# Mesolabe Compass and Square Roots - Numberphile

TLDRIn this fascinating Numberphile video, the spotlight is on Hippocrates of Chios, a mathematician whose contributions have been overshadowed by his namesake, the famous physician Hippocrates of Kos. The video delves into two of Hippocrates' mathematical innovations: the mesolabe compass and a geometric method for finding square roots. The mesolabe compass is a tool that allows for multiplication and division using only lines, demonstrated through the example of multiplying 3 by 4. The geometric square root method involves drawing lines and curves to find the square root of any number, illustrated by finding the square root of 9. The video emphasizes the elegance of these ancient Greek mathematical principles and their significance in the history of geometry. It also highlights how these methods were later recognized by René Descartes in his book 'The Geometry,' which contributed to the development of calculus. The video concludes with a nod to the educational platform Brilliant, which offers a wealth of interactive content on similar mathematical topics.

### Takeaways

- 😀 Hippocrates of Chios, not to be confused with Hippocrates of Kos, the medic, was a significant mathematician whose contributions have been overlooked.
- 📏 The mesolabe compass is an ancient Greek tool that allowed for multiplication and division using lines, showcasing the Greeks' focus on geometry over numeracy.
- 🔢 Multiplication and division can be visually represented using the mesolabe compass by drawing lines and creating parallel lines to find the product or quotient.
- 📐 To multiply numbers, one would find the numbers on the lines of the mesolabe compass and draw a line through them to get the product.
- 📏 Division with the mesolabe compass involves drawing a line through the dividend and divisor and creating a parallel line to find the quotient.
- 🤔 The simplicity of the mesolabe compass' principle highlights the Greek preference for geometrical methods over arithmetical ones.
- 🔍 Hippocrates of Chios also devised a method to find the square root of any number using lines and a semicircle, demonstrating the power of geometrical reasoning.
- 📏 To find a square root, one would mark a line, add one, and use it as the radius of a semicircle to find the square root through geometric similarity.
- 📐 The square root method relies on the properties of similar triangles formed within a semicircle, as first noted by Thales.
- 👀 René Descartes acknowledged Hippocrates' work in his book 'The Geometry', indicating the importance of understanding angles and line lengths in geometry.
- 🌟 Despite Descartes' claim that geometry could calculate anything, the development of calculus was necessary for measuring change, showing the evolution of mathematical thought.

### Q & A

### Who is Hippocrates of Chios and what is his contribution to mathematics mentioned in the script?

-Hippocrates of Chios was a mathematician who lived on the island of Chios, distinct from the more well-known Hippocrates of Kos, the medic. His significant contribution mentioned in the script is the invention of the mesolabe compass, a tool that allowed for the multiplication and division of any two numbers using lines, predating the use of numeral systems.

### How did the mesolabe compass work for multiplication?

-The mesolabe compass worked for multiplication by using two lines at any angle, marking them off like a ruler. To multiply two numbers, one would find the numbers on the lines and draw a line through them, which would then visually represent the product.

### Can you explain the process of division using the mesolabe compass as described in the script?

-Division with the mesolabe compass involved drawing a line through the dividend and the divisor on the marked lines. A parallel line was then drawn through the unit on the bottom line. This parallel line would intersect with the product line at the quotient.

### What is the second idea from Hippocrates of Chios mentioned in the script, and how does it relate to finding square roots?

-The second idea from Hippocrates of Chios mentioned in the script is a method for finding the square root of any number using lines and curves. This method involves drawing a line, marking it off, and using a semicircle with a set square to find the square root by creating right-angled triangles.

### How does the method for finding square roots using the mesolabe compass relate to the Pythagorean theorem?

-The method for finding square roots using the mesolabe compass creates a right-angled triangle, which is a key element in the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is used to determine the square root.

### What is the significance of the 3-4-5 triangle in the context of the script?

-The 3-4-5 triangle is a famous Pythagorean triple, which is a set of three positive integers that satisfy the Pythagorean theorem. In the script, it is used to illustrate the relationship between the sides of the triangle and the square root calculation, showing that 5 squared equals 25, which is the sum of 4 squared (16) and 3 squared (9).

### How did Thales' theorem about semi-circles contribute to Hippocrates' method for finding square roots?

-Thales' theorem states that in a semi-circle, any two lines drawn from the endpoints of the diameter to any point on the curve will always form a right angle. This principle is used in Hippocrates' method to create similar right-angled triangles, which allows for the calculation of square roots.

### What is the connection between the script's discussion on geometry and the development of calculus?

-The script discusses how the principles of geometry, as demonstrated by the mesolabe compass and Hippocrates' methods, laid the groundwork for more advanced mathematical concepts. It mentions that René Descartes and Pierre de Fermat began looking for ways to measure change, which eventually led to the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.

### What is the significance of René Descartes' book 'The Geometry' in relation to the script's content?

-René Descartes' book 'The Geometry' is significant because it recognized and documented the work of Hippocrates of Chios, including his methods for multiplication, division, and finding square roots using geometry. The book also contributed to the understanding of how geometry could be used to calculate various mathematical problems.

### How does the script suggest that learning through interactive activities can enhance mathematical understanding?

-The script suggests that interactive learning, such as engaging with the content on Brilliant's website, can be an effective way to deepen one's understanding of mathematical concepts. It encourages viewers to not just watch videos but to actively participate in learning through courses, quizzes, and other interactive content.

### Outlines

### 📏 Mesolabe Compass: Ancient Greek Geometry for Multiplication and Division

The script introduces Hippocrates of Chios, a mathematician whose contributions have been overshadowed by his namesake, the medic Hippocrates of Kos. It discusses the mesolabe compass, an ancient Greek geometric tool used for multiplying and dividing numbers using lines. The process involves marking lines like a ruler and drawing parallel lines to perform calculations. An example is given where multiplying 3 times 4 is demonstrated using this method. The script also touches on the Greeks' preference for geometry over numeracy and introduces a second idea by Hippocrates for finding square roots using lines and curves.

### 📐 Hippocrates' Method for Finding Square Roots and the Influence on Descartes

This paragraph delves into Hippocrates of Chios' method for finding the square root of any number using simple geometric constructions. It describes a process involving marking a line, creating a semicircle with a set radius, and using perpendicular lines to find the square root. The explanation is illustrated with the square root of 9, resulting in a 3-centimeter line. The script also connects this method to the Pythagorean theorem and acknowledges the impracticality for very large numbers but emphasizes the sound theory behind it. It concludes by discussing the historical significance of Hippocrates' work, its recognition by René Descartes in 'The Geometry,' and the broader impact on the development of calculus by Leibniz and Newton.

### Mindmap

### Keywords

### 💡Mesolabe compass

### 💡Hippocrates of Chios

### 💡Multiplication

### 💡Division

### 💡Square root

### 💡Geometry

### 💡Parallel lines

### 💡Pythagorean triangle

### 💡Thales

### 💡René Descartes

### 💡Calculus

### Highlights

Introduction to Hippocrates of Chios, a mathematician whose contributions have been overlooked in history.

The mesolabe compass, an ancient Greek tool for multiplication and division using lines.

The process of using the mesolabe compass to multiply numbers like 3 and 4.

The ability to perform division with the mesolabe compass, as demonstrated with dividing 12 by 4.

Hippocrates of Chios' second idea for finding square roots of any number using lines and curves.

A step-by-step guide on using geometric methods to find the square root of 9.

The theoretical application of the method to find square roots of non-square numbers and fractions.

The impracticality of using the method for very large numbers like a billion due to the scale of the lines.

The connection between the geometric method and the Pythagorean theorem demonstrated with a 3-4-5 triangle.

Thales' theorem and its relevance to the geometric method for finding square roots.

The concept of similar triangles and their role in the geometric method for square roots.

An explanation of how squaring and square rooting are related in the geometric method.

Recognition of Hippocrates of Chios' work by René Descartes and its inclusion in 'The Geometry'.

Descartes' perspective on the power of geometry for calculation and its limitations before calculus.

The influence of Descartes and Fermat on the development of calculus by Leibniz and Newton.

A call to action for viewers to explore Brilliant.org for interactive learning on square roots and other mathematical topics.

A special offer for Numberphile viewers to get 20% off a premium membership on Brilliant.org.