THE SYMBOL  FOR SMILAR AND CONGRUEND

Our familiar signs, in geometry, for similar (on the left), and for congruent (on the right) are due to Leibniz (1646-1715.) (Eves p253) 

Leibniz made important contributions to the notation of mathematics. In Leibnizian manuscripts occurs this symbol (on the left) for “similar,” and this symbol (on the right) for “equal and similar” or “congruent.” (Cajori p211)

EVES, HOWARD "An introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976 

CAJORI,FLORIAN "A History of Mathematics", The Macmillan Company 1926

THE SYMBOL  FOR ANGLE AND RIGHT ANGLE

In 1923, the National Committee on Mathematical Requirements, sponsored by the Mathematical Association of America, recommended this symbol (on the left) as standard usage for angle in the United States. Historically, Pierre Herigone, in a French work in 1634, was apparently the first person to use a symbol for angle. He used both the symbol above as well as this symbol on the right, which had already been used to mean "less than." The standard symbol survived, along with other variants, as follows. 

These appeared in England circa 1750.


During the 19th century in Europe these forms were used to designate the angle ABC, and the angle between a and b , respectively. 


This symbol, representing the arc on the angle, first appeared in Germany in the latter half of the 19th century.

THE SYMBOL FOR PI

This symbol for pi was used by the early English mathematicians William Oughtred (1574 -1660), Isaac Barrow (1630-1677), and David Gregory (1661-1701) to designate the circumference , or periphery, of a circle. The first to use the symbol for the ratio of the circumference to the diameter was the English writer, William Jones, in a publication in 1706. The symbol was not generally used in this sense, however, until Euler (1707-1783) adopted it in 1737. (Eves p99)

Oughtred's notation was the forerunner of the relation pi = 3.14159..., first used by William Jones in 1706 in his Synopsis palmariorum matheseos. Euler first used pi = 3.14159... in 1737. In his time, the symbol met with general adoption. (Cajori p158)

This symbol for pi was used by Oughtred in an expression to represent the ratio of the diameter to the circumference. Isaac Barrow, from 1664, used the same symbolism. David Gregory used pi in an expression to represent the ratio of the circumference to the radius in 1697. The first to use pi definitely to stand for the ratio of circumference to diameter was an English writer William Jones. He used it to symbolize the word "periphery." Euler adopted the symbol in 1737, and since that time it has been in general use. (Smith p312)

The number pi is the ratio of the circumference of a circle to its diameter. It is also the ratio of the area of a circle to the area of the square on its radius. The adoption of the symbol for pi for this ratio is essentially due to the usage given it by Leonhard Euler from 1736 on. In the 1730's, Euler first used p and c for the circumference -to-diameter ratio, then adopted this symbol for pi. However, he is not the originator of the symbol.

An actual ratio symbol as illustrated here on the right had been used by William Oughtred in 1647 and by Isaac Barrow in 1664 to indicate the ratio of the diameter of a circle to it's circumference or periphery.


David Gregory, nephew of Scottish mathematician James Gregory (1638-1675), used this symbol on the left for the ratio of circumference to radius in 1697. In 1706 the English writer William Jones, in a work that gave the 100-place approximation of John Machin, first used the single symbol for pi. This computation of pi to a large number of places by means of various series representations was aided by the use of such relations as pi/4 = 4 arctan (1/5) - arctan (1/239), as given by Machin in 1706. (NCTM p148,152)

EVES, HOWARD "An Introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976.

CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926

SMITH, D.E. "History of Mathematics" volume II. Dover Publications 1958

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom". National Council of Teachers of Mathematics (USA) 1969

THE SYMBOL FOR PERCENT

Percent has been used since the end of the fifteenth century in business problems such as computing interest, profit and loss, and taxes. However, the idea had its origin much earlier. When the Roman emperor Augustus levied a tax on all goods sold at auction, centesima rerum venalium, the rate was 1/100. Other Roman taxes were 1/20 on every freed slave and 1/25 on every slave sold. Without recognising percentages as such, they used fractions easily reduced to hundredths.

In the Middle Ages, as large denominations of money came to be used, 100 became a common base for computation. Italian manuscripts of the fifteenth century contained such expressions as "20 p 100" and "x p cento" to indicate 20 percent and 10 percent. When commercial arithmetics appeared near the end of that century, use of percent was well estasblished. For example, Giorgio Chiarino (1481) used "xx. per .c." for 20 percent and "viii in x perceto" for 8 to 10 percent. During the sixteenth and seventeenth century, percent was used freely for computing profit and loss and interest. (NCTM p146,147}

In its primitive form the per cent sign is found in the 15th century manuscripts on commercial arithmetic, where it appears as this symbol after the word "per" or after the letter "p" as a contraction for "per cento." The use of the per cent symbol can be seen in this extract from an anonymous Italian manuscript of 1684 (Smith p250)

The percent sign, %, has probably evolved from a symbol introduced in an anonymous Italian manuscript of 1425. Instead of "per 100," "P cento," which were common at that time, this author used the symbol shown. 

By about 1650, part of this symbol had been changed to the form shown on the right. Finally, the "per" was dropped, leaving this symbol to stand alone, and this in turn became %. (NCTM p147)

The solidus form (%) is modern. (Smith p250)

This symbol stands for "per thousand". (Hogben p92)

It is natural to expect that percentage will develop into per millage, and indeed this has not only begun, but it has historic sanction. Bonds are quoted in New York using this symbol on the right, and so in other commercial lines. At present, indeed, the symbol above (Hogben) is used in certain parts of the world, notably by German merchants, to mean "per mill," a curious analogue to % developed without regard to the historic meaning of the latter symbol.(Smith p250)

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

HOGBEN, LANCELOT "The Wonderful World of Mathematics", Macdonald and Company 1968