UNDER MULTIPLICATION

HISTORY OF MULTIPLICATION SIGN
X was used by William Oughtred (1574-1660) in the Clavis Mathematicae (Key to Mathematics), composed about 1628 and published in London in 1631 (Smith). Cajori calls X St. Andrew's Cross.

X actually appears earlier, in 1618 in an anonymous appendix to Edward Wright's translation of John Napier's Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.

The dot (·) was advocated by Gottfried Wilhelm Leibniz (1646-1716). According to Cajori (vol. 1, page 267):

The dot was introduced as a symbol for multiplication by G. W. Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli: "I do not like X as a symbol for multiplication, as it is easily confounded with x; ... often I simply relate two quantities by an interposed dot and indicate multiplication by ZC · LM. Hence, in designating ratio I use not one point but two points, which I use at the same time for division."

The dot was used earlier by Thomas Harriot (1560-1621) in Analyticae Praxis ad Aequationes Algebraicas Resolvendas, which was published posthumously in 1631, and by Thomas Gibson in 1655 inSyntaxis mathematica. However Cajori says, "it is doubtful whether Harriot or Gibson meant these dots for multiplication. They are introduced without explanation. It is much more probable that these dots, which were placed after numerical coefficients, are survivals of the dots habitually used in old manuscripts and in early printed books to separate or mark off numbers appearing in the running text" (Cajori vol. 1, page 268).

However, Scott (page 128) writes that Harriot was "in the habit of using the dot to denote multiplication." And Eves (page 231) writes, "Although Harriot on occasion used the dot for multiplication, this symbol was not prominently used until Leibniz adopted it."

The asterisk (*) was used by Johann Rahn (1622-1676) in 1659 in Teutsche Algebra (Cajori vol. 1, page 211).

By juxtaposition. In a manuscript found buried in the earth near the village of Bakhshali, India, and dating to the eighth, ninth, or tenth century, multiplication is normally indicated by placing numbers side-by-side.

Multiplication by juxtaposition is also indicated in "some fifteenth-century manuscripts" . Juxtaposition was used by al-Qalasadi in the fifteenth century (Cajori vol. 1, page 230).

According to Lucas, Michael Stifel (1487 or 1486 - 1567) first showed multiplication by juxtaposition in 1544 in Arithmetica integra.

In 1553, Michael Stifel brought out a revised edition of Rudolff's Coss, in which he showed multiplication by juxtaposition and repeating a letter to designate powers.