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Sanchez: Demonstrating Infinity 1
Running head: DEMONSTRATING INFINITY
Demonstrating Infinity within a Projection
Mathematics
Design
_______________________________________Signature of Sponsoring Teacher
_______________________________________Signature of School Science Fair Coordinator
Octavio Sanchez145 S. Campbell Avenue Phoenix Military AcademyChicago, IL 60612Grade 12
Sanchez: Demonstrating Infinity 2
Table of Contents
Title Page ……………………………………………………………………………….1
Table of Contents ……………………………………………………………………….2
Acknowledgments ……………………………………………………………………….3
Purpose and Hypothesis ……………………………………………………………….4
Review of Literature …………………………………………………………………….…5-13
Materials and Methods of Procedure ..……………………………………………..…….14-15
Results ……..……………………………………………………………………..……….16-26
Data Analysis ………………………………………………………………………27
Conclusions ………………………………………………………………………………28
References ……………………………………………………………………….……..29-32
Sanchez: Demonstrating Infinity 3
Acknowledgments
I want to start by acknowledging a person that does no give up on any of the science fair
participants, and makes sure that our papers, projects, and presentation are top notch. That
person is Ms. Tobias and I do not want her hard work to go unnoticed. A huge thank you to Dr.
Jaji for looking at our paper and telling us what needed to be described more in a casual point of
view for people with out the background to still understand. Mr. Surina was a big help when we
needed someone to watch as we typed our papers at eight in the afternoon or when we needed a
tech guru. Finally, I want to thank the person the made countless calculations with me, the
person that sat down and looked at this projected upside down and inside out with me, the one
teacher that pushed me when I felt like giving in, Mr. Carroll.
Sanchez: Demonstrating Infinity 4
Purpose and Hypothesis
Purpose
The purpose behind this experiment is to find the best fit line with in different planes.
Hypothesis
The smaller the hyperbolic sphere the greater the curvature will be which will indicate
that the radius will be smaller.
Dependent Variable
The radii of the curvature will change.
Independent Variable
The distance between the projector and the plane, sphere.
Control
The triangle that is being displayed which is an equilateral triangle.
Sanchez: Demonstrating Infinity 5
Review of Literature
Space
Space is made up of three different dimensions that are known by the human mind, but
cannot be perceived physically. The three different dimensions are up/down, left/right, and
forward/backwards. In a visual perception, you can imagine a coordinate plane with an X and Y
axis, but to see the third dimension you will have to imagine the Z axis which crosses between
the origin coming out breaking a second-dimension perspective making it into a third-dimension
perspective, i.e. a cube, pyramid, tetrahedral, Trigonal bipyramidal. Old mathematicians started
to examine space, they started to use Geometry that was not Euclidean, but instead Non-
Euclidean to get a better idea on how everything is shaped, a better angle to look at things.
Time
When talking about time there can be many ideas that come into mind and those ideas
can range from Star Wars and Star Trek to Albert Einstein’s brilliant Special Theory of
Relativity. Speaking of this Special Theory of Relativity he had two important postulates:
“1. The speed of light (abut 300,000,000 meters per second) is the same for all observers
whether or not they’re moving.
2. Anyone moving at a constant speed should observe the same physical law.” (Fuller,
“How Warp Speed Works”, 2008).
All of this can be compiled to one point, and that is that time represents the fourth dimension
when talking about what is perceived in space-time continuum. In space-time continuum time is
used as an organizer making everything be in sequence from past to present to the future. A point
of view that is very important is that time is something very fundamental to our universe. This
fundamental is known to be a dimension independent of events, or is known as the prominent
Sanchez: Demonstrating Infinity 6
fourth dimension. This fundamental is also referred to as Newtonian Time (Rynasiewicz, 2004).
The contrasting viewpoint to the one that was previously mention is that time is not referred to as
a container in which events and objects simply move through. Time is used to organize events
and compare them. The final third point of view is that, time is not a thing or an event which
indicated that it cannot be measured nor can it be traveled (Mattey, 1997) Time is one of the
seven fundamentals in SI units and in the International System Quantities.
Dimensions
When talking about dimensions it can be viewed from a mathematical and physics point
of view. Most of the time that you are trying to find the dimensions of an object it is basically
corresponding to the coordinate points of it. In a sense, you are looking for the 1st Dimension or
the dimensions of the shape to locate the best attainable information.
1st Dimension
In physics and mathematics, a sequence of numbers can be understood as a location in n
dimensional space. When “n” equals one, the set of all locations is called one-dimensional space.
An example of a one-dimensional space is the number line, where the position of each point can
be described by a single number.
2nd Dimension
In physics and mathematics, the 2nd dimensional space is a geometrical model in the
coordinate plane in which length and width lie on. In addition, length and width are commonly
called two dimensions.
Sanchez: Demonstrating Infinity 7
3rd Dimension
A three-dimensional figure can be viewed differently to a two-dimensional figure. The
reason that is can be viewed differently is because the three-dimensional figure has length, width,
and height (two additional terms to describe the figure are depth and breadth).
4th Dimension
There really is not much data about this dimension besides the fact that it is consider
holding a shape such as the tesseract and it can be imagined as a shape within the shape. It is
commonly looked at as a square with in a square and it moves, so it incorporates time and lets
the smaller square become the bigger square as time passes by.
Non-Euclidean Geometry
According to Donna Roberts, “Non-Euclidean Geometry is any form of geometry
that contains a postulate (axiom) which is equivalent to the negation of the Euclidean parallel
postulate” (Roberts, 1998). In Non-Euclidean Geometry, there are different postulates and
different types of geometry within it. The following are the five postulates that are known to
Non-Euclidean Geometry:
1.) A straight line can be drawn from any point to any point.
2.) A finite straight line can be produced continuously in a straight line.
3.) A circle may be described with any point as center and any distance as a radius.
4.) All right angles are equal to one another.
5.) If a transversal falls on two lines in such a way that the interior angles on one side of the
transversal are less than two right angles, then the lines meet on the side on which the angles are
less than two right angles (Non-Euclidean Geometry, 2014).
Sanchez: Demonstrating Infinity 8
The types of Geometry that were used in this project were Riemannian geometry, known as
Elliptic Geometry. The second geometry used was Hyperbolic Geometry, known as Monkey
Saddle Geometry.
Elliptical Geometry
"Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces
the parallel postulate with the statement "through any point in the plane, there exist no
lines parallel to a given line." To achieve a consistent system, however, the basic axioms of
neutral geometry must be partially modified. Most notably, the axioms of betweenness are no
longer sufficient (essentially because betweenness on a great circle makes no sense, namely if
and are on a circle and is between them, then the relative position of is not uniquely
specified), and so must be replaced with the axioms of subsets.” (Wolfram, "Elliptic Geometry",
2017). In simple terms, this is trying to indicate that the plane that is being represented has a
curvature to it, like a sphere, which is why when there is a triangle it will be greater than 180
degrees.
Hyperbolic Geometry
“A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having
constant sectional curvature . This geometry satisfies all of Euclid's
postulates except the parallel postulate, which is modified to read: For any infinite straight line
and any point not on it, there are many other infinitely extending straight lines that pass
through and which do not intersect . In hyperbolic geometry, the sum of angles of
a triangle is less than , and triangles with the same angles have the same areas.
Furthermore, not all triangles have the same angle sum.” (Wolfram, "Elliptic Geometry", 2017).
Sanchez: Demonstrating Infinity 9
Euclidean Geometry
According to the “Shorter Oxford English Dictionary”, “Euclidean means the geometry
of ordinary experience based on axioms of Euclid, esp. the one stating that parallel lines do not
meet. This also is known to be a flat surface. Euclidean Geometry also has five known postulates
those are as following:
1) Any two points describe a line.
2) A line is infinitely long.
3) A circle is uniquely defined by its center and a point on its circumference.
4) Right angles are all equal.
5) Given a point and a line not containing the point, there is one and only one parallel to the line
through the point (Postulates, 2014).
Space-Time Continuum
Einstein believes that space and time were necessary for each other and without each
other they will fade and cease to exist. Space-time is also described to be what our world is in. In
addition, this is also viewed by a Euclidean space. Regarding the fact that we know that space is
made from three dimensions and time as one we consider them to make the fourth dimension.
Curved Space
When talking about curved surfaces we can relate this to Einstein’s General relativity
because it will be said that space is not flat, like how it was assumed to be but instead it is
curved. The notion of Anti-De Sitter Space, which contains no matter what so ever but does have
a negative energy density. The fact that it is curved shows that there is a force being used, that
force is gravity, but gravity is not like any other force. Gravity does not allow for Earth to move
Sanchez: Demonstrating Infinity 10
because Earth is in a curved space, but space allows for Earth to move in that space because it
follows the nearest object in a straight path in a curved surface, this is called Geodesic (Hawking,
2005).
Gravity
Gravity is the attraction between two items. Everything in this universe has gravity and
everything in this universe is attracted to something. Most of the time the force is equal to the
mass of the object. According to “Gravity and Gravitation”, Thus, F = Gm1m2/r2, where m1 is
the mass of the first object, m2 is the mass of the second object, r is the distance between their
centers, and G is a fixed number termed the gravitational constant. (If m1 and m2 are given in
kilograms and r in meters, then G = 6.673 × 10−1N m2/kg2.)” This can be related
back to Stephen Hawking’s book, “A Briefer History of Time”, thus meaning
that it is like something called geodesic. This also means that gravity is calculated by
finding the differences between distances, and using the mass of the certain thing that is being
calculated by the gravity (Gravity and gravitation, 2014).
Newtonian Gravity
Newton’s universal law says that all the objects that have gravity attract one another
(Gravity and gravitation, 2014). The way that it works is by making sure that if you want to
make the gravity between each one of the objects stronger you must bring them closer and the
same goes to making the attraction weaker (Gravity and gravitation, 2014). This means that
Einstein’s law of general relativity is being used, somehow. For example, when the law says that
gravity is the force that makes the earth to stay in place while making sure that it allows it to
chase after the nearest thing, or object, but in a curved space (Hawking, 2005). This law also
says that the mass and weight are not the same because something can weigh less in a different
Sanchez: Demonstrating Infinity 11
gravity field and still have the same mass it had anywhere else. Take a brick of gold for example,
it will weigh less on the moon or mercury compare to the earth, but it will still have the same
mass.
Theory of relativity
According to the article, Einstein’s Special Theory of Relativity, it states that, “Relativity
is that area of physics that must do with how observers in motion with respect to the
phenomenon observed can account for their observations given that two different frames of
reference (that of the observer and that of what is observed) are involved” (Einstein’s Special
Theory of Relativity, 2001). This proves that time and position truly matters to the point where in
reality it is the perception of someone. Just as if someone dropped something from a building the
person can see the object curve while the person that dropped the object sees it falling straight.
The way this relates to the project being presented is simple since a concave to observer one can
be completely something different.
Quantum Mechanics
Quantum Mechanics can be a way to study the natural world in a way that everything
being observed is through the way of energy waves. According to UXL Encyclopedia of
Science, “For example, physicists normally talk about light as if it were some form of wave
traveling through space. Many properties of light—such as reflection and refraction—can be
understood if light is thought of as waves bouncing off an object or passing through the object”
(UXL Encyclopedia of Science, 2015). Something else very important to quantum mechanics is
that sometimes the waves can travel as matter and sometimes it travels as a wave, this is
considered the principal of dualist.
Sanchez: Demonstrating Infinity 12
Convex and Concave
Two important terms that were used in this paper were “concave” and “convex”. These
two terms are important because they are describing the sides of the triangle and they describe
how the space is distorted. Most of the times these terms are used when talking about lenses, but
in this instance, it describes the triangle. When talking about lenses that are convex it means that
the lens attracts the light towards the middle so the lens is outwards, in context this means that
the side of the triangle was coming out. Concave is the complete opposite, so when speaking
about a lens that is concave it means that the lens is inwards, in context it means that the side of
the triangle was coming inside (UXL Encyclopedia of Science, 2015).
Perception
Perception is very important in this experiment since it can change the whole results.
Perception is the way an organism organizes information. The way it organizes that information
is by using the five senses. That way when the information reaches the brain it can sort it out and
make sure that the information is processed correctly (UXL Encyclopedia of Science, 2015). The
way perception plays a role in this experiment is simple; the person observing the distortion can
notice different things from a different observer.
Metaphysics
Metaphysics dates to the ancient and medieval times, were philosophers thought that the
idea of metaphysics had to do with chemistry and or astrology. Although they were somewhat
correct, the one philosopher to correct this was Aristotle. Metaphysics was the “science” that
studied “being as such” or “the first causes of things” or “things that do not change”, but
metaphysics cannot be defined that way anymore because of Aristotle. Although Aristotle made
Sanchez: Demonstrating Infinity 13
a huge impact to the understanding of this word, he did not know what this word exactly meant.
Aristotle also had a collection of books dedicated to this science of philosophy; there were
fourteen books to be exact (Inwagen & Sullivan, "Metaphysics", 2007).
Sanchez: Demonstrating Infinity 14
Materials and Methods of Procedure
Materials
The following is a list of the materials that were used in the experiment.
Overhead Projector
Transparency Film
Camera
Measuring tape, 200 cm long
A volleyball
Half a Basket
White paint
Expo Marker
String
Methods of Procedure
The following is the procedure that was used in the experiment
1. Make equilateral triangles inside of each other, but as you add more you make them
smaller.
2. Print out on a transparency film.
3. Put the transparency film underneath the overhead projector
4. Project the transparency onto the hyperbolic field and be 100cm away from the ball
5. Once it is projected take a picture and print it out on a second transparency.
6. Use the 200 cm measure tape and stick it on a board
7. Tie a string at the end of the expo marker and have someone hold the string at 0cm
Sanchez: Demonstrating Infinity 15
8. As they hold it at 0cm make sure that you draw a radii, most of this will be guess and
check.
9. Repeats steps 3-8, but as you continue different observations make sure that you change
the plane that you are projecting on
Sanchez: Demonstrating Infinity 16
Results
Figure 1. Conversions and Scaling
Figure 2. Example of how we measured the Radii
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Figure 3. Calculating and compiling
Figure 4. The volleyball is the first hyperbolic plane that is being used.
Sanchez: Demonstrating Infinity 18
Figure 5. Different angle on the volleyball.
Figure 6. Top view of the volleyball plane.
Sanchez: Demonstrating Infinity 19
Table 1
Observation #1: Volleyball
(100cm away)
Scaled Radius of Curvature Scaled Sides of Equilateral
Triangle
Triangle 1 3cm 4cm
Triangle 2 4cm 7cm
Triangle 3 5cm 11cm
Triangle 4 6cm 15.5cm
Triangle 5 7cm N/A
Triangle 6 N/A N/A
Triangle 7 N/A N/A
Triangle 8 N/A N/A
Triangle 9 N/A N/A
Note. Corresponds to figures 4-6
Sanchez: Demonstrating Infinity 20
Figure 7. The half basket is being used as the plane during the second observation
Figure 8. A view of the half circle at a different angle.
Sanchez: Demonstrating Infinity 21
Table 2
Observation #2: Half Circle
(100cm away)
Scaled Radius of Curvature Scaled Sides of Equilateral
Triangle
Triangle 1 2cm 5cm
Triangle 2 3cm 8cm
Triangle 3 4cm 11cm
Triangle 4 5cm 15.5cm
Triangle 5 6cm N/A
Triangle 6 7cm N/A
Triangle 7 8cm N/A
Triangle 8 N/A N/A
Triangle 9 N/A N/A
Note. Corresponding to figures 7-8
Sanchez: Demonstrating Infinity 22
Figure 9. The first yoga ball, smaller one, is now the plane for third observation.
Figure 10. A different Angle on the first yoga ball.
Sanchez: Demonstrating Infinity 23
Figure 11. Top view of the first yoga ball.
Sanchez: Demonstrating Infinity 24
Table 3
Observation #3: Yoga Ball 1
(100cm away)
Scaled Radius of Curvature Scaled Sides of Equilateral
Triangle
Triangle 1 1cm 4cm
Triangle 2 2cm 7cm
Triangle 3 3cm 10cm
Triangle 4 4cm 13cm
Triangle 5 5cm 16cm
Triangle 6 6cm N/A
Triangle 7 7cm N/A
Triangle 8 8cm N/A
Triangle 9 8cm N/A
Note. Corresponding to figures 9-11.
Sanchez: Demonstrating Infinity 25
Figure 12. The second yoga ball is now the new plane that is being observed.
Figure 13. A different angle on yoga ball 2.
Sanchez: Demonstrating Infinity 26
Table 4
Observation #4: Yoga Ball 2
(100cm away)
Scaled Radius of Curvature Scaled Sides of Equilateral
Triangle
Triangle 1 0cm 3cm
Triangle 2 0cm 4cm
Triangle 3 0cm 6.5cm
Triangle 4 0cm 8cm
Triangle 5 4cm 11cm
Triangle 6 4cm 13.5cm
Triangle 7 4cm 16cm
Triangle 8 4cm 17.5cm
Triangle 9 4cm 18.5cm
Note. Corresponding to figures 12-13
Sanchez: Demonstrating Infinity 27
Data Analysis
Figure 14. Linear Regression t-Test
Figure 15. Linear Regression t-Test Cont.
Y=a+bx
Y=1+3x
T=1x10^99
P=0
Df=3
R=1
R²=1
This all indicates that the results are close the infinity which means that further test should be
conducted to make sure that the radius of the curve is equal to zero.
Sanchez: Demonstrating Infinity 28
Conclusions
Through multiple trials and different observations, it has been concluded that the
observation with the best data is observation number 3, the first yoga ball. This has the best
information because it has a constant growth. Not only does it have a constant growth, but it
gives the best of both scaled lengths of the triangles and the scaled radius of the curvature. This
indicates that the probability that the data does not yield a linear graph is almost zero and both R
and R² are equal to one so the fit is nearly linear. In addition, it allows metaphysical questions to
be asked and let there be room for improvement. For example, the philosopher Emmanuel Kant
used introspection and data analysis to question the unknown since most of the studies conducted
in the metaphysical world only exist in the mind of the observer, so the observation can be bias
(Inwagen & Sullivan, "Metaphysics", 2007).
Sanchez: Demonstrating Infinity 29
Reference List
Einstein's Special Theory of Relativity. (2001). In J. S. Baughman, V. Bondi, R. Layman, T.
McConnell, & V. Tompkins (Eds.), American Decades (Vol. 1). Detroit: Gale. Retrieved
from http://ic.galegroup.com/ic/suic/ReferenceDetailsPage/ReferenceDetailsWindow?fail
Over Type=&query=&prodId=SUIC&windowstate=normal&contentModules=&display-
query=&mode=view&displayGroupName=Reference&limiter=&currPage=&disableHig
hlighting=false&displayGroups=&sortBy=&search_within_results=&p=SUIC&action=e
&catId=&activityType=&scanId=&documentId=GALE
%7CCX3468300276&source=Bookmark&u=cps&jsid=d4c66e91ed58583e5348aff61490
4e58
Guinn, J. (2014). Gravity and gravitation. In K. L. Lerner & B. W. Lerner (Eds.), The Gale
Encyclopedia of Science (5th ed.). Farmington Hills, MI: Gale. Retrieved from
http://ic.galegro up.com/ic/suic/ReferenceDetailsPage /ReferenceDetailsWindow ?
failOverType=&query=&prod Id=SUIC&windowstate=normal&content
Modules=&display-query=&mode=view&displayGroupName=Reference&limiter=
&currPage=&disableHighlighting=false&displayGroups=&sortBy=&search_within_resu
lts=&p=SUIC&action=e&catId=&activityType=&scanId=&documentId=GALE
%7CCV2644031037&source=Bookmark&u=cps&jsid=3aa3c5c2a0188c176aeec41fded6
0399
Hawking, S., & Mlodinow, L. (2005). A briefer history of time. New York: Bantam Books.
Inwagen, P. V., & Sullivan, M. (2007, September 10). Metaphysics. Retrieved March 14, 2017,
from https://plato.stanford.edu/entries/metaphysics/
Lens. (2015). In A. H. Blackwell & E. Manar (Eds.), UXL Encyclopedia of Science (3rd ed.).
Sanchez: Demonstrating Infinity 30
Farmington Hills, MI: UXL. Retrieved from http://ic.galegroup.com/ic/suic/Reference
Details Page/ReferenceDetailsWindow?failOverType=&query=&prodId=SUIC&window
state=normal&contentModules=&display-query=&mode=view&displayGroupName=
Reference&limiter=&currPage=&disableHighlighting=false&displayGroups=&sortBy=
&search_within_results=&p=SUIC&action=e&catId=&activityType=&scanId=&docum
entId=GALE
%7CCV2644300598&source=Bookmark&u=cps&jsid=e8b2b1085a097aaf44b46b6c4f66
413c
Maddocks, J. R. (2014). Postulate. In K. L. Lerner & B. W. Lerner (Eds.), The Gale
Encyclopedia of Science (5th ed.). Farmington Hills, MI: Gale. Retrieved from
http://ic.gale group.com/ic/suic/ReferenceDetailsPage/ReferenceDetailsWindow ?
failOverType=&query=&prodId=SUIC&windowstate=normal&contentModules=&displ
ay-query=&mode=view&displayGroupName=Reference&limiter=&currPage
=&disableHighlighting=false&displayGroups=&sortBy=&search_within_results=&p=S
UIC&action=e&catId=&activityType=&scanId=&documentId=GALE
%7CCV2644031780&source=Bookmark&u=cps&jsid=9435055c93d80fc0b71839b705d
b75db
Markosian, N. (2002, November 25). Time. Retrieved January 3, 2016, from http://plato.stan
ford.edu/entries/time/
Miller, G. H. (2014). Non-Euclidean geometry. In K. L. Lerner & B. W. Lerner (Eds.), The Gale
Encyclopedia of Science (5th ed.). Farmington Hills, MI: Gale. Retrieved from
http://ic.gale group.com/ic/suic/ReferenceDetailsPage/ReferenceDetailsWindow ?
failOverType=&query=&prodId=SUIC&windowstate=normal&contentModules=&displ
Sanchez: Demonstrating Infinity 31
ay-query=&mode=view&displayGroupName=Reference&limiter=&curr
Page=&disableHighlighting=false&displayGroups=&sortBy=&search_within_results=&
p=SUIC&action=e&catId=&activityType=&scanId=&documentId=GALE
%7CCV2644031547&source=Bookmark&u=cps&jsid=910b7abff6d2d310764547769a8
6b966
Perception. (2015). In A. H. Blackwell & E. Manar (Eds.), UXL Encyclopedia of Science (3rd
ed.). Farmington Hills, MI: UXL. Retrieved from http://ic.galegroup.com/ic/suic/
Reference DetailsPage/ReferenceDetailsWindow?failOverType=&query=&prodId=SUIC
&windowstate=normal&contentModules=&display-query=&mode=view&displayGroup
Name=Reference& limiter=&currPage=&disableHighlighting=false&displayGroups=
&sortBy=&search_within_results=&p=SUIC&action=e&catId=&activityType=&scanId
=&documentId=GALE
%7CCV2644300738&source=Bookmark&u=cps&jsid=23cdebc19e052ce4c4d8db5c02c
7568d
Quantum Mechanics. (2015). In A. H. Blackwell & E. Manar (Eds.), UXL Encyclopedia of
Science (3rd ed.). Farmington Hills, MI: UXL. Retrieved from http://ic.galegroup.
com/ic/suic /ReferenceDetailsPage/ReferenceDetailsWindow?failOverType=&query
=&prodId=SUIC&windowstate=normal&contentModules=&display-query=&mode
=view&displayGroupName= Reference&limiter=&currPage=&disableHighlighting=
false&displayGroups=&sortBy=&search_within_results=&p=SUIC&action=e&catId=&
activityType=&scanId=&documentId=GALE
%7CCV2644300800&source=Bookmark&u=cps&jsid=bd00d5cf65f276462bab45d1614
4cb68
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Shorter Oxford English Dictionary on Historical Principles: A-M. (n.d.). Retrieved January 8,
2016, from https://books.google.com.au/books/about/Shorter_Oxford_English_Dictio
nary _on_His.html?id=BmWfSAAACAAJ
Roberts, D. (1998). Non-Euclidean geometries. Retrieved December 8, 2015, from
http://regentsprep.org/regents/math/geometry/gg1/euclidean.htm
Rynasiewicz, R. (2004, August 12). Newton's Views on Space, Time, and Motion. Retrieved
January 3, 2016, from http://plato.stanford.edu/entries/newton-stm/
UC Davis Philosophy 175 Lecture Notes on Kant: Practical Reason. (n.d.). Retrieved January 3,
2016, from http://hume.ucdavis.edu/mattey/phi175/practicallechead.html