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MODERN MATHEMATICS ALGEBRA 29 

Arabic phrase which means restoration and reduction the 
first word referring to the fact that the same magnitude may 
be added to or subtracted from both sides of an equation, and 
the last word meaning the process of simplification. The 
science of algebra was brought among the Arabs by Mohammed 
ben Musa (Mahomet the son of Moses), better known as 
Alkarismi, in a work written about 830 A.D., and was certainly 
derived by him from the Hindoos. The algebra of Alkarismi 
holds a most important place in the history of mathematics, 
for we may say that the subsequent Arab and the early 
medieval works on algebra were founded on it, and also that 
through it the Arabic or Indian system of decimal numera- 
tion was introduced into the West. It seems that 
the Arabs were quick to appreciate the work of others 
notably of the Greek masters and of the Hindoo mathe- 
maticians but, like the ancient Chinese and Egyptians, 
they did not systematically develop a subject to any con- 
siderable extent. 

Algebra was introduced into Italy in 1202 by Leonardo of 
Pisa (about 1175-1230) in a work based on Alkarismi's 
treatise, and into England by Robert Record (about 1510- 
1558) in a book called the Whetstone of Witte published in 
1557. Improvements in the method or notations of algebra 
were made by Record, Albert Girard (1595-1632), Thomas 
Harriot (1560-1621), Descartes, and many others. 

In arithmetic we use symbols of number. A symbol is any 
sign for a quantity which is not the quantity itself. If a man 
counted his sheep by pebbles, the pebbles would be symbols of 
the sheep. At the present day, when most of us can read and 
write, we have acquired the convenient habit of using marks on 
paper, 1, 2, 3, 4, and so on, instead of such things as pebbles. 
Our 1 + 1 is abbreviated into 2, 2 + 1 is abbreviated into 3, 
3 + 1 into 4, and so on. When " 1," " 2," " 3," &c., are used 
to abbreviate, rather improperly, " 1 mile," "2 miles," "3 miles," 
&c., for instance, they are called signs for concrete numbers. 
But when we shake off all idea of " 1," " 2," &c., meaning one, 
two, &c., of anything in particular, as when we say, " six and 
four make ten," then the numbers are called abstract numbers. 
To the latter the learner is first introduced in treatises on