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 authors were brought to the doors of all classes in the cheapest variety of forms. In consequence of the Civil War, the high price of labour, and the restrictive duties laid on in order to protect native industry, coupled with the frequent intercourse with England, a great change took place, and American publishers and booksellers, while there was still no international copyright, made liberal offers for early sheets of new publications. Boston, New York and Philadelphia still retained their old supremacy as bookselling centres. Meanwhile, the distinct publishing business also grew, till gradually the conditions of business became assimilated to those of Europe.

In the course of the 16th and 17th centuries the Low Countries for a time became the chief centre of the bookselling world, and many of the finest folios and quartos in our libraries bear the names of Jansen, Blauw or Plantin, with the imprint of Amsterdam, Utrecht, Leiden or Antwerp, while the Elzevirs besides other works produced their charming little pocket classics. The southern towns of Douai and St Omer at the same time furnished polemical works in English.

 BOOLE, GEORGE (1815–1864), English logician and mathematician, was born in Lincoln on the 2nd of November 1815. His father was a tradesman of limited means, but of studious character and active mind. Being especially interested in mathematical science, the father gave his son his first lessons; but the extraordinary mathematical powers of George Boole did not manifest themselves in early life. At first his favourite subject was classics. Not until the age of seventeen did he attack the higher mathematics, and his progress was much retarded by the want of efficient help. When about sixteen years of age he became assistant-master in a private school at Doncaster, and he maintained himself to the end of his life in one grade or other of the scholastic profession. Few distinguished men, indeed, have had a less eventful life. Almost the only changes which can be called events are his successful establishment of a school at Lincoln, its removal to Waddington, his appointment in 1849 as professor of mathematics in the Queen’s College at Cork, and his marriage in 1855 to Miss Mary Everest, who, as Mrs Boole, afterwards wrote several useful educational works on her husband’s principles.

To the public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications which have become standard works. His earliest published paper was one upon the “Theory of Analytical Transformations,” printed in the Cambridge Mathematical Journal for 1839, and it led to a friendship between Boole and D. F. Gregory, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Boole’s memoirs and detached papers, both on logical and mathematical topics, will be found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on Differential Equations, edited by Isaac Todhunter. To the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, Boole contributed in all twenty-two articles. In the third and fourth series of the Philosophical Magazine will be found sixteen papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy, in the Bulletin de l’Académie de St-Pétersbourg for 1862 (under the name G. Boldt, vol. iv. pp. 198-215), and in Crelle’s Journal. To these lists should be added a paper on the mathematical basis of logic, published in the Mechanic’s Magazine for 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications.

Only two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work. These treatises are valuable contributions to the important branches of mathematics in question, and Boole, in composing them, seems to have combined elementary exposition with the profound investigation of the philosophy of the subject in a manner hardly admitting of improvement. To a certain extent these works embody the more important discoveries of their author. In the 16th and 17th chapters of the Differential Equations we find, for instance, a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole’s chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivalled skill and power in tracing out these results.

During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition; and part of his last vacation was spent in the libraries of the Royal Society and the British Museum. But this new edition was never completed. Even the manuscripts left at his death were so incomplete that Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and wisely printed them, in 1865, in a supplementary volume.

With the exception of Augustus de Morgan, Boole was probably the first English mathematician since the time of John Wallis who had also written upon logic. His novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. Speculations concerning a calculus of reasoning had at different times occupied Boole’s thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities (1854), should alone be considered as containing a mature statement of his views. Nevertheless, there is a charm of originality about his earlier logical work which no competent reader can fail to appreciate. He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy between the symbols of algebra and those which can be made, in his opinion, to represent logical forms and syllogisms, that we can hardly help saying that logic is mathematics restricted to the two quantities, 0 and 1. By unity Boole denoted the universe of thinkable objects; literal symbols, such as x, y, z, v, u, &c., were used with the elective meaning attaching to common adjectives and substantives. Thus, if x = horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraical symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, 1−x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1−x) (1−y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.

Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, which formed