We start our consideration of this topic with a short trip through some trigonometry and analytic geometry. As illustrated in figure 6.1, suppose that you have a circular cone made of cheese on the table in front of you. With the blade of your knife or saw parallel to the table top, you slice through the cone of cheese. The cross-section of the cut cone is a circle. Your second slice, with the knife blade held at a small angle to the plane of the table, yields an ellipse. The third slice, cut at an angle parallel to the side of the cone, gives a parabola. And your fourth slice, with the knife blade at a steep or even a vertical angle, produces a hyperbola.
FIG. 6.1
The conic sections.
FIG. 6.2
The ellipse.
These shapes—the circle, ellipse, parabola, and hyperbola—are the “conic sections.” They have been studied and analyzed by mathematicians for more than two thousand years. Incidentally, if you do not happen to have a cone-shaped piece of cheese around, you can use a flashlight and a nearby wall to view the conic sections. This methodology is easier, less expensive, and involves fewer calories.
For our uphill-flowing river problem, we are interested in only the second of these conic sections: the ellipse shown in figure 6.2. This is an extremely important curve; it appears in a great many problems of mathematics and science. For example, it describes the paths that the planets and most comets follow in their orbits around the sun. It is utilized as the profile for some types of arch bridges and other structures. It provides the basis for an important subject in mathematics called elliptic functions.
From analytic geometry we establish that the equation of the ellipse is
where x and y are the coordinates of a particular point on the curve, a is half the length of the major (longer) axis of the ellipse, and b is half the length of the minor (shorter) axis. In the special case a = b = r0, equation (6.1) reduces to this is the equation of the circle. From equation (6.1) we obtain the expression
which says that for every value of the quantity x, there are two corresponding values of the quantity y. Suppose that a = 5 and b = 3. For the value of x = 4, for example, we determine from equation (6.2) that y = + 1.8 and – 1.8.
Back to equation (6.1) and figure 6.2. If this ellipse is rotated about the x-axis we generate a solid of revolution that looks somewhat like a football or watermelon; it is called a prolate ellipsoid. Alternatively, if the ellipse is rotated about the y-axis, the solid of revolution is termed an oblate ellipsoid; it resembles a slightly flattened pumpkin or the planet earth.
That is correct: the planet earth. As we know, the earth makes a complete rotation about its axis every 24 hours. This rotation is sufficiently fast to produce a velocity of about 460 meters per second at the equator. Like everything else that rotates about an axis, the earth creates centrifugal forces. For the rotating earth, these forces are large enough to significantly distort its shape from a perfect sphere to a slightly flattened pumpkin, that is, an oblate ellipsoid.
According to Abramowitz and Stegun (1965), the equatorial radius of the earth is a = 6,378,388 m and the polar radius is b = 6,356,912 m. The difference between these two distances indicates that the earth's diameter measured between two points on the equator on opposite sides of the earth is about 43 kilometers greater than the diameter from the north pole to the south pole.
We are now at the threshold of our “river flowing uphill” problem. Indeed, here is the gimmick (a bit sneaky, but never mind): What major rivers of the world have their mouths further from the center of the earth than their sources? As you eagerly endeavor to find answers to the question, you will be forced to combine these topics of mathematics with some topics of geography.
First we cast our mathematics into a more useful form. Refer-ring to figures 6.2 and 6.3 and the relationships
where r is the distance to the center of the earth and ø is the latitude of a particular point on the surface, we convert equation (6.1) into the form
FIG. 6.3
Definition sketch for the coordinates of the earth considered as an oblate ellipsoid.
TABLE 6.1
Since this is somewhat awkward to use for computation, we utilize the expression sin2 ø + cos2 ø = 1, to obtain
Substituting a = 6,378,388 m and b = 6,356,912 m into this equation gives
Table 6.1 lists values of r for various values of latitude, ø. It also indicates the amount of radial distortion (r – b) due to centrifugal force.
Now what rivers are likely to be uphill flowing? First, we exclude those flowing mostly west to east or east to west, since, in these cases, there is little or no change in latitude. Evidently we seek north to south or south to north rivers. Obviously, you will need some maps and tables to proceed with your analysis. A good deal of information can be obtained from the excellent atlas of Rand McNally (1978). Contour maps are needed to determine elevations of sources.
A good first guess is: the Mississippi River. This river flows almost directly north to south. It has its source in Lake Itasca in northern Minnesota at a latitude ø1 = 47.5°, and an elevation h1 = 450 m above mean sea level. From equation (6.6), the distance from the center of the earth to sea level at this latitude is r1 = 6,366,687 m. The source is 450 meters higher. So d1 = r1 + h1 = 6,367,137 m is the distance from the source of the Mississippi to the center of the earth.
The mouth of the Mississippi is about 120 kilometers southeast of New Orleans in the Gulf of Mexico at a latitude ø2 = 29.0°. At sea level we have h2 = 0. From equation (6.6) we compute r2 = 6,373,321 m, and since h2 = 0, the distance from the mouth of the river to the center of the earth has the value d2 = r2 + h2 = 6,373,321. The difference between these two distances is Δ d = d2 – d1 = +6,184 m. This result indicates that the mouth of the Mississippi River is 6,184 meters further from the center of the earth than is its source. Accordingly, this major river “flows uphill.”
The Missouri River, with its source in Madison County, Montana, and mouth in the Mississippi River near Saint Louis, also flows uphill. In this case, Δd = 2,365 m. The Rio Grande starts in San Juan County, Colorado, at an elevation h1 = 1,830 m and latitude ø1 = 37.7°. Its mouth is in the Gulf of Mexico with ø2 = 26.0° and h2 = 0. So, with Δd = 2,080 m, the Rio Grande flows uphill. The Colorado-Green River also flows uphill with Δd = 650 m.
Are there other uphill-flowing rivers in North America? Undoubtedly. Try the Arkansas and the Ohio; possibly the Connecticut, Hudson, and Sacramento. However, the Columbia, Saint Lawrence, and Yukon Rivers, with small latitude changes, probably do not qualify. The Mackenzie River, though it flows mainly south to north, flows the wrong way to be a candidate for uphill flowing.
On this latter point, the Nile River flows almost due north from its source in Lake Victoria (ø1 = 2.0°, h1 = 1,135 m) to its mouth in the Mediterranean (ø2 = 31.5°, h2 = 0). Accordingly, d1 = 6,379,498, d2 = 6,372,503, and the difference Δd = d2 – d1 = – 6,995 m. The minus sign means that the Nile is flowing downhill in two senses: (a) it goes from a high place to a low place on the earth's surface and (b) its mouth is closer to the earth's center than is its source. Too bad; the Nile does not even come close.
The important criterion to be “uphill flowing” should now be clear: the river must have its source at high latitude and its mouth at low latitude.
PROBLEMS As best you can, calculate Δd for some of these rivers: Amazon, Mekong, Volga, Yenisei, Danube, Rhine, Yangtse, and Zaire.
An Epilogue: The Shape of the Earth
Two comments: First, we have covered some topics involving mathematics and geography in our deliberations about rivers flowing uphill. Let's go a step further and include a bit of history in our considerations. Second, though we label this section an “epilogue” to our chapter, it is distinctly a “prologue” to the entire matter.
A word of explanation: As recently as the early eighteenth century, there was not even agreement among the world's scientists about the shape of the earth. The French believed it to be a prolate (watermelon-shaped) e
llipsoid; the British believed it to be an oblate (pumpkin-shaped) ellipsoid.
In order to settle the matter, in the spring of 1735, the French Academy of Sciences organized two scientific expeditions to measure arcs of the earth's surface: one near the equator and the other as close as possible to the North Pole. Soon after, two teams were deployed, one to Ecuador in South America, the other to northern Sweden in Europe. Though the team to Sweden returned to France a year later, ten years passed before the Ecuadorian team finally got back to Paris.
An extremely interesting story of the experiences and main achievements of the two scientific expeditions is given by Fernie (1991, 1992) in a series of three fascinating articles entitled “The Shape of the Earth,” which are highly recommended reading. By the way, as we all know, the earth is oblate.
7
A Brief Look at π, e, and Some Other Famous Numbers
The professor went to the board one day
And asked his class, just for the fun
What is the only conceivable way
To combine e, i, π, zero and one?
A brilliant young scholar, he proved quite a hero
Who knew the professor just loved to tease
Replied: eiπ + 1 = 0
Then he requested: the next question, please.
Many years ago, the writer received a magnificent $15 for submitting this poem, which, not long afterward, appeared in the mathematical nursery rhyme corner of a trade journal. Though the poem is pathetic enough, the real misfortune is that space did not allow acknowledgment to the famous Swiss mathematician, Leonhard Euler. In any event, the episode serves as a prologue to our next endeavor: an examination of several of the important numerical constants of mathematics.
Without question, the most famous of these “numbers” is the one we call π; it has the approximate numerical value π = 3.14159. Although its basic definition relates the circumference of a circle to its diameter, π appears in a large number of mathematical problems that have nothing whatsoever to do with circles.
Another extremely important number is the one identified as e. Its approximate value is e = 2.71828. It serves as the basis for so-called natural logarithms and also for things like exponential growth in demography, radioactive decay in physics, and bell-shaped curves in probability theory. Another famous number is the one called the golden ratio, ø = 1.61803. This number shows up in the strangest places, including the architecture of the Parthenon in Greece and the Great Pyramids of Egypt, the geometry of five-pointed stars and logarithmic spirals, and the shape of sunflower blossoms and Nautilus sea shells.
Yet another important number is Euler's constant. Its numerical value is γ = 0.57721. Again, this number appears in many problems of mathematics including the theory of heat conduction and the theory of extreme value distribution in statistics. Our fifth and final number is also the newest one: the Feigenbaum number, δ = 4.6692. This numerical constant makes its appearance in the relatively new area of mathematics called chaos theory.
In addition to these mathematical constants, there are also a great many physical constants; more than thirty of them are listed by Abramowitz and Stegun (1965). The ones we use in our various analyses are the following:
Gravitational acceleration, g = 9.82 m/s2 (in fact, g changes slightly with latitude, varying from 9.78 at the equator to 9.83 m/s2 at the poles)
Gas constant, R* = 8.314 joule/°K mol
Gravitational constant, G = 6.673 × 10-11 newton m2/kg2
Speed of light, c = 2.998 × 108 m/s
The Most Famous Number of All: π
The most ancient, most familiar, and most important number in all of mathematics—indeed in all of human civilization—is the one we designate with the symbol π. It represents the ratio of the circumference of a circle and its diameter and has the approximate value π = 3.14159. Thus, if C and D are, respectively, the circumference and diameter of a circle, then C = πD. Also, if A is the area of a circle and R = D/2 is its radius, then A = πR2.
More than 4,000 years ago, the Babylonians had established an approximate value for this important number: π = 25/8 (that is, 3.125). At about the same time, the Egyptians had determined that π = 256/81 (3.1605). The greatest mathematician of antiquity, Archimedes of Syracuse (287-212 B.C.), knew that π was more than 223/71 (3.14085) but less than 22/7 (3.14286).
Surprisingly accurate numerical values of π were also known in ancient Chinese, Hindu, and Mayan civilizations. The Bible, however, missed it by quite a bit. A verse in the Old Testament, 1 Kings 7:23, implies the value π = 3. Now perhaps that was an excusable mistake but the following one was not. In 1897, a bill was introduced in the Indiana state legislature which required that π = 3. The bill passed the House of Representatives by a 63 to 0 vote. Fortunately, a mathematics professor from Purdue University arrived on the scene in the nick of time. Thanks to his intervention, the π = 3 bill was withdrawn from deliberation by the Indiana Senate and, to date, has not been considered further.
It turns out that π is an irrational number, that is, it cannot be expressed as a ratio of two integers, like 22/7. In addition, it is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, such as x2 – 7x + 12 = 0. The consequence of this transcendental property of π is that the numbers to the right of the decimal point (i.e., 3.14159…) go on forever and ever without any apparent order or pattern.
It has long been a kind of contest or competitive game among mathematicians—and now computer specialists—to enlarge the number of decimal places of π. In the year 1560, it had been established that π = 3.141592, that is, it was known to an accuracy of six numbers to the right of the decimal point. By the end of the sixteenth century, π had been calculated to thirty decimal places. A summary of the growth of our knowledge of the number of known decimal places of π prior to the twentieth century is presented in table 7.1.
TABLE 7.1
With the invention and very fast development of electronic computers during the twentieth century, the number of known decimal places of π has increased rapidly and enormously. In 1947, π was known to 808 places. By 1957, the number had increased to over 10,000 and by 1967 to 500,000. Twenty years later, in 1987, the number had grown to 25 million. In 1997, Professor Kanada and his colleagues at the University of Tokyo computed π to an incredible 51.5 billion decimal places.
One wonders why they want to have all this information about the number of known decimal places of π. Well, for mathematicians involved in number theory, the extensive list of decimal places provides very useful information concerning patterns, distributions, randomness, and other properties and features of number sequences.
Over the years, many analytical methods and mathematical equations have been developed and utilized to calculate π. For example, in the past the following expression has been employed to determine the value of π:
This kind of equation is called an infinite series. Another well-known expression utilized for the computation of π is
You might want to try calculating π from equations (7.1) and (7.2). You will quickly discover that they are very slow in producing an answer. Indeed, a good many terms must be employed to obtain even a rough estimate of π. An equation that is much more suitable is the one employed by the noted German mathematician Carl Friedrich Gauss (1777-1855):
With regard to the use of infinite series for calculating the value of π, a remarkable advance was made in 1995 when the following expression was given by Bailey et al. (1997):
Although this equation is only slightly more complicated than the preceding expressions, it yields the value of π much more quickly. You might want to convince yourself that π is correctly computed to six decimal places by using simply the terms corresponding to k = 0,1,2,3 in equation (7.4).
A charming little book by Beckmann (1977) gives a brief history of π and descriptions of numerous interesting things about it. In addition, the book contains various mnemonic devices for π. As
we know, such devices help us remember things. For example, the mnemonic BASMOQ PN3T2 gives the provinces and territories of Canada. There are a great many mnemonics for π. The following one helps us remember its numerical value to fourteen decimal places. The number of letters in each word gives the respective number in the sequence (i.e., 3.14159 26535 8979).
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
If you prefer a shorter mnemonic device for π, here is one that will give you seven decimal places (3.1415926):
May I have a large container of coffee?
A substantial contribution was made to mathematics literature with the publication of Pi: A Source Book by Berggren, Borwein, and Borwein (1997). This voluminous work presents the history of this important number over the past 4,000 years. Included in its contents are seventy representative documents on the subject. Most of the contents, of course, deal seriously with the mathematical and computational aspects. For example, the contributions of the Indian mathematical genius Srinivasa Ramanujan (1887-1920) are included in the book.
Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Paperbacks) Page 5
