Bows, Arrows, and What Will Become Sine
Why Read This
What Makes This Article Worth Your Time
Summary
What This Article Is About
Dilip D’Souza imagines the great 6th-century Indian mathematician Aryabhata narrating, in the first person, the intellectual journey that gave birth to the concept now known as the sine function in trigonometry. Speaking as Aryabhata, the narrator watches his archer friend practice and becomes obsessed with the geometry of the bow and bowstring. He frames the bow as an arc of a circle, names the straight line connecting its ends “jya” (Sanskrit for bowstring), and works out 24 values of the half-chord — the ardhajya — corresponding to different degrees of bowstring pull. This table of values is effectively the world’s first sine table, constructed 1,400 years before the word “sine” existed in European mathematics.
The article then traces the extraordinary etymological journey of the word. Arabian traders carry the concept westward, where mathematicians including Muhammad ibn Musa al-Khwarizmi render it phonetically as “jiba,” written in vowel-less Arabic as “jb.” In the 12th century, Gherardo of Cremona translates al-Khwarizmi’s Arabic texts into Latin, misreading “jb” as “jaib” — meaning “fold” or “pocket” — and translates it as the Latin “sinus.” From sinus came the modern English word sine. Through a playful, first-person narrative, D’Souza reveals that one of the most fundamental concepts in mathematics has its roots not in a Greek theorem but in the arc of an Indian bowman’s weapon.
Key Points
Main Takeaways
The Bow Inspired the Concept
Aryabhata’s observation that the depth of an archer’s bowstring pull determines the arrow’s range led him to model the bow geometrically as an arc of a circle.
Jya Was the Original Sine
Aryabhata named the half-chord of a circle “jya” (Sanskrit for bowstring), compiled 24 values of it, and produced what is effectively the world’s first trigonometric sine table.
A Translation Error Gave Us “Sine”
When Gherardo of Cremona translated Arabic mathematical texts into Latin in the 12th century, he misread the word “jb” as “jaib” (meaning pocket) and rendered it as the Latin “sinus” — the direct origin of the English word “sine.”
Mathematics Travels Across Cultures
The concept of jya moved from Sanskrit India to Islamic mathematics through Arabian traders, then from Arabic to Latin through 12th-century European translators — illustrating how mathematical knowledge crosses civilisations.
“Arc” Also Derives from Archery
The article notes that the mathematical term “arc” derives from the Latin word “arcus,” meaning bow — making the archery-mathematics connection even deeper than the etymology of sine alone.
Narrative Form Illuminates History of Science
D’Souza uses a first-person historical persona to show that mathematical abstraction often grows from lived, physical observation — grounding abstract ideas in sensory, human experience.
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Article Analysis
Breaking Down the Elements
Main Idea
Mathematical Concepts Have Human, Physical Origins — and Words Carry History
The article’s central argument is twofold: first, that abstract mathematical concepts like the sine function grew out of concrete, embodied human problems — in this case, archery; and second, that the words we use in mathematics carry centuries of intercultural transmission within them. The etymology of “sine” is not a trivial footnote — it is evidence that Indian mathematical thought, transmitted through the Islamic world, is foundational to modern mathematics.
Purpose
To Illuminate and Recover a Hidden History of Mathematics
D’Souza writes to recover and celebrate a piece of mathematical history that is widely unknown — even to people who use the sine function daily. The purpose is both educational and corrective: to show that modern mathematics is not solely a Western inheritance, and that the transmission of knowledge across cultures, languages, and centuries is as fascinating as the mathematics itself. The playful narrative device makes that history feel intimate rather than academic.
Structure
Embedded Persona → Geometric Abstraction → Etymological Journey → Reveal
The article is structured as a slow reveal. It opens with a first-person persona (Aryabhata speaking as a narrator), uses the physical act of archery to motivate geometric abstraction, constructs the concept of jya through close observation, and then fast-forwards through centuries of linguistic transmission to deliver the payoff — that “sine” is jya in disguise. The movement is Concrete → Abstract → Historical → Revelatory, with the final line functioning as a satisfying intellectual punchline.
Tone
Playful, Erudite & Quietly Revelatory
The tone is gently whimsical — the narrator casually mentions he “has the gift of seeing into the future” and refers to pizza — while remaining intellectually rigorous. D’Souza wears his scholarship lightly, never lecturing, always inviting the reader to enjoy the journey. The prevailing feeling is one of quiet wonder: the delight of discovering that something as apparently dry as a trigonometric ratio has a warm, human, cross-civilisational origin story.
Key Terms
Vocabulary from the Article
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Tough Words
Challenging Vocabulary
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A tough fibrous cord of tissue connecting muscle to bone; historically used to strengthen weapons such as bows, adding elasticity and power.
“It is made of wood but strengthened with sinews. The combination makes it firm, supple and elastic.”
Pulled or stretched tight; having no slack or looseness. Used here to describe the condition of a bowstring when it is under tension and ready for use.
“At rest, the string is taut, straight, and twangs a healthy, satisfying note when idly plucked.”
To establish a mutual relationship or connection between two or more things, such that a change in one corresponds in a predictable way to a change in another.
“Is there some way to correlate the length of the pull to how much force the arrow gets, to how far it travels when released?”
Bending and moving easily and gracefully; flexible without cracking or breaking. In a bow, suppleness allows it to store and release energy efficiently when drawn.
“It is made of wood but strengthened with sinews. The combination makes it firm, supple and elastic.”
In a manner relating to the sounds of speech; when a word is borrowed phonetically, it is reproduced based on how it sounds rather than its original spelling or meaning.
“Mathematicians like al-Khwarizmi call it ‘jiba’, the closest they can get, phonetically, to the sound of the word I use.”
Involving only the most basic and elementary facts or principles; undeveloped or primitive in form. Used here to describe a simple early technology before any advanced engineering.
“Take something as rudimentary as a small wooden raft that one of our ancestors used to float across a raging river.”
Reading Comprehension
Test Your Understanding
5 questions covering different RC question types
1According to the article, the mathematical term “arc” derives from a Latin word that also means “bow.”
2Why did the narrator (Aryabhata) choose a circle with a radius of 3438 units specifically for his jya calculations?
3Which sentence most directly explains the crucial role the Arabic language’s lack of written vowels played in the eventual creation of the word “sine”?
4Evaluate each of the following statements based on the article.
Aryabhata defines the jya as the full chord connecting both ends of the bow — not the half-chord.
The narrator says that a jya of 3438 units corresponds to the bow at rest — i.e. when no arrow is being drawn.
The narrator compiled a total of 24 jya values, ranging from 3438 units down to 225 units (called “makhi”).
Select True or False for all three statements, then click “Check Answers”
5Based on the article’s account of jya’s journey from Sanskrit to Latin, what can we most reasonably infer about the nature of mathematical knowledge transmission across cultures?
FAQ
Frequently Asked Questions
Aryabhata was a pioneering Indian mathematician and astronomer who lived in the 5th and 6th centuries CE. He is best known for his work Aryabhatiya, in which he compiled tables of what we now call sine values (called jya), worked on approximations of pi, and made significant contributions to algebra and astronomy. The article presents him as the originator of the concept that eventually became the modern trigonometric sine function.
Al-Khwarizmi was a 9th-century Islamic mathematician who incorporated Indian mathematical ideas — including Aryabhata’s jya — into Arabic mathematical texts. He rendered “jya” phonetically as “jiba,” which in vowel-less Arabic script was written as “jb.” His texts became the bridge through which Indian trigonometric knowledge entered the European mathematical tradition via Latin translators in the 12th century. Al-Khwarizmi is also the figure from whose name the word “algorithm” derives.
This is a playful literary device used by D’Souza to make the first-person narrative work. Since Aryabhata lived in the 6th century CE but references concepts — like the word “arc,” Latin “sinus,” and even pizza — that did not exist until centuries later, the narrator winks at the reader by claiming he can see the future. The device allows D’Souza to reveal the full etymological journey within a single, continuous first-person voice without breaking the narrative frame.
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This article is rated Advanced. While the prose is accessible and often charming, it demands several sophisticated reading competencies simultaneously: tracking a first-person historical persona whose reliability is intentionally playful, following multi-step etymological chains across four languages (Sanskrit, Arabic, Latin, English), understanding basic geometric concepts (chord, arc, radius), and inferring the author’s deeper cultural and historical argument from beneath the narrative surface. Readers must also catch subtle humour and anachronism as literary devices rather than errors.
Dilip D’Souza is an Indian writer and journalist known for combining mathematical and scientific curiosity with literary storytelling. He writes regularly for publications including 3 Quarks Daily, exploring the history and philosophy of mathematics, data, and science for general audiences. His approach — using narrative, humour, and historical personas to illuminate abstract ideas — is characteristic of the “narrative non-fiction of mathematics” genre, making technical concepts genuinely pleasurable to encounter.
The Ultimate Reading Course covers 9 RC question types: Multiple Choice, True/False, Multi-Statement T/F, Text Highlight, Fill in the Blanks, Matching, Sequencing, Error Spotting, and Short Answer. This comprehensive coverage prepares you for any reading comprehension format you might encounter.
