Problem 33 - A History of Mathematics: An Introduction

Question

Use al-B?r?ni's nontrigonometric procedure for calculating distances on the earth to find the great circle distance. between New York (latitude \(41^{\circ} \mathrm{N}\), longitude \(74^{\circ} \mathrm{W}\) ) and London (latitude \(52^{\circ} \mathrm{N}\), longitude \(0^{\circ}\) ). Assume that the circumference of the earth is 25,000 miles.

Step-by-Step Solution

Answer

Answer: The great circle distance between New York and London is approximately 6507.19 miles.

Step 1: Compute the difference in latitudes and longitudes

We first find the difference in latitudes and longitudes between the two locations. The absolute difference in latitudes is \(|52^{\circ} - 41^{\circ}| = 11^{\circ}\). The absolute difference in longitudes is \(|0^{\circ} - (-74^{\circ})| = 74^{\circ}\).

Step 2: Convert degrees to radians

To make the calculations, we need to convert the differences in latitudes and longitudes from degrees to radians. We can use the formula: radians = (degrees / 180) * pi Latitude difference in radians: \(11^{\circ} * \frac{\pi}{180} = \frac{11\pi}{180}\) Longitude difference in radians: \(74^{\circ} * \frac{\pi}{180} = \frac{37\pi}{90}\)

Step 3: Calculate the total meridian arc

Now we need to determine the total meridian arc between the two locations. The meridian arc is the distance covered in the meridional direction, along the Earth's surface. We use the formula: meridian arc = Earth's circumference * (latitude difference in radians) / (2pi) Total meridian arc: \(25,000\,\text{miles} * \frac{\frac{11\pi}{180}}{2\pi} = \frac{11}{180}*25,000\,\text{miles} = 1527.78\,\text{miles}\)

Step 4: Calculate the distance on the parallel circle

Now we have to calculate the distance on the parallel circle (at the latitude of London) between the two points. The circumference of a parallel circle is given by: C_parallel = Earth's circumference * cos(latitude) Since we need to find the length of the arc on the circle, ratio of the difference in longitudes (expressed in radians) and the parallel circumference, which is given by: arc_distance_London = C_parallel * (longitude difference in radians) / (2pi) C_parallel = \(25,000\,\text{miles} * \cos{(52^{\circ})}\) = 25,000 * 0.61566 = 15,391.50 miles arc_distance_London = \(15,391.50\,\text{miles} * \frac{\frac{37\pi}{90}}{2\pi} = \frac{37}{90} * 15,391.50\,\text{miles} = 6341.47\,\text{miles}\)

Step 5: Calculate the great circle distance

Finally, we can compute the great circle distance using the Pythagorean theorem, considering the total meridian arc and arc distance along the parallel circle as the two perpendicular sides of a right triangle. Great circle distance = \(\sqrt{(\text{total meridian arc})^2 + (\text{arc distance along parallel circle})^2}\) Great circle distance = \(\sqrt{(1527.78\,\text{miles})^2 + (6341.47\,\text{miles})^2} = 6507.19\,\text{miles}\) Thus, the great circle distance between New York and London is approximately 6507.19 miles.

Key Concepts

Al-Biruni's Method

Al-Biruni, a Persian mathematician and astronomer, devised a clever way to measure distances on Earth's surface without relying on complex trigonometry. His method combines geographic measurements and simple geometry to calculate distances efficiently.
By understanding the curvature of the Earth, Al-Biruni's technique involves breaking down distances into measurable arcs, such as meridian and parallel circle arcs. This approach provides an alternative way of distance calculation on Earth, influencing the way we look at geographical measurements today.
With Al-Biruni's method, one can determine the distance between two points based on their latitudinal and longitudinal differences, alongside Earth's circumference. This procedure is especially useful in historical contexts when detailed trigonometric tables were unavailable or cumbersome.

Meridian Arc

A meridian arc is an essential concept in geographical calculations. It represents the curve on the Earth's surface that follows a constant longitude, connecting two points from north to south. Essentially, it's the north-south arc of a great circle.
When calculating the distance between two points on Earth, such as New York and London, determining the meridian arc is a fundamental step. This involves computing the arc based on the latitudinal difference converted into radians.

• The Earth's circumference is used to derive the meridian arc.
• The formula for the arc is: \(meridian\,arc = Earth's\,circumference \times \frac{latitude\,difference\,in\,radians}{2\pi}\)

This method gives an approximation of the distance traveled along the Earth's curvature from one latitude to another.

Parallel Circle Arc

The parallel circle arc is the distance that lies along a latitude line, running east-west between two points of different longitudes.
When calculating distances like between New York and London, the parallel circle arc has to be considered in addition to the meridian arc for a comprehensive distance assessment. This arc takes into account the Earth's rotation and curvature.
For the calculation:

• Determine the parallel circle circumference at the given latitude using: \(C\_parallel = Earth's\,circumference \times \cos(latitude)\)
• Compute the arc based on the longitudinal difference in radians, giving the formula: \(arc\_distance = C\_parallel \times \frac{longitude\,difference\,in\,radians}{2\pi}\)

These calculations together illustrate how the Earth's spherical shape impacts distance estimation.

Distance Calculation on Earth

Calculating distances on Earth involves considering both meridian and parallel circle arcs, bringing us closer to understanding great circle distance. This distance is the shortest path between two points on the Earth's surface.
The process involves the following steps:

• First, calculate the meridian arc to assess the north-south distance using latitudinal differences.
• Second, compute the parallel circle arc to account for the east-west distance based on longitudinal differences.
• Lastly, apply the Pythagorean theorem to these two orthogonal components to find the great circle distance: \[Great\,circle\,distance = \sqrt{(meridian\,arc)^2 + (parallel\,circle\,arc)^2}\]

This comprehensive approach ensures that all angular and spatial distances are accounted for, providing accurate geographic measurements.