A Square and A Cube · Lesson 3 of 4
A Pinch of History
“Trace the historical ideas and terminology behind squares, cubes and roots.”
• Describe how ancient people used tables of squares and cubes. • Explain the meanings of the Sanskrit terms varga, ghana and mula in this chapter. • Connect geometric squares and cubes with the powers used in arithmetic. • Identify the historical roles of Aryabhata and Brahmagupta mentioned in the chapter. • Explain why the mathematical word root is connected with the idea of an origin or basis.
Squares, cubes and roots are very old mathematical ideas. The chapter notes that Babylonian clay tablets from about 1700 BCE contain lists of perfect squares and perfect cubes. Such lists were useful when people needed numerical results for land measurement, building and other geometric calculations.
Problem
Why would a prepared list of squares have been useful?
- 1.Suppose a land-measurement problem required a square root.
- 2.A known table could help match an area with a nearby square number.
- 3.This saved repeated multiplication and made practical calculations faster.
In ancient Sanskrit mathematical writing, the word varga was used both for a square figure or its area and for the square of a number. The word ghana was similarly used for a solid cube and for the product of a number multiplied by itself three times. The chapter notes that these terms were used in India at least from the third century BCE.
A Sanskrit term used for a square figure or its area, and also for the square power of a number.
A Sanskrit term used for a solid cube and also for the cube power of a number.
Problem
How does the word varga connect geometry with arithmetic?
- 1.A geometric square with side 5 has area 5 × 5 = 25.
- 2.The arithmetic square 5² is also 25.
- 3.The same word could therefore describe both the square figure and the square power.
Aryabhata, writing in 499 CE, described a square figure with four equal sides and also connected the term varga with the product of two equal quantities. This shows how the numerical idea of squaring grew naturally from the geometry of a square.
The chapter also explains the history of the word root. The Sanskrit word mula can mean the root of a plant, but also basis, cause or origin. In mathematics, varga-mula referred to the origin of a square — what we now call a square root — and ghana-mula referred to a cube root.
A Sanskrit word meaning root, basis, cause or origin. It was used in expressions for square root and cube root.
Problem
Why is 7 called the square root of 49?
- 1.49 comes from 7 × 7.
- 2.So 7 is the number from which the square 49 originates.
- 3.This idea of an origin or basis matches the historical meaning of mula.
The chapter notes that the use of plant-root words for mathematical roots later appeared in Arabic and Latin as well. It also mentions Brahmagupta, writing in 628 CE, who described the root of a square as the quantity of which the number is a square.
Quiz
What were Babylonian lists of squares and cubes useful for?
In the chapter, the Sanskrit word varga is connected with:
What does ghana refer to in the historical discussion?
Which meaning of mula helps explain the mathematical word root?
Which mathematician is mentioned in the chapter with the year 499 CE?
Practice Problems
- Explain in your own words why ancient lists of perfect squares and cubes would have been useful before modern calculating tools.
- Describe the connection between the geometric meaning of a square and the arithmetic meaning of squaring a number.
- What do the terms varga and ghana mean in the historical discussion of this chapter?
- Explain why the word root is a sensible name for the inverse operation of squaring or cubing.
- Write a short timeline using these four points from the chapter: Babylonian lists around 1700 BCE, ancient Indian use of varga and ghana, Aryabhata in 499 CE, and Brahmagupta in 628 CE.
Key Takeaways
• Lists of squares and cubes were used thousands of years ago for practical calculations. • The Sanskrit term varga connected a square figure with the square power of a number. • Ghana connected the solid cube with the cube power of a number. • Mula means root, basis, cause or origin and was used for mathematical roots. • Aryabhata and Brahmagupta are among the Indian mathematicians mentioned in the chapter’s historical account. • Mathematical language changed across cultures, but the underlying ideas of squares, cubes and roots remained.