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“THE SQUARE OF the hypotenuse is equal to the sum of the squares of the other two sides.” Apart from giving us the pun about squaws sleeping on various hides, including hippopotamus, this word rendition of the Pythagorean Theorem is certainly less concise than c2 = a2 + b2.

That is, mathematics has evolved an elegant language, many of its symbols recognizable even to non-math types. Foremost among these is the symbol “=.” Tidbits follow, today and tomorrow in Parts 1 and 2, both mathwise and otherwise.

Its Mathematical Definition. The symbol “=” indicates that the two things on either side of it share the same mathematical value. A = B. However, it wasn’t always so easy to show this.

Earliest mathematics was largely a words game. Welsh physician and mathematician Robert Recorde was the first to employ the symbol “=.” With his The Whetstone of Witte, 1557, he reasoned that these two parallel lines were the most equal things in existence.

The Whetstone of Witte can be read online in its original entirety, all 332 pages of it. Be prepared to decipher mid-sixteenth-century typography and to learn about coßik numbers.

Coßike Nombers. Italians were among the first Europeans to adopt Arabic methods of algebra. Around 820 A.D., Arabic mathematician Muhammad ibn Musa al-Kwārizmī (thus our word “algorithm”) wrote a book titled Kitab al-Mukhtasar fi hisab al-Jabr wal-muqabala, “The Compendious Book on Calculation by Completion and Balancing.” This work and its translations were used throughout Europe well into the sixteenth century.

A page from al-Kwārizmī’s text.

In Mathematics in Historical Context, Jeff Suzuki writes, “Italian algebras of the time referred to the unknown quantity as the cosa (‘thing’ in Italian), so outside Italy algebra was known as the cossick art, its practitioners were cossists.…”

More than 40 percent of The Whetstone of Witte is devoted to The Arte of Cossike Nombers.

The Whetstone of Witte, page 238–239. Note, contrasted with modern practice, Recorde chose to elongate his +, –, and = signs. Image from

Inequalities. The mathematical symbol ≠ describes things being bluntly not equal. There are also more informative symbols meaning “less than” or “greater than.” Curiously, the coding language of this website precludes their employment in text. My workaround is illustrated here:

On the left, “less than.” On the right, “greater than.” Image by bad from

Note, with these inequality symbols, the pointy part always points to the smaller value; the bigger part of the symbol always abuts the larger value.

According to, the symbols first appeared in 1631 in Artis Analyticæ Praxis ad Aequationes Algebraicas Resolvendas. The book was the work of British mathematician Thomas Harriot who died ten years prior to its being published.

Sciencing notes, “The symbols were invented by the book’s editor. Harriot initially used triangular symbols…. Interestingly, Harriot also used parallel lines to denote equality. However, Harriot’s equal sign was vertical (||) rather than horizontal (=).”

And combining the symbolism, ≤ and ≥, means exactly what you would think: “less than or equal to” and “greater than or equal to,” respectively. French mathematician Pierre Bouguer first used these symbols in 1734. British logician and mathematician John Wallis had a similar idea in 1670, but he put the single horizontal line above, not below.

This may not seem like a big deal. However, tomorrow in Part 2 we’ll learn the power of “=,” how it’s used culturally, and how it’s enclosed geometrically. ds

© Dennis Simanaitis,, 2020


  1. sabresoftware
    March 23, 2020

    The following link has the coding to show what you want:

    • Dennis
      March 23, 2020

      Kewl. I suspect it would also handle the oddity of certain numerical strings being incorrectly interpreted as phone numbers.
      Thanks sincerely for expanding my meager HTML. — Dennis

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