代数的缩写The syncopation of algebra

85° The syncopation of algebra. In 1842, G. H. F. Nesselmann characterized three stages in the historical development of algebraic notation. First we have rhetorical algebra, in which the solution of a problem is written, without abbreviations or symbols, as a pure prose argument. Then comes syncopated algebra, in which stenographic abbreviations are adopted for some of the more frequently recurring quantities and operations. Finally, as the last stage, we have symbolic algebra, in which solutions largely appear in a mathematical shorthand made up of symbols having little apparent connection with the entities they represent. It is fairly accurate to say that algebra prior to the time of Diophantus was rhetorical. One of Diophantus's major contributions to mathematics was the syncopation of Greek algebra. Rhetorical algebra, however, persisted pretty generally in the rest of the world, with the exception of India, for many hundreds of years. Specifically, in western Europe, most algebra remained rhetorical until the fifteenth century. Symbolic algebra made its first appearance in western Europe in the sixteenth century, but did not become prevalent until the middle of the seventeenth century. It is not often realized that much of the symbolism of our elementary algebra textbooks is not more than three hundred years old.

Diophantus had abbreviations for the unknown, powers of the unknown up through the sixth, subtraction, equality, and reciprocals.

Our word "arithmetic" comes from the Greek word arithmetike, a compound of the words arithmos for "number" and techne for "science." It has been rather convincingly pointed out by T. L. Heath that Diophantus's symbol for the unknown was probably derived by merging the first two Greek letters, α and ρ, of the word arithmos. This came, in time, to look like the Greek final sigma ζ. While there is doubt about this, the meaning of the notation for powers of the unknown is quite clear. Thus "unknown squared" is denoted by Δ^Y, the first two letters of the Greek word dunamis (ΔΥΝΑΜΙΣ) for "power." Again, "unknown cubed" is denoted by K^Y, the first two letters of the Greek word kubos (ΚΥΒΟΣ) for "cube." Explanations are easily furnished for the succeeding powers of the unknown, Δ^YΔ (square-square), ΔK^Y (square-cube), and K^YK (cube-cube). Diophantus's symbol for "minus" looks like an inverted V with the angle bisector drawn in. This has been explained as a compound of Λ and I, letters in the Greek word leipsis (ΛΕΙΨΙΣ) for "lacking." All negative terms in an expression are gathered together and preceded by the minus symbol. Addition is indicated by juxtaposition, and the coefficient of any power of the unknown is represented by the alphabetic Greek numeral following the power symbol. If there is a constant term then Ṁ, an abbreviation of the Greek word monades (ΜΟΝΑΔΕΣ) for "units," is used, with the appropriate number coefficient. Thus x³ + 13x² + 5x and x³ − 5x² + 8x − 1 would appear as

K^YαΔ^Yιγςε and K^Yαϛη∧Δ^YεṀα,

which can be read literally as

unknown cubed 1, unknown squared 13, unknown 5

and

(unknown cubed 1, unknown 8) minus (unknown squared 5, units 1).

It is thus that rhetorical algebra became syncopated algebra.