David Eugene Smith: Algebra

David Eugene Smith was American mathematician. Explore interesting quotes on algebra.
David Eugene Smith: 66   quotes 0   likes

“In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science.”

David Eugene Smith

This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.
History of Mathematics (1923) Vol.1

History , Literature , Mathematics , Appearance

“The first epoch-making algebra to appear in print was the Ars Magna of Cardan”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p.384
Context: The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra.

Appearance

“Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p. 392

“The first noteworthy attempt to write an algebra in England was made by Robert Recorde, whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde.
The first Italian textbook to bear the title of algebra was Bombelli's work of 1572.
By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. …this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675).
So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p. 386, Ch. 6: Algebra,-->

School , Time , Perfection

“It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B. C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position… and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. …Sun-tzï solved such problems by analysis and was content with a single result…
The Chinese certainly knew how to solve quadratics as early as the 1st century B. C., and rules given even as early as the K'iu-ch'ang Suan-shu… involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should…
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. …numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner”

David Eugene Smith

1819
Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Way , Question , Problem , Appearance

“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

David Eugene Smith

David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906)

Mathematics , Study , Universe , Time

“With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra…”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Science

“The first writer on algebra whose works have come down to us is Ahmes. He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; Brahmagupta, whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic… Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)… extends the work through quadratic equations.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)… which characterized in a careless way the knowledge… thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations.
The noteworthy work… and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence.
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work… to appear in print was the General Trattato of Tartaglia…”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p. 384; Ch. 6: Algebra

Way , Knowledge , Appearance , Science

“When we speak of the early history of algebra it is necessary to consider… the meaning of the term. If… we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the the science was known about 1800 B. C., and probably still earlier.<”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra, p. 378

History , Problem , Science