Mesolabe Compass and Square Roots - Numberphile

The video script discusses the mathematical contributions of Hippocrates of Chios, a figure often overshadowed by his namesake, the medic Hippocrates of Kos. It highlights two of Hippocrates' major innovations: the mesolabe compass for multiplication and division using geometric lines, and a method for calculating square roots through lines and semicircles. The script emphasizes the elegance and simplicity of these ancient Greek geometric techniques, which predate the common use of numeracy. It also mentions how René Descartes acknowledged Hippocrates' work in his book 'La Géométrie,' illustrating the enduring impact of these ideas on mathematical thought.

Takeaways

• 📐 The mesolabe compass was invented by Hippocrates of Chios, a mathematician who lived on the island of Chios.
• 📏 With the mesolabe compass, multiplication and division of any two numbers could be performed using lines, without the need for numerical calculations.
• 🔢 To multiply numbers using the mesolabe, one would locate the numbers on the marked lines and draw a line through the points to visually represent the product.
• 📏 Dividing numbers with the mesolabe involves drawing a line through the dividend and divisor, and then creating a parallel line to find the quotient.
• 🤔 The method is ancient, dating back to before Christ, and demonstrates the Greeks' interest in geometry over numeracy.
• 🔍 Hippocrates of Chios also devised a method for finding the square root of any number using lines and curves.
• 📏 To find the square root, one would mark a line with the number plus one, use a semicircle with a specific radius, and raise a perpendicular to find the square root.
• 🔢 This geometric method for square roots would yield results in fractions for non-square numbers, illustrating the Greeks' use of geometry for mathematical problems.
• 📚 René Descartes acknowledged Hippocrates' work in his book 'The Geometry', highlighting the importance of angles and line lengths in calculations.
• 🧐 Descartes believed that with a few angles and line lengths, one could calculate anything, a belief that was later expanded upon with the discovery of calculus.
• 🎓 The script also promotes the educational platform Brilliant, which offers courses and quizzes on a variety of mathematical topics, including square roots.

Q & A

Who was Hippocrates of Chios known for?

Hippocrates of Chios was known as a mathematician, not to be confused with Hippocrates of Kos, the medic. He is credited with the invention of the mesolabe compass and methods for finding square roots using geometrical constructions.

What is the mesolabe compass?

The mesolabe compass is a geometric tool used by Hippocrates of Chios to perform multiplication and division using lines, without the need for numerical calculations.

How did the mesolabe compass allow for multiplication and division?

The mesolabe compass allowed for multiplication and division by marking lines and using them like a ruler to find the product or quotient of two numbers through geometric constructions.

What is the geometric method for finding the square root of a number as described by Hippocrates of Chios?

The method involves drawing a line, marking it off with units, adding one to the length, and then drawing a semicircle with the marked length as the radius. A perpendicular is raised from a chosen point on the line, and the length of the line segment from the center of the semicircle to where the perpendicular meets the curve is the square root of the original number.

Why did the geometric method for finding square roots become impractical for very large numbers?

The method becomes impractical for very large numbers because it would require marking an extremely long line with billions of units, which is not feasible in practice, although the theoretical concept still holds.

How is the concept of similarity used in finding square roots with the geometric method?

The concept of similarity is used by recognizing that the triangles formed by the geometric construction are similar, meaning their sides are in proportion. This proportion is used to determine the length of the line segment, which represents the square root.

What is the significance of the Pythagorean triangle in the geometric method for finding square roots?

The Pythagorean triangle, specifically the 3-4-5 triangle, is used to demonstrate that the square of the hypotenuse (radius of the semicircle) is equal to the sum of the squares of the other two sides, which helps validate the geometric method for finding square roots.

Who recognized the work of Hippocrates of Chios and included it in their own work?

René Descartes recognized the work of Hippocrates of Chios and included his geometric constructions in his book 'The Geometry', highlighting the importance of understanding angles and lengths for calculations.

What was Descartes' view on the power of geometry in calculations?

Descartes believed that with an understanding of a few angles and lengths of lines, one could calculate absolutely anything. However, he was proven wrong with the discovery of calculus, which allowed for the measurement of changing quantities.

How does the concept of right angles relate to the geometric method for finding square roots?

The concept of right angles is crucial as it is used to establish that the triangles formed in the geometric construction are similar. This similarity is based on the property that in a semi-circle, any two lines that meet on the curve create a right angle.

What is the educational value of the geometric methods described in the script?

The geometric methods described offer a historical perspective on mathematical problem-solving and highlight the elegance of geometric constructions for performing calculations. They also serve as an introduction to more complex mathematical concepts and the evolution of mathematical thought.

How does the script encourage active learning and engagement with mathematical concepts?

The script encourages active learning by inviting the reader to visualize and understand the geometric constructions for multiplication, division, and finding square roots. It also suggests exploring these concepts further through interactive platforms like Brilliant, which offers courses and quizzes on similar topics.

Outlines

00:00

📏 Mesolabe Compass: Ancient Mathematical Tool

The script introduces Hippocrates of Chios, a mathematician whose contributions have been overshadowed by the more famous Hippocrates of Kos. The focus is on the mesolabe compass, an ancient Greek tool that enabled the multiplication and division of any two numbers using lines. The process involves marking lines like a ruler and using a set square to create parallel lines to find the product or quotient. The script also demonstrates how to find the square root of a number using lines and a semicircle, illustrating the geometric approach to mathematics prevalent in ancient Greece. The simplicity and elegance of these methods highlight the ingenuity of Greek mathematicians and their preference for geometry over numeracy.

05:00

📐 Geometry and Square Roots: Hippocrates' Legacy

This paragraph delves into another geometric method attributed to Hippocrates of Chios for finding square roots, inspired by the principles established by Thales. The method involves drawing lines and using the properties of similar triangles formed within a semicircle to deduce the square root of a number. The script explains how the proportions of these triangles can be used to calculate square roots, even for non-square numbers, which would result in fractions. The historical significance of these methods is underscored by their mention in René Descartes' book 'The Geometry,' which influenced the development of calculus. The paragraph concludes with a modern reference to Brilliant.org, a resource for learning more about mathematics, and an acknowledgment of their sponsorship for the video content.

Keywords

Mesolabe compass

The mesolabe compass is an ancient Greek geometric tool used for performing multiplication and division using lines. In the video, it is demonstrated how to use this tool to multiply numbers like 3 times 4 by drawing lines and using a set square to create parallel lines. The concept is central to the video's theme of showcasing ancient mathematical techniques.

Hippocrates of Chios

Hippocrates of Chios is an ancient Greek mathematician who is credited with the invention of the mesolabe compass. The video script emphasizes his contribution to geometry and suggests that he deserves a more prominent place in history. His work is the focus of the video, illustrating the historical significance of his mathematical discoveries.

Multiplication

Multiplication is a mathematical operation that combines two numbers to find their product. In the context of the video, multiplication is performed using the mesolabe compass by drawing lines and creating parallel lines to represent multiples of a number. For instance, multiplying 3 by 4 is demonstrated by drawing lines and using a set square to find the product.

Division

Division is the mathematical operation of dividing a number into equal parts to find the quotient. The video script explains how division can be achieved using the mesolabe compass by drawing lines and creating parallel lines to divide numbers. An example given is dividing 12 by 4, which is shown to result in the answer 3.

Geometry

Geometry is a branch of mathematics concerned with the properties and relationships of points, lines, angles, surfaces, and solids. The video highlights the importance of geometry in ancient Greek mathematics, as demonstrated by the use of the mesolabe compass and the method for finding square roots, which are both geometric in nature.

Square root

A square root is a value that, when multiplied by itself, gives the original number. The video script describes an ancient method for finding square roots using lines and a semicircle. For example, finding the square root of 9 is illustrated by drawing a line and a semicircle, and using perpendicular lines to determine the square root.

Pythagorean triangle

A Pythagorean triangle is a right-angled triangle where the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the video, a 3-4-5 triangle is used to illustrate the relationship between the sides of the triangle and the square root method, showing the connection between geometry and the square root calculation.

Thales

Thales of Miletus was an ancient Greek philosopher and mathematician known for his work in geometry. The video script references Thales' theorem, which 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. This principle is used to explain the method for finding square roots.

Similar triangles

Similar triangles are triangles that have the same shape but different sizes, with corresponding angles and sides in proportion. The video explains how the method for finding square roots relies on the concept of similar triangles, as the angles in the triangles formed by the semicircle and perpendicular lines are equal, making the triangles similar.

René Descartes

René Descartes was a French philosopher, mathematician, and scientist who is considered the father of modern philosophy. The video mentions Descartes in the context of his recognition of Hippocrates of Chios' work and his own contributions to geometry, specifically his book 'La Géométrie,' which influenced the understanding of geometric principles.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The video script briefly touches on calculus as a more advanced mathematical tool that was not available in ancient times but is necessary for measuring changing quantities, contrasting it with the static geometric methods of the past.

Highlights

Hippocrates of Chios, a mathematician often missed in history, deserves a bigger place for his contributions.

Introduction to the mesolabe compass, an ancient Greek tool for multiplication and division using lines.

The mesolabe compass allows for the multiplication of any two numbers using a simple geometric approach.

Division can be achieved with the mesolabe compass by drawing parallel lines through given numbers.

Hippocrates' method predates Christ and showcases the ancient Greeks' interest in geometric representations of math.

A demonstration of how to multiply numbers using the mesolabe compass with the example of 3 times 4.

Hippocrates' second idea involves finding the square root of any number using lines and curves.

A step-by-step guide on using geometric shapes to find the square root of a given number.

The method involves drawing a line, marking it off, and using a semicircle to find the square root.

An explanation of how the square root method works for any number, including non-square numbers.

The impracticality of using the method for very large numbers due to the scale of the lines involved.

The theoretical soundness of the method and its practical application in smaller scales.

A connection between the square root method and the Pythagorean theorem through a 3-4-5 triangle.

Thales' theorem and its role in the square root method, showing the right angles formed in a semi-circle.

The concept of similar triangles and their proportions in the context of finding square roots.

Hippocrates of Chios' ideas were recognized by René Descartes and featured in his book 'The Geometry'.

Descartes' perspective on geometry's ability to calculate anything before the discovery of calculus.

The historical significance of Hippocrates' methods and their influence on the development of calculus.

A call to action for viewers to explore Brilliant's resources on square roots and related mathematical concepts.

An offer for a 20% discount on Brilliant's premium membership for viewers interested in further mathematical exploration.