Rackham. Loeb Classical Library 264. Cambridge, MA: Harvard University Press.
Aurelius, Marcus. 1944. Meditations. Translated by A.S.L Farquharson. Oxford: Clarendon Press.
Cicero, Marcus Tullius. 1913. De Officiis . Translated by Walter Miller. London: William Heinemann.
Cicero, Marcus Tullius. 1842. Laws . In The Political Works of Marcus Tullius Cicero , translated by Francis Barham. London: Edmond Spettigue.
Cicero, Marcus Tullius. 1939. Orator . Translated by G.L. Hendrickson and H.M. Hubbell. Loeb Classical Library 342. Cambridge, MA: Harvard University Press.
Epictetus. 1890. Discourses . In The Works of Epictetus: Consisting of His Discourses, in Four Books, the Enchiridion, and Fragments , translated by Thomas Wentworth Higgenson. 2 vols. Boston: Little, Brown, and Company.
Plato. 1892. Laws. In The Dialogues of Plato , Vol. V. Translated by Benjamin Jowett. New York: Macmillan & Co.
Plato. 1925. Philebus . In Statesman, Philebus, Ion , translated by H.N. Fowler and W.R.M. Lamb, 202-399. Loeb Classical Library 164. Cambridge, MA: Harvard University Press.
Plato. 1969. Republic. In Plato in Twelve Volumes , Vols. 5 & 6. Translated by Paul Shorey. Cambridge, MA: Harvard University Press.
Plato. 1925. Statesman . In Statesman, Philebus, Ion , translated by H.N. Fowler and W.R.M. Lamb, 4-195. Loeb Classical Library 164. Cambridge, MA: Harvard University Press.
Plato. 1925. Symposium. In P lato in Twelve Volumes, Vol. 9 . Translated by Harold N. Fowler. Cambridge, MA: Harvard University Press.
Plato. 1925. Timaeus. In Plato in Twelve Volumes , Vol. 9. Translated by W.R.M. Lamb. Cambridge, MA: Harvard University Press.
- In what follows, (i), (ii), (iii), and (iv) in brackets will refer back to this rubric. ↵
- References to classical texts are given by the author, title, and locators that are consistent across editions and translations. Where quotations from English translations are given in the chapter, see “Translations of classical texts” at the end of the chapter for the specific English translations used or adapted by the author. ↵


6.7: Sage Exercises
The following exercises are less about cosets and subgroups, and more about using Sage as an experimental tool. They are designed to help you become both more efficient, and more expressive, as you write commands in Sage. We will have many opportunities to work with cosets and subgroups in the coming chapters. These exercises do not contain much guidance, and get more challenging as they go. They are designed to explore, or confirm, results presented in this chapter or earlier chapters.
Important
: You should answer each of the last three problems with a single (complicated) line of Sage that concludes by outputting
True
. A “single line” means you will have several Sage commands packaged up together in complicated ways. It does not mean several Sage commands seperated by semi-colons and typed in on a single line. Be sure include some intermediate steps used in building up your solution, but using smaller ranges of values so as to not overwhelm the reader with lots of output. This will help you, and the grader of your work, have some confidence that the final version is correct.
When you check integers below for divisibility, remember that
range()
produces plain integers, which are quite simple in their functionality. The
srange()
command produces Sage integers, which have many more capabilities. (See the last exercise for an example.) And remember that a list comprehension is a very compact way to examine many possibilities at once.
Use
.subgroups()
to find an example of a group \(G\) and an integer \(m\text{,}\) so that (a) \(m\) divides the order of \(G\text{,}\) and (b) \(G\) has no subgroup of order \(m\text{.}\) (Do not use the group \(A_4\) for \(G\text{,}\) since this is in the text.) Provide a single line of Sage code that has all the logic to produce the desired \(m\) as its output. (You can give your group a simple name on a prior line and then just reference the group by name.) Here is a very simple example that might help you structure your answer.
Verify the truth of Fermat's Little Theorem (either variant) using the composite number \(391=17\cdot 23\) as the choice of the base (either \(a\) or \(b\)), and for \(p\) assuming the value of every prime number between \(100\) and \(1000\text{.}\)
Build up a solution slowly — make a list of powers (start with just a few primes), then make a list of powers reduced by modular arithmetic, then a list of comparisons with the predicted value, then a check on all these logical values resulting from the comparisons. This is a useful strategy for many similar problems. Eventually you will write a single line that performs the verification by eventually printing out
True
. Here are some more hints about useful functions.
Verify that the group of units mod \(n\) has order \(n-1\) when \(n\) is prime, again for all primes between \(100\) and \(1000\text{.}\) As before, your output should be simply
True
, just once, indicating that the statement about the order is true for all the primes examined. As before, build up your solution slowly, and with a smaller range of primes in the beginning. Express your answer as a single line of Sage code.
Verify Euler's Theorem for all values of \(0\lt n\lt 100\) and for \(1\leq a \leq n\text{.}\) This will require nested
for
statements with a conditional. Again, here is a small example that might be helpful for constructing your one line of Sage code. Note the use of
srange()
in this example.
The symmetric group on \(7\) symbols, \(S_7\text{,}\) has \(7! = 5040\) elements. Consider the following questions without employing Sage, based on what we know about orders of elements of permutation groups (Exercise \(5.4.13\)).
- What is the maximum possible order?
- How many elements are there of order \(10\text{?}\)
- How many elements are there of order \(1\text{?}\)
- How many elements are there of order \(2\text{?}\)
- What is the smallest positive integer for which there is no element with that order?
These questions will be easier if you are familiar with using binomial coefficients for counting in similarly complex situations. But either way, give some serious thought to each question (and maybe a few of your own) before firing up Sage.
Now, compute how many elements there are of each order using the
.order()
method, and then embed this into a list comprehension which creates a single list of these counts. You can check your work (or check Sage) by wrapping this list in
sum()
and hopefully getting \(5040\text{.}\)
Comment on the process of studying these questions first without any computational aid, and then again with Sage. For which values of \(n\) do you think Sage would be too slow and your mind quicker?