Start of Symbolism

Students often ask why there are so many symbols in algebra. There is the plus sign, minus sign, division and multiplication symbols, parentheses, brackets, exponents, logarithms, letters for variables, and the list goes on. Surprisingly most of these symbols have only been around for a little more than four hundred years.

In 1842 G.H.F. Nesselman categorized the historical development of algebraic symbolism into three stages, rhetorical, syncopated, and symbolic algebra. Rhetorical algebra writes the solution of a problem without any abbreviations or symbols. Syncopated algebra uses shorthand abbreviations for some of the more frequently used operations, quantities, and relations. Symbolic algebra writes the solutions to problems in a type of mathematical shorthand made up of symbols, some with less than obvious connections to the ideas and things they represent.

Before the time of Diophantus of Alexandria (c. A.D. 250), all algebra was rhetorical. Diophantus introduced some symbolism to Greek algebra, but the wordy rhetorical algebra endured in most of the world for many centuries, except in India. Not until the fifteenth century did some type of syncopation begin to appear in Western Europe, and in the sixteenth century symbolic algebra started to be used there. Its development was slow and not wide spread until around the middle of the seventeenth century.

Below are some examples from the Palantine Anthology, compiled about A.D. 500 by Metrodorus. These are quite similar to problems found in algebra texts today. They are referred to as "distribution," "work," and "mixture" problems. Try to write a solution to these in purely rhetorical algebra, no symbols for numbers or anything. Everything must be written out in words.

1. [a "distribution" problem] How many apples are needed if four persons of six receive one-third, one-eighth, one-fourth, and one-fifth, respectively, of the total number, while the fifth receives ten apples, and one apple remains left for the sixth person?

2. [a "work" problem] Brickmaker, I am in a hurry to erect this house. Today is cloudless, and I do not require many more bricks, for I have all I want but three hundred. Thou alone in one day couldst make as many, but thy son left off working when he had finished two hundred, and thy son-in-law when he had made two hundred and fifty. Working all together, in how many days can you make these?

3. [a "mixture" problem] Make a crown of gold, copper, tin, and iron weighing 60 minae: gold and copper shall be two thirds of it; gold and tin three fourths of it; and gold and iron three fifths of it. Find the weights of gold, copper, tin, and iron required.

Unknown	Unknown Squared
Unknown Cubed	Unknown to the 4th
Unknown to the 5th	Unknown to the 6th
Minus	Constant

As stated earlier, Diophantus and the Hindus were some of the first to use some type of shorthand or symbols in their algebra. In Diophantus's Arithmetica there are abbreviations for the unknown up through the sixth power, subtraction, equality, and reciprocals. Some of these are shown in the table on the right. The unknown looks looks like the Greek final sigma. The "unknown squared" is denoted by the first two letter of the Greek word dunamis for "power." "Unknown cubed" was the first two letters of the Greek word kubos for "cube." The succeeding powers of the unknown are listed in the table. Diophantus's "minus" symbol looks like a compound of an upside down V and an I, letters in the Greek word leipis for "lacking." Addition was exressed by juxtaposition. All negative terms in an expression are gathered together following the minus symbol. Numerical coefficients follow the power of the unknown and are symobolized by the corresponding alphabetic Greek numeral. A capital M with a small circle placed directly above it is used with the appropriate numerical coefficient to represent a constant term. It is an abbreviation of the Greek word monades for "units."

Here is an example:

would be

x3 + 2x - 3.

The Hindus syncopation was quite similar to that of Diophantus. Brahmagupta (c. A.D. 628) invented his own system of abbreviations. Additions was indicated by juxtaposition. Subtraction indicated by a dot over the subtrhed. Multiplication was shown by writing bha (the first syllable of bhavita, "the product") after the factors, division by writing the divisor under the dividend, square root by writing ka (from the word karana, "irrational") before the quantity. The unknown was ya [note a bar above the a, and also on some of the a's in the word to follow] (from yavattavat, "so much as"). Known integers were prefixed by ru [bar on u] (from rupa[bar on u], "the absolute number"). Any other unknowns were defined by the initial syllables of words for different colors. An example is: