BLAISE PASCAL

HEXAGON THEOREM

AND

CYCLOID CONTEST

ANGIE CLARK

MICHELLE WHITT

GROUP PROJECT #2

SOURCES:

www1: www-grsups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html

www2: www.wcsu.ctstateu.edu/~gay002/geometry.htm

www3: www-history.mcs.st-andrews.ac.uk/history/Curves/Cycloid.html

www4: pascal-central.com/blaise.html

Boyer, Carl B. A History of Mathematics , second edition, John Wiley & Sons Inc.: 1991, p361-367.

Finney, Ross and Thomas, George. Calculus and Analytical Geometry , eighth edition, Addison-Wesley Publishing Company: 1992, p349, 652-653.

History of Blaise Pascal

Blaise Pascal was born June 19, 1623 in Clermont-Ferrand, France. He was the only son, and the third of four children. Pascal was home taught by his father; however, mathematics was forbidden until Pascal turned fifteen. At the age of twelve Pascal began to study geometry on his own. By fourteen, Pascal's father relented his previous conditions, and Pascal began to study mathematics and attending Mersenne meetings, a religious order of the Minims. By the time Pascal turned sixteen, he established enough mathematical interest to present at on of the Mersenne meetings a piece of paper that contained many projective geometry theorems, including Pascal's hexagon theorem.
Pascal published his first work, Essay on Conic Sections , in February 1640. He invented the Pascaline, the first digital calculator that looked like a, "mechanical calculator of the 1940's," (www1), in 1645 to help his father with his tax collection work. Sometime in 1646 Pascal's father was injured and Pascal was looked after by a religious movement; which led Pascal to become very religious.
In 1647 Pascal proved, "to his satisfaction," that a vacuum existed (www1). And in 1648, he observed, "that the pressure of the atmosphere decreases with height," which led to his deduction that the vacuum existed above the atmosphere (www1). By 1653, Pascal had written, Treatise on the Equilibrium of Liquids , which explained his law of pressures. Throughout 1648,1653, and 1654, Pascal worked on The Generation of Conic Sections , which considered the conic generated by a central projection of a circle.
In October 1654, Pascal was affected psychologically by a carriage accident, and later that year devoted his life to Christianity. Pascal's most famous work in philosophy, written during 1656-1658, was Pensees , a collection of human suffering and faith in God.
Pascal's last work was on the cycloid, "a curve traced by a point on the circumference of a rolling circle," (www1). He came up with a contest that dealt with finding the solutions to problems of a cycloid. Only two people submitted answers, and neither one of them won. Pascal published his solutions to the cycloid contest problems in Letters to Carcavi . Afterwards, Pascal gave up science and began to again devote his life to religion. He died at the age of thirty-nine, on August 19, 1962 from a, "malignant growth in his stomach that grew to his brain," (www1).

Pascal's Hexagon Theorem

Pascal's hexagon theorem states, If all six vertices of a hexagon lie on a circle and the three pairs of opposite sides intersect, then the three points of intersection are collinear," (www2). It is interesting helpful to note that, "nobody knows how Pascal proved his theorem since his original proof has been lost for centuries. It is known to have existed because it was seen and proved by the famous mathematician G.W. Leibinz," (www2).

Examples

> p := theta -> [2*cos(theta),sin(theta)];

> p := proc (theta) options operator, arrow; [2*cos(theta), sin(theta)] end;

> p(0);

> op(p(theta));

> ellipse:=plot([op(p(theta)),theta=0..2*Pi],scaling=constrained):

>

> line1:=plot([[-1.8,0.44],[0.2,-0.99]],color=red): line2:=plot([[0.2,-.99],[1.8,0.44]],color=blue): line3:=plot([[1.8,0.44],[-1.4,-0.714]],color=turquoise): line4:=plot([[-1.4,-0.714],[0.2,0.99]],color=magenta): line5:=plot([[0.2,0.99],[1.4,-0.714]],color=green): line6:=plot([[-1.8,0.44],[1.4,-0.714]],color=yellow): line7:=plot([[-2.2,-0.2],[2.2,-0.22]],color=black,linestyle=3):

> plots[display](ellipse,line6,line3,line1,line2,line4,line5,line7,axes=none);

The drawing above is an example of the Hexagon Theorem inscribed in a conic. In this case,the conic used was an ellipse. The black line represents the Pascal Line, connecting the three collinear points. The collinear points occur at each of the following intersections:

red and magenta

yellow and turquoise

blue and green.

Proof of Pascal's Hexagon Theorem:

> with(plottools):
c := circle([0,0], 1, color=blue):

Warning, new definition for ellipse

> l1:=plot([[0.9,0.44],[-1,0]],color=turquoise):

> l2:=plot([[-1,0],[0,1]],color=red):

> l3:=plot([[0,1],[0.6,-0.8]],color=magenta):

> l4:=plot([[0.6,-0.8],[-0.7,0.71]],color=violet):

> l5:=plot([[-0.7,0.71],[-0.7,-0.71]],color=green):

> l6:=plot([[-0.7,-0.71],[0.9,0.44]],color=yellow):

> txt:=plots[textplot]([0.94,0.47,`A`],color=yellow):

> txt2:=plots[textplot]([-1.04,0.03,`B`],color=turquoise):

> txt3:=plots[textplot]([0.04,1.05,`C`],color=red):

> txt4:=plots[textplot]([0.64,-0.84,`D`],color=magenta):

> txt5:=plots[textplot]([-0.74,0.75,`E`],color=violet):

> txt6:=plots[textplot]([-0.74,-0.75,`F`],color=green):

> plots[display](c,l1,l2,l3,l4,l5,l6,txt,txt2,txt3,txt4,txt5,txt6,scaling=constrained,axes=none);

= the intersection of (turquoise line) and (violet line)

= the intersection of (magenta line) and (yellow line)
= the intersection of (red line) and (green line)

Extend lines and to make the triangle

> l7:=plot([[-0.7,0.71],[-0.7,3.2]],color=green,linestyle=3):

> l8:=plot([[0,1],[-0.7,3.2]],color=magenta,linestyle=3):

> txt7:=plots[textplot]([-0.75,0.1,`V`]):

> txt8:=plots[textplot]([-0.75,0.3,`N`]):

> txt9:=plots[textplot]([-0.3,0.19,`L`]):

> txt10:=plots[textplot]([0.3,0.22,`W`]):

> txt11:=plots[textplot]([0.1,-0.15,`M`]):

> txt12:=plots[textplot]([-0.74,3.24,`U`]):

> plots[display](c,l1,l2,l3,l4,l5,l6,l7,l8,txt,txt2,txt3,txt4,txt5,txt6,txt7,txt8,txt9,txt10,txt11,txt12,scaling=constrained,axes=none);

>

>

>

From Menelaus' theorem:

* * = 1

* * = 1

* * = 1

Using the theorem that states, "If two lines through a point P meet at a circle at points and , and and respectively, then * = * ," (www2)
Then:

* * = 1

which shows that the points L, M, and N are collinear (www2).

Pascal's Cycloid Contest

In 1658, after ignoring mathematics because of his devotion to religion, Pascal began to contemplate cycloid problems. A cycloid, as stated before, "is the curve traced out by a point on the circumference of a circular loop which rolls along a straight line," (www4). He solved various problems of the cycloid including; the problem of the area of any segment of the cycloid, the center of gravity of any segment, and the problems of surface area and volume of the solid of revolution formed by rotating the cycloid about the x-axis (www3).
Pascal developed questions like the ones he previously solved, and held a contest for other mathematicians to solve them. Only two people, Wallis and Lalouere, entered, and neither were successful. Sluze, Ricci, Huygens, Wren and Fermat submitted their discoveries to Pascal. Pascal published his solutions and some of the other's discoveries in Letters to Carcavi.

Examples

A wheel, which is used in this example, is a circle with radius h.

> with(plottools):
wheel:=plot([cos(t),sin(t),t=0..2*Pi],scaling=constrained,axes=none,color=blue):

> h:=plot([[0,0],[0,-1]],color=blue):

> txt:=plots[textplot]([0.1,-0.5,`H`],color=blue):

> plots[display](wheel,h,txt);

A flashlight placed on a certain point of the wheel will trace the path of the cycloid when put into horizontal motion. The "stationary path" of the wheel is shown below.

> frame:=Phi->plot([-cos(t),sin(t),t=0..Phi],color=gold):

> plots[display]([seq(frame(i*Pi/20),i=0..40)],insequence=true,scaling=constrained,axes=none);

A cycloid is the shape traced out as a wheel rotates along a horizontal path. The following is an example of what a cycloid looks like.

In the equation for this cycloid, the variables are defined in the following manner:

a = speed of rotation (defined, in this problem, as )

h = radius of the wheel (defined, in this problem, as 1)

> g:= (a,h,t) -> [a*t+h*cos(a*t/h+Pi/2),h-h*sin(a*t/h+Pi/2)]:

> plot({[op(g(2*Pi,1,t)),t=0..Pi]},color=blue,scaling=constrained);

If the wheel is put into horizontal motion, the flashlight path forms a cycloid.

> frame:=Tau->plot({[op(g(2*Pi,1,t)),t=0..Tau]},color=gold):

> plots[display]([seq(frame(i*Pi/20),i=0..13)],insequence=true,scaling=constrained);

P roblem

Find the arclength of the following cycloid.

Solution

The arclength of a cycloid, whose radius (a) is 1, is equal to the following:

> f:=(t)->((1-cos(t))^(1/2)):

> Int(f(t),t=0..2*Pi)=int(f(t),t=0..2*Pi);