The above expression in modern notation is . This use of a vinculum appears to be the earliest use of a grouping symbol of any kind mentioned by Cajori.

**Vinculum above.** According to Cajori, the first use of the
vinculum above the parts affected was by Frans van Schooten (c.
1615-1660), who "in editing Vieta's collected works, discarded the
parentheses and placed a horizontal bar above the parts affected." In
Van Schooten's 1646 edition of Vieta, is used to represent *B*(*D*^{2}
+ *BD*).

Ball (page 242) says the vinculum was introduced by Francois Vieta (1540-1603) in 1591. This information may be incorrect.

**Grouping expressed by letters.** In the late fifteenth century
and in the sixteenth century various writers used letters or words to
indicate grouping. The earliest use of such a device mentioned by
Cajori (vol. 1, page 385) is the use of the letter *v* for
*vniversale* by Luca Paciolo (or Pacioli) (c. 1445 - prob. after
1509) in his *Summa* of 1494 and 1523.

**Parentheses.** Parentheses ( ) are "found in rare instances as
early as the sixteenth century" (Cajori vol. 1, page 390). Apparently
the earliest work Cajori names in which round parentheses are found
is *General trattato di numeri e misure* by Nicolo Tartaglia (c.
1506-1557) in 1556. Round parentheses occur once in *Ars magna*
by Cardan, as printed in *Opera* (1663) (Cajori vol. 1, page
392; Cajori does not indicate whether the parentheses occur in the
original 1545 edition).

Cajori (vol. 1, page 391) says that Michael Stifel (1487 or 1486 - 1567) does not use parentheses as signs of aggregation in his printed works, but that they are found in one of his handwritten marginal notes. Cajori expresses the opinion that these parentheses are actually punctuation marks rather than mathematical symbols.

Kline says parentheses appear in 1544. He presumably refers to
*Arithmetica integra* by Michael Stifel.

**Brackets.** Brackets [ ] are found in the manuscript
edition of *Algebra* by Rafael Bombelli (1526-1573) from about
1550 (Cajori vol. 1, page 391).

Ball (page 242) and Lucas say brackets were introduced by Albert Girard (1595-1632) in 1629. This information appears to be inaccurate.

Kline says square brackets were introduced by Vieta (1540-1603). He
presumably refers to the 1593 edition of *Zetetica,* which
according to Cajori uses both braces and brackets.

**Braces.** Braces { } are found in the 1593 edition of
Francois Vieta's *Zetetica* (Cajori vol. 1, page 391).

**Grouping symbols in numeration.** In the writing of large
numbers, various methods have been used to separate numerals into
groups, including dots, vertical bars, commas, arcs, colons, and
accent marks.

In 1202, Leonardo of Pisa in *Liber Abaci* directs that the
hundreds, hundred thousands, hundred millions, etc., be marked with
an accent mark above, and that thousands, millions, thousands of
millions, etc., be marked with an accent below (Cajori vol. 1, page
58).

The earliest example of the modern system of simply separating the
numeral into groups of three with commas shown by Cajori is in 1795
in the article "Numeration" in *Mathematical and Philosophical
Dictionary* by Charles Hutton.

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