The Rounding of the Earth

The Rounding of the Earth

An examination of the early Greeks' theory that the Earth is round, and some of the consequences resulting from that theory.

by John Schwarz

August, 2004

Ancient Man's Concept of the Earth's Shape >>>

Ancient Man's Concept of the Earth's Shape

Many thousands of years ago, common sense led man to believe that the Earth was flat. How could it be otherwise? If the Earth were round, for example, things would fall off the underside. Furthermore, since nothing in the heavens appeared to be as large as the Earth, they assumed it to be the dominant body in the universe. So they pictured the Earth as a platform in space with the sun and the moon and the stars all revolving around it. (See Fig. 1.)

Not having electric lights and various modern-day distractions, these ancient people were much more aware of the positions and motions of the moon and the stars in the night ski, and the movement of the sun in the daytime, then most of us are today. So it was only natural that when they wanted to tell traders and travelers how to get from one place to another, they used the stars and the sun as direction markers. They made use of the fact that the sun came up in the morning in a certain direction (which we call the east)*, and went down in the evening in the opposite direction (which we call the west). And then there was this star (we call it the North Star) that appeared at a fixed distance above the horizon throughout the night, and which (unlike other stars) never changed its position in the sky relative to a fixed viewer. (We call the direction in which this star is seen as north.)

So with the sun rising and falling in the east-west direction, and the North Star shining in a direction at right angles to that (thus defining a north-south direction), a map of the flat Earth could be thought of as a set of horizontal east-west lines (now called latitudes), and a set of perpendicular north-south lines (now called longitudes). If a traveler wanted to know how to get to city A from city B, for example, he could be told to travel for X number of days in the east or west direction (using the sunrise and sunset positions as a guide), and then for Y number of days in the north or south direction (using the North Star as a guide).

*Throughout this paper, the slight angle of the earth's rotational axis with respect to it's orbital axis around the sun has been neglected. This was done to simplify the understanding and explanation of the basic concepts involved.

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Doubts About the Flat Earth Concept >>>

Doubts About the Flat Earth Concept

As people progressed and began to travel further and further distances, some of the things they observed were very puzzling. For one, while the North Star always appeared at a fixed height above the horizon throughout the night, this height increased as you traveled northward. Conversely, this height decreased as you traveled southward, disappearing from the sky altogether if you went far enough south. If the Earth was flat, how could this be? In addition, if the Earth was flat, the height of the North Star should change if you traveled in an east-west direction. But it didn't. The height remained the same no matter how far east or west you went. It was also difficult to explain why you could look directly at the North Star throughout the night without changing your head position, whereas just about all of the other stars changed their position in the sky as the night went on.

Another thing that puzzled travelers was the height of the sun in the sky at its highest point during the day. As you traveled northward, this high point got noticeably lower and lower in the sky. Conversely, if you traveled southward, the sun's high-point moved higher and higher each day, eventually passing directly overhead. This could be partially explained if the sun were close enough to the Earth. In that case, the suns rays would reach the Earth at various angles. (A false assumption, see Appendix A1.) Even so, the extent to which this phenomena occurred was incompatible with a flat Earth. It was also puzzling why the height of the noonday sun (like the height of the North Star) remained the same as you traveled in an east-west direction. This would not be the case if the Earth was flat.

A flat Earth also had a problem explaining the movement of the stars. At evening time, for example, stars slowly became visible at various places in the sky, and then proceeded to move to other positions throughout the night. Some moved quite far during that time, others barely moved at all. Nevertheless, the next evening they all reappeared at the same place they were on the preceding evening. It was difficult to envision an orbital path for each star such that its travel during the day caused it to end up in the same spot each evening.

Other things may have bothered these early people also. For example, if the Earth were a flat platform, how come nobody could see the edge? And when ships sailed out to sea, why did the hull disappear from sight before the masts? And if this meant that the ship was about to go over the edge, how come it was able to return? And when it did return, why did the masts appear before the hull? (For more on the line-of-sight vision distance from an observer on the shore to a ship at sea, see Appendix A2.)

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The Greeks Propose a "Spherical Earth" Theory >>>

The Greeks Propose a "Spherical Earth" Theory

About 2000 years ago, the Greeks concluded that most of the unanswered questions and puzzles that arose out of the belief that the Earth was flat, could be answered if it was assumed that the Earth was a sphere, not a flat platform. After all, the sun and the stars appeared to be spherical in shape, so why not the Earth? They also reasoned that if the circumference of the Earth was very large, the curvature over a line-of-sight area would be so small that it would appear to be flat, even though in reality it was very slightly curved.

The answer to the question of why things didn't fall off the underside remained unclear, but perhaps these Greeks asked the question "If a sphere is in a space that appears to be boundless in all directions, how do you know which is the underside?" Besides, when a person jumps off the ground, why doesn't he fly off into space? Couldn't whatever causes him to fall back to Earth on the top side also cause him to fall back to Earth on the bottom side?

The rationale for a spherical Earth is shown in Fig. 2. At point \(C\), light from the North Star makes an angle of \(C_{1}\) degrees with respect to the horizon. (The horizon is defined by a line tangent to the Earth at point \(C\).) Thus an observer at point \(C\) sees the North Star as being \(C_{1}\) degrees above the horizon. Furthermore, this same angle is observed all along that latitude because the North Star lies directly above the north pole. At point \(D\), an observer sees the North Star as appearing only \(D_{1}\) degrees above the horizon. And again, this angle is the same at any point along that latitude. (It can be seen that at any latitude below the equator, the North Star is never visible.) A spherical Earth therefore explained why the North Star always appeared at a fixed height above the horizon throughout the night, and why this height increased as you traveled northward, and disappeared altogether if you went far enough south.

Another justification for a round Earth was the angle of the noonday sun above the horizon. (Refer again to Fig. 2.) At point \(A\), light from the sun makes an angle of \(A_{1}\) degrees with respect to the horizon. (The horizon again being defined by a line tangent to the Earth at point \(A\).) Thus an observer at point \(A\) sees the noonday sun (that is, the sun at its highest point in the sky) as being \(A_{1}\) degrees above the horizon. Furthermore, this same noonday angle is observed all along that latitude. At point \(B\), an observer sees the sun as appearing only \(B_{1}\) degrees above the horizon. And again, this angle is the same at any point along that latitude. Thus the observations of the sun's behavior was in complete agreement with the spherical Earth theory.

Then there was the question of why the stars moved about during the night, but then reappeared at the same starting position each evening. This conundrum could be easily resolved if it was assumed that the day-night cycle was caused by the Earth rotating about its north-south axis, and not by the sun rotating around the Earth. This implied that man was an observer on a rotating Earth watching stars that were in a fixed position in the sky, and not an observer on a fixed Earth watching stars move through the night sky. The revolving Earth theory readily explained why the stars reappeared in the same position each evening, and then appeared to move in the directions observed during the night. It should be noted that this theory, while it required the Earth to be spherical, did not necessarily imply that the Earth circled the sun. All it implied was that the sun circled the Earth at a slower rate than the day-night cycle, like the moon for example (which circles the Earth every 29.5 days). Most of these Greeks at that time did, in fact, still believe that the sun circled the Earth.

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Determining the Earth's Circumference >>>

Determining the Earth's Circumference

Convinced that the Earth was round, the Greeks set out to determine its circumference. It is not surprising that their approach involved measuring the length of shadows inasmuch as the sundial (which relies on the sun's shadow) was already a well-established tool for telling the time of day. The Greeks' method of determining the circumference was as follows.

As depicted in Fig. 2, the angle of the noonday sun with respect to the horizon increases as one goes from north to south on the same longitude. The Greeks reasoned that if the change in this angle was measured over a known distance, the result could be extrapolated to determine the distance (in this case, the circumference) over a full \(360^{\circ}\) change. This was achieved as follows:

Referring to Fig. 3, a stake (labeled stake \(A\)) was driven into the ground so that it was perpendicular to the Earth in all directions. (Probably done by making it parallel to a weight hanging at the end of a cord.) When the sun was determined to be at its highest point in the sky, the angle \(A\) was measured. This could be achieved by measuring angle \(A\) directly, or (more likely), by measuring the height of the stake and the length of it's shadow, and from these determining the magnitude of angle \(A\). The process was repeated with stake \(B\) at a second site on the same latitude as stake \(A\), and at a known distance from it.

As shown in the Fig.3 diagram, dash-line extensions of stakes \(A\) and \(B\) intersect at point \(O\), the center of the Earth. A line, \(\overline{OX}\), is constructed so as to be parallel to the sun's rays. By virtue of the geometric theorem on a transverse line intersecting parallel lines, angles \(A\) and \(B\) at the top of the stakes are equal to angles \(A\) and \(B\) shown at the earth's center. Angle (\(B-A\)) is therefore the angular distance between stake \(A\) and stake \(B\), and is the same fraction of \(360^{\circ}\) as the distance between the stakes is to the circumference of the Earth. Mathematically, this can be expressed as

\begin{align*} \frac{\angle(B-A)}{360^{\circ}}&=\frac{\text{distance between stakes}}{\text{circumference of the Earth}} \\ &\text{or,} \\ \text{circumference of the Earth}&=(\text{distance between stakes})\frac{360^{\circ}}{\angle(B-A)} \end{align*}

It was a dicey calculation because both distance and angles were difficult to measure accurately, and even small errors would result in a much larger error in the calculated circumference. To the Greeks' credit, they were only off by less than \(20\%\), calculating the circumference to be about \(21,000\text{ miles}\), whereas it is actually about \(25,000\text{ miles}\).

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The Circumference of the Earth, Marco Polo, Christopher Columbus >>>

The Circumference of the Earth, Marco Polo, Christopher Columbus: A Connection?

In the year 1271, an Italian merchant named Marco Polo left Venice, Italy, for a journey to the Far East, going all the way through China to the edge of the Pacific ocean. Along the way, he kept a careful log of the many wondrous things he saw, as well as the time it took to travel from one place to another. At that time, China was technologically more advanced than the West in many ways, and therefore had many items of trade that were of great interest to the West. Upon his return to Italy many years later, he published a book on his travels. It became very popular and heightened the desire to find better trade routes to the Far East. It also inspired many later explorers, including one named Christopher Columbus.

For the next 200 years, merchants sought in vain to find a better trade route to China than the long and tedious one used by Marco Polo. Finally, in their frustration, they began to consider a different approach. Feeling certain of the evidence indicating that the Earth was round, they wondered if it wouldn't be simpler to reach China by traveling westward across the Atlantic ocean. They set out to calculate if this was reasonable. They began by analyzing Marco Polo's log books, calculating the distances covered on a more or less day by day basis. This was a difficult task because the travel data was given in days, not distance. Nevertheless, they came up with a number for the distance from Italy to China's ocean coast. Adding to this the distance from Italy to Spain's Atlantic coast, and subtracting the total from the Greeks' calculation of the earth's circumference, they concluded that China lay about 3000 to 4000 miles west of Spain. The calculation contained huge errors of course, not the least of which was the accounting of the zig zag nature of Marco Polo's journey, the days-to-distance conversion imbedded in the calculation of distance, and the Greeks' calculation of the earth's circumference. In fact, as it turned out, the magnitude of the error (in miles) was roughly the width of the Pacific ocean plus the width of the Americas. Nevertheless, the work was sufficient to bolster Columbus's argument that China could be reached by sailing 3000 to 4000 miles westward across the Atlantic.

As history records, Columbus sailed westward from Spain, and after 3000 to 4000 miles --- Presto! There was land, just as the theory predicted. Is it any wonder that he could be so adamant that they were islands off China's coast.

It is interesting to note that in this case an incorrect theory (that China lay immediately west of Europe), coupled with erroneous data, led to a dramatic new discovery. Advancements in science sometimes take strange paths.

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The Significance of the Greeks' Work >>>

The Significance of the Greeks' Work

The Greeks' theory that the Earth was round led to its being mapped into a set of longitude (north pole to south pole) lines, and a set of circular latitude lines perpendicular to these. (See Fig. 4.) A map might be drawn showing 18 longitude lines, for example, which would then be spaced \(20^{\circ}\) apart around the globe. Arbitrarily choosing one of the lines as the \(0^{\circ}\) line, the next line after that would be the \(20^{\circ}\) line, the next after that the \(40^{\circ}\) line, etc. Circular latitude lines were then drawn around the Earth at right angles to these. Since the North Star is \(90^{\circ}\) above the horizon at the North Pole, the circle at the North Pole was arbitrarily labeled the \(90^{\circ}\) circle. If the latitude circles were spaced at \(20^{\circ}\) intervals also, the next circle down would be the \(70^{\circ}\) circle, meaning that anywhere on this circle the North Star would be \(70^{\circ}\) above the horizon. Similarly, the next circle down would be the \(50^{\circ}\) circle, etc. The equator would be the \(0^{\circ}\) circle (indicating that the North Star is \(0^{\circ}\) above the horizon all along this circle). The South Pole would be the \(-90^{\circ}\) circle. (The North and South Pole "circles" would be mere points.) While only lines spaced \(20^{\circ}\) apart might be shown on a particular map, interpolation could be used to obtain any desired degree of longitude or latitude.

Oddly, the spherical representation of perpendicular north-south and east-west lines bears a close resemblance to the north-south, east-west lines used to represent the flat Earth concept. (Refer to Fig. 1.) Over small areas, for example, the two mappings are virtually identical. But over large areas, the spherical mapping results in much smaller distances between longitude lines in the far northern and southern regions. For long sea and land journeys, it soon became apparent to travelers that the spherical maps made long distance travel much more predictable.

Spherical mapping led to the location of cities and places being defined by the intersection of their latitude and longitude lines. If a city was designated as being at longitude \(15^{\circ}\), latitude \(45^{\circ}\), for example, it meant that it was located at the intersection of the \(15^{\circ}\) north-south longitude line, and the \(45^{\circ}\) latitude circle. Its north-south position could be confirmed by noting that the North Star was \(45^{\circ}\) above the horizon. There was no way to confirm its east-west angular position using celestial observations because the noonday sun and North Star angles above the horizon remained the same all along the east-west latitude line.

If a traveler wanted to go to a city that was \(5^{\circ}\) north and \(5^{\circ}\) west of the above city, he could do so by first heading due north (i.e., by heading directly toward the North Star) until the angle of the North Star was at \(50^{\circ}\) with respect to the horizon. In theory, he could measure the distance covered by measuring the angle of the North Star above the horizon on different nights. Using the techniques described earlier, the distance covered (\(D\)) would be given by the formula

\begin{align*} D=\frac{(\text{circumference of the Earth})(\angle A-\angle B)}{360} \end{align*}

where angle \(A\) and angle \(B\) were the above-horizon angles measured on different nights.

Having reached the correct northern latitude, the traveler would then head due west by aiming directly at the setting sun, which would keep him on the same latitude circle. However, as explained above, there was no way to use the stars to measure angular distance in the east-west direction. This became a huge problem, especially for navigators at sea, and it took nearly 2000 years to come up with a solution. Here is how it came about.

During the eighteenth century, and even long before then, the inability to know how far west a ship had sailed became an especially serious problem. One reason was that pirate ships were routinely raiding English, French, Spanish and other cargo vessels as they traveled north and south along the shoreline. (Traveling within sight of the shoreline was necessary so that the vessels would know where they were.) The pirate ships would wait at shore until a victim vessel was sighted, and then proceed to overwhelm the vessel and plunder its contents. This could be avoided if the cargo vessels sailed far enough west of the coast line so that they could not be sighted from shore, but they had no accurate way of determining how far west they had sailed.

Another problem was the catastrophic loss of ships due to collisions with rock formations just below the surface in coastal waters. These could be avoided if ships could sail further away from shore, but again, there was the problem of knowing how far from shore they were with no land in sight to guide them.

The problem became so serious that the British government offered a huge reward for anyone who could find a solution, and assembled a committee of the leading scientists of the time to analyze proposed solutions. It was more or less assumed that the solution would be based on the position and movement of the stars and other celestial bodies. One of the proposed solutions was made, not by a scientist, but by an uneducated watchmaker!

The watchmaker reasoned that every point on a latitude line traversed a \(360^{\circ}\) circle every 24 hours; in other words, each point rotated \(15^{\circ}\) every hour. So if a ship left shore at sunrise at a certain time of day and sailed due west (by following the setting sun), and then found that sunrise the next morning occurred one hour earlier, the ship would have sailed \(15^{\circ}\) from its starting point on the latitude circle. Therefore all that was needed to determine a ship's westward position was an accurate clock that could measure the difference in sunrise times between where the ship was, and when sunrise occurred on shore. This was easier said then done because clocks at that time were pendulum clocks which did not operate reliably in choppy seas. But the watchmaker had invented a clock based on a different timing mechanism, one that would operate on the open seas. To demonstrate his solution, he went to sea on a cargo vessel that was one of several ships carrying trading goods. On their north-south journey, the ships ran into a violent storm and were not sure how far west they were. The watchmaker had never been to sea before and became quite seasick. Nevertheless, even in his misery he was able to convince the captain of his vessel that they were not far enough west to avoid a dangerous set of shoals that extended well out from the coastline. The other ships maintained their course, ran into the shoals, and sank. After that, the watchmaker's solution was soon adopted by the maritime industry.

Sadly, the committee of distinguished scientists would not recognize the watchmaker's solution because it was not celestial-based. (Even brilliant scientists can be biased.) Many years later the British government did award the watchmaker the grand prize, despite the committee's reluctance.

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Ptolemy, Copernicus, Newton >>>

Ptolemy, Copernicus, Newton --- The Interplay of Facts and Theory

About A.D. 150, a brilliant Greek mathematician and astronomer named Ptolemy made an extensive study of the movement of the stars and the planets. He concluded that all the heavenly bodies, including the sun, rotated about the Earth. He even traced out the motions of the planets and worked out a theory that was consistent with the observed data. The work was so impressive (and in agreement with wishful thinking) that it went unquestioned for the next 1400 years.

Then, in 1543, an astronomer named Copernicus looked at more or less the same data and concluded that the Earth and the planets all circled the sun! How could two brilliant scientists come to opposite conclusions after studying essentially the same information? Here are some reasons.

Ptolemy lived at a time when common sense said that the Earth was flat, even though he was sure that it was round. To convince people of this, Ptolemy had to find an answer to the question "If the Earth is round, why don't objects fall off the underside?". Ptolemy's answer was that in a universe that extends out in all directions, there is no such thing as up or down, or topside or underside, so there must be a force at the center of the Earth that pulls objects anywhere on Earth toward that center. It is this force that holds objects on the surface, and pulls them back to the surface if they are dropped from above it.. Since this force must reside at the center of the Earth for this theory to be true, it seemed logical to him that the force extended outward in all directions and determined the positions and movements of all objects in the universe. Ptolemy did not have a mathematical basis for his theories, only that they were plausible. His explanation of the paths of the sun and the planets as they orbited the Earth may have been plausible, but could hardly be provable because it was based on the erroneous assumption of a universally controlling force within the Earth.

When Copernicus studied the data 1400 years later, he wasn't trying to justify a central force, he was trying to find a theory that explained all the data. Copernicus not only showed convincingly that the data only made sense if the planets circled the sun, but that the resulting model of our solar system was a picture of simplicity compared to the complicated model required by Ptolemy's theory. Nevertheless, the idea was not well received. For one thing, it was an affront to man's ego that man was not the center of the universe. It was also unlikely to many that something could be wrong for so long a time without the error being discovered sooner. So the Copernicus theory was largely ignored.

About 120 years later, another brilliant scientist named Newton studied the problem. He concluded that Corpernicus was correct, that the sun was the center solar system, but at the same time he was in agreement with Ptolemy that there must be a force within the Earth that attracted objects to its surface. But Newton wondered if this mysterious force might simply be a mutual force of attraction between any two separate objects, one of the objects, in this case, being the Earth. If this were true, it seemed reasonable that the more massive the objects were, the greater the force of attraction would be. Thus if a large stone was picked up, the force of attraction between the Earth and the stone would be larger than if a smaller stone was picked up, thus making one feel lighter than the other.

Newton's hypothesis implied that if there is an attractive force field between any two objects, there must be a force field between any two planetary objects, such as the Earth and the moon, for example. But if this is the case, why don't the Earth and the moon crash into each other? Newton's explanation was that they don't because one is rotating around the other. To understand this, picture a ball tied to the end of a fairly long string. Holding the free end of the string in one hand, swing the ball around over your head so that it rotates in a circular path. The ball doesn't fly off into space because the force in the string keeps it in its orbit. If there were an attractive force between your hand and the ball equal to the force experienced in the string, there would be no need for the string. If a shorter string was used, it would be found that the ball has to be rotated faster in order to keep it in orbit, and that the force in the string is now larger. Newton reasoned that this is what happens in our solar system. In other words, that the orbital radius and speed of the planets are determined by the force of attraction between the sun and each of the various planets.

Newton concluded that the force of attraction between two bodies was proportional to the product of the mass of each of the bodies. (Thus the larger the masses, the larger the force). But the amount of this force should also depend on the distance between the bodies, because if they were far enough apart, the force would approach zero (one body wouldn't know that the other even existed). So Newton proposed the following formula for calculating the magnitude of the mutual force of attraction between any two bodies:

\begin{align*} F=\frac{(M_{1})(M_{2})}{R^{2}} \end{align*}

where \(M_{1}\) is the mass of one body, \(M_{2}\) the mass of the second body, and \(R\) is the distance between them. The distance \(R\) is squared which reflects the fact that as the force field spreads out over space it becomes diminished by the amount of area it is covering, and this area is proportional to the square of the distance from the source.

Newton used his force formula and other established physics formulas to explain the motion of the planets and their moons, the paths of comets, the action of tides, and other heretofore unexplainable motions in the heavens. He had at last succeeded in uniting the universe with a single set of laws. And so, nearly 2000 years after the early Greeks first began driving stakes in the ground to measure the length of their shadows, the mystery of our solar system's motions was finally resolved.

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Appendix A1 >>>

Appendix A1

The Nature of Rays of Light

When a star radiates light into space, it does so in all directions from all points on the star. Fig. A1 illustrates this, showing radiation from three points (A, B, C) on a star. Only a tiny fraction of this light will fall on a planet such as Earth, which will then be able to "see" the star. Although our star (the sun) is enormous compared to Earth, its distance from the Earth is also enormous. Because the distance between the Earth and the sun is many times larger than the diameter of the sun, the rays that fall on Earth are virtually parallel. (See Fig. A1.) Knowing that the rays of light that hit the Earth from any star are essentially parallel, greatly simplifies the ability to understand and analyze the relative motions of the Earth and the stars that are being observed.

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Appendix A2 >>>

Appendix A2

Line of Sight Vision

A viewer is standing at the shoreline of a very large body of water and watching a ship sail out to sea in a direction perpendicular to the shoreline. How far will the ship travel before the top of the ship is no longer visible?

Refer to Fig. A2 in which:
\(R=\text{ the radius of the Earth}\)
\(h_{1}=\text{ the eye-level height of the viewer}\)
\(h_{2}=\text{ the height of the top of the ship above the water}\)
\(w_{1}=\text{ the distance from the eye-level of the viewer to the horizon}\)
\(w_{2}=\text{ the distance from the horizon to the top of the ship}\)
\(W=\text{ the distance from the viewer to the ship at which the ship is no longer visible }=w_{1}+w_{2}\)

From the Pythagorean theorem,

\begin{align*} (R+h_{1})^{2}&=R^{2}+w_{1}^{2}\tag{1}\label{eq:1} \\ \text{and }(R+h_{2})^{2}&=R^{2}+w_{2}^{2}\tag{2}\label{eq:2} \end{align*} \begin{align*} \text{From }\eqref{eq:1}~w_{1}&=\left[(R+h_{1})^{2}-R^{2}\right]^{\frac{1}{2}} \\ &=\left[h_{1}(2R+h_{1})\right]^{\frac{1}{2}} \\ &=\left[(2R)(h_{1})\right]^{\frac{1}{2}}\text{ (approx.)} \end{align*} \begin{align*} \text{From }\eqref{eq:2}~w_{2}&=\left[(R+h_{2})^{2}-R^{2}\right]^{\frac{1}{2}} \\ &=\left[h_{2}(2R+h_{2})\right]\frac{1}{2} \\ &=\left[(2R)(h_{2})\right]^{\frac{1}{2}}\text{ (approx.)} \end{align*} \begin{align*} \text{But }W&=w_{1}+w_{2} \\ \text{Therefore }W&=\left[(2R)(h_{1})\right]^{\frac{1}{2}}+\left[(2R)(h_{2})\right]^{\frac{1}{2}} \end{align*}

As an example, choose:
\(R=4000\text{ miles}\)
\(h_{1}=6\text{ feet}\)
\(h_{2}=20\text{ feet}\)

This results in \(W=8.5\text{ miles}\).

In other words, when the ship is \(8.5\text{ miles}\) from shore, the top of it will no longer be visible.

If the top of the hull of the ship is \(6\text{ feet}\) above the water line, it will disappear from view when the ship is \(6\text{ miles}\) from shore.

<<< Appendix A1