THE immense circulation of WALKINGAME'S TUTOR'S ASSISTANT, even in its original form, is sufficiently evinced by the very extensive and uniformly increasing demand which the Proprietors of the present Edition have for many years experienced.

To advance the utility of a work held in such high estimation among Conductors of Schools; by simplifying the Rules, correcting and modernizing the antiquated phraseology, supplying deficiencies where there was a paucity of Examples, and incorporating with its original matter such emendations and additions as appear to be called for by the present improved state of Arithmetical Science; will, it is presumed, be rendering an acceptable service to the Public.

Amongst the various improvements introduced in this Edition, may be enumerated, a more intelligible elucidation of the system of Notation; of Direct, Inverse, and Compound Proportion, Practice, Interest, Progression, &c.; more perspicuous illustrations of the theory and practice of Vulgar and Decimal Fractions, Evolution, Duodecimals, &c.; the substitution of the new Arithmetical and Commercial Tables; the insertion of many additional Examples (particularly in the elementary Rules) adapted to exercise and improve the judgment of the Learner; also of Rules for the particular cases in Profit and Loss, of Involution, of Theorems for the solution of all the possible cases in Arithmetical and Geometrical Progression, Superficial Mensuration, a number of useful Supplemental Questions, and a Compendium of Book-Keeping.

Such are the attempts that have been made to enhance the real worth of this popular Treatise on Arithmetic. How far the intention may have been judiciously or successfully executed, must be left to a candid Public to determine.

With a confident reliance, however, on the favourable consideration of those whose judgment and experience most essentially qualify them to discriminate between realities and specious pretensions to improvement, and duly to appreciate the difficulties of such an undertaking; the work is respectfully submitted to a trial before the tribunal of its legitimate judges, with an anxious and hopeful anticipation of obtaining their verdict of approval.

Derby, February 7, 1827.

Advertisement to the Sixth Edition.

WHEN the Editor first undertook the task of modernizing and improving this work, with an anxious desire of rendering it more commensurate with the progress of scientific information, and better adapted to the present improved systems of instruction, thereby to facilitate the arduous labours of the Teacher in the communication,

and of the Pupil in the acquisition of a branch of knowledge so useful and indispensable; he was limited materially in the execution of his views by the necessity of conforming the whole of the arrangement as well as a considerable part of the methods of operation that were susceptible of improvement, to those of FALCONAR'S KEY to the original work. The same necessity compelled him to retain every example contained in the old editions, however obsolete, useless, or absurd the subject might have become, from the innovations which time invariably introduces in the management of commercial transactions. Hence the student was subjected to the double inconvenience of referring, for some important parts of the additional matter, to the Addenda in the Arithmetic, and to an Appendix to the Key. This impediment to improvement is now removed: for having constructed a new KEY with correspondent improvements, he has been enabled to complete his intentions by the regular incorporation of the newly introduced matter, by an arrangement more rational and more consistent with the modern practice of instruction, by the expunction of some useless subjects, and the substitution of others of more real utility.

Besides the emendations and additions enumerated in the preceding Preface, the worked examples under each Rule are now exhibited in a more convenient form, considerably amplified, and illustrated by copious explanatory notes;-the theory and practice of Circulating Decimals are comprehensively and clearly explained;considerable improvements have been effected in that part which treats of the Doctrine of Compound Interest and Annuities, the Tables of which have been greatly extended; and in order to correct the errors that abound in the Tables of this description contained in several scientific publications which have been collated, the accuracy of these has been verified by actual calculation. Some useful additions have also been made in the Mensuration and Bookkeeping.

Three years have now elapsed since the IMPROVED TUTOR'S ASSISTANT was first issued from the press. That it has been greatly approved, has been manifested by the most unequivocal and satisfactory proof, the sale of several large impressions. Under such auspicious symptoms of encouragement, therefore, the Editor feels an increased confidence in the prospect of obtaining the sanction of an enlightened public.

Derby, February 1, 1830.

THE public will, no doubt, be surprised to find there is another attempt made to publish a book of ARITHMETIC, when there are such numbers already extant on the same subject, and several of them that have so lately made their appearance in the world; but I flatter myself, that the following reasons which induced me to compile it, the method, and the conciseness of the Rules, which are laid down in so plain and familiar a manner, will have some weight towards its having a favourable reception.

Having some time ago drawn up a set of Rules and proper Questions, with their Answers annexed, for the use of my own school, and divided them into several books, as well for more ease to myself, as the readier improvement of my scholars, I found them, by experience, of infinite use; for when a master takes upon him that laborious (though unnecessary) method of writing out the Rules and Questions in the children's books, he must either be toiling and slaving himself after the fatigue of the school is over, to get ready the books for the next day, or else must lose that time which would be much better spent in instructing and opening the minds of his pupils. There was, however, still an inconvenience which prevented them from giving me the satisfaction I at first expected; i. e. where there are several boys in a class, some one or other must wait till the boy who first has the book finishes the writing of those rules or questions he wants, which detains the others from making that progress they otherwise might, had they a proper book of Rules and Examples for each boy; to remedy which, I was prompted to compile one, in order to have it printed, which might not only be of use in my own school, but in other schools, where the instructers wish their scholars to make a quick progress. It will also be of great use to such persons as have acquired some knowledge of numbers at school, to make them the more perfect; likewise to such as have completed themselves therein, it will prove, after an impartial perusal, on account of its great variety and brevity, a most agreeable and entertaining Exercise Book. I shall not presume to say any thing more in favour of this work, but beg leave to refer the unprejudiced reader to the remark of a certain Author,* concerning compositions of this nature. His words are as follow:

"And now, after all, it is possible that some who like best to tread the old beaten path, and to toil at their business, when they may do it with pleasure, may start an objection against the use of this wellintended Assistant, because the course of Arithmetic is always the same; and therefore say, that some boys, lazily inclined, when they see another at work upon the same question, will be apt to make his operation pass for their own. But these little forgeries are soon detected by the diligence of the Tutor: therefore, as different questions to different boys do not in the least promote their improvement, so neither do the same questions impede it. Neither is it in the power of any master (in the course of his business) now full of spirits so

* Dilworth.

ever he may be, to frame new examples at pleasure in any Rule; but the same question will frequently occur in the same Rule, notwithstanding his greatest care and skill to the contrary.

"It may also be farther objected, that to teach by a printed book is an argument of ignorance and incapacity; which is no less trifling than the former. He, indeed (if such a one there be) who is afraid his scholars will improve too fast, will, undoubtedly, decry this method: but that master's ignorance can never be brought in question, who can begin and end it readily; and, most certainly, that scholar's non-improvement can be as little questioned, who makes a much greater progress by this, than by the common method."

To enter into a long detail of every Rule would tire the reader, and swell the preface to an unusual length; I shall, therefore, unly give a general idea of the method of proceeding, and leave the rest to speak for itself; which, I hope, the reader will find to answer the title, and the recommendation given it. As to the Rules, they follow in the same manner as the table of contents specifies, and in much the same order as they are generally taught in schools. I have gone through the four fundamental Rules in Integers first, before those of several denominations: in order that they being well understood, the latter will be performed with much more ease and despatch, according to the rules shown, than by the customary method of dotting. In Multiplication I have shown both the beauty and use of that excellent Rule, in resolving most Questions that occur in merchandising; and have prefixed to Reduction several Bills of Parcels, which are applicable to real business. In working Interest by Decimals, I have added tables to the Rules, for the more ready calculating of Annuities, &c. and have not only shown the use, but the method of making them: likewise a Table calculated for finding the Interest of Money for any number of days, at any rate per cent. by Multiplication and Addition only; which may also be applied to the calculation of Incomes, Salaries, or Wages, for any number of days; and I may venture to say, I have gone through the whole with so much plainness and perspicuity, that there is none better extant.

I have nothing farther to add, but a return of my sincere thanks, to all those gentlemen, schoolmasters, and others, whose kind approbation and encouragement have now established the use of this book in almost every school of eminence throughout the kingdom: but I think my gratitude more especially due to those who have favoured me with their remarks; though I must still beg of every candid and judicious reader, that if he should, by chance, find a transposition of a letter, or a false figure, to excuse it; for, notwithstanding there has been great care taken in correcting, yet errors of the press will inevitably creep in; and some may also have slipped my observation : in either of which cases, the admonition of a good-natured reader will be very acceptable to his

much obliged,

and most obedient humble servant,

F. WALKINGAME.