What is Algebra? (Historically)

At the beginning of my second year, I thought it would be interesting to explore definitions of Algebra in historical texts: both how the author described Algebra and also what topics were included in Algebra. Unfortunately, I have not had time to do a more comprehensive search, but I found five British and American texts that were interesting and spanned about a century. (I chose to focus on these countries because the texts are readily available in digital form and also because I am limited in my language).

Please see below for additional details, but I list their definitions here:

  • Hutton (1831) wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”
  • Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”
  • Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”
  • Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”
  • Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”

Charles Hutton

Charles Hutton had been a professor of Mathematics for the Royal Military Academy. You can see additional details about his life here: https://en.wikipedia.org/wiki/Charles_Hutton

Slide2

In his “Course of Mathematics,” he included many Algebra topics, starting from basic operations, moving through roots and powers, to arithmetic and geometric progression, to some equations, and finishing with interest formulas.

Slide3

Because his definition of Algebra depended on his definition of Mathematics, I include both here. He wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”

Slide4

Frances H. Smith, A.M.

Again, find additional details at https://en.wikipedia.org/wiki/Francis_Henney_Smith

Slide5

Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”

Slide6

Charles Davies, LL.D.

Find additional information about Charles Davies at the MAA website.

Slide7

Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”

Slide8

J. M. Taylor, A.M.

James Morford Taylor served 57 years at Colgate University as a professor of Mathematics.

Slide9

Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”

Slide10

George A. Wentworth

Slide11Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”Slide12

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