lieve, it is confessed by all that it is a task too hard for children to be made complete masters of arithmetic ; and therefore the best way of instructing them in it, is most certainly, first to give them a general notion of it, in the and next to enlarge upon it afterwards if there be time: otherwise it must be done by themselves, as their increase in years and growth in understanding will periit. * For arithmetic is the more valuable, as it is the more exact, easy, and short; and the art lies in giving as few rules as possible, and clearly explaining them, and not confounding principles toges ther, and then diversiying them into several rules, when they are built me reason, which has not only inade arithmetic seemn difficult of access, but has hindered many from being accomptants." To enter into a detail of the following particulars, would be tedious, and well this preface beyond its just limits; hut that the kind reader may not be wholly at a loss, I shall beg leave to speak as follows, viz. 1. That the whole is divided into five parts, as the title page expresses it. 2. That the rules and examples are contrived in the plainest manner, and the whole put in such an easy method as is no where else extant. 3. I have omitted reduction of foreign coins, partly because, all those tables which I have met with, which shew the value of foreign coins in English money, are very erroneous, but principally because all such questions as relate to the turning of the money of one country into that of another, are much better answered under the head of exchange. For the value of foreign species, (such I mean as relate only to exchange) both of gold and silver, in every country is unsettled, and therefore such coins are subject to vary in their prices, as the merchants find an opportunity to profit by them. Hence proceed the various courses of exchange; and from them again, the particular worth of any quantity of foreign coin in English money, which is sometimes inore, sometimes less, according as the course of exchange runs at that time, when such foreign coins become due. Add to this the agio or advance money usually paid abroad on the changing current money into exchange or bank money, which is two, three, or more per cent, in payment, according to what the exchange or bank money is worth more than the current money, and this cannot be done otherwise than by the rule of three. 4. In interest, &c. hy decimals, I have followed Mr. Ward's method, by which means the rule is drawn into a much narrower compass; and appears inore beautiful to the eye, than in words at length. 5. In all places where it could be done conveniently, I have given directions for varying the examples by way of proof ; because it not only discovers the reason of the operation, but at the same time both produces a new question, and prove the old one. And sure I am, that the varying the question, when it may be done under the same rule, contributes very much towards a thorough understanding of it, and making a good accomptant, as every one's experience will teach him. 6. I have thrown the subject of the following pages into a catechetical form, that they may be more instructive ; for children can betterjudge of the force of an answer, than following reason through a chain of consequences. Hence also it proves a very good examining book; for at any time, in what place soever the scholar appears to be defective, he can inmediately he put back to that place again without the formal way of beginning every thing apew. 7. lo order to make the progress still quicker, every example to be wrought hath its answer arnexed to it; so that they who do not chuse to have every operation proved by varying the question, may know without it whether the work he right or not. 8. Concerning contractions in numbers, which some are very fond of, I have said very little, and my reason is this : contractions are no farther valnable than they are useful ; hence if in order to lessen the number of figures in an operation, there is not only more time spent than in the ordinary way, but those contractions are also more liable to error, such contractions ought to he rejected. And now, after all, it is possible that some who like best to tread the old beaten path, and to sweat at their business when they might do it with pleasure, may start an objection against the use of this well intended assistant; be. cause the course of arithinetic 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 this 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 hinder it. Neither is it in the power of any master, (in the course of his business) how full of spirits soever he be, to frame new questions at pleasure in any rule, but the same questions will frequently occur in the saine rule, notwithstanding his greatest care and skill to the contrary. • It may also be further 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 any such there be) who is afraid his scholars will improve too fast, will undoubtedly decry this inethod. But that master's ig. norance can never be brought in question, who can hegin 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 he possibly can by the common method. As to the order of the rules, I can hadly find two masters follow it alike; some liking best to teach that rule first, which another thinks more convenient to teach afterward; while a third looks upon it as a matter quite indifferent among some rules, which he teaches first. But this need be no hindrance to the use of this book. For however the rules are placed here, every man may turn to that rule first, which he likes should be taught first, and if a master has a mind to teach vulgar fractions immediately after reduction of whole numbers, as some do, he may do it as easily as in the order they now lie. To the eleventh edition, and which is continued in this, I have added duodecimals, commonly called cross multiplication ; wherein I have largely treated of twat sort of arithmetic, in every branch; shewing how the same may be proved by varying the operations ; by whole numbers, by vulgar fractions and by decimals; and lastly by a particular sort of division, wherein the divisor, dividend, and quotient are each of them of several denominations, just as the factors and products are in multiplication, without reducing thein into the lowest térm or denomination mentioned. And as Duodecimals by all the writers that I have seen, except Mr. Hawney, have only been superficially treated of, I think, I may venture to say, without any breach: of modesty, that this is the completest piece of that kind extant. As a further improvement of this compendium, I have considerably enlarged the rule of exchange, and among others, have given a variety of examples of real bills of exchange to be wrought by the pupil, in order to shew him, in a more particular manner, the necessity of knowing how to turn one country into the money of another country, value for value, where the Inerchant happens to be engaged in foreign trade. I have also taken the lilae erty to put the double rule of three after exchange, which in most of the forIper editions stood before it, to the end that all the mercantile rules in whole pombers might stand together; and likewise that the pupil might, at the end of change, enter upon a course of book-keeping, if there should not be time for him to go through the whole compendiuin first. I should have been very glad to have seen an attempt of this nature, stampt by the authority of some person of distinction and of better abilities ; but since no abler band has undertaken it, I hope its homely appearance w pot lessen its usefulness. A 2 The printer's errors, as well as my own defects, I hope will candidly be over, looked; but because a man's failings are so familiar to himself, that he can scarce discern them; therefore the kind admonitions of a good natured reader, shall always be very acceptable. I have nothing more to add, but my repeated thanks for favours received, together with iny earnest desire that you may be prosperous in your several mdertakings, and to beg this additonal favour of being esteemed, GENTLEMEN, Your most humble, and Most obedient Servant, | THOMAS DILWORTH. ON HIS COMPENDIUM OF ARITHMETIC, ENTITLED, THE SCHOOLMASTER'S ASSISTANT. WHILE some, seducive of the rising Age, Thy cares, how worthy of the Good and Wise, Thy labours, Friend, have found their just success, MOSES BROWN. TO NR. THOMAS DILWORTH, ON HIS SOHOOLMASTER'S ASSISTANT.' Learning, the Glory of Britanvia's Isle, WILLIAM DEAN. PART I..OF WHOLE NUMBERS. Page Of Notation . .. .: 2 Y Simple Fellowship ... 73 Of Addition ....... 4 Compound Fellowship . . . 75 Of Subtraction .... 2! f Exchange i . . . . . 76 Oi Multiplication . . . . 2pi the Comparison of · 341f the Double Rule of Three 92 L Of Conjoined Proportion . . 93 s Of Allegation Medial . . . 94 --Inverse . . . . . 471 Alternate . . . . 96 f Practice . . . . . . ; 49 or Single Position -' .-- 102 Of Simple Interest . . . . 62 of Double Position -- - 103 for days . . . . . 67 of Comparative Arithmetic - 104. Of Compound Interest ... 68 of Progressive Arithmetic - Of Equation of Payments the Of Permutation, or chang. JF Notation .. l of Division , . . . . Of Reduction . . . . . . 1 Di the Single Rule of Three O f Notation ...... 123 a general Rule for extracting 1250f Annuities and Pepsions in Of Division . . . . . 126 Arrears . . . . . 149 Of Reduction . . i 126or The Present Worth of Direct . .. • 129 of Annuities and Leages in of the Square Root .:..1320 Simple Interest for Days, Of a Vulgar Fraction. 1330f Rebate or Discount . . Of a mixt Number . . 133101 Equation of Payments the -Of a Vulgar Fraction. 139 of Compound Interest ... 159 Of the Biquadrate Root .. 140° Of the Sursolid Root . . . 1404 Arrears . . . . . . 161 Of the Square Cube Root . . 142 of the present Worth of Of the second Sursolid Root . 142 Annuities . ..... 168 Of the Square Biquadrate JOf Annuities and Leases io of the Cubed Cube Root :: 143 of Freehold Estates in Re- . Of the third Sursolid Root . 143 version . . . . . . 168 |