3Blue1Brown - Measuring the earth with Terence Tao
In the 10th century, an Arab mathematician named Aloni measured the Earth's radius with remarkable accuracy, within 1% of its actual value. He achieved this by climbing a mountain and measuring the angle of declination of the horizon, which is not perfectly horizontal when viewed from a height. By applying basic trigonometry, he formed a right-angle triangle where the hypotenuse was the Earth's radius plus the mountain's height. Using the angle of declination, he calculated the Earth's radius through a trigonometric formula involving cosine. This method highlights the ingenuity of early scientists in measuring large distances using simple tools and mathematical principles. The discussion is part of a broader conversation about historical methods of measuring cosmic distances, from the Earth to distant galaxies.
Key Points:
- Aloni measured Earth's radius within 1% accuracy using trigonometry.
- He used the angle of declination from a mountain to form a right-angle triangle.
- The hypotenuse was the Earth's radius plus the mountain's height.
- Trigonometric formula involving cosine was used for calculation.
- Highlights historical ingenuity in measuring cosmic distances.
Details:
1. 🌍 Ancient Earth Radius Calculation
- In the 10th Century, Aloni, an Arab mathematician, calculated the Earth's radius with an accuracy of 1%.
- Aloni's method involved climbing a mountain and measuring the horizon, which appeared tilted due to Earth's curvature.
- This technique demonstrates early scientific understanding and measurement accuracy, highlighting ingenuity in historical scientific methods.
- Aloni's calculation exemplifies the use of observational techniques to achieve precise results without modern technology.
2. 📐 Trigonometry and Horizon Measurements
- Standing on a mountain, the horizon's apparent angle is influenced by Earth's curvature, forming a right-angle triangle with the hypotenuse as the sum of Earth's radius and the mountain's height.
- Using trigonometric principles, the angle (Theta) of the horizon can be calculated, offering practical applications in fields like surveying and navigation.
- For example, if a mountain is 1,500 meters high and Earth's radius is approximately 6,371 kilometers, trigonometry helps determine the horizon's angle, which is crucial for accurate horizon distance calculations.
3. 🔢 Determining Earth's Radius
- To calculate Earth's radius, use the height of a mountain and an angle related to it with a trigonometric approach.
- The formula R = r + H * cosine(Theta) is used, where R is Earth's radius, r is a reference radius, H is the mountain's height, and Theta is the angle.
- Understanding each variable: R is the sought-after Earth's radius, r is a known or estimated reference radius, H is the measurable height of a mountain, and Theta is the angle of elevation from a specific point to the mountain top.
- This method illustrates practical application of trigonometry in geophysics, making it accessible and relevant for real-world calculations.
4. 🗺️ Exploring Cosmic Distances
- Trigonometry, initially used for terrestrial measurements like mountain heights, has been adapted for space to measure distances between celestial bodies.
- Early methods focused on determining Earth's size, laying the groundwork for more advanced techniques to measure vast cosmic distances.
- Modern astronomy employs sophisticated technologies, such as parallax and redshift, to accurately gauge the distances to stars and galaxies.
- The development of measurement techniques reflects humanity's growing understanding of the universe, spanning from basic geometric methods to complex space-based observations.