The work, the first volume of which is now offered to the public, was designed in the first instance to be a second edition of a Treatise of Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.
I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now the press, will embrace the principle of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as many make them component parts of one uniform and connected system.
In the preface to my former Treatise I have given a general exposition of my reasons for distinguishing arithmetical from symbolical algebra, and of my views of the just relations which their principles bear to each other, through I did not then consider it necessary to separate the exposition of one science altogether from the other. A more matured consideration of the subject, however, has convinced me of the expediency of this separation; for it is extremely difficult, when the two sciences are treated simultaneously, to keep their principles and results apart from each other, and to obviate the confusion, obscurity
Algebra become more and more abstracted over time, starting out as the use of letters to replace numbers, and slowly losing the geometric, and procedural arithmetic. As mathematicians tried to work out the laws of how numbers worked in Algebra, Algebra became more about the logical relations between objects.
"Most of the symbolism found in our elementary algebra textbooks is less than 400 years old. Francois Viete (1540-1603) was the greatest French Mathematician of the sixteenth century. His most famous work is his In artem analyticem isagoge (1591) in which he developed a great deal of algebraic symbolism. He used vowels for unknown quantities and consonants for known ones. Viete also qualified coefficients of polynomials and used + and - symbols. He had no symbol for equals."[Wilson]Peacock and DeMorgan, [i]n "trying to overcome the last reservations about the legitimacy of the negative and complex numbers, these mathematicians suggested that algebra be conceived as a purely formal, symbolic language, irrespective of the nature of the objects whose laws of combination it stipulated...The British tradition of symbolic algebra was instrumental in shifting the focus of algebra from the direct study of objects (numbers, polynomials, and the like) to the study of operations among abstract objects." [Encyclopedia Britannica]