**March 11, 2020.** *Bla*

### Contents

## 1. Introduction

Simple measurements, combined with clever physical reasoning, can reveal deep facts about the world around us. We give three examples from our solar system: the size of the earth; the mass of the sun; and the temperature of Pluto.

## 2. Eratosthenes’ method

Eratosthenes (276–195 BC) was head librarian at the Library of Alexandria and one of the great thinkers of the ancient world. The library was destroyed by invading Roman emperors, and most of Eratosthenes’ work along with it. Thankfully, his elegant method for calculating the size of the earth, using only the shadow of a stick, survives. As we will see, you can use this method to easily estimate latitude!

### 2.1. A trigonometry refresher

On the summer solstice, locals in the Egyptian city of Syene noticed that, at noon, the sun hit the bottom of a deep well. Eratosthenes inferred that the sun must be directly overhead. He also knew from Egyptian surveyors that Syene was roughly $5000$ stadia ($\approx 850 \,\mathrm{km}$) away from Alexandria. Eratosthenes performed a single experiment. At noon on the summer solstice, he measured the shadow of a vertical rod in Alexandria. He found it was roughly $1/8$ of the length of the rod.