Exploring the Methods of Differential Calculus through the ...

Courtney Burris
Mattanawcook Academy
Upward Bound Regional Math Science
University of Maine
Mentors: Matt Dube, Megan Aydelott, and Adam Duncan
Exploring the Methods of Differential Calculus through the
Brachistochrone Problem
Abstract
The focus of this paper is exploring the methods, Newton and Leibniz’, of differential calculus to
solve the Brachistochrone Problem. The Brachistochrone Problem was proposed by Johann
Bernoulli. Modern concavity tests are used to provide insight into this problem from a physics
perspective. The objective is to find the type of curves that will allow a bead to be dropped from
point A to point B and arrive there in the shortest time based on the function’s concavity. The
problem is explored by comparing three functions, their accelerations, speeds, and velocities.
Based on these criteria, the decreasing parts of concave up class of functions is the best suited
function for this problem.
Introduction
Differential calculus is a method for finding the rate of change of a curve. When it was
discovered in the late seventeenth century, there was a controversy as to who actually discovered
it: Isaac Newton, an English physicist known for his pioneering studies of gravity (Newton,
1687), or Gottfried Wilhelm von Leibniz, a German philosopher with mathematical interests.
Newton discovered it first between 1665 and 1666, but Leibniz published his accounts first in
1684 in the treatise Nova methodus pro maximis et minimis, itemque tangentibus, quae nec
fractas nec irrationales quantitates moratur et singulare pro illis calculi genus(Rickey, 1994).
Newton’s work was not published until after his death. Leibniz was accused of plagiarizing
Newton’s work because on a mission to London he was shown Newton’s work by Oldenburg
and Collins, members of the Royal Society (Cooke, 1997).
To settle the calculus controversy, Johann Bernoulli, a Swiss mathematician, proposed the
Brachistochrone Problem in the Acta Eruditorum, the same place where Leibniz’s contributions
were published (Bernoulli, 1696). In general, the problem is about constructing the optimal ramp
that can get a bead from one fixed point to another in the shortest amount of time (Rickey, 1994).

The goal of this project is to determine which classes of curves will make a bead arrive from
point A to point B in the shortest time, as proposed by Bernoulli in the Brachistochrone problem.
This problem will be explored through the concepts of derivatives and concavity, pioneered by
the work of Newton and Leibniz and interpreted through knowledge of physics relationships
momentum and stopping distance.
Brachistochrone Problem
In 1697, Johann Bernoulli posed a problem to his brother Jacob, Leibniz, and Newton. The
problem stated:
To determine the curved line joining two given points, situated at different
distances from the horizontal and not in the same vertical line, along which a
mobile body, running down by its own weight and starting to move the upper
point, will descend most quickly to the lowest point (Rickey, 1994).
“Brachistochrone” came from the greek words βράχιστος χρόνος, meaning shortest time.
Bernoulli was not the first person to propose this problem. Previously Galileo, in trying to solve
this problem, found that a circular arc was better, meaning that it got the bead from point A to
point B faster, than a straight line, but he was wrong in thinking that a circular arc was the
answer because the answer is not a circular arc, it is an inverse cycloid. (Rickey, 1994)
Bernoulli had proposed this problem once before in a journal, Acta Eruditorum (O’Connor,
2002). He set a six month deadline, but in those six months no one sent him the correct solution.
Leibniz thought he had the right answer, and he indeed had part of it, but he had not solved the
whole problem. Bernoulli had not received an answer from Newton and thus concluded that
Newton could not solve it. In actuality, Newton had never seen the problem. Leibniz convinced
Bernoulli to extend the deadline until Easter. Leibniz had hoped that if he himself could
successfully solve the problem, it would put the controversy to rest and people would side with
him and his calculus over Newton’s. This hope, however, did not happen. Newton received the
problem in the mail on New Year’s Day, and rumor has it that he solved it at four o’clock the
next morning. Leibniz however never successfully finished the problem (Rickey, 1994).
In modern times, mathematicians have resorted to geometric methods of solving this problem
using results such as Thales’ Theorem (Boute, 2012). There are in reality many ways of solving
the problem. Newton’s solution contrived from the length of the curve segment and the normal
and tangential vectors is one way of viewing the problem, but geometric arguments can also be
used.
Methods and Definitions
For this exploration, the concepts of the differential calculus are employed to provide insight into
the classes of curves that would perform favorably for the tasks of the Brachistochrone Problem.
The differential calculus is the study of mathematics specifically concerned with the

determination of slope: a measure of the change in a single dependent variable relative to the
change in a single independent variable. Both Newton and Leibniz contrived different ways for
exploring the idea of slope as changes in the independent variable became increasingly smaller.
This notion fits into mathematical history as Newton’s predecessors-John Wallis and Isaac
Barrow-were two of the pioneers of a revolutionary concept: infinity. Rather than to look at the
magnitude of infinity, they treated the impacts of changes that were infinitely small. These
infinitely small changes served as a reciprocal view of infinity.
Figure 1. Newton’s Method (Newton, 1736) for calculating the slope of curve VV at point V.
Figure 1 is an example of Newton’s Method. The green lines are secant lines drawn from the
curve intersecting at point V. Newton’s method simply reduces the distance between the
intersection points of the secant lines (shown in descending shades of green) and curve VV. If
that distance were infinitesimally small, then the blue line is the tangent line at point V.

Figure 2. Leibniz’ method (Leibniz, 1684) of characteristic triangles for calculating slope at
a point.
Figure 2 is an example of Leibniz’ method. This figure was constructed by first drawing the axis
AX and curve VV. A line is then drawn perpendicular to the axis from point V, this is called the
ordinate v. A tangent line is then drawn tangent to curve VV at point V and connected to axis AX
at point B. Line XB is called an absissa. The ordinate, absissa, and tangent line connect to form a
characteristic triangle.
The more modern view of calculus unifies both of these definitions of slope under the concept of
a limit. The limit is the value that a dependent variable approaches as the independent variable
approaches a specified value. The modern definition of the infinitesimal slope (i.e., the
derivative) has become completely colored by this approach. This approach is shown as
equation 1:
This method clearly follows Newton’s method precisely. More importantly, it also follows
Leibniz’s method in a simple manner; by construction, Leibniz had already limited the
calculation geometrically by choosing to calculate the slope of the tangent line directly, as
opposed to Newton’s approach of secant lines.
Slope calculations are important for determining whether curves are increasing or decreasing and
how linear interpolations of curves behave relative to the original curve. Leibniz refers to this as
a curve’s concavity (Leibniz, 1684). A curve is concave up (Figure 3) if a linear interpolation of
the curve sits above the real curve; similarly, a curve is concave down (Figure 4) if a linear
interpolation of the curve sits beneath the curve. Newton’s method, when seen graphically,
makes concavity obvious.
(1)

Figure 3. A concave up curve.
Figure 4. A concave down curve.
Successful consideration of the Brachistichrone problem also requires knowledge of two
important concepts in physics: momentum and stopping distance.
Momentum is the amount of potential energy that an object moving at speed has. This energy is
directly related to the mass of the object and the velocity of the object by the equation
(Henderson, 2012):
The stopping distance of an object has nothing to do with its size, but everything to do with its
velocity. The stopping distance is the amount of distance that it takes for an object moving at a
particular speed to return to a state of no momentum. A positive value of stopping distance
implies that velocity has an effect after the moment at which it is measured. Stopping distance
can be modeled by the equation (Nave, 2012):
(2)

Biographies
Sir Isaac Newton was born prematurely on Christmas, 1642. He went to Cambridge University,
where his focus was mainly in chemistry. He also studied algebra and analytic geometry. He
worked under Isaac Barrow, who discovered infinitesimal calculus. In 1665 there was an
outbreak of the Great Plague which caused Newton to return home to Woolsthrope, England
(Cooke, 1997). He remained there for two years and while Cambridge was closed he studied his
own mathematics and physics research. It was during this time that he created the Binomial
Theorem. Newton died in 1727, at the age of 85, in England (Struik, 1948).
Gottfried Wilhelm von Leibniz was born in Leipzig, Germany in 1646. He was a Roman
Catholic and his religion was a large part of his life (Struik, 1948). At fifteen, Leibniz entered the
University of Leipzig, where he studied law, following in the footsteps of Descartes, Fermat and
Viète (Cooke, 1997). When he graduated at the age of 20 he was deemed too young to receive
his doctorate in law. After college he served as a diplomat for the Elector of Mainz. He
eventually became the diplomat for the Electors of Hanover, where he worked for four decades.
It was during a mission to Paris in 1672 that Leibniz first became interested in mathematics. The
following year he went to London, where he met members of the royal society, Henry Oldenburg
and James Collins (Cooke, 1997). Oldenburg and Collins played major roles in the calculus
controversy. It was said that they told Leibniz of Newton’s advances in calculus (Rickey, 1994).
He discovered his calculus between 1673 and 1676. He was influenced by Huygens and the
studies of Pascal and Descartes. Leibniz died in 1716, in Germany (Struik, 1948).
Results
There are three basic curves that could be the solution to the Brachistochrone problem: a concave
up curve, a straight line, and a concave down curve. The first candidate is the concave up curve;
the second candidate is the straight line; the third candidate is the concave down curve. These
candidate ramps are graphed in Figure 5.
Candidate Ramp One:
Candidate Ramp Two:
Candidate Ramp Three:
(3)

Figure 5. Graphs of the three candidate ramps.
In Figure 5 (and all other figures in this section), distance stands for the horizontal distance, not
the distance along the ramp. This graph represents the three candidate ramps. The bottom line is
the first candidate; the middle line is the second candidate, and the top line is the third candidate.
All three candidates ramps start and end at the same point but don’t have any other common
points. Candidate one starts off with a steep decline and then flattens out. Candidate two has a
steady decline from point A to point B. Candidate three starts off with a slowly declining and
then ends with a steep decline.
The next step is to take the first derivative of each candidate ramp. The first derivative will be
the velocity attained by the bead at that point on the ramp. A graph of the velocity curves is
presented in Figure 6.
Candidate Ramp One:
Candidate Ramp Two:
Candidate Ramp Three:

Figure 6. Velocity curves for each candidate ramp.
Candidate one has the lowest velocity at the start. The velocity of candidate one increases and
has the highest ending velocity. Candidate three has the highest starting velocity. The velocity of
candidate three decreases and has the lowest ending velocity. There is a point where each pair of
candidate ramps have intersecting velocity curves, implying that at one point those two ramps
produce the same velocity. In fact, two of the intersections are guaranteed by the Mean Value
Theorem because candidate ramp two is a linear approximation between the two endpoints of
both candidate ramp one and candidate ramp two.
The next step is to find the absolute value of each of the velocity curves. The absolute value of
the velocity curve represents the speed induced by the ramp. Once the absolute value is found,
the three speeds are compared graphically in Figure 7.
Candidate Ramp One:
Candidate Ramp Two:
Candidate Ramp Three:

Figure 7. Speed curves for the three candidate ramps.
The concave up curve starts out with the highest speed but the speed declines and it ends with the
lowest speed. On the straight line, the speed is constant. The concave up curve starts with the
slowest speed, but the speed increases and it ends with the highest speed. There is a point in this
graph at which the curves of the first and third candidates intersect, meaning that they have the
same speed at that instant. Also at that point the speed of candidate two is higher than that of
candidate one and three. These intersections correspond precisely to the intersections of the
velocity curves.
After the three speeds are compared the final step is to find the second derivatives of each
candidate ramp, representing the acceleration curve of the candidate ramps. The acceleration
curves are calculated and shown graphically in Figure 8.
Candidate Ramp One:
Candidate Ramp Two:
Candidate Ramp Three:

Figure 8. Acceleration curves of the three candidate ramps.
Candidate three has positive acceleration everywhere, meaning it is picking up speed as
horizontal distance increases. Candidate two has zero acceleration, implying that it is traveling at
a constant speed. Candidate one has negative acceleration values everywhere, meaning that it is
slowing down as horizontal distance increases. It is however of note that the acceleration curves
for both Candidate one and candidate three are increasing. In candidate one’s case, that implies
that it is slowing down at a decreasing rate and in the case of candidate three, this observation
implies that the ramp induces greater increases in speed as the bead progresses.
Discussion
Recall from equation (2) the momentum is proportional to the mass and the velocity. In this case,
the bead has a fixed mass, meaning that the effect of mass on momentum can be ignored. Thus
only velocity (Figure 6) can affect the momentum of the bead. The velocities of the bead for the
cases studied are all negative. Since the velocities are all negative, the absolute value, speed
(Figure 7), is used because it does not matter in which direction the bead travels, rather the rate
of motion that it exhibits. This observation is how the speed of the bead affects the momentum of
said bead. Having increased velocities earlier in the process allows for the stopping distance to
contribute to achieving the target location in a quicker time.
Recall from equation (3) that stopping distance is assumed to be for a flat surface, meaning that
acceleration due to gravity would not have an effect on the stopping distance; however, the
candidate ramps are not flat surfaces, so acceleration does have an effect on the stopping distance

of the bead in that it can only increase the stopping distance. Thus having acceleration prolongs
the effect of velocity, and by proportionality, momentum.
The optimal curve is in the concave up class of curves, but only the decreasing portion of the
curve. These curves are optimal because they have a high initial velocity which is directly
proportional to momentum, meaning that the concave up curve will have a high momentum and
has a high speed, that will cause the bead to arrive from point A to point B in the shortest amount
of time. The decreasing portion is the only portion that would work for the Brachistochrone
problem because when the curve would start to increase again the force of gravity would be
pushing down against it, decreasing its momentum, speed, and acceleration, which would cause
the bead to arrive from point A to point B in a slower time than the other curves.
Figure 9. The inverse cycloid, the real solution to the Brachistochrone problem (Slinker, 1992).
References
Boute, R. (2012). The brachistochrone problem solved geometrically: A very elementary
approach. 193.
Cooke, R. (1997). The history of mathematics: A brief course. New York: John Wiley & Sons,
Inc.
Dictionary. (2012, July 09). Retrieved from http://dictionary.reference.com/
Henderson, T. (2012). The impulse-momentum change theorem. Retrieved from
http://www.physicsclassroom.com/class/momentum/u4l1a.cfm
Johnson, N. (2004). The brachistochrone problem. 192-197.

Khan, S. (Performer) (n.d.). Newton leibniz and usain bolt : Why we study differential
calculus. KhanAcademy. [Video podcast].
O'Connor, J., & Robertson, E. (2002). The brachistochrone problem. Retrieved from
http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html
O'Connor, J., & Robertson, E. (n.d.). Johann bernoulli. Retrieved from http://wwwhistory.mcs.st-and.ac.uk/Biographies/Bernoulli_Johann.html
Nave. (2012). Stopping distance for auto. Retrieved from http://hyperphysics.phyastr.gsu.edu/HBASE/crstp.html
Newton, I. (1687). Philosophia naturalis principia mathematica.
Rickey, V. (1994). Isaac newton: Man, myth, and mathematics. In F. Swetz (Ed.), From Five
Fingers to Infinity A Journey through the History of Mathematics (pp. 483-508). Peru,
Illinois: Open Court Publishing Company.
Rickey, V. (1996). The history of the brachistochrone. In Historical Notes for the Calculus
Classroom (pp. 63-67). Retrieved from
http://www.math.usma.edu/people/rickey/hm/CalcNotes/brachistochrone.pdf
Schrader, D. (1994). The newton-leibniz controversy concerning the discovery of the calculus. In
F. Swetz (Ed.), From Five Fingers to Infinity A Journey through the History of
Mathematics (pp. 509-520). Peru, Illinois: Open Court Publishing Company.
Schultz, P. (2000, August 24). Lecture 19 leibniz' invention of calculus. Retrieved from
http://school.maths.uwa.edu.au/~schultz/L19Leibniz.html
Slinker, G. (1992). Inbetweening using a physically based model and nonlinear path
interpolation. Retrieved from http://home.comcast.net/~gslinker/thesis/Thesis.html
Storr, A. (1985). Isaac newton. 1779-1884.
Struik, D. (1948). A concise history of mathematics. New York: Dover Publications, Inc.
Acknowledgements
Many people have helped me throughout this project, whether they provided me with
information, answered my numerous questions, or helped calm me down when I felt

overwhelmed. First of all, I would like to thank Kelly Ilseman for giving me the opportunity this
summer to do this projects. Next I would like to thank Adam Duncan and Megan Aydelott for
being such wonderful mentors and guiding me throughout this project. I would like to give a big
thank you to Chelsea Mucciarone for helping me stay sane and letting me vent to her when I was
so frustrated that I didn’t think I could keep going with this project. I would especially like to
thank Jeremy Swist for helping me organize my paper and for his help in choosing a topic. I also
really appreciate his efforts in translating Leibniz’s manuscripts from Latin to English. Last, but
certainly not least I would like to give a big thank you to Matt Dube for going out of his way to
track down resources for me. Also, he deserves an even bigger thank you for explaining it all to
me.
Autobiography
My name is Courtney Burris. I was born in Milford, Connecticut on March 1st, 1995. My family
moved to Maine in August of 2000. I am currently a senior at Mattanawcook Academy, in
Lincoln. I moved to Lincoln in 2004. I am also an Upward Bound student, and this is my first
summer here. I am the only girl on my wrestling team, my passion is acting, and I enjoy writing
poetry. If I could be anything I would be an actress. When I graduate I plan to move to California
to take acting classes and audition for some roles. In the fall of 2013 I hope to attend the
University of Pennsylvania where I will study finance and business. I plan to be a chief financial
officer for a large corporation, unless the acting thing works out for me.