Matematiksel Semboller-4


The symbols of elementary arithmetic are almost wholly algebraic, most of them being transferred to the numerical field only in the 19th century, partly to aid the printer in setting up a page and partly because of the educational fashion then dominant of demanding a written analysis for every problem. When we study the genesis and development of the algebraic symbols of operation, therefore, we include the study of the symbols in arithmetic. Some idea of the status of the latter in this respect may be obtained by looking at almost any of the textbooks of the 17th and 18th centuries. Hodder in 1672 wrote "note that a + (plus) sign doth signifie Addition, and two lines thus = Equality, or Equation, but a X thus, Multiplication," no other symbols being used. His was the first English arithmetic to be reprinted in the American colonies in Boston in 1710. Even Recorde (c1510-1558), who invented the modern sign of equality, did not use it in his arithmetic, the Ground of Artes (c1542), but only in his algebra, the Whetstone of witte (1557). (Smith p395)

There is some symbolism in Egyptian algebra. In the Rhind papyrus we find symbols for plus and minus. The first of these symbols represents a pair of legs walking from right to left, the normal direction for Egyptian writing, and the other a pair of legs walking from left to right, opposite to the direction for Egyptian writing. [Eves 1, p42]

The earliest symbols of operation that have come down to us are Egyptian. In the Ahmes Papyrus (c1550 B.C.) addition and subtraction are indicated by these symbols on the left and right above respectively. 

The Hindus at one time used a cross placed beside a number to indicate a negative quantity, as in the Bakhshali manuscript of possibly the 10th century. With this exception it was not until the 12th century that they made use of the symbols of operation. In the manuscripts of Bhaskara (c1150) a small circle or dot is placed above a subtrahend as illustrated for -6, or the subtrahend is enclosed in a circle to indicate 6 less than zero.

The early European symbols for plus are listed opposite. The word plus, used in connection with addition and with the Rule of False Position is not known before the latter part of the 15th century.

The use of the word minus as indicating an operation occurred much earlier, as in the works of Fibonacci (c1175-1250) in1202. The bar above the letter simply indicated an omission. In the 15th century, this third symbol was also often used for minus, but most writers preferred the other variations.

In the 16th century the Latin races generally followed the Italian school, using the letters p and m, each with the bar above it, or their equivalents, for plus and minus. However, the German school preferred these symbols, neither of which is found for this purpose before the 15th century. In a manuscript of 1456, written in Germany, the word "et" is used for addition and is generally written so that it closely resembles the modern symbol for addition. There seems little doubt that the sign is merely a ligature for "et", much in the same way that we have the ligature "&" for the word "and."
The origin of the minus sign has been more of a subject of dispute. Some have thought that it is a survival of the bar above the three symbols for minus as listed above. It is more probably that it comes from the habit of early scribes of using it as a shorthand equivalent of "m." Thus Summa became Suma with the bar above the letter u, and 10 thousand became an X with ther bar above the letter. It is quite reasonable to think of the dash (-) as a symbol for "m" (minus), just as the cross (+) is a symbol for "et." Other forms of minus are here illustrated.

There were other various written forms for plus and minus, as in piu (Italian), mas (Spanish), plus (French) and et (German) for plus and as in de or men (Italian), menos (Spanish), moins (French) for minus. Examples of such usage include:

Pacioli (1494), Italian de or m for minus
Tartaglia (1556) and Catanes (1546), Italian, piu and men
Santa-Cruz (1594), Spanish, mas and menos
Peletier (1549), French, plus and moins
Gosselin (1577), P and M
Trenchant (1566), + and -

The expression "plus or minus" is very old, having been in common use by the Romans to indicate simply "more or less". It is often found on Roman tombstones, where the age of the deceased is given as illustrated to indicate "94 years, more or less".

These signs first appeared in print in an arithmetic, but they were not employed as symbols of operation. In the latter sense they appear in algebra long before they do in arithmetic.They appeared in Johann Widman's (c1460-?) arithmetic published in Leipzig in 1489, the author saying: "Was - ist / das ist minus...vnd das + das ist mer." He then speaks of "4 centner + 5 pfund," and also of "4 centner - 17 pfund," thus showing the excess or deficiency in the weight of boxes or bales. (Smith p395 to 399)

Observe that Francis Vieta (1540-1603) employed the Maltese cross (+) as the shorthand for addition, and the (-) for subtraction. These two characters had not been in very general use before his time. The introduction of the + and - symbols seems to be due to the Germans, who, although they did not enrich algebra during the Renaissance with great inventions, as did the Italians, still cultivated it with great zeal. The arithmetic of John Widmann, brought out in 1489 in Leipzig, is the earliest printed book in which the + and - symbols have been found, and the facsimile shown is from the Augsburg edition of his work, dated 1526. The + sign is not restricted by him to ordinary addition; it has the more general meaning "et" or "and" as in the heading, "regula augmenti + decrementi." The - sign is used to indicate subtraction, but not regularly so. The word "plus" does not occur in Widmann's text; the word "minus" is used only two or three times. The symbols + and - are used regularly for addition and subtraction, in 1521, in the arithmetic of Grammateus, the work of Heinrich Schreiber, a teacher at the University of Vienna. His pupil Christoff Rudolff, the writer of the first text book on algebra in the German language (printed in1525) employs these symbols. So did Michael Stifel, who brought out an improved second edition of Rudolff's book on algebra Die Coss in 1553. Thus, by slow degrees, the adoption of the + and - symbols became universal. Several independant paleograhic studies of Latin manuscripts of the fourteenth and fifteenth centuries make it almost certain that the + sign comes from the Latin et, as it was cursively written in manuscripts just before the time of the invention of printing. The origin of the sign - is still uncertain. (Cajori p139)

The first one to make use of these signs in writing an algebraic expression was the Dutch mathematician Vander Hoecke, who in 1514 gave this illustration (on the left) for radical three quarters minus radical three fifths, and for radical 3 add 5 he gave the sign as shown on the right.

These symbols seem to have been employed for the first time in arithmetic, to indicate operations, by Georg Walckl in 1536. The illustration on the left indicates the addition of one third of 230, and the one on the right indicates the subtraction of one fifth of 460. From this time on the two symbols were commonly used by both German and Dutch writers, the particular signs themselves not being settled until well into the 18th century.

England adopted the Teutonic forms, and Robert Recorde (c1510-1558) wrote (c1542) "thys fygure +, whiche betoketh to muche, as this lyn, - plaine without a cross lyne, betokeneth to lyttle". As symbols of operation most of the English writers of this period reserved the + and - signs for algebra. Thus Digges (1572) in his treatment of algebra: "Then shall you ioyne them with this signe + Plus", and Hylles (1600) says: "The badg or signe of addition is +," stating the sum of 3 and 4 as "3 more 4 are 7," and writing 10___3 for "10 lesse 3." (Smith p399-402)

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

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


William Oughtred (1574-1660) contributed vastly to the propagation of mathematical knowledge in English by his treatises, the Clavis Mathematicae, 1631, published in Latin (English edition 1647), Circles of Proportion, 1632, and Trigonometrie, 1657. Among his most noted pupils are the mathematician John Wallis (1616-1703) and the astronomer Seth Ward. 

Oughtred laid extraordinary emphasis upon the use of mathematical symbols : altogether he used over 150 of them. Only three have come down to modern times, namely the cross symbol for multiplication, :: as that of proportion, and the symbol for "difference between." The cross symbol, on the left, occurs in the Claris, but the letter X, seen on the right, which closely resembles it, occurs as a sign of multiplication in the anonymous "Appendix to the Logarithmes" in Edward Wright's translation of John Napier's Descriptio, published in 1618. This appendix was most probably written by Oughtred.

Leibniz (1646-1715) objected to the use of Oughtred's cross symbol because of possible confusion with the letter X. On 29 July 1698 he wrote in a letter to John Bernoulli : "I do not like (the cross) 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." 

Through the aid of Christian Wolf (1679-1754) the dot was generally adopted in the 18th century as a symbol for multiplication. Wolf was a professor at Halle, and was ambitious to figure as a successor of Leibniz. Presumably Leibniz had no knowledge that Harriot in his Artis analyticae praxis, 1631, used a dot for multiplication, as in aaa__3.bba=+2.ccc. Harriot's dot received no attention, not even from Wallis. (Cajori p157)

The common symbol as illustrated was developed in England about 1600. It was not a new sign, having long been used in cross multiplication, in the check of nines, where Hylles (1600) speaks of it as the "byas crosse" in connection with the multiplication of terms in the division or addition of fractions, for the purpose of indicating the corresponding products in proportion, and in the "multiplica in croce" of algebra as well as in arithmetic.

The symbol was not readily adopted by arithmeticians, being of no practical value to them. In the 18th century some use was made of it in numerical work, but it was not until the second half of the 19th century that it became popular in elementary arithmetic. On account of its resemblance to x it was not well adapted to use in algebra, and so the dot came to be employed, as in 2 . 3 = 6 (Europe) as well as in America. This device seems to have been suggested by the old Florentine multiplication tables; at any rate Adriaen Vlacq (c1600-1667), the Dutch computer (1628), used it in some of his work, thus:

factores---- 7 . 17


although not as a real symbol of operation. In his text he uses a rhetorical form, thus; "3041 per 10002 factus erit 30416082."

Christopher Clavius (1537-1612), a Jesuit of Rome, wrote in 1583 using the idea of a dot for multiplication, as in 3/5.4/7 for 3/5 X 4/7; and Thomas Harriot (1560-1621) in a posthumous work of 1631 actually used the symbol in a case like = 2a cubed. The first writer of prominence to employ the dot in a general way for algebraic multiplication seems to


On the topic of mathematical symbols.....

"Every meaningful mathematical statement can also be expressed in plain language. Many plain-language statements of mathematical expressions would fill several pages, while to express them in mathematical notation might take as little as one line. One of the ways to achieve this remarkable compression is to use symbols to stand for statements, instructions and so on."

Lancelot Hogben

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