# Mesolabe Compass and Square Roots - Numberphile

TLDRThe video script discusses the innovative mathematical methods of Hippocrates of Chios, a Greek mathematician whose contributions have been largely overlooked. It explains how he used a mesolabe compass to perform multiplication and division with lines, and a geometric approach to find square roots of numbers using lines and a semicircle. The script highlights the elegance and simplicity of these ancient Greek geometric techniques and their influence on later mathematicians like René Descartes, who emphasized the power of geometry in calculation.

### Takeaways

- 😀 Hippocrates of Chios, not the medic, was a significant mathematician whose work has been largely overlooked.
- 📏 The mesolabe compass is an ancient tool used by Hippocrates for multiplication and division using lines.
- 🔢 Multiplication can be visualized by drawing lines and using a set square to create parallel lines, representing the product.
- ✂️ Division is achieved by drawing lines and creating parallel lines in the opposite direction to find the quotient.
- 📐 Hippocrates also devised a method for finding square roots using lines and a semicircle.
- 🔍 The square root method involves drawing a line, marking it, and using a semicircle to find the length representing the square root.
- 🤔 The practicality of the method decreases with very large numbers due to the impracticality of drawing extremely long lines.
- 📚 Descartes acknowledged Hippocrates' work in his book 'The Geometry' and saw it as an introduction to geometry.
- 📉 Descartes believed that with a few angles and line lengths, one could calculate anything, not accounting for the need for calculus.
- 🌐 The script highlights the historical significance of geometry in mathematics and its evolution over time.
- 🎓 The video transcript promotes the use of interactive learning platforms like Brilliant for a deeper understanding of mathematical concepts.

### Q & A

### Who was Hippocrates of Chios and what is his significance in the history of mathematics?

-Hippocrates of Chios was a mathematician who lived on the island of Chios, distinct from the famous physician Hippocrates of Kos. He is significant for his contributions to geometry, particularly for his invention of the mesolabe compass, which allowed for multiplication and division using lines, and his method for finding square roots.

### What is a mesolabe compass and how was it used to perform multiplication and division?

-A mesolabe compass is a geometric tool invented by Hippocrates of Chios. It consists of two lines marked off like a ruler. By using this tool, one could multiply or divide any two numbers by drawing lines and using a set square to create parallel lines, which would then visually represent the result of the multiplication or division.

### Can you provide an example of how the mesolabe compass was used to multiply numbers?

-An example given in the script is multiplying 3 by 4. You would find the number 3 on one line and 1 on the bottom line, draw a line through them, and then use a set square to draw a parallel line that passes through the number 4 on the bottom line. The intersection points would visually represent the multiplication result, which is 12 in this case.

### How did the mesolabe compass allow for division?

-Division with the mesolabe compass is performed by reversing the multiplication process. For example, to divide 12 by 4, you would draw a line through the 12 and the 4 on the marked lines. Then, draw a parallel line through the 1, which would intersect at the point representing the result, which is 3 in this case.

### What is the method Hippocrates of Chios used to find the square root of any number?

-Hippocrates used a method involving lines and curves to find the square root of any number. One would start by marking a line with the number plus one, use it as the radius of a semicircle, and then raise a perpendicular from the number. The length of the line where the perpendicular meets the semicircle represents the square root of the original number.

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

-The method for finding square roots is related to the Pythagorean theorem because it involves creating right-angled triangles within the semicircle. The similarity of these triangles allows for the calculation of the square root through geometric proportions.

### Why did René Descartes mention Hippocrates of Chios in his book 'The Geometry'?

-René Descartes mentioned Hippocrates of Chios because he recognized the significance of his geometric methods. Descartes believed that with an understanding of angles and line lengths, one could calculate anything, highlighting the importance of geometry in mathematical calculations.

### What is the connection between the mesolabe compass and the concept of numeracy among the ancient Greeks?

-The mesolabe compass demonstrates that the ancient Greeks were more focused on geometric methods rather than numeracy for calculations. This tool allowed them to perform arithmetic operations visually through geometry, reflecting their preference for shapes and forms over numerical operations.

### What is the significance of the Pythagorean triangle in Hippocrates' method for finding square roots?

-The Pythagorean triangle, specifically the 3-4-5 triangle, is significant because it provides a practical example of how the method works. The relationship between the sides of the triangle demonstrates the geometric principles that underlie the calculation of square roots.

### Why did the script mention Thales and his contribution to geometry?

-Thales was mentioned because his theorem about right angles in a semi-circle is foundational to Hippocrates' method for finding square roots. The script uses Thales' theorem to explain how the angles in the geometric construction create similar triangles, which is key to the square root calculation.

### How does the script relate the ancient Greek's geometric methods to the development of calculus?

-The script suggests that the geometric methods of the ancient Greeks, like those of Hippocrates of Chios, laid the groundwork for later mathematical developments. It points out that Descartes and Fermat were looking for ways to measure change, which eventually led to the invention of calculus by Leibniz and Newton, allowing for the measurement of changing quantities.

### Outlines

### 📏 The Mesolabe Compass: Ancient Mathematical Tool

This paragraph introduces Hippocrates of Chios, a mathematician often overshadowed by the more famous Hippocrates of Kos. The focus is on the mesolabe compass, an ancient Greek mathematical tool used for multiplication and division through geometric means. The explanation demonstrates how to use two marked lines to perform calculations, such as multiplying 3 by 4, using a set square to create parallel lines that intersect with the marked lines. The method is illustrated with a step-by-step guide, showcasing the simplicity and elegance of ancient Greek geometrical mathematics. The paragraph also touches on the Greeks' preference for geometry over numeracy.

### 📐 Discovering Square Roots with Geometry: Hippocrates of Chios' Method

Continuing the theme of Hippocrates of Chios' contributions, this paragraph delves into his method for finding the square root of any number using simple geometric constructions. The process involves drawing a line, marking it off, and using a semicircle with a set square to find the square root of a given number, exemplified by finding the square root of 9. The explanation highlights the practicality and theoretical soundness of the method, despite its impracticality for very large numbers. The paragraph also connects this geometric approach to square roots with the Pythagorean theorem, demonstrating the relationship through a 3-4-5 triangle. The historical significance of Hippocrates' work is underscored by its mention in René Descartes' 'The Geometry' and its influence on the development of calculus by Leibniz and Newton.

### 🌐 Thales' Theorem and the Geometry of Square Roots

This paragraph builds upon the previous discussion of geometric methods for mathematical operations, specifically focusing on the relationship between Thales' theorem and the calculation of 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 circumference will form a right angle. By creating two similar triangles within a semi-circle, the paragraph explains how the proportions of these triangles can be used to determine square roots. The explanation is illustrated with examples of how changing the lengths of the lines affects the calculation, emphasizing the versatility of this geometric approach. The connection between squaring and square rooting through geometry is also highlighted.

### 🎓 The Legacy of Hippocrates and the Evolution of Mathematical Thought

The final paragraph reflects on the historical oversight of Hippocrates of Chios' contributions to mathematics and how his work was rediscovered and appreciated by later mathematicians like René Descartes. Descartes' book 'The Geometry' is mentioned as a source that acknowledges Hippocrates' ideas, which were foundational to the development of calculus by Leibniz and Newton. The paragraph also transitions to a modern context, promoting Brilliant.org as a resource for interactive learning in mathematics, offering a discount for viewers interested in exploring the topics discussed in the video. The paragraph concludes by encouraging viewers to actively engage with mathematical concepts rather than passively consuming content.

### Mindmap

### Keywords

### 💡Hippocrates of Chios

### 💡Mesolabe compass

### 💡Multiplication

### 💡Division

### 💡Square root

### 💡Pythagorean triangle

### 💡Thales

### 💡Similar triangles

### 💡René Descartes

### 💡Calculus

### 💡Brilliant

### Highlights

Hippocrates of Chios, a mathematician often overlooked in history, introduced the mesolabe compass.

The mesolabe compass allows for multiplication and division using lines, regardless of the angle.

Multiplication can be visualized by finding numbers on the lines and drawing through them.

Ancient Greeks used geometry for mathematical operations, predating Christ.

Division is performed by drawing a line through the numbers and creating a parallel line.

Hippocrates also devised a method to find the square root of any number using lines and curves.

To find the square root, a line is marked off and a semicircle is drawn with a set square.

The length of the line where it meets the semicircle represents the square root.

This method works for any number, including those not perfect squares, resulting in fractions.

The impracticality of marking extremely large numbers does not diminish the theoretical validity.

The method is based on the Pythagorean theorem and the properties of similar triangles.

Thales' theorem about right angles in semi-circles is key to understanding the square root method.

Similar triangles are used to establish proportions and find square roots.

Hippocrates of Chios' methods have been overlooked in many history books.

René Descartes acknowledged Hippocrates' work in his book 'The Geometry'.

Descartes believed understanding angles and line lengths could calculate anything, before calculus.

The development of calculus by Leibniz and Newton expanded the ability to measure changing quantities.

Brilliant.org offers courses and quizzes on square roots and related mathematical concepts.

A premium membership on Brilliant provides access to a wealth of mathematical resources.