## Reviews of the 1940 edition of Peacock's *Algebra*

In 1940 George Peacock's

*Treatise on Algebra*was republished in two volumes: Volume I.*Arithmetical Algebra*; Volume II.*On Symbolical Algebra and Its Applications to the Geometry of Position*. The first volume was a reprint of the 1842 edition, while the second volume was a reprint of the 1845 edition. It is remarkable that Peacock's book would be republished 100 years after it was first published. Below we present extracts from two reviews of the 1940 edition. The reasons for republishing the work are explained in these extracts:**1. Review by: May Margaret Beenken.**

*National Mathematics Magazine*

**16**(5) (1942), 269-270.

Peacock's

*Treatise on Algebra*enjoys the distinction of being named by Mortimer J Adler in*How to Read a Book*in his chapter on*The Great Books*. It is mentioned along with Hilbert's*Foundations of Geometry*and Dedekind's*Theory of Numbers*as a book which even the layman might dare to read. The book is outstanding for its clarity of exposition; for indeed Peacock was a teacher of the first class, a distinction rare among British contemporary mathematicians. Peacock did much to reform the teaching of algebra and to place it on a scientific basis through his*Treatise on Algebra*, both the earlier volume published in 1830 and the present treatise of 1842 and 1845. The significance of this work may be further appreciated when we realize that it was written at a time when books appeared in Britain actually protesting the use of negative numbers. What appears today quite commonplace to one who is not familiar with any algebra except the elementary type, was a novelty when Peacock published his work. Indeed we owe much to Peacock for our modem conception of algebra. It was he who developed algebra as an abstract system of symbols to be combined according to operations that conform with pre-assigned postulates, and dispensed with the then current idea that the symbols of algebra had to represent the numbers of ordinary arithmetic. In his attempt to place algebra on a strictly logical basis, Peacock divided the science of algebra into two parts, Arithmetical Algebra, treated in Volume I, and Symbolical Algebra in Volume II. He described arithmetical algebra as follows: "In arithmetical algebra, we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions (whether expressed or understood) as in common arithmetic: The signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so, in case they were replaced by digital numbers: thus in expressions, such as*a*+*b*, we must suppose*a*and*b*to be quantities of the same kind: in others, like*a*-*b*, we must suppose*a*greater than*b*, and therefore homogeneous with it; in products and quotients, like ab and*a*/*b*, we must suppose the multiplier and divisor to be abstract numbers: all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations, must be rejected as impossible, or as foreign to the science." ... The terminology employed throughout the treatise is typical of the age in which it was written and in many cases differs from that employed in modem textbooks on college algebra and theory of equations. But the exposition is everywhere clear and the book possesses a mathematical elegance more characteristic of an original work than of a mere textbook.**2. Review by: Louis Charles Karpinski.**

*Amer. Math. Monthly*

**49**(4) (1942), 254-255.

The re-publication today of a textbook on algebra, with a total of 880 pages, originally issued in London about a century ago (vol. I, 1842; vol. II, 1845) is an amazing achievement which inspires somewhat critical examination of the various circumstances which have made this possible. The suggestion of the publication was made by the authorities of St John's College in Annapolis, Maryland, where the book is used as a text. Not only in mathematics but in other fields, also, this college has maintained the desirability of the serious use as textbooks by the young students of the college of historical classics of science. The publication of Peacock's

*Treatise*was effected by the collaboration of St John's College with*Scripta Mathematica*. The recent changes in higher algebra might seem to indicate that as preparation for algebra so old a text would be useless. However, much more than half of this work, including much arithmetical computation, corresponds precisely to material presented in the elementary textbooks and in the algebra texts used widely in colleges for the freshman work in algebra, and considerable material given in a second course in college algebra for juniors. In the first place, Peacock presents not only a large amount of work on the theory of equations but he includes what amounts to a systematic course in plane trigonometry. Peacock believes in numerical problems and computation to an extent not attempted in American texts. A student who masters these two volumes will have adequate preparation in trigonometry and algebra, including series, for the great body of elementary applications to physics. There are many teachers of mathematics who believe that this material represents much that cannot be by-passed in any sound preparation for the use of mathematics in science. The success of this text in its day and its feasibility as a text even today is dependent upon the author's brilliant and harmonious historical development of the foundations of algebra. ... Such a work as that of Peacock helps us to understand the developments of Hamilton and Boole and that possibly some of the modern abstractions have been brewing more than a hundred years. Pedagogically too, the insistence on numerical illustration and computation seems to have a message today for many who omit this work to advance into abstract fields before the pupil is able to master the groundwork.