falfe Ones; to difpute ftrenuously, as well as judge folidly "with a Readiness of Invention, a Juftnefs of Method, and Clearnefs of Expreffion.

"In like Manner, thefe Difciplines do inure and corroborate the "Mind to a conftant Diligence in Study, to undergo the Trouble of an attentive Meditation, and chearfully contend with fuch "Difficulties as lie in the Way.

66

They wholly deliver us from a credulous Simplicity, most ftrongly fortify us against the Vanity of Scepticifm, effectually "restrain us from a rafh Prefumption, most easily incline us to a due Affent, perfectly fubject us to the Government of right Rea"fon, and infpire us with Refolution to wrestle against the unjust Tyranny of falfe Prejudices. If the Fancy be unftable and fluctuating, it is at it were poifed by this Ballast, and steadied by this Anchor; if the Wit be blunt, it is fharpened by this Whet«ftone; if luxuriant, it is pared by this Knife; if headstrong, "it is reftrained by this Bridle; and, if dull, it is roused by this Spur. The Steps are guided by no Lamp more clearly through "the dark Mazes of Nature, by no Thread more furely thro' "the intricate Turnings of the Labyrinths of Philofophy; nor, laftly, is the Bottom of Truth founded more happily by any other Line. I will not mention with how plentiful a Stock of Knowledge the Mind is furnished from thefe, with what whole"fome Food it is nourished, and what fincere Pleasure it enjoys.

66

But, if I fpeak further, I fhall neither be the only Perfon, nor "the first who affirm it, that, while the Mind is abstracted and ele vated from fenfible Matter, diftinctly views pure Forms, conceives the Beauty of Ideas, and inveftigates the Harmony of "Proportions, the Manners themselves are fenfibly corrected and improved, the Affections compofed and rectified, the Fancy calmed and fettled, and the Understanding raised and excited to more divine Contemplations. All which I might defend by the "Authority, and confirm by the Suffrages of the greatest Philo "fophers.My Time, my Speech, my Breath, would fail me,

even but lightly, to run over the principal Heads of these Things: "For no Words can juftly defcribe the Limits, can fully contain "the Advantages, can perfectly exhauft the Praises of that Sci"ence, which runs thro' and encompaffes the Heavens, the Earth, "and the Seas."

Notwithstanding the Usefulness of Mathematical Learning is fo very great, that the Conveniencies of Life and Commerce cannot be had without it, nor the Mind be fo extenfively improved in the Nature of Things, as hath been juft fhewn; yet it is well known what Difficulties the Students in thefe Sciences meet with, for Want of a regular Introduction to all the most useful Branches, according to the modern Improvements; and it being notorious that there is no fuch Book extant in our own Language (whatever may be in any other) I am inclined, to think, that I cannot employ fome of my leifure Hours more to the Service of the Public, than in attempting a compendious Courfe of the Mathematics, and Natural Philofophy,

fophy, according to the modern Improvements, in fuch a Manner as may be not only neceffary for Learners, but also agreeable to Men who are well verfed in thefe fublime Sciences.

I am fenfible that the Field now opened is large enough for much greater Geniuses than mine; therefore, in Order to have this Treatife as perfect as may be, I defire all ingenious Perfons, who have made Improvements in any Part of Mathematical Knowledge, to communicate their Discoveries (Poft paid) for encouraging a Work, which, if well executed, will be of great Service to Students in thefe Sciences.

Perhaps fome may afk, what will be the Bulk of these Essays, and whether what will be contained in them is new? To which take this for Answer, that if any Perfon would pretend to publish a Treatife of the Nature of this, with nothing but what is his own Invention, it would be a poor Performance; therefore, I shall only fay, that I do not in the leaft doubt of their containing fome Things new, befides the Method of handling them: As to the Number of Volumes, all that can be faid is, that it fhall be my Endeavour to make this Work, not fo compendious as fome, nor fo pompous as others; by carefully avoiding total Obfcurity on one Hand, and a tedious Prolixity on the other. All I have to add further, is to defire the Critics candidly to excufe fome Inaccuracies of Stile, and to affure the Public, that, if it meets with due * Encouragement, my Diligence shall not be wanting.

BENJAMIN DONN

Biddeford July 5th 1756.

* In fuch Cafe we propofe to give the Elements of Algebra, Altimetry, Architecture, Arithmetic, Aftronomy, Book-keeping, Catoptrics, Chance, Chronology, Conics, Decimals, Dialling, Dioptrics, Embattling, Exponentials, Fluxions, Fortification, Gauging, Geography, Geometry, Gunnery, Hydraulics, Hydrography, Hydroftatics, Infinite Series, Levelling, Longimetry, Marine Architecture or Shipbuilding, Menfuration, Motion Laws, of; Mufic, Theory of; Navigation, Perspective, Planometry, Pneumatics, Projectiles, Series, Statics, Surveying, Trigonometry, &c. With the Conftruction of the feveral Mathematical and Philofophical Inftruments. Alfo the Rationale of Chymistry, a Sketch of the Animal Oeconomy, &c.

THE

PREFACE

ΤΟ ΤΗΕ

ARITHMETIC.

Tis highly reafonable to fuppofe, that fome Method of Numbering was used by Adam and Eve in Paradife, for communicating their Ideas to each other, of so many, or fo much, &c. but in the Beginning, whilft the Manner of Men's Living was fimple, and Things were in a manner common, as there was not then any great Skill requifite or neceffary in Numbers, an Enquiry into the Nature or Properties of them was certainly much neglected.

Though History neither acquaints us with the Author, or Time of the Invention of Arithmetic; yet is it natural to suppose, that, when Commerce first began in the World, then, as fome kind of Computation was abfolutely neceffary, Men began, in good Earneft, to apply themselves to study the Properties of Numbers, and to reduce them into a kind of Art.

All Arts and Sciences have had their happy Ages, in which they have appeared with greater Splendor, and caft a ftronger Light; but this Splendor, this Light, and thofe Times of Knowledge, have been, many Times, not only of fhort Continuance, but fucceeded by long, very long, Ages of Ignorance and Obfcurity.

For the more regular Shewing the Progrefs of Arithmetic, we fhall endeavour to trace it through the feveral Ages of Learning. And, as we are in the Dark concerning the Affairs of Mankind before the Deluge, the firft Age of Arithmetic, and many other Arts and Sciences, may be computed from thence, (about 2 or 3000 Years before Chrift, for Chronologifts are not agreed in this Point) to the Time that the Greeks travelled into Egypt and Babylon, (a

bout

bout 660 Years before Chrift) for the Improvement of themselves in Literature.

As

* "the Phænicians were the firft Navigators, having no Rivals for many Ages," it is natural to fuppofe, that Arithmetic muft either have been invented by them, or have received great Improvements from them. For, in Order to carry on Mercantile Business, there is a Neceffity of Computations, and Neceffity is the Mother of Invention: And this Suppofition is confonant with Hiftory, which acquaints us, that, if it did not receive its Origin from them, it was greatly improved, and applied by them to Mercantile, Nautical, Architectonical, and Fabrical Úfes. From hence, Arithmetic paffed into Greece; for the Greeks had no Letters, until Cadmus, King of Baotia, brought them, and Arithmetic, and Navigation, and Commerce from Phænicia; and fet up Schools, and taught Arithmetic, Trade, and Navigation (about 1440 Years before Chrift.) From Phenicia alfo † Trade and Arithmetic were carried into Egypt.-Jofephus fays that, when Abraham was in Egypt, he taught Arithmetic to the Egyptians,

Though Arithmetic, like other Arts, was no Doubt at first very rude, and improved by Degrees, and in this firft Age fell far fhort of the modern Systems; yet they had laid a very good Foundation for an Art of Computation; for they were not ignorant of that Method of Notation still used by us, and called the Arabian. For which excellent Notation, the Arabians acknowledged themfelves beholden to the Genius of the Indians. This is all we know concerning the Progrefs of Arithmetic in the first Age of Learning, and, therefore, we pafs on to the fecond Age of Arts and Sciences; which is for the moft Part included amongst the Greeks, beginning with Pericles (about 460 Years before Chrift) and ending with the Death of Alexander's first Succeffors, (viz. about 300 Years before Chrift.)

The fecond Age of Arts.

In this Age flourished a Number of learned Men, viz. Thales, Plato, Ariftotle, Pythagoras, the Inventor of the Multiplication Table, &c. who travelled into Egypt and various Parts of Afia, in Order to acquire a greater Degree of Knowledge in Arithmetic, Geometry, Aftronomy, &c. which they brought into Greece, and taught their Countrymen. The Greeks made Ufe of two Methods of Notation; the First of which was of the fame Nature as the Roman Notation, (which we shall explain presently;) the other and best Method was thus: The firft nine Letters of their Alphabet, A, B, г, A, E, Z, H, 0, 1, expreffed the first Numbers from 1 to 9; the next nine Letters K, A, M, N, ¤, O, II, 3, 2, represented any Number of Tens, from one to nine, viz. 10, 20, 30, 40, 50, 60, 70, 80, 90, respectively. Any Number of Hundreds they expreffed by other Letters, fupplying what was wanting by other Characters; and thus

* Universal History, Arithmetic,

they

↑ Malcolm's Hitory of Arithmetic, in his

they proceeded, ufing the fame Letters with fome other Marks to represent Thousands, Tens of Thousands, &c.

*From the Greeks, Arithmetic was handed to the Romans. Which Obfervation naturally brings us to the third Age of Arts and Sciences, which commenced at the Birth of our Saviour, and continued to the Deftruction of the Roman Empire by the Goths, (about the Year 410.)

The third Age of Sciences.

+ The Roman Method of Notation, as it is ftill in Ufe amongst us, we shall just describe: The Letters made Use of by the Romans for this Purpose were, I. V. X. L. C. D. M. The Letter I was to denote a Unit; V=5, X=10, L=50, C=100, D-500, M=1000. As they had no different Value for their Places, the intermediate Numbers were expreffed by a Repetition of fome of thefe Letters; the Letter, denoting the greateft Value, being generally placed to the left Hand: Thus II 2, III=3, &c. VI=5+1=6, VII= 5+1+1=7, &c. XI=10+1=11, XII=10+1+1=12, &c. XX=10+10=20, XXX=10+10+10=30, &c. LX=50+10 =60, LXVIII=50+10+5+1+1+1=68, &c. DX=500+10 =510, DC=500+100=600, DCCLXXXVII= 500 + 100+ 100+50+10+10+10+5+1+1=787, &c.

But, to exprefs themfelves more compendiously, they fometimes wrote a Letter denoting a leffer Value before a greater, and in fuch Cafe their Difference is to be understood: Thus, IV=5—1=4, IX=10-1=9,IIX=10-1-18, XL=50-10=40, CD-500 -100-400, &c. When a Number is expreffed by more than two Letters, and Part of it is expreffed by this laft Method, it is proper to diftinguish it from the Letters on the left Hand by a Comma, or Point; thus, for Inftance, 149 may be thus expreffed, C,XL,IX, inftead of CXXXXVIIII. Again, 449 may be thus expreffed, CD,XL,IX, inftead of CCCCXXXXVIIII. They had also other Peculiarities in their Notation as follows. For D=500, they fometimes write I; and the Addition of every Ɔ made a Number 10 Times as much; thus, IO=5000, 100=50000, &c. Alfo, for M1000, they fometimes wrote CIO; and, by placing Cand on each Hand, they expreffed 10 times as much as before. Thus CC=10000; 100000. But, for expreffing any Number of Thousands, they had yet a more compendious Way; viz. by making a Dash (-) over any Letter denoting a Number lefs than a Thoufand; thus, V: 5000, VI−6000, X=10000, LXX=70000, C=100000, andMa thousand Thousands— aMillion MM 2000000,&c. Hence it appears, that though fome Numbers are more compendiously expreffed, according to this (Roman) Method of Notation, than by the common Method; yet, they are but few, if compared with what are otherwife. And if we consider that there is not fuch a regular Progreffion in the Value of the

* Supplement to Harris's Lexicon, + Malcolm's Arithmetic.

fimple