2.2: Trade-Offs
As suggested in the previous section, accounting can play a role in reducing both adverse selection and moral hazard. However, because these two problems relate to two different user needs (i.e., the need to predict future investment performance and the need to evaluate management stewardship), it is unlikely that accounting information will always be perfectly and simultaneously useful in alleviating these problems. For example, information about the current values of assets may help an investor better predict the future economic prospects of the company, particularly in the short term. However, current values may not reveal much about management stewardship, as managers have very little control over market conditions. Similarly, the depreciated historical cost of property, plant, and equipment assets can reveal something about management's decision-making processes regarding the purchase and use of these assets, but historical costs provide very little value in estimating future returns. Accounting standard setters recognize that any specific disclosure may not meet all users' needs, and as such, trade-offs are necessary in setting standards. Sometimes trade-offs between different user purposes are required, and sometimes the trade-off is simply a matter of evaluating the cost of producing the information compared with the benefit received. Because of these trade-offs, accounting information must be viewed as an imperfect solution to the problem of information asymmetry. Still, those who set accounting standards attempt to create the framework for the production of information that will be useful to all readers, in particular to the primary user groups of investors, lenders, and creditors.


10.1: Why It Matters- Microsoft PowerPoint Intermediate Skills
Why learn to complete intermediate skills in Microsoft PowerPoint?
In the last module, we focused on some basic functionality in PowerPoint. But this program is capable of a lot more, including incorporating images and videos as well as allowing you to precisely position items.
Before we move into the mechanics of each of these, let us consider for a moment the types of information you may wish to convey in a presentation.
- You want to show a clip from an advertising campaign.
- You want to add information to every slide
All of these questions can be answered with the valuable tools available in PowerPoint.
Contributors and Attributions
CC licensed content, Original
- Why It Matters: Microsoft PowerPoint Intermediate Skills. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


33.3: Accelerators Create Matter from Energy
Learning Objectives
By the end of this section, you will be able to:
- State the principle of a cyclotron.
- Explain the principle of a synchrotron.
- Describe the voltage needed by an accelerator between accelerating tubes.
- State Fermilab’s accelerator principle.
Before looking at all the particles we now know about, let us examine some of the machines that created them. The fundamental process in creating previously unknown particles is to accelerate known particles, such as protons or electrons, and direct a beam of them toward a target. Collisions with target nuclei provide a wealth of information, such as information obtained by Rutherford using energetic helium nuclei from natural \(α\) radiation. But if the energy of the incoming particles is large enough, new matter is sometimes created in the collision. The more energy input or \(ΔE\), the more matter \(m\) can be created, since \(m=ΔE/c^2\). Limitations are placed on what can occur by known conservation laws, such as conservation of mass-energy, momentum, and charge. Even more interesting are the unknown limitations provided by nature. Some expected reactions do occur, while others do not, and still other unexpected reactions may appear. New laws are revealed, and the vast majority of what we know about particle physics has come from accelerator laboratories. It is the particle physicist’s favorite indoor sport, which is partly inspired by theory.
Early Accelerators
An early accelerator is a relatively simple, large-scale version of the electron gun. The Van de Graaff (named after the Dutch physicist), which you have likely seen in physics demonstrations, is a small version of the ones used for nuclear research since their invention for that purpose in 1932 (Figure \(\PageIndex{1}\)). These machines are electrostatic, creating potentials as great as 50 MV, and are used to accelerate a variety of nuclei for a range of experiments. Energies produced by Van de Graaffs are insufficient to produce new particles, but they have been instrumental in exploring several aspects of the nucleus.
Another, equally famous, early accelerator is the cyclotron , invented in 1930 by the American physicist, E. O. Lawrence (1901–1958). For a visual representation with more detail, see Figure \(\PageIndex{2}\). Cyclotrons use fixed-frequency alternating electric fields to accelerate particles. The particles spiral outward in a magnetic field, making increasingly larger radius orbits during acceleration. This clever arrangement allows the successive addition of electric potential energy and so greater particle energies are possible than in a Van de Graaff. Lawrence was involved in many early discoveries and in the promotion of physics programs in American universities. He was awarded the 1939 Nobel Prize in Physics for the cyclotron and nuclear activations, and he has an element and two major laboratories named for him.
A synchrotron is a version of a cyclotron in which the frequency of the alternating voltage and the magnetic field strength are increased as the beam particles are accelerated. Particles are made to travel the same distance in a shorter time with each cycle in fixed-radius orbits. A ring of magnets and accelerating tubes, as shown in Figure \(\PageIndex{2}\), are the major components of synchrotrons. Accelerating voltages are synchronized (i.e., occur at the same time) with the particles to accelerate them, hence the name. Magnetic field strength is increased to keep the orbital radius constant as energy increases. High-energy particles require strong magnetic fields to steer them, so superconducting magnets are commonly employed. Still limited by achievable magnetic field strengths, synchrotrons need to be very large at very high energies, since the radius of a high-energy particle’s orbit is very large. Radiation caused by a magnetic field accelerating a charged particle perpendicular to its velocity is called synchrotron radiation in honor of its importance in these machines. Synchrotron radiation has a characteristic spectrum and polarization, and can be recognized in cosmic rays, implying large-scale magnetic fields acting on energetic and charged particles in deep space. Synchrotron radiation produced by accelerators is sometimes used as a source of intense energetic electromagnetic radiation for research purposes.
Modern Behemoths and Colliding Beams
Physicists have built ever-larger machines, first to reduce the wavelength of the probe and obtain greater detail, then to put greater energy into collisions to create new particles. Each major energy increase brought new information, sometimes producing spectacular progress, motivating the next step. One major innovation was driven by the desire to create more massive particles. Since momentum needs to be conserved in a collision, the particles created by a beam hitting a stationary target should recoil. This means that part of the energy input goes into recoil kinetic energy, significantly limiting the fraction of the beam energy that can be converted into new particles. One solution to this problem is to have head-on collisions between particles moving in opposite directions. Colliding beams are made to meet head-on at points where massive detectors are located. Since the total incoming momentum is zero, it is possible to create particles with momenta and kinetic energies near zero. Particles with masses equivalent to twice the beam energy can thus be created. Another innovation is to create the antimatter counterpart of the beam particle, which thus has the opposite charge and circulates in the opposite direction in the same beam pipe. For a schematic representation, see Figure \(\PageIndex{3}\).
Detectors capable of finding the new particles in the spray of material that emerges from colliding beams are as impressive as the accelerators. While the Fermilab Tevatron had proton and antiproton beam energies of about 1 TeV, so that it can create particles up to 2 \(TeV/c^2\), the Large Hadron Collider (LHC) at the European Center for Nuclear Research (CERN) has achieved beam energies of 3.5 TeV, so that it has a 7-TeV collision energy; CERN hopes to double the beam energy in 2014. The now-canceled Superconducting Super Collider was being constructed in Texas with a design energy of 20 TeV to give a 40-TeV collision energy. It was to be an oval 30 km in diameter. Its cost as well as the politics of international research funding led to its demise.
Figure \(\PageIndex{4}\)
: This schematic shows the two rings of Fermilab’s accelerator and the scheme for colliding protons and antiprotons (not to scale).
In addition to the large synchrotrons that produce colliding beams of protons and antiprotons, there are other large electron-positron accelerators. The oldest of these was a straight-line or linear accelerator, called the Stanford Linear Accelerator (SLAC), which accelerated particles up to 50 GeV as seen in Figure. Positrons created by the accelerator were brought to the same energy and collided with electrons in specially designed detectors. Linear accelerators use accelerating tubes similar to those in synchrotrons, but aligned in a straight line. This helps eliminate synchrotron radiation losses, which are particularly severe for electrons made to follow curved paths. CERN had an electron-positron collider appropriately called the Large Electron-Positron Collider (LEP), which accelerated particles to 100 GeV and created a collision energy of 200 GeV. It was 8.5 km in diameter, while the SLAC machine was 3.2 km long.
Figure \(\PageIndex{5 }\)
:
The Stanford Linear Accelerator was 3.2 km long and had the capability of colliding electron and positron beams. SLAC was also used to probe nucleons by scattering extremely short wavelength electrons from them. This produced the first convincing evidence of a quark structure inside nucleons in an experiment analogous to those performed by Rutherford long ago.
Example \(\PageIndex{1}\):Calculating the Voltage Needed by the Accelerator Between Accelerating Tubes
A linear accelerator designed to produce a beam of 800-MeV protons has 2000 accelerating tubes. What average voltage must be applied between tubes (such as in the gaps in Figure) to achieve the desired energy?
Strategy
The energy given to the proton in each gap between tubes is \(PE_{elec}=qV\) where \(q\) is the proton’s charge and \(V\) is the potential difference (voltage) across the gap. Since \(q=q_e=1.6×10^{−19}C\) and \(1 eV=(1 V)(1.6×10^{−19}C)\), the proton gains 1 eV in energy for each volt across the gap that it passes through. The AC voltage applied to the tubes is timed so that it adds to the energy in each gap. The effective voltage is the sum of the gap voltages and equals 800 MV to give each proton an energy of 800 MeV.
Solution
There are 2000 gaps and the sum of the voltages across them is 800 MV; thus,
\(V_{gap}=\frac{800 MV}{2000}=400 kV\).
Discussion
A voltage of this magnitude is not difficult to achieve in a vacuum. Much larger gap voltages would be required for higher energy, such as those at the 50-GeV SLAC facility. Synchrotrons are aided by the circular path of the accelerated particles, which can orbit many times, effectively multiplying the number of accelerations by the number of orbits. This makes it possible to reach energies greater than 1 TeV.
Summary
- A variety of particle accelerators have been used to explore the nature of subatomic particles and to test predictions of particle theories.
- Modern accelerators used in particle physics are either large synchrotrons or linear accelerators.
- The use of colliding beams makes much greater energy available for the creation of particles, and collisions between matter and antimatter allow a greater range of final products.
Glossary
- colliding beams
- head-on collisions between particles moving in opposite directions
- cyclotron
- accelerator that uses fixed-frequency alternating electric fields and fixed magnets to accelerate particles in a circular spiral path
- linear accelerator
- accelerator that accelerates particles in a straight line
- synchrotron
- a version of a cyclotron in which the frequency of the alternating voltage and the magnetic field strength are increased as the beam particles are accelerated
- synchrotron radiation
- radiation caused by a magnetic field accelerating a charged particle perpendicular to its velocity
- Van de Graaff
- early accelerator: simple, large-scale version of the electron gun


9.6: Solve Applications of Quadratic Equations
By the end of this section, you will be able to:
- Solve applications modeled by quadratic equations
Before you get started, take this readiness quiz.
-
The sum of two consecutive odd numbers is \(−100\). Find the numbers.
If you missed this problem, review Example 2.18. -
Solve: \(\frac{2}{x+1}+\frac{1}{x-1}=\frac{1}{x^{2}-1}\).
If you missed this problem, review Example 7.35. -
Find the length of the hypotenuse of a right triangle with legs \(5\) inches and \(12\) inches.
If you missed this problem, review Example 2.34.
Solve Applications Modeled by Quadratic Equations
We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications.
Let’s first summarize the methods we now have to solve quadratic equations.
Methods to Solve Quadratic Equations
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula
As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.
Use a Problem-Solving Strategy
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Solve the equation using algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
We have solved number applications that involved consecutive even and odd integers, by modeling the situation with linear equations. Remember, we noticed each even integer is \(2\) more than the number preceding it. If we call the first one \(n\), then the next one is \(n+2\). The next one would be \(n+2+2\) or \(n+4\). This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
\(\begin{array}{cl}{}&{\text{Consecutive even integers}}\\{}& {64,66,68}\\ {n} & {1^{\text { st }} \text { even integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive even integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive even integer }}\end{array}\)
\(\begin{array}{cl}{}&{\text{Consecutive odd integers}}\\{}& {77,79,81}\\ {n} & {1^{\text { st }} \text { odd integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive odd integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive odd integer }}\end{array}\)
Some applications of odd or even consecutive integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
The product of two consecutive odd integers is \(195\). Find the integers.
Solution :
Step 1 : Read the problem
Step 2 : Identify what we are looking for.
We are looking for two consecutive odd integers.
Step 3 : Name what we are looking for.
Let \(n=\) the first odd integer.
\(n+2=\) the next odd integer.
Step 4 : Translate into an equation. State the problem in one sentence.
“The product of two consecutive odd integers is \(195\).” The product of the first odd integer and the second odd integer is \(195\).
Translate into an equation.
\(n(n+2)=195\)
Step 5 : Solve the equation. Distribute.
\(n^{2}+2 n=195\)
Write the equation in standard form.
\(n^{2}+2 n-195=0\)
Factor.
\((n+15)(n-13)=0\)
Use the Zero Product Property.
\(n+15=0 \quad n-13=0\)
Solve each equation.
\(n=-15, \quad n=13\)
There are two values of \(n\) that are solutions. This will give us two pairs of consecutive odd integers for our solution.
\(\begin{array}{cc}{\text { First odd integer } n=13} & {\text { First odd integer } n=-15} \\ {\text { next odd integer } n+2} & {\text { next odd integer } n+2} \\ {13+2} & {-15+2} \\ {15} & {-13}\end{array}\)
Step 6 : Check the answer.
Do these pairs work? Are they consecutive odd integers?
\(\begin{aligned} 13,15 & \text { yes } \\-13,-15 & \text { yes } \end{aligned}\)
Is their product \(195\)?
\(\begin{aligned} 13 \cdot 15 &=195 &\text{yes} \\-13(-15) &=195 & \text { yes } \end{aligned}\)
Step 7 : Answer the question.
Two consecutive odd integers whose product is \(195\) are \(13,15\) and \(-13,-15\).
The product of two consecutive odd integers is \(99\). Find the integers.
- Answer
-
The two consecutive odd integers whose product is \(99\) are \(9, 11\), and \(−9, −11\).
The product of two consecutive even integers is \(168\). Find the integers.
- Answer
-
The two consecutive even integers whose product is \(128\) are \(12, 14\) and \(−12, −14\).
We will use the formula for the area of a triangle to solve the next example.
Area of a Triangle
For a triangle with base, \(b\), and height, \(h\), the area, \(A\), is given by the formula \(A=\frac{1}{2} b h\).
Recall that when we solve geometric applications, it is helpful to draw the figure.
An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can only have an area of \(120\) square feet and the architect wants the base to be \(4\) feet more than twice the height. Find the base and height of the window.
Solution :
| Step 1 : Read the problem. Draw a picture. | |
| Step 2 : Identify what we are looking for. | We are looking for the base and height. |
| Step 3 : Name what we are looking for. |
Let \(h=\) the height of the triangle. \(2h+4=\) the base of the triangle. |
|
Step 4 : Translate into an equation. We know the area. Write the formula for the area of a triangle. |
\(A=\frac{1}{2} b h\) |
| Step 5 : Solve the equation. Substitute in the values. | \(120=\frac{1}{2}(2 h+4) h\) |
| Distribute. | \(120=h^{2}+2 h\) |
| This is a quadratic equation, rewrite it in standard form. | \(h^{2}+2 h-120=0\) |
| Factor. | \((h-10)(h+12)=0\) |
| Use the Zero Product Property. | \(h-10=0 \quad h+12=0\) |
| Simplify. | \(h=10, \quad \cancel{h=-12}\) |
| Since \(h\) is the height of a window, a value of \(h=-12\) does not make sense. | |
| The height of the triangle \(h=10\). | |
|
The base of the triangle \(2h+4\). \(2 \cdot 10+4\) \(24\) |
|
|
Step 6 : Check the answer. Does a triangle with height \(10\) and base \(24\) have area \(120\)? Yes. |
|
| Step 7 : Answer the question. | The height of the triangular window is \(10\) feet and the base is \(24\) feet. |
Find the base and height of a triangle whose base is four inches more than six times its height and has an area of \(456\) square inches.
- Answer
-
The height of the triangle is \(12\) inches and the base is \(76\) inches.
If a triangle that has an area of \(110\) square feet has a base that is two feet less than twice the height, what is the length of its base and height?
- Answer
-
The height of the triangle is \(11\) feet and the base is \(20\) feet.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
We will use the formula for the area of a rectangle to solve the next example.
Area of a Rectangle
For a rectangle with length, \(L\), and width, \(W\), the area, \(A\), is given by the formula \(A=LW\).
Mike wants to put \(150\) square feet of artificial turf in his front yard. This is the maximum area of artificial turf allowed by his homeowners association. He wants to have a rectangular area of turf with length one foot less than \(3\) times the width. Find the length and width. Round to the nearest tenth of a foot.
Solution :
| Step 1 : Read the problem. Draw a picture. | |
| Step 2 : Identify what we are looking for. | We are looking for the length and width. |
| Step 3 : Name what we are looking for. |
Let \(w=\) the width of the rectangle. \(3w-1=\) the length of the rectangle |
| Step 4 : Translate into an equation. We know the area. Write the formula for the area of a rectangle. | |
| Step 5 : Solve the equation. Substitute in the values. | |
| Distribute. | |
|
This is a quadratic equation; rewrite it in standard form. Solve the equation using the Quadratic Formula. |
|
| Identify the \(a,b,c\) values. | |
| Write the Quadratic Formula. | |
| Then substitute in the values of \(a,b,c\). | |
| Simplify. | |
| Rewrite to show two solutions. | |
|
Approximate the answers using a calculator. We eliminate the negative solution for the width. |
|
| Step 6 : Check the answer. Make sure that the answers make sense. Since the answers are approximate, the area will not come out exactly to \(150\). | |
| Step 7 : Answer the question. | The width of the rectangle is approximately \(7.2\) feet and the length is approximately \(20.6\) feet. |
The length of a \(200\) square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden, to the nearest tenth of a foot.
- Answer
-
The length of the garden is approximately \(18\) feet and the width \(11\) feet.
A rectangular tablecloth has an area of \(80\) square feet. The width is \(5\) feet shorter than the length.What are the length and width of the tablecloth to the nearest tenth of a foot?
- Answer
-
The length of the tablecloth is approximately \(11.8\) feet and the width \(6.8\) feet.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
Pythagorean Theorem
In any right triangle, where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse, \(a^{2}+b^{2}=c^{2}\).
Rene is setting up a holiday light display. He wants to make a ‘tree’ in the shape of two right triangles, as shown below, and has two \(10\)-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be?
Solution :
| Step 1 : Read the problem. Draw a picture. | |
| Step 2 : Identify what we are looking for. | We are looking for the height of the pole. |
| Step 3 : Name what we are looking for. |
The distance from the base of the pole to either stake is the same as the height of the pole.
Let \(x=\) the height of the pole.
Each side is a right triangle. We draw a picture of one of them. |
|
Step 4 : Translate into an equation.
We can use the Pythagorean Theorem to solve for \(x\).
|
\(a^{2}+b^{2}=c^{2}\) |
| Step 5 : Solve the equation. Substitute. | \(x^{2}+x^{2}=10^{2}\) |
| Simplify. | \(2 x^{2}=100\) |
| Divide by \(2\) to isolate the variable. | \(\frac{2 x^{2}}{2}=\frac{100}{2}\) |
| Simplify. | \(x^{2}=50\) |
| Use the Square Root Property. | \(x=\pm \sqrt{50}\) |
| Simplify the radical. | \(x=\pm 5 \sqrt{2}\) |
| Rewrite to show two solutions. | \(x=5 \sqrt{2}, \quad \cancel{x=-5 \sqrt{2}}\) |
| If we approximate this number to the nearest tenth with a calculator, we find \(x≈7.1\). | |
| Step 6 : Check the answer. Check on your own in the Pythagorean Theorem. | |
| Step 7 : Answer the question. | The pole should be about \(7.1\) feet tall. |
The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is \(20\) feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth.
- Answer
-
The length of the flag pole’s shadow is approximately \(6.3\) feet and the height of the flag pole is \(18.9\) feet.
The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.
- Answer
-
The distance between the opposite corners is approximately \(7.2\) feet.
The height of a projectile shot upward from the ground is modeled by a quadratic equation. The initial velocity, \(v_{0}\), propels the object up until gravity causes the object to fall back down.
The height in feet, \(h\), of an object shot upwards into the air with initial velocity, \(v_{0}\), after \(t\) seconds is given by the formula
\(h=-16 t^{2}+v_{0} t\)
We can use this formula to find how many seconds it will take for a firework to reach a specific height.
A firework is shot upwards with initial velocity \(130\) feet per second. How many seconds will it take to reach a height of \(260\) feet? Round to the nearest tenth of a second.
Solution :
| Step 1 : Read the problem. | |
| Step 2 : Identify what we are looking for. | We are looking for the number of seconds, which is time. |
| Step 3 : Name what we are looking for. | Let \(t=\) the number of seconds. |
| Step 4 : Translate into an equation. Use the formula. | |
| Step 5 : Solve the equation. We know the velocity \(v_{0}\) is \(130\) feet per second. The height is \(260\) feet. Substitute the values. | |
| This is a quadratic equation, rewrite it in standard form. Solve the equation using the Quadratic Formula. | |
| Identify the values of \(a, b, c\). | |
| Write the Quadratic Formula. | |
| Then substitute in the values of \(a,b,c\). | |
| Simplify. | |
| Rewrite to show two solutions. | |
| Approximate the answer with a calculator. | |
| Step 6 : Check the answer. The check is left to you. | |
| Step 7 : Answer the question. | The firework will go up and then fall back down. As the firework goes up, it will reach \(260\) feet after approximately \(3.6\) seconds. It will also pass that height on the way down at \(4.6\) seconds. |
An arrow is shot from the ground into the air at an initial speed of \(108\) ft/s. Use the formula \(h=-16 t^{2}+v_{0} t\) to determine when the arrow will be \(180\) feet from the ground. Round the nearest tenth.
- Answer
-
The arrow will reach \(180\) feet on its way up after \(3\) seconds and again on its way down after approximately \(3.8\) seconds.
A man throws a ball into the air with a velocity of \(96\) ft/s. Use the formula \(h=-16 t^{2}+v_{0} t\) to determine when the height of the ball will be \(48\) feet. Round to the nearest tenth.
- Answer
-
The ball will reach \(48\) feet on its way up after approximately \(.6\) second and again on its way down after approximately \(5.4\) seconds.
We have solved uniform motion problems using the formula \(D=rt\) in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula \(D=rt\) assumes we know \(r\) and \(t\) and use them to find \(D\). If we know \(D\) and \(r\) and need to find \(t\), we would solve the equation for \(t\) and get the formula \(t=\frac{D}{r}\).
Some uniform motion problems are also modeled by quadratic equations.
Professor Smith just returned from a conference that was \(2,000\) miles east of his home. His total time in the airplane for the round trip was \(9\) hours. If the plane was flying at a rate of \(450\) miles per hour, what was the speed of the jet stream?
Solution :
This is a uniform motion situation. A diagram will help us visualize the situation.
We fill in the chart to organize the information.
We are looking for the speed of the jet stream. Let \(r=\) the speed of the jet stream.
When the plane flies with the wind, the wind increases its speed and so the rate is \(450 + r\).
When the plane flies against the wind, the wind decreases its speed and the rate is \(450 − r\).
|
Write in the rates.
Write in the distances. Since \(D=r⋅t\), we solve for \(t\) and get \(t=\frac{D}{r}\). We divide the distance by the rate in each row, and place the expression in the time column. |
|
|
We know the times add to \(9\)
and so we write our equation. |
\(\frac{2000}{450-r}+\frac{2000}{450+r}=9\) |
| We multiply both sides by the LCD. | \((450-r)(450+r)\left(\frac{2000}{450-r}+\frac{2000}{450+r}\right)=9(450-r)(450+r)\) |
| Simplify. | \(2000(450+r)+2000(450-r)=9(450-r)(450+r)\) |
| Factor the \(2,000\). | \(2000(450+r+450-r)=9\left(450^{2}-r^{2}\right)\) |
| Solve. | \(2000(900)=9\left(450^{2}-r^{2}\right)\) |
| Divide by \(9\). | \(2000(100)=450^{2}-r^{2}\) |
| Simplify. |
\(\begin{aligned}200000&=202500-r^{2} \\ -2500&=-r^{2}\\ 50&=r\end{aligned}\)\ The speed of the jet stream is \(50\) mph. |
|
Check: Is \(50\) mph a reasonable speed for the jet stream? Yes. If the plane is traveling \(450\) mph and the wind is \(50\) mph, Tailwind \(450+50=500 \mathrm{mph} \quad \frac{2000}{500}=4\) hours Headwind \(450-50=400 \mathrm{mph} \quad \frac{2000}{400}=5\) hours The times add to \(9\) hours, so it checks. |
The speed of the jet stream was \(50\) mph.
MaryAnne just returned from a visit with her grandchildren back east. The trip was \(2400\) miles from her home and her total time in the airplane for the round trip was \(10\) hours. If the plane was flying at a rate of \(500\) miles per hour, what was the speed of the jet stream?
- Answer
-
The speed of the jet stream was \(100\) mph.
Gerry just returned from a cross country trip. The trip was \(3000\) miles from his home and his total time in the airplane for the round trip was \(11\) hours. If the plane was flying at a rate of \(550\) miles per hour, what was the speed of the jet stream?
- Answer
-
The speed of the jet stream was \(50\) mph.
Work applications can also be modeled by quadratic equations. We will set them up using the same methods we used when we solved them with rational equations.We’ll use a similar scenario now.
The weekly gossip magazine has a big story about the presidential election and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes \(12\) hours more than Press #2 to do the job and when both presses are running they can print the job in \(8\) hours. How long does it take for each press to print the job alone?
Solution :
This is a work problem. A chart will help us organize the information.
We are looking for how many hours it would take each press separately to complete the job.
|
Let \(x=\) the number of hours for Press #2 to complete the job.
Enter the hours per job for Press #1, Press #2, and when they work together. |
|
|
The part completed by Press #1 plus the part completed by Press #2 equals the amount completed together.
Translate to an equation. |
|
| Solve. | |
| Multiply by the LCD, 8\(x(x+12)\). | |
| Simplify. | |
| Solve. | |
| Since the idea of negative hours does not make sense, we use the values \(x=12\). | |
| Write our sentence answer. | Press #1 would take \(24\) hours and Press #2 would take \(12\) hours to do the job alone. |
The weekly news magazine has a big story naming the Person of the Year and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes \(6\) hours more than Press #2 to do the job and when both presses are running they can print the job in \(4\) hours. How long does it take for each press to print the job alone?
- Answer
-
Press #1 would take \(12\) hours, and Press #2 would take \(6\) hours to do the job alone.
Erlinda is having a party and wants to fill her hot tub. If she only uses the red hose it takes \(3\) hours more than if she only uses the green hose. If she uses both hoses together, the hot tub fills in \(2\) hours. How long does it take for each hose to fill the hot tub?
- Answer
-
The red hose take \(6\) hours and the green hose take \(3\) hours alone.
Access these online resources for additional instruction and practice with solving applications modeled by quadratic equations.
Key Concepts
-
Methods to Solve Quadratic Equations
- Factoring
- Square Root Property
- Completing the Square
- Quadratic Formula
-
How to use a Problem-Solving Strategy.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
-
Area of a Triangle
- For a triangle with base, \(b\), and height, \(h\), the area, \(A\), is given by the formula \(A=\frac{1}{2}bh\).
-
Area of a Rectangle
- For a rectangle with length,\(L\), and width, \(W\), the area, \(A\), is given by the formula \(A=LW\).
-
Pythagorean Theorem
- In any right triangle, where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse, \(a^{2}+b^{2}=c^{2}\).
-
Projectile motion
- The height in feet, \(h\), of an object shot upwards into the air with initial velocity, \(v_{0}\), after \(t\) seconds is given by the formula \(h=-16 t^{2}+v_{0} t\).


1.1: What is Aesthetics?
It is a notorious characteristic of philosophy that any attempt to define it raises more questions than it answers: if this is true of philosophy more broadly, it is perhaps even more true of that branch known as aesthetics. Though in some respects the modern discipline as we know it today is traceable to eighteenth century European philosophy, the important work done in that century was not isolated from many centuries of work prior. In addition, this is to say nothing of the long tradition of aesthetical work in China and Japan, for example, which can trace its origins at least as far back as the European tradition (and, as we shall see, there are certain similarities of origin). Finally, though aesthetics is often taken today to be concerned with works of art, this is both an overstatement today and at odds with much of historical aesthetics.
The question, then, is not an easy one. In the face of such a dilemma, it is perhaps best to start etymologically: what does the word “aesthetic” mean on its own, and where does it come from? Though it was first brought into common use with the work of the German philosopher Alexander Baumgarten ([1735] 1954), the word is Greek in origin, from the word αἰσθητικός (aisthetikos: Liddell & Short 1940), which refers to the perception and experience of the senses. On this understanding, then, the study of aesthetics is the study of something sensed , in a broad understanding of that word, rather than something imagined or reasoned. That is, the object of study in aesthetics must be, at least in part, sensorial. Of course, one might think that this is true of science, but the difference is crucial: science is the study of the material world in itself, whereas aesthetics—in its most fundamental sense—is about the experience of things in that world. In particular, aesthetics is about their level of pleasantness, as in asking whether a particular experience is pleasant or not.
At this point we begin to arrive not only at a working definition of aesthetics, but also a statement of its most important questions. Perhaps most importantly, we can arrive at an explanation of why its questions are worth asking and why it is a useful discipline to undertake. Our definition, then, might be this: aesthetics is a sub-branch of philosophy that examines questions of the pleasantness of our experiences concerning things in the world (where pleasantness is taken in a broad sense to include, for example, the intellectual pleasure of being challenged or confronted). This is still quite general, but it gives us a framework from which to build a deeper understanding; though, as suggested at the beginning, any hopes of narrowing it down further may be futile. Certainly, the immediate benefit of this definition is that it highlights quite nicely a tension that resides at the heart of all aesthetic work: the tension between personal, subjective experiences and more universal, objective experiences. If we place all experiences on a spectrum, those at the subjective extreme, such as a personal enjoyment of swimming or celery are clearly experiences unique to a particular individual: though of course many people like swimming (and, apparently, celery), we do not expect anyone else to share in this enjoyment. At the other end of the spectrum we find objective experiences, which are so universal as to be applicable to humanity in general—experiences such as hunger, thirst, laughter, physical attraction, tiredness, physical pain, the experience of colour, the experience of feeling the water on one’s skin while swimming, and so on. Objective experiences are not concerned with pleasantness; although we might find the experience of swimming (for example) to be either pleasant or not, nevertheless the experiences that make up the overall concept of swimming, such as the experience of feeling the water on one’s skin, are not in themselves experiences of pleasantness, and so lie outside the discipline of aesthetics. But so, too, do subjective experiences; although a personal like or dislike of eating celery, for example, certainly has to do with pleasantness, it has to do with pleasantness for you , and nobody else. Certainly, one could ask if there is anything that ties together all people who like to enjoy celery, but if the answer is physical, then it’s a question for physics, and if mental, psychology.
If we eliminate the experiences at either extreme, we find in between certain experiences that hold tension between being subjective and objective, personal and universal: experiences like listening to a song, a symphony, or the sound of the waves; looking at a beautiful sunset, a painting by Turner or Tensho Shubun, a sculpture, a piece of graffiti or a dance; or reading a novel or a poem. What’s interesting about these experiences is that they are undoubtedly personal, and yet, unlike the case of liking celery, we expect these experiences to be universal, shared by others. Unlike eating celery, which is either pleasant or not, these other experiences involve a kind of judgment, like “this is beautiful,” making it much closer to an objective experience like “this is yellow.” And, just as we would expect others to agree that a yellow object really is yellow, and think their perceptions wrong or faulty if they disagreed, so too with experiences such as looking at a beautiful sculpture such as the Winged Victory of Samothrace, we expect others to agree that it is beautiful—in fact, at times we expect them to agree even if they don’t like it , allowing a tension between saying “this is a good book, but I don’t personally like it.” And yet, at the same time, these experiences remain deeply personal, subjective. And so we hear and use phrases like “this piece speaks to me about. . . .” It is these kinds of experiences which are the central focus of aesthetics, and so we call these experiences “aesthetic experiences.” This tension between the personal and the universal, then, is the driving principle of the study of aesthetics.
If aesthetics is concerned with experiences such as these, then it becomes clear that to restrict it to any one type of experience or to one tradition is unjustifiable, even ridiculous. And so, though much of the work done by contemporary aestheticians has its roots in only the last few centuries, the ancient world was no stranger to aesthetics. Plato (428/427–348/347 BCE) famously thought the impact that the experience of art could have on people was so powerful as to be dangerous, and that art did not have anything to offer philosophy since it merely imitates reality, whereas philosophy seeks true reality ([380 BCE] 1974, bk. X, 595a–605c). Thus, art is a form of deception, so to speak. [1] In contrast to this, the Epicurean philosopher Philodemus (c. 110-30 BCE) wrote a work dedicated to examining the philosophical import of the Homeric corpus (see Asmis 1991), [2] and Augustine (354–430 CE) ([386–87] 2007) claimed that the study of poetry was an important introductory step into philosophy (2007). In China, Confucius (551–479 BCE) shared Plato’s suspicion of art, yet he valued appreciation of beauty for the sensibilities of the self and for its moral qualities also (1938), while his contemporary in India, Bharata, [3] taught a theory of rasa as the end of the arts, a concept not too dissimilar from the Aristotelian notion of catharsis [4] (1950-1961; see Gerow 2002).
This brief overview of the kinds of experiences we call aesthetic, however, raises another issue that is often overlooked. Put simply, it suggests that the usual restriction of aesthetics to artworks and to natural phenomena is incomplete. After all, it is not uncommon for a mathematical equation to be termed “beautiful,” or for aesthetic concepts and terms to be used in contexts such as social interactions, military maneuvers, and even politics.
Aesthetics as an 18th century discipline
Nevertheless, it is a fact that, as I have said above, the discipline as we know it today has its origins largely in eighteenth century Europe, and so a brief overview of this lineage is not out of place. This section, therefore, provides an historical overview of the origins of aesthetics as a modern philosophical subject in the 18th century, and notes its journey through engagement with fine arts to modern interest in pop culture. The discussion here is not meant to be an exhaustive historical outline, but a demonstration of the central questions of aesthetics through the last three hundred years. This will provide the impetus for a discussion of aesthetics as the study of beauty.
In Paul Guyer’s (2005, 25) turn of phrase, aesthetics
was not baptized until 1735, when the twenty-one-year-old Alexander Gottlieb Baumgarten, in his dissertation “Philosophical Meditations on some matters pertaining to Poetry,” introduced the term to designate “the science for directing the inferior faculty of cognition or the science of how something is to be sensitively cognized.”
Baumgarten, however, was himself working in a field begun some twenty years earlier, with the work of the Earl of Shaftesbury ( Characteristiks of Men, Manners, Opinions, Times, 1711), and his two followers Joseph Addison (“The Pleasures of the Imagination” in The Spectator , 1712) and Frances Hutcheson ( An Inquiry into the Original of our Ideas of Beauty and Virtue, 1725) in Britain, and the work of Jean-Baptiste Du Bos ( Critical Reflections on Poetry, Painting, and Music, 1719) in France. Shaftesbury (1671–1713) made the important distinction, still upheld today, between enjoying something for the benefit it brings one—whether that be physical, mental, emotional, or any other kind of benefit—and enjoying something for its own sake, simply because it is worthy of being enjoyed ([1711] 1999, 318–319).
Shaftesbury’s answer to the fundamental question of aesthetics—how is it that our experience is both subjective and yet in some sense objective and universal—claimed, in a rather Platonic fashion, that the beauty of the natural world and the created works of humanity lead one’s mind “higher,” to an appreciation of the beauty of the entirety of creation, and ultimately to its creator, the source of all beauty (Shaftesbury [1711] 1999, 322ff). This explains how it is we make aesthetic judgments, since we have an objective standard of beauty to which we can refer, though we can only come to know this standard through our experience of its instantiations, thus leading the way to a need for refinement. David Hume, though he discarded the notion of a creator of beauty and instead argued that we move with the imagination to a recognition of some form of utility—whether real or not ([1739–40] 2009, 463–470)—understood the need for some kind of standard to explain our use of aesthetic judgments, and so introduced the idea of an ideal critic whose senses were perfectly refined to the reception of aesthetic experiences (Hume [1757] 2000). [5]
Another important influential distinction of the eighteenth century was made by the British philosopher and statesman, Edmund Burke (1729–1797), who distinguished between the beautiful and the sublime. For Burke ([1757] 2005), beauty is a social quality, “where women and men, and not only they, but when other animals give us a sense of joy and pleasure in beholding them (and there are many that do so), they inspire us with sentiments of tenderness and affection towards their persons” (part 1, sec. 10). The sublime, on the other hand, is the deeper experience, the more profound, “the strongest emotion of which the mind is capable of feeling” (part 1, sec. 7). The sublime is oriented towards what is beyond our comprehension, whereas the beautiful, for Burke, has no apparent end. So, for example, if, in listening to “If Love’s a Sweet Passion” by Henry Purcell, one is moved to a surge of emotion, even to tears, Burke would consider this a sublime experience, because of its power to call up strong and passionate emotions. What is notable about this distinction is that Burke’s concept of the sublime allows for “negative” aesthetic experiences, such as the experience of Jordan Wolfson’s virtual-reality artwork “Real Violence,” to be considered sublime, and therefore positively appraised. Such an artwork is capable of inducing “the strongest emotions” which, for Burke, can ultimately lead us beyond the artwork to something greater, and thus the experience of it is sublime.
Probably the most important philosophical work on aesthetics in the eighteenth century, however, was written by the German philosopher Immanuel Kant (1724–1804), namely the Critique of Judgement (1790). As is evident from the title of his work, Kant took the question of aesthetic judgment as paramount, making it the focus of the first half of his book. A complete discussion of Kant’s work is outside the scope of this chapter, but a few points are worthy of mention here.
First, Kant’s formulation of the faculty of judgment is influenced by Shaftesbury’s and Hume’s, with its most well-known characteristic being a disinterestedness in the object of judgment. What this means is that the observer, the person having the aesthetic experience, has no vested interests in the thing experienced, and so the judgment is outside of any benefit to them (Kant [1790] 2015, sec. 2).
Kant kept Burke’s distinction between the beautiful and the sublime, but modified it in a way that draws together threads from Shaftesbury as well. For Kant, beauty is present when we discern the intelligibility of what we experience without any apparent ultimate purpose. Thus beauty is present, for Kant, at a paradox of being purposive—that is, appearing to have been in some way designed—and being without an actual apparent purpose. As an example, when looking at a flower that we call beautiful, its beauty seems to be designed, to have a purpose. And yet no particular purpose is apparent, no clear concept of “what this beauty is for.” Similarly with a sunset, we may wonder at its beauty, and feel it to be purposive , but there is no clear, definite purpose—after all, what purpose could the beauty of a sunset have? The sublime, on the other hand, comes into play when we stand in the face of something so truly awe-inspiring that it rejects all attempts to understand, and we simply stand in its presence, as it were (Kant [1790] 2015, sec. 23–29). The American Jewish poet and singer-songwriter, Leonard Cohen, expressed this quite nicely when explaining the sentiment of his most famous song, Hallelujah :
This world is full of conflicts and things that cannot be reconciled, but there are moments when we can transcend the [binaries] … and reconcile. … Regardless of how impossible the situation [seems], there is a moment when you open your mouth, throw open your arms and embrace the whole mess … and you just say “Hallelujah! Blessed is the name.” (Cohen 1988)
For Cohen, the song was about acknowledging that there are some things in our world that are so big they are beyond us, and when we glimpse that bigger picture, even a little, our response is to cry out, in Cohen’s words, “Hallelujah!” Cohen’s formulation is particularly fitting because, for Kant (as for Shaftesbury), it is through aesthetic experiences such as these that we come to know the ultimate source of beauty or sublimity.
Kant answers the tension between the personal and universal in aesthetic experiences by linking the experience of the aesthetic with the fundamental nature of rational beings (Kant [1790] 2015, sec. 5). For Kant, it is intrinsic and unique to rationality to be able to see things as valuable in themselves—it is, indeed, the basis of his theory of morality in the Critique of Practical Reason . This ability, however, can be used in two ways: pragmatically or what I shall call “aesthetically.” In the former, we only use this ability with regards to purely practical (especially moral) reasoning, and thus the ability to see something as intrinsically valuable is itself a purely pragmatic ability. As an example, imagine someone comes to you wanting funding for a music preschool. You could reason to yourself that music is intrinsically valuable, and so worth the financial burden of funding the school, and this would be a fair thought process. But notice in this example that the ability to see something as intrinsically valuable is subject to the larger, practical question of “should I fund this music preschool?” This use of intrinsic value as a tool for reasoning is even more common in moral reasoning, where one might reason that it is wrong to hurt an animal because life itself is intrinsically valuable and therefore is worth protecting. Notice again that there is a “and therefore x action should be done.” Clearly, the ability to see something as valuable in itself can become a purely pragmatic ability, that is, a useful skill, but not itself intrinsically valuable. This is because if we only use our ability to see things as valuable in themselves to help us with making decisions, then essentially we are only treating this skill as a tool to be used to improve our decision making about what to do or not to do. Just as our ability to see space (i.e., our ability for depth perception) is a tool which helps us move about the physical world, so, too our ability to see things as valuable in themselves is, if used exclusively for practical and moral reasoning, simply a tool to help us move about the moral world.
In these examples of “pragmatic intrinsic valuing,” though the approach may be uniquely rational, it is still practical; but if we put all practical thoughts to the side, and stand observing something in its intrinsic worth—nature as a whole is the most perfect object of this for Kant (see Kant [1790] 2015, sec. 6)—then we engage in the most uniquely rational activity of all (sec. 49). And, if this is the case, then it follows from the fact that it is uniquely rational that it is also, for Kant, a form of freedom for the rational being, in which rationality is not bound by the necessity to choose or deliberate, but can purely experience the value of something simply because it is valuable. Thus, for Kant, aesthetics becomes the most uniquely personal—even the most uniquely human—activity, since it is the function and expression of rationality to experience aesthetically.
These themes of 18th century aesthetics draw out that tension at the heart of aesthetics, the tension between the personal and the universal. In particular, Kant’s notion of the aesthetic experience as uniquely, even supremely, rational draws out this same tension. It does this by highlighting the uniquely rational element—which is, of course, universally human—and the uniquely personal element of standing in the presence of the source of that experience, coupled with its role (for Kant) as instigator of a personal journey from the beautiful or sublime thing to beauty and sublimity as such. Though, for Kant, such experiences were largely (though not exclusively) found in the natural world, the cause—i.e., whether the object of aesthetic experience is natural or created by humanity—is not important for our discussion. What is important is the connection between the 18th century discipline and that fundamental tension which I have noted earlier. Thus it can truly be said that aesthetics is an 18th century discipline, for it is here that we find the most influential approach to that tension which is at its heart.
Aesthetics as the study of beauty
As the foregoing discussion has highlighted, the origins of modern philosophical aesthetics in the eighteenth century has tended to focus on the question of beauty (and its correlatives, such as sublimity, ugliness, and so on). This immediately raises the question, of course, of what is meant by beauty, for this is not a simple property like redness or squareness. Rather, beauty is a quality, intangibly constituted by different features in different edges, and what is beautiful in one thing might not be in another—for example, hard edges may look attractive on a building, but not on a cat.
So the first question is, what makes something beautiful? While this topic is discussed in great detail in due course, it may be pointed out here that if it is true that aesthetic experiences are those that hold tension between the personal and the universal, as I have argued in this chapter, then it stands to reason that some aspect of what it is that makes something beautiful, which we might call “objectively pleasant,” [6] must speak to this very tension. Of course, as we have seen, this is the fundamental question of aesthetics, so this is perhaps unsurprising. Nevertheless, it’s worth taking a moment to explore the relationship between beauty and the tension between the personal and the universal. Raising this question does lead us, however, to expand the concept of beauty and deformity (as Hume would call it) or ugliness (as we might say today), to be something of a placeholder for any and all experiences which we might tend to insist upon universalising. This is because it is clear that if aesthetics is the study of beauty–as it is so often said to be–it can only be so if beauty is taken to encompass far more than simply what is agreeable.
If we return to the working definition of aesthetics presented at the beginning of this chapter, we understand that pleasantness can not be synonymous with pleasure as opposed to pain, for this would fail to take into account the “pleasantness” of looking at Utagawa Toyokuni’s (1769–1825) hanging scroll, “Courtesan in Her Boudoir,” which portrays a courtesan putting herself together after having sex, seen fixing her hair with her clothes still partially open. The picture is not a happy one, and to derive enjoyment of it in a way that ignores the quiet sadness of the picture seems perverse, certainly out of place. Instead, we enjoy this picture precisely because of its portrayal of a situation tinged with sadness. Or again, the experience of a mathematical equation one has struggled for hours to achieve may have a certain intellectual pleasure at having overcome the difficulties presented by the equation, but has nothing to do with the pleasantness of the equation as such. Instead the pleasantness is to be found in the elegance and simplicity of the equation, the originality of thought, and so on, in spite of the pain, struggle, frustration, and tiredness experienced in grappling with it. The experience of reading Thomas Hardy’s poem “The Walk,” written after his wife’s death, is another example of this distinction:
The Walk
You did not walk with me
Of late to the hill-top tree
By the gated ways,
As in earlier days;
You were weak and lame,
So you never came,
And I went alone, and I did not mind,
Not thinking of you as left behind.
I walked up there to-day
Just in the former way;
Surveyed around
The familiar ground
By myself again:
What difference, then?
Only that underlying sense
Of the look of a room on returning thence.
This poem is saturated with sorrow, and when we read it we feel that same sorrow, and it would be wrong to describe ourselves as finding pleasure in Hardy’s sorrow. And yet the poem has pleasantness— that is, beauty—in its ability to capture, contain, and convey that emotion.
This “objective pleasantness” that we find in these aesthetic experiences, then, is a pleasantness that seems to be divorced from the question of our enjoyment and appraisal of the cause of that experience. This explains how it is that we can be expected to appreciate a book, painting, sculpture, piece of music, and so on even if we are not expected to like it, because the pleasantness of the aesthetic experience—which we might call our appreciation of it—seems to be assumed to be separate from the enjoyment and approval of the cause of that experience. If it is possible to appreciate an experience—that is, to have the appropriate response to it—and yet still not like it, then there seems to be two elements to an individual’s experience: one purely personal, and thus not aesthetic as such, and the other personal-yet-universal. It is this latter element that constitutes the individual’s aesthetic experience proper. This might then explain why, despite its significance in the eighteenth century, Burke and Kant’s distinction between beauty and the sublime is not much used today, beauty instead becoming the overriding concept for all experiences that are universal yet personal, and which we believe have “pleasantness.” Thus we find our answer to the question “what is beauty?” in this unique kind of pleasantness found in aesthetic experiences, devoid of their “goodness” or personal pleasantness.
As these examples show, beauty is not a “one-size-fits-all” concept—or if it is, it looks so radically different in different sizes that it is only in these different forms that we can talk about it in any detail. And yet, expanding the notion of beauty in this way does not thereby render it useless. Though it seems to cover a wide range of experiences, and apply to a diverse—and at times contradictory—range of qualities, beauty has a role as the determining factor in aesthetic judgments. When we have an aesthetic experience, we feel words like “beautiful” are uniquely appropriate: we describe as beautiful not only the awesome, the inspiring, and the joy-filled, but also those experiences saturated with sorrow and desperation. Even when the experience seems too bleak, or what is portrayed in an artwork is too confronting or disturbing for us to be comfortable with calling it “beautiful” directly, it is still not uncommon to hear of such a work of art as being “beautifully put together.” Beauty, then, still remains a powerful and useful concept in the study of aesthetics.
Why Aesthetics?
Drawing the different threads together, we are now in a position to reconsider and provide a more complete answer for the question of why aesthetics is worth pursuing. So far we have spoken of the experience of a tension between the personal and the universal as the main focus of aesthetics, but, of course, the experience cannot be had without someone to experience it. And so the individual is a crucial element in the equation of an aesthetic experience. The example above of Toyokumi’s hanging scroll suggests two important aspects of this individual element.
The first aspect is to do with “proper response,” or “correct pleasantness,” as one might say. In looking at Toyokumi’s “Courtesan in Her Boudoir” it would seem out of place to enjoy the painting because it includes a naked breast, for example: to look on the image in this way fails to do justice to the image as a work of art, certainly, but more than that it denigrates it as the object of an aesthetic experience. Likewise, to enjoy it because one enjoys the idea of a woman sold into the life of a courtesan, usually suffering not only venereal diseases but also lead poisoning from the make-up she wore, then this too would be a grossly inappropriate response, missing the point of the artwork and missing out on the aesthetic experience altogether. A similar point may be made about John Steinbeck’s novella, Of Mice and Men , in which George must kill his best friend Lenny; we rightly feel for George, and find the book pleasant in its tragedy and its highlighting of a number of injustices, such as the injustice of a society that fails to care for its most vulnerable, the injustice of Lenny not being cared for by anyone except George, the injustice of George being put in a situation in which he thinks he has no option but to kill his best friend, and so on. We call the book beautiful, eye-opening, and we recommend it to others. Yet if we were to enjoy the book because we like the idea of shooting our friend or of killing someone with a disability, then again we have failed to have the correct aesthetic response. So the first aspect of the individual element in an aesthetic experience is the question of an appropriate response.
This is inextricably entwined with the second aspect, which is the question of the development and cultivation of appreciation and appropriate response. If the appropriate response can be expected (irrespective of enjoyment), then one naturally turns to the question of aesthetic education, or how this appropriate response comes about, and how one develops the disposition from which such an appropriate response arises. It stands to reason that if judgments about aesthetic experiences are to be universal—that is, if we expect, as indeed we do, someone else to agree with our judgment—then we can only do this because we believe they are capable of making the same appropriate response (since, obviously, we assume our own judgment to be appropriate). This is because we cannot expect them to have a response that agrees with our own if that response is random, or purely based on personality—recalling our discussion at the beginning of the chapter, the experience must be “objectively personal,” that is, personal yet universal. This leaves only two options: 1) everybody is born with the exact same disposition towards having an appropriate response that does not change as they grow ; or 2) everybody’s disposition towards having an appropriate response changes and is affected by the circumstances of each person’s life and experience. The problem with the first option is trying to accommodate those who do not have an appropriate response: the only way to accommodate them is to say they have an innate abnormality. But in this case we would be unable to judge them for it. After all, we don’t judge someone born blind for not agreeing with us that the object in front of us is yellow: it is simply not possible for them to agree or disagree, since they are physically incapable of experiencing the colour yellow. Likewise, if we say that all humanity is innately disposed to appropriate responses towards aesthetic experiences, then those who do not have the appropriate response are “let off the hook,” as it were.
It seems, then, that the only option is to acknowledge that our disposition towards an appropriate response changes and develops over time, and thus acknowledge the possibility of aesthetic education, that is, education in developing appropriate responses. And, though the main focus of this chapter is the individual aesthetic experience, it’s worth noting here that this shift in disposition towards an appropriate response that happens over time is true both at the individual and also at the cultural/societal level. [7] So, for example, for an Ancient Greek, revulsion at disproportionality was deemed an appropriate response, whereas in contemporary Western society, notwithstanding that this may well be the response of some people, disproportionality is culturally acceptable, and at times even the most praiseworthy feature of a work of art (one immediately thinks of Picasso, for example). To return to the subject of changing disposition towards appropriate responses, in taking up this option in conjunction with the assertion that a response to an aesthetic experience can be either appropriate or inappropriate (though there may be more than one appropriate or inappropriate response), then we immediately acknowledge the role of the individual’s own disposition as a factor in aesthetic experiences, and thus in the study of aesthetics. Aesthetic experiences are, after all, the source of the personal in the tension between the personal and universal that drives aesthetics as a discipline.
This then suggests an amended answer to the question of why one ought to study aesthetics, if we take aesthetics to be the study of aesthetic experiences (as we have defined them here) involving both the object and subject of that experience. Under this definition, aesthetics is a discipline worthy of study because it examines and seeks to explain the myriad of experiences that make up a large part of the human experience, in which we respond to something on a personal, subjective level, and yet seek to universalise it on an objective level. Its subject matter lies on the threshold between the uniqueness of the individual and the shared experience of humanity, and looks to settle disputes about whether and why we can expect others to share a particular experience. Thus aesthetics can be taken to be a philosophical study of beauty (or lack thereof) and our reaction to it— in a word, taste.
References
Aristotle. (335 BCE) 1996. Poetics . Translated by Malcolm Heath. London and New York: Penguin.
Asmis, Elizabeth. 1991. “Philodemus’s Poetic Theory and On the Good King According to Homer’.” Classical Antiquity 10, no. 1: 1–45. https://doi.org/10.2307/25010939 .
Augustine of Hippo. (386–87) 2007. On Order [De Ordine] . Translated by Silvano Borruso. South Bend, IN: St. Augustine’s Press.
Baumgarten, Alexander Gottlieb. (19735) 1954. Reflections on Poetry . Translated by Karl Aschenbrenner and William B. Holther. Berkeley: University of California Press.
Bharata. 1950–1961. The Nāṭyaśāstra: A Treatise on Hindu Dramaturgy and Histrionics ascribed to Bharata-Muni . 2 vols. Bibliotheca Indica 272–272A. Translated by Manomohan Ghosh. Calcutta: Asiatic Society.
Burke, Edmund. (1757) 2005. Philosophical Inquiry into the Origin of Our Ideas of the Sublime and Beautiful. In The Works of the Right Honourable Edmund Burke, Vol. I . Salt Lake City: Project Gutenberg. https://www.gutenberg.org/files/15043/15043-h/15043-h.htm#A_PHILOSOPHICAL_INQUIRY
Cohen, Leonard. 1988. “How the heart approaches what it yearns: Interview with Leonard Cohen presented by John McKenna”, Part 1. RTE Ireland, May 9 & 12, 1988. https://www.leonardcohenfiles.com/rte.html
Confucius. 1938. The Analects of Confucius . Translated by A. Waley. New York: Random House.
Gerow, Edwin. 2002. “Rasa and Katharsis: A Comparative Study, Aided by Several Films.” Journal of the American Oriental Society 122, no. 2, Indic and Iranian Studies in Honor of Stanley Insler on His Sixty-Fifth Birthday (April/June): 264–277.
Guyer, Paul. 2005. “History of Modern Aesthetics.” In The Oxford Handbook of Aesthetics , edited by Jerrold Levinson, 25–60. Oxford: Oxford University Press.
Hume, David. (1739–40) 2009. A Treatise of Human Nature: Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects . Auckland: The Floating Press.
Hume, David. (1757) 2000. “Of The Standard Of Taste.” Infomotions, Inc. http://infomotions.com/etexts/philosophy/1700-1799/hume-of-740.htm.
Kant, Immanuel. (1790) 2015. Critique of Judgement . Translated by J.H. Bernard. Salt Lake City: Project Gutenberg. https://www.gutenberg.org/ebooks/48433.
Levin, Susan. 2001. Ancient Quarrel Between Philosophy and Poetry Revisited: Plato and the Greek Literary Tradition . Oxford: Oxford University Press.
Liddell, Henry George and Robert Short. 1940. “αἰσθητικός.” A Greek-English Lexicon . http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dai)sqhtiko%2Fs.
Pappas, Nickolas. 2012. “Plato on Poetry: Imitation or Inspiration?” Philosophy Compass (October): 669–678.
Planinc, Zdravko. 2003. Plato Through Homer: Poetry and Philosophy in the Cosmological Dialogues . Columbia: University of Missouri Press.
Plato. 1963. The Collected Dialogues of Plato Including the Letters . Translated and edited by Edith Hamilton and Huntington Cairns. Princeton: Princeton University Press.
Shaftesbury, Earl of (Anthony Ashley Cooper). (1711) 1999. Characteristics of Men, Manners, Opinions, Times . Edited by L. E. Klein. Cambridge: Cambridge University Press.
Sushytska, Julia. 2012. “On the Non-Rivalry Between Poetry and Philosophy: Plato’s Republic, Reconsidered.” Mosaic: A Journal for the Interdisciplinary Study of Literature 45, no. 1 (March): 54–70.
Westenberg, Thomas. 2015. “Epicurean Arts: The Aesthetic Theory of Philodemus of Gadara.” Masters Thesis, Macquarie University.
- There have been a number of recent arguments, however, that Plato has been strongly misinterpreted on this point (Levin 2001; Planinc 2003; Pappas 2012; Sushytska 2012). ↵
- For further discussion of the Epicurean view that the arts could be philosophical, see Westenberg (2015). ↵
- Estimates of Bharata’s life range from 500 BCE to 500 CE, but most put him between 500 and 200 BCE. ↵
- Catharsis is a notion famously introduced in Aristotle’s discussion of tragedy. Simply put, it is the purgation or purification of one’s emotions, achieved through a quasi-experience of those emotions during the performance of the tragedy. See Aristotle’s Poetics ([335 BCE] 1996, 1449b21–29). ↵
- Hume here takes aesthetic experiences to be experiences of works of art. ↵
- Meaning we expect a certain level of universal appreciation for the object of our experience. People often use this concept naturally when they say, for example, “I don’t like it, but I can appreciate it.” ↵
- One may wonder, if a society’s position on aesthetics can change, how it can be considered universal. The answer lies in the fact that the society as a whole changes because someone (or a group of someones) challenges and “educates” (for lack of a better word) the society in a new way of thinking. We might think of the Impressionist movement, which challenged the prevailing realism in painting, was initially rejected, and later became widely recognised. In essence, there is no distinction here between the individual whose disposition changes, yet maintains a conviction that aesthetic experiences are universal, and the society or culture which does the same. ↵


1.2: Chemicals Compose Ordinary Things
Chemistry is the branch of science dealing with the structure, composition, properties, and the reactive characteristics of matter. Matter is anything that has mass and occupies space. Thus, chemistry is the study of literally everything around us – the liquids that we drink, the gasses we breathe, the composition of everything from the plastic case on your phone to the earth beneath your feet. Moreover, chemistry is the study of the transformation of matter. Crude oil is transformed into more useful petroleum products such as gasoline and kerosene by the process of refining. Some of these products are further transformed into plastics. Crude metal ores are transformed into metals, that can then be fashioned into everything from foil to automobiles. Potential drugs are identified from natural sources, isolated and then prepared in the laboratory. Their structures are systematically modified to produce the pharmaceuticals that have led to vast advances in modern medicine. Chemistry is at the center of all of these processes and chemists are the people that study the nature of matter and learn to design, predict and control these chemical transformations. Within the branches of chemistry you will find several apparent subdivisions. Inorganic chemistry, historically, focused on minerals and metals found in the earth, while organic chemistry dealt with carbon-containing compounds that were first identified in living things. Biochemistry is an outgrowth of the application of organic chemistry to biology and relates to the chemical basis for living things. In the later chapters of this text we will explore organic and biochemistry in a bit more detail and you will notice examples of organic compounds scattered throughout the text. Today, the lines between the various fields have blurred significantly and a contemporary chemist is expected to have a broad background in all of these areas.
In this chapter we will discuss some of the properties of matter, how chemists measure those properties and we will introduce some of the vocabulary that is used throughout chemistry and the other physical sciences.
Let’s begin with matter. Matter is defined as any substance that has mass. It’s important to distinguish here between weight and mass. Weight is the result of the pull of gravity on an object. On the Moon, an object will weigh less than the same object on Earth because the pull of gravity is less on the Moon. The mass of an object, however, is an inherent property of that object and does not change, regardless of location, gravitational pull, or whatever. It is a property that is solely dependent on the quantity of matter within the object.
Contemporary theories suggests that matter is composed of atoms . Atoms themselves are constructed and characterized from the subatomic particles called neutrons, protons and electrons that the atom possesses. Atoms will be studied closer in chapter 4 of this text. There are at least 116 different types of atoms recognized by scientists today, each type having a different number of protons in its nucleus. These different types of atoms are called elements .
As different elements were discovered and named, abbreviations of their names were developed to allow for a convenient chemical shorthand. The abbreviation for an element is called its chemical symbol. A chemical symbol consists of one or two letters, and the relationship between the symbol and the name of the element is generally apparent. Thus helium has the chemical symbol He, nitrogen is N, and lithium is Li. Sometimes the symbol is less apparent but is decipherable; magnesium is Mg, strontium is Sr, and manganese is Mn. Symbols for elements that have been known since ancient times, however, are often based on Latin or Greek names and appear somewhat obscure from their modern English names. For example, copper is Cu (from cuprum), silver is Ag (from argentum), gold is Au (from aurum), and iron is (Fe from ferrum). Throughout your study of chemistry, you will routinely use chemical symbols and it is important that you begin the process of learning the names and chemical symbols for the common elements.  Table \(\PageIndex{1}\) contains a starter list of common elements that you should begin learning now, as they will be the ones you are expected to know for this course.
| Element | Chemical Symbol | Element | Chemical Symbol |
|---|---|---|---|
| Aluminum | Al | Gold | Au |
| Beryllium | Be | Argon | Ar |
| Boron | B | Potassium | K |
| Carbon | C | Calcium | Ca |
| Fluorine | F | Zinc | Zn |
| Helium | He | Sulfur | S |
| Hydrogen | H | Phosphorus | P |
| Lithium | Li | Chlorine | Cl |
| Magnesium | Mg | Iodine | I |
| Manganese | Mn | Arsenic | As |
| Mercury | Hg | Barium | Ba |
| Neon | Ne | Bromine | Br |
| Nitrogen | N | Iron | Fe |
| Oxygen | O | Copper | Cu |
| Silicon | Si | Lead | Pb |
| Sodium | Na | Silver | Ag |
| Tin | Sn | Chromium | Cr |
Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
-
Paul R. Young , Professor of Chemistry, University of Illinois at Chicago, Wiki: AskTheNerd; PRY askthenerd.com - pyoung uic.edu; ChemistryOnline.com
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Henry Agnew (UC Davis)


8.3: Circles in the Sky
Immediately after the big bang, the universe was so hot that the usual constituents of matter could not form. Photons could not move freely in space, as they were constantly bumping into free electrons. Eventually, about \(350,000\) years after the big bang, the universe had expanded and cooled to the point that light could travel unimpeded. This free radiation is called the cosmic microwave background (CMB) radiation, and much of it is still travelling today. The universe has cooled and expanded to the point that this radiation has stretched to the microwave end of the electromagnetic spectrum, having a wavelength of about \(1\) or \(2\) millimeters.
The CMB radiation is coming to us from every direction, and it has all been travelling for the same amount of time - and at the same speed. This means that it has all traveled the same distance to reach us at this moment. Thus, we may think of the CMB radiation that we can detect at this instant as having come from the surface of a giant \(2\)-sphere with us at the sphere's center. This giant \(2\)-sphere is called the last scattering surface (LSS).
It is perhaps comforting to think that everyone in the universe has their own last scattering surface, that everyone's LSS has the same radius, and that this radius is growing in time.
The CMB radiation coming to us has a temperature that is remarkably uniform: it is constant to a few parts in \(100,000\), which makes the temperature of the radiation in the LSS very nearly perfectly uniform. As Craig Hogan points out in [14] , this is much smoother than a billiard ball. Nonetheless, there are slight variations in the temperature. These variations, due to slight imbalances in the distribution of matter in the early universe, were predicted well before they were finally found (when our instruments became sensitive enough to detect them). These very slight temperature differences might reveal the shape of the universe.
Imagine our universe is a giant \(3\)-torus. Assume a fundamental domain for the universe is a rectangular box as shown in Example \(8.1.1\), and that this box is our Dirichlet domain (we're at the center of this box). We may tile \(\mathbb{R}^3\) with copies of this fundamental domain, placing ourselves in the same position of each copy of the fundamental domain. Now, imagine our last scattering surface in the fundamental domain. In fact, there will be a copy of our last scattering surface surrounding each copy of us in each copy of the fundamental domain.
If our last scattering surface is small relative to the size of the fundamental domain, as in Figure \(8.3.1(a)\), then it will not intersect any of its copies. However, if the last scattering surface is large relative to the size of the fundamental domain, as in Figure \(8.3.1(b)\), then it will intersect one or more of its copies. Moreover, adjacent copies of the LSS will intersect in a circle. In this happy case, our last scattering surface will contain circles with matching temperature distributions. Look again at Figure \(8.3.1(b)\). We have three copies of our fundamental domain pictured as well as three copies of the LSS (only one of which is shaded to make the situation less cluttered). Two vertical circles of intersection appear in the figure. From our point of view at the center of the LSS, the two images of the circle will be directly opposite one another in the sky. Since these circles are one and the same, the temperature distribution around the two circles will agree. Therein lies the hope. Scan the temperature distribution in the last scattering surface for matching circles.
This strategy for detecting a finite universe is called the circles-in-the-sky method, which, in cosmic topology, has advantages over the cosmic crystallography method. In theory, the circles-in-the-sky method can be used to detect any compact manifold, regardless of the geometry it admits. Also, the search for matching circles is independent of a metric. One doesn't need to make a claim about the geometry of the universe to detect a finite universe.
This method is computationally very intensive. The search for matching circles on this giant \(2\)-sphere involves the analysis of a six parameter space: the center \((\theta_1,\phi_1)\) of one circle on the LSS, the center of the second circle \((\theta_2,\phi_2)\text{,}\) the common angular radius \(\alpha\) of the two circles (since these circles are copies of the same circle they will have the same radius), and the relative phase of the two circles, say \(\beta\text{.}\) (See the diagram that follows.) In general, \(\beta \neq 0\) if the face identifications in the \(3\)-manifold involve rotations. It remains for us to analyze whether a statistically significant correlation exists between the temperatures as we proceed around the circles.
When comparing the size of the LSS relative to the size of space, it is convenient to define the following length dimension. The injectivity radius at a point in a manifold, denoted \(r_{inj}\text{,}\) is half the distance of the shortest closed geodesic path that starts and ends at that point. A necessary condition, then, for detecting matching circles in the LSS at our location is that our observable radius \(r_{obs}\) exceeds our injectivity radius \(r_{inj}\text{.}\)
If the universe is a \(3\)-torus, and our LSS has diameter larger than some dimension of the \(3\)-torus, then the LSS will intersect copies of itself, and the matching circles would be diametrically opposed to one another on the LSS. Suppose our Dirichlet domain in a \(3\)-torus universe is an \(a\) by \(b\) by \(c\) box in \(\mathbb{R}^3\text{,}\) where \(a \lt b \lt c\text{.}\) Our LSS is then centered at the center of the box. The injectivity radius of the universe is \(\dfrac{a}{2}\text{.}\) In the following figure, we assume that \(r_{obs}\text{,}\) the radius of the LSS, is greater than \(\dfrac{a}{2}\) but less than \(\dfrac{b}{2}\text{.}\) In this case, the circles-in-the-sky method would detect one pair of matching circles in the temperature distribution of the LSS. From the Earth \(E\text{,}\) we would observe that circle \(C_1\text{,}\) when traced in the counterclockwise direction, matches circle \(C_2\) when traced in the clockwise direction, with no relative phase shift.
If the size is right, all six compact orientable Euclidean \(3\)-manifolds would have matching circles that are diametrically opposed to one another on the LSS. The phase shift on these matching circles will be non-zero if the faces are identified with a rotation.
If we live in a Poincaré dodecahedral space and our LSS is large enough, we might see six pairs of matching circles, each pair consisting of diametrically opposed circles in the sky with matching temperature distributions after a relative phase shift of \(36^{\circ}\text{.}\) The following figure indicates the matching circles that would arise from the identification of the front face and rear face of the dodecahedron. From the Earth \(E\text{,}\) we would observe that circle \(C_1\) when traced in the counterclockwise direction matches circle \(C_2\) when traced in the clockwise direction, with a phase shift of \(36^{\circ}\text{.}\)
In general, the matching circles we (might) see can depend not only on the shape of the universe, but also on where we happen to be in the universe. This follows because the Dirichlet domain can vary from point to point (see Example \(7.7.7\) for the two-dimensional case). Of the \(10\) Euclidean \(3\)-manifolds, only the \(3\)-torus has the feature that the Dirichlet domain is location independent. Some (but not all) elliptic \(3\)-manifolds have this feature, and in any hyperbolic \(3\)-manifold, the Dirichlet domain depends on your location. Thus, if we do observe matching circles, it can not only reveal topology but also the Earth's location in the universe.
Searches to date have focused on circles that are diametrically opposed to one another on the LSS (or nearly so). This restriction reduces the search space from six parameters to four. Happily, most detectable universe shapes would have matching circles diametrically opposite one another, or nearly so. At the time of this writing, no matching circles have been found, and this negative result places bounds on the size of our universe. For instance, an article written by the people who first realized a small finite universe would imprint itself on the LSS [19] , concludes from the absence of matching circles that the universe has topology scale (i.e., injectivity radius \(r_{inj}\)) bigger than \(24\) gigaparsecs, which works out to \(24 \times 3.26 \times 10^9 \approx 78\) billion light-years. So, a geodesic closed path trip in the universe would be at least \(156\) billion light-years long.
This stupendous distance is hard to fathom, but it appears safe to say that we might abandon the possibility of gazing into the heavens and seeing a distant image of our beloved Milky Way Galaxy.
In addition to [19] , other accessible papers have been written on the circles-in-the-sky method, as well as the cosmic crystallography method. (See [23] , [26] , and [24] .) Jeff Weeks also discusses both programs of research in The Shape of Space , [12] .


2.6: Dots, Ties, and Borrowed Divisions
- In written music, standard note lengths are always halves (or halves of halves, and so on) of other note lengths. To get any other note length, dots, ties, or borrowed divisions must be used.
A half note is half the length of a whole note; a quarter note is half the length of a half note; an eighth note is half the length of a quarter note, and so on. (See Duration:Note Length.) The same goes for rests. (See Duration: Rest Length.) But what if you want a note (or rest) length that isn't half of another note (or rest) length?
Dotted Notes
One way to get a different length is by dotting the note or rest. A dotted note is one-and-a-half times the length of the same note without the dot. In other words, the note keeps its original length and adds another half of that original length because of the dot. So a dotted half note, for example, would last as long as a half note plus a quarter note, or three quarters of a whole note.
Make groups of equal length on each side, by putting a dotted note or rest in the box.
Solution
A note may have more than one dot. Each dot adds half the length that the dot before it added. For example, the first dot after a half note adds a quarter note length; the second dot would add an eighth note length.
Tied Notes
A dotted half lasts as long as a half note plus a quarter note. The same length may be written as a half note and a quarter note tied together. Tied notes are written with a curved line connecting two notes that are on the same line or the same space in the staff. Notes of any length may be tied together, and more than two notes may be tied together. The sound they stand for will be a single note that is the length of all the tied notes added together. This is another way to make a great variety of note lengths. Tied notes are also the only way to write a sound that starts in one measure and ends in a different measure.
Ties may look like slurs, but they are not the same; a slur connects to notes with different pitches and is a type of articulation.
Borrowed Divisions
Dots and ties give you much freedom to write notes of varying lengths, but so far you must build your notes from halves of other notes. If you want to divide a note length into anything other than halves or halves of halves - if you want to divide a beat into thirds or fifths, for example - you must write the number of the division over the notes. These unusual subdivisions are called borrowed divisions because they sound as if they have been borrowed from a completely different meter . They can be difficult to perform correctly and are avoided in music for beginners. The only one that is commonly used is triplets, which divide a note length into equal thirds.
Some Borrowed Divisions
Borrowed Duplets
Notes in jazzy-sounding music that has a "swing" beat are often assumed to be triplet rhythms, even when they look like regular divisions; for example, two written eighth notes (or a dotted quarter-sixteenth) might sound like a triplet quarter-eighth rhythm. In jazz and other popular music styles, a tempo notation that says swing usually means that all rhythms should be played as triplets. Straight means to play the rhythms as written.
Some jazz musicians prefer to think of a swing rhythm as more of a heavy accent on the second eighth, rather than as a triplet rhythm, particularly when the tempo is fast. This distinction is not important for students of music theory, but jazz students will want to work hard on using both rhythm and articulation to produce a convincing "swing".
Swing Rhythms