1. Cash Book.—Every entry is posted from this book, but not all to the ledger as in double entry " Charges " being posted to the day book. It is not journalised, and is in itself a ledger, as it contains the bank account, and reports its own cash balance. On the other hand, it is unlike the "Cash" of single entry, be cause every entry is posted somewhere, whereas by the latter system only personal accounts are carried to the ledger.
2. Day Book.—This book also exhibits a marked difference be tween the journal of double entry and the day book of single entry. The journal is simply a posting medium, and when its use is served is almost valueless. The single entry day book, on the other hand, is only a posting medium to a certain extent, as it does not embrace all transactions ; but in this system the day book unites the characteristics of journal and ledger, and also becomes in itself a profit and loss account, as by deducting the amount of charges from the amount of the business fees (say for solicitors books) the profit on said business is shown.
3. The Ledgers. These books also lose their completeness under the mixed method. It has already been shown that in double entry every amount must appear in the ledger, and in single entry that only personal accounts are posted in it. By this system not only are all personal accounts included, but those applicable to "Capital," to "Banks," "Bills," &c.; whilst, on the other hand, such accounts as "Profit and Loss," "Charges," and "Cash" are excluded.
It would be out of place here to dwell on the many intricacies of this subject, or on the difficulties which are constantly presenting themselves even to the most prac tical men. With a thorough knowledge of the art, how ever, and that patience and perseverance so essential to the calling of a book-keeper, the gravest impediments are overcome, and everything becomes simple and plain. Our sole object having been to show the utility of book-keeping as a science, and the peculiar features of existing systems with their advantages and disadvantages, it is unnecessary to enter more minutely into details by describing subsidiary books or forms of accounts, as these are only so many materials out of which the fabric of book-keeping is erected, and can be seen in any counting-house or mercantile esta blishment where regular systems are adopted.
BOOLE, George, one of the most original logicians and mathematicians whom England has produced, -was born in Lincoln on the 2d of November 1815. His father was a tradesman of limited means, but of studious char acter and active mind. Being especially interested in mathematical science the father gave his son early instruc tion in the rudiments of the science he was so greatly to advance ; but it is remarkable that the extraordinary mathematical powers of George Boole did not manifest them selves in early life. The classical languages formed at first the favourite subject of his studies. Not until the age of seventeen years 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 appoint ment in 1849 as professor of mathematics in the Queen s College at Cork, and his marriage in 1855 to Miss Mary Everest.
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 1 844. 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 Iloyal Society, and in the supplementary volume on Differential Equations, edited by Mr Todhunter. To the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, Boole con tributed in all twenty-two articles. In the third and fourth series of the Philosophical Magazine will be found sixteen papers. The Iloyal 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 I Academic de /St Pctersbourg 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 have become the standard text-books on the important branches of mathe matics 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 pro ducing a second edition of his Differential Eqiiations much more complete than the first edition ; and part of his last vacation was spent in arduous study in the libraries of the Iloyal Society and the British Museum for the purpose of acquiring a complete knowledge of the less accessible original memoirs on the subject. It must be always a matter of regret that this new edition was never completed. Even the manuscripts left at his death were so incomplete that Mr 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.
pure mathematics, his writings on logic may be considered as still more original. With the exception of De Morgan, he was probably the first English mathematician since the time of Wallis who had also written upon logic ; and his vrholly novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. From the preface to his Mathematical Analysis of Logic, printed as a separate tract in 1847, we learn that speculations con cerning a calculus of reasoning had at different times occupied Boole s thoughts, but it was not till the spring of 1847 that a memorable logical controversy led him to put his ideas into a definite form. Boole afterwards regarded
this pamphlet as a hasty and imperfect exposition of his