1

thematics, it would be neceffary to have fome Treatife of Arithmetic to render that Course compleat; and therefore this alone would be a fufficient Reason for Writing on this Subject: But this it is hoped will be found to be the least of my Reasons in Vindication of thefe Effays.

Long before I had any Thoughts of publishing a Treatife of Arithmetic, I looked into feveral Arithmetical. Writers for the Demonstrations and Reasons of all the Rules; but did not meet with any Book that contained them. The greatest Part of the Treatifes that are moft efteemed, are fo very defective that they do not fo much as contain the Demonftration of a fingle Rule. The best that came to my Sight were Ward's Introduction to the Mathematics and Kerfey upon Wingate; but the Rules are few in Number that thefe have demonftrated. In short, it is notorious, that mòft Writers, * Teachers of Arithmetic, have too much neglected this valuable Part, without which a Perfon can only work by Rote, juft vas a Parrot is taught to prate; not having feen the Reafon or Inveftigation of the Rules he makes Ufe of. Hence fuch a Perfon, being afked how he knows the Rules to be true, can only answer fuch an Author, or fuch a Master told me fo: How liable then is fuch a one to be mifled and to make Ufe of falfe Rules? Farther, though the Method of teaching the Rules without Demonftrations may ferve the Purpofe of the Idle, and the Worthlefs, who, if they can be taught fome practical Rules, by which they may ftumble through fome Bufinefs or Poft, have as much as they defire, and more than they deferve; yet how can the inquifitive and rational Mind be fatisfied, without feeing the Nature and Reafon of Things? Certainly no Ferfon, who has not tailed it, can be sensible of that exquifite Pleasure, (Pleasure much more delightful than the greatest Enjoyment of the Senfes) which fuch Minds poffefs on any Difcovery of the Nature and Reafon of Things. It is this Tafte, this Curiofity, and Ability of enquiring into the Nature of Things, which is the principal Charafteriic that diftinguishes Man fo much fuperior to the Brute; for neither our Form or fenfual Pleasures are a fufficient Diftinétion; for who can affirm that our fenfual Pleafures are greater than thofe enjoyed by fome of the brute Creati

*Having, in the General Preface, fald fomething on the Advantage of hay ing a good Mafer, it may be of fome Ule to caution Parents to be careful in the Choice of a Preceptor for their Children; for there are Quacks in Mathematics as well as in Phytic, and, as Mr. Webfier jufly obferves, in his Effay, on Education, when a Man has tried all Shifts and itill failed, if he can but --- compute the Minutes in a Year, or the Inches in a Mile, he fets up for a Teacher of Arithmetic, and, by the Buit of low Prices, perhaps gathers a Number of Scholars, and, thus impofing on the inconfiderate Patents, both robs them of their Money, and the more unhappy Children of their Time.

The Ignorance of to many Matters is really the Fault of Parents, for what is faid in N°. 313 of Mr. Addifon's Spectator is strictly true, viz. "We often fee "20 Parents, who, though cach expects his Son fhould be made a Scholar, are not contented all together to make it worth While for any Man of liberal "Education to take upon him the Care of their Instruction." The Confequence of which is, that, "for Want of Encouragement in the Country, we have many "a promifing Genius fpoiled and abufed in thofe little Seminaries."

tion? And, as to the Form of our Bodies, how nea ly do fome of the Ape kind approach unto it, particularly the Oran-Otan, which is very little different in Shape, &c. from the Human Species?

t. Upon these Confiderations, I was determined, for my own Ufe and that of my Pupils, to attempt to demonftrate all the Rules of Practical Arithmetic; not having then any Thoughts of making them public, but now, for the Reafons mentioned in the General Preface, I have yielded to the Publicationof them.

The rough Draught of thefe Effays was almoft finished, before I heard any Thing of Mr. Malcolm's Arithmetic; in which, when I first heard its Character, I thought that ingenious Gentleman might have done what was my Intention; but, upon Perufing it, found it could not be any material Objection to the publishing these Essays, as may appear from the following Reafons.

1. Because this is defigned as the firft Volume of an intended Treatife of the Mathematics.

2. Though Mr. Malcolm's is both an ingenious and laborious Treatife, and abounds with many Things not ufeful in Practical Arithmetic, and which more properly belong to Treatifes of Algebra, yet there are feveral ufeful Rules omitted, particularly Single and Double Pofition; Rules, without which, many curious and ufeful Questions could not be eafily folved by thofe Perfons who do not understand Algebra; and I may venture to affirm, that Double Pofition may be of Ufe even to *Algebraists, in folving fome adíected and exponential Equations, c. more expeditiously than the Method

of Series.

3- Laftly, The Bulk will deter many from Perufing it; for very few Learners would be willing to ftudy 526 Quarto Pages, before they arrive at the Golden Rule,

Hence, after all that hath been hitherto done by others, I flatter myfelf with Hopes, that a compendious Treatife of Practical Arithmetic, with fo much of the Theory as is ufeful in Demonftrating the Practice, will meet with a favourable Reception; having fhewn to all impartial Judges, that the great Number of Writels on this Subject cannot be any great Objection to the Publithing of thefe Effays.

As to the Method made Use of in treating thefe Effays, the Reader will first mind the neceffary Definitions and Axioms; then the Rule with its Illuftration by Examples, and its Demonstration generally in a Note. And, being of Opinion that it is beft to treat every Subject diftinct and entire by itself, to prevent embarraffi ng the Learner, he will find the Doctrine of Vulgar Fractions at the End of Common Arithmetic; and, for the fanie Reason, Decimal Arithmetic is treated of in a separate Effay. The Demonstrations

Notwithstanding the Affertion of fome ingenious Authors, who, having ac quired fome Knowledge in the Algebraic Art, entirely contemn and flight th's Rule, and fay it is only useful to fuch as have no Acquaintance with Algebra, and fit only for the Ignorant and Indolent; yet, had they been well acquainted with the Application of the Method of Trial and Errors (as fome call this Rule) they would have been convinced, that it may be of great Ufe even to the molt expert Algebraifts,

b 3

are

are Algebraical, because that Method is both the easiest, and shorteft; and, without it, many Things could not be well demonstrated at all; and the Algebra, here given, will not only ferve for this Purpofe, but also be an ufeful Introduction to the Algebraic Art, which will be more fully treated of in its proper Place. The Al gebraical Demonftrations are given by Way of Notes, because, if the Learner hath not already acquired fome Knowledge in the Practical Part of Arithmetic, his beit Way of Studying will be to get first fome Acquaintance with that Part, omitting the Algebraical Part in his firft Reading; and then, in his fecond Reading, to take in the Algebraic Part as it occurs.

In the Courfe of this Work there are feveral Questions in Verfe, taken from the Annual Diaries, &c. Not on Account of the Accurateness of the Verfification, (for in that many are deficient) but for thefe two Reafons, 1. That no Solutions, at least by Arithmetic only, have been published of them; 2. That they have been found by Experience to contain a kind of Charm, to entice young Students to folve them.

For a more particular Account of these Eflays, the Reader is de fired to look into the Book itself; for I fhall not detain him here, with any Thing more in its Favour, well knowing that, if it has not Merit fufficient to recommend itself, all that I could here fay would be of no Service. Therefore, fuch as it is, I fubmit it to the Examination of all impartial Judges and Lovers of Truth; defiring them either to país by, or candidly acquaint me with, what they find amifs; for I fhall not be afhamed to retract, or alter, whatever shall appear to me to be not confiflent with the Nature and Reafon of Things.

BENJAMIN DONN.

As the Author at firft had an Inclination to publish these Elays by Subfcription, he thinks himself obligated to return. Thanks to the Gentlemen who offered to encourage that Defign.

1

ADVERTISEMENT.

N the General Preface, we have hinted, that, if thefe Eflays meet with a favourable Reception from the Public, we intend to attempt the other Parts of the Mathematics, as a new and compendious Courfe of Mathematical Learning is, in our Opinion, a Thing much wanted: And of the fame Opinion is a learned Correfpondent, as will appear by the following Extract of a Letter to the Author :

<<WHITBY, Feb. 5, 1757.

"SIR,

-fincerely wish that what you are now printing may anfwer your Expectation, and meet with the Approbation "of the Public. As foon as I fee your Eflays advertised, I intend 86 to be one of the Purchasers. I have long thought a compleat Syftem of Mathematical Learning to be a Thing much wanted, and which, if well executed, could not fail of meeting with proper Encouragement-As for Oughtred, Ozanam, Leybourn, "and fome others, which were looked upon in their Age as good Things, they are now grown quite obfolete, and are hardly "worth the Notice of our prefent Mathematicians, any more than "the voluminous Performances we meet with in Harris, Chambers, and other Dictionary Writers.

"I am, with Efteem,

"SIR,

"Your most humble Servant,

LIONEL CHARLTON."

In Works of this Nature it is impoffible but Errata will happen but, if they are more in Number than fome might expect, the candid Reader will attribute them to the Author's being, at the Time of Printing this Volume, about 200 Miles diftant from the Prefs. Note, The Letter b. added after the Number of the Line fignifies to count from the Bottom.

Page 81. 116. for having r. have; p. 27, in the Operation in Art. 69. the Line between the Numbers 115 and 23 fhould be between 23 and 34500000; p. 32 in the Margin, for 8350768, r. 88350768; p. 34 1. 11, for unlikely r. likely; p. 40 l. 21, for b x = n, r. bx cn; p. 60 l. 20, for I remaining r. 41 remaining; p. 61 1. 20, for in r. into; p. 64 1. 8, for Duantities r. Quantities; ibid. 1. 9. for a, b, c, c. the Divifor r. a x bx c &c.

:

51

*

the Divifor; ibid. 1. 11, for Qnext uote r. next Quote; ibid. 1. 12, after by r. dividing; p. 66 in the Note 1. 5 b. for bt r. b; p. 96 1. 4b, for anfwered 41. r. anfwered 21; p. 98, 1. laft, for 167 r. 163; p. 104, in the two last Lines, for 10, oz r. 1, cz; p. 112 in Queftion fecond, for 8 r. 5; p. 122 1. 2cb. for here r. there;' p. 133. Art. 226, for less than r. lefs; p. 153, the marginal References fhould be 306 and † 305; p. 156 1.7, for 19 r. 9; p. 157 1.86. for 293 r. 312; p. 1581. 12, for of a Bill r. off a Bill; p. 159, 1. 11, for 299 r. 318; ibid. in the marginal References, for 293 r. 312, and for † 296 r. † 315; p. 161 1. 12, for Men r.: 40 Men; p. 164, for Month and Months r. Week and Weeks refpectively; p. 1741. 2, for 589 r. 598; p. 1771. 10, for 10s. x 10s. r. 10s. + 10s.; p. 179 1. 23 for 349 r. 87, and for 2. p. 185 1. 20 b. for Rule r. Rate; p. 1871. 17, for in that r. in that; ibid. 1. 18, for in that r. in that; p. 1911. 19. between Flemish and 19 place:; p. 197 1. 2. of Art. 405, blot out not; p. 201, Example 2, for 6d. per lb r. 10d. per lb; p. 203. Art. 418, the Numbers 48 and 72 should be linked, and not 40 and 48; p. 204 1. 10 b. for intended r. intend; p. 205 in the firft Stating, for -r.:; ibid. 1. 46. for Alligat Aionlternate r. Alligation Alternate; p., 207 1. 14, for last r. leaft; p. 233 1. 21, for either 1 or r. neither Inor; p. 235 1. 10 b. for b r. b2 237 1.9, for 8 r. 62; ibid. 1. laft, for 2rx x x x r. 2 r + xxx; p. 240, for a + b2 г. a+b)2; ibid. 1. 3 b. for nr r. n— ; P 240, at the Beginning of 1. 12 b. prefix the Afterifm*; p. 247, at the Fnd of Art. 485, write and the leffer Root; ibid. at the End of the Note for Squares r. Cubes; p. 265 1. 14b. for Profit r. Proof; p. 27615 for Jupp fedr. Juppofe. p. 294 1. 18, for Eyes r. Ears; p. 300 for 1. 5, r. 3; p. 303 1. 8. of the Note, after Quote write ; p. 308 1.7, for Abbreviation r. abbreviated; ibid. 1. 15 and 26, for xr. p. 324 1. 4, for 1 D, o's x 1 with d, ơ's r. I with with d, c's; p. 348 in the laft Line of the Note, for

D, o's

xcr.

; P.

be

a

× c; p. 355 in Note 1. 2 b. for cr. d; p. 356

at the End of the Note, for b being. being = b.

The