imple Characters, &c. in the Roman Method as there is in the Arabian, it will plainly follow, that the Roman Method is not fo convenient for Computations.

*A Sexagefimal Notation was invented in the fecond Century of Chriftianity, as is fuppofed, by Claudius Ptolomeus. In this Notation, every Unit was fuppofed to be divided into 60 equal Parts, and each of thefe Parts fubdivided into 60 Parts, &c. and hence the Parts were called Sexagefimal Fractions; and, to make the Com putation eafter, the Progreffion in the whole Numbers was alfo Sexagefimal. From 1 to 59 was expreffed in the common Way; then 60 was called a Sexagena prima, and was denoted by 1 with a Dash over it; thus, 160; II twice 60-120, &c. Sixty-times 60 or a Sexagena fecunda was thus expreffed, II, &c. In Sexagefimal Fractions, Numerators lefs than 60 were expreffed by the per Letters, and their Denominators by one or more Dashes, fet either over the Numerator on the left Hand, or under it on the right Hand; thus, the Fraction was wrote thus X or X. For the readier Performing of Multiplication and Divifion, the following Tables were made Ufe of:

pro

And continued after this Manner fo far as 60 by 60

60.00. Then,

for knowing the Names of the Product, the following Table was

But this Method of Multiplying, &c. even if we have fuch Tables at hand, is fo very troublefome and perplexed, that it is no Wonder to find, that, as the Arabian Notation gained, the Sexagefimal loft Ground; and is now entirely out of Ufe. The Sexagena Integrorum went first out of Ufe; but the Sexagefimal Fractions were continued until the Invention of Decimals.

* We cannot find the Romans made any great Improvement in Arithmetic.

† By the Moors Arithmetic was brought into Spain; from Spain it was carried into feveral other Parts of Europe by feveral learned Men, who went there to ftudy the Arabic Learning, (and even the Greek Learning from the Arabic Verfions, before they got the Originals themselves) imported by the Saracens.

During the long Period of about 1100 Years, from the End of the third Age of Arts and Sciences to the Commencement of the fourth, almoft all Europe lay in a State of Slavery, attended with the Ravages of Superftition and Ignorance. In fuch a State, the very Principles of Science must be almoft buried in Oblivion; however, this is the most ancient State, in which we can trace Arithmetic in Europe. Dr. Wallis has fhewn by good Authorities, that it was known in Europe before the Year 1000. He has particularly fhewn, that Gilbertus a Monk, afterwards known by the Name of Pope Silvester the Second, (who died in the Year 1003) was acquainted with this Art, and carried it from Spain into France long before his Death. He has alfo fhewn, that it was known in Britain before the Year 1150, and far advanced in common Ufe before the Year 1250; as is evident by the Arithmetic of Joannes de Sacro Bosco, who died about the Year 1256.

As to the Antiquity of Numeral Figures, the faid Dr. Wallis thinks their Ufe in Europe was as old at least as the Time of Hermanus Contra&us, (who lived about the Middle of the eleventh Ċentury) and, if not frequent in ordinary Affairs, yet at least in Mathematical Things, and especially in Aftronomical Tables. He gives us an Account of a Mantle-tree of the Parlour-chimney at the Dwelling-House of Mr. William Richards, Rector of Helmden in North

Both the Letters and Figures are of an antique Form, agreeing well enough with that Age; hence the Dr. concludes, that the Ufe of fuch Figures here, (in England) even on ordinary Occafions, is at leaft

*Supplement to Harris's Lexicon. + Malcolm's Hiftory of Arithmetic. Philofophical Tranfactions, No. 154.

leaft as ancient as the Year 1133; and judges it to have been fomewhat more ancient, because the Shape of the Figures, though not come juft to the Shape we now use, was even then confiderably varied from the Arabic; which argues, that they had then been some Time in Ufe; fuch Change of Shape in Figures and Letters coming on gradually with Time.

That the Arabic Numerals were in common Ufe in England, about this Time, may be alfo fupported by the Date on a Chalice in the Church of Welch Bicknor in Monmouthshire, which, Mr. Green fays, is 1176; which, by the Make of the Veffel and Mode of the Figures, feems to be genuine.

The fourth Age of Arts and Sciences begins with the Art of Printing; that is, about the Middle of the fifteenth Century; and continues to the present Time.

Now, we are come to that happy Age of Learning, (and may it long continue through the Bleffing of God, and Encouragement of great Men!) in which not a Century, nor a fingle Year, paffes, without a Difcovery of fomething of Moment, fome great Improvement in the Sciences.

To particularize the Discoveries and Improvements of this Age would require a Volume; we fhall, therefore, only take Notice, in this Place, that Decimal Arithmetic was invented about the Year 1550. The firft that ufed Decimals, in extracting the Square and Cube Roots, was (if we are not mistaken) our Countryman Buckleus; but the first who writ an exprefs Treatife of Decimals was Simon Stevinus, about the Year 1585. As to Circulating Decimals and Loggarithims, we shall give their Hiftory hereafter; and fhall only here add, for the Sake of the Curious, a Lift of Arithmetical Writers:

Pfellus wrote in the Year 1008. In the Year 1503. Nemorarias -1504. Carolus-1513. Blafius-1514. Boethius, Siliceus-1515. Lax-1520. Suiffet-1522. Tonfal-1526. Cirvello-15 30. Bradwarden-1532. Aventinus-1536. Morfianus, Peurbachius-1537. Andreas--1539. Bronchorft, Cardanus, Glareanus, Ringelbergius-1540. Scheubelius, Willichius-1542. Finaus-1544. Vulpius, Welpius, Caius, Elias-1549. Vicar-1550. Alberti, Anatolius, Flicker→→→ $552. Poftellus-1553. Faber-1554. Camerarius, Gerafenus, Stifelius-1555. Cuno, Ramus-1556. Nabod-1557. Archimedes1558. Peletarius, Rhæticus-1559. Monfon, Scalichius — 1560. Barres-1563. Beda-1564. Thierfelders, Ulmans-1565. Euclid, Kaltenbrunners, Nefius, Stenius, Strigelius 1566. Massen,

Munnos, Pencerus, Segura 1568. Steinmet1573. Beaufardus, Kopffers 1575. Diophantus-1576. Lagne 1577. Hammelius, Hobelius, Saligniacus-1580. Riefens (Jacobus)—1581. Lonicerus1585. Schrectenbergers, Stevinus-1587. Poppins-1591. Kundlers, Meureras (Joan.) Xylander-1592. Pifcator-1593, Suevius 1595. Helmreich 1596. Fiffeld, Snellius 1598. SchleutnerusHelmreich-1596. 1599. Buteo-1600. Gletfmannus, Reinhards, Reymers, Schey, Sculken, Taf-1601. Goffens, Urfi-1602. Johnson-1603. Caraldus, *Gentleman's Magazine for May, 1756. b

Glichtovaus,

-1618.

Clichtovæus, Daviden, Richters, Riefens (Ifaac.) Romanus, Waferus, Wurfifus 1604. Cortes, Frifius, Landus, Sectgerwick—1605. Dibvadius-1606. Chauvet, Hofflins, Ruinellus-1607. Clavius, Meurerus (Chrift.)1609. Barlaamus, Engelbertus, Henifchius, Peckoldts, Wildgovels, Zuichetta-1610. Loffius, Noviomagus, Sotters1611. Campanella, Herwart, Longomontanus, Metius, Refenius, Severinus-1612. Brandis, Kaufungers, Laurembergius, Wittekind-1613. Alftedius, Brunus, Jacobs, Latomus, Neudorffers, Wilhelmus-1614. Krafften, Monhemius, Mullerus (Chrift.) Neper-1615. a-Culen, Finckius, Grunewald, Lucius-1616. Daetrus, Rudolffs. 1617. Faulhabern, Hainkelmans, Heern, Langius, Vitalis, ZonfenBungus, Cappaus, Geigers, Lucas, Stephanus, Urfinus, Wirth-1619 Lants, Olaus, Remmelinus, Strubius 1620. Bevern, Caffini, Hennings, Kandler, Malapertuis, Micylus, Roffels, Van-Zefen1621. Bechmanus, Lavus, Mulichs, Spikers-1622. Follinus-1623 Riefens (Adam.) Wagner-1624. Brigs, Bufcherus, Cafar, Kepler, Neufville, Katen-1625, Servin-1627. Veronenfis-1628. Taccius, Ulacq.1629. Malleolus-1630. Mullinghaufen-1631. Mullerus (Facob) Oughtred-1633. Row or Roe-1635. Bartfchius, Kruger-1636. Crugerus-1638. Van-Schoten-1640. Chavin, Salmafius, Johnson -1641. Curtius-1644. Bullialdus, Launay, Smirnaus-1646. Micralius 1647. Schmidts, Trenchant 1648. Middendorpius, Renaldus-1650. Frommius, Meierus, Mengolus, Nottragelius, Weberus

--

16-2. Pierantonius-1653. Meenenaer, Moller1655. Tacquet --1656. Wallis, Willisford1658. Dee, Gendre, Hoffman, Mellis, Record-1660. Bebm, Leotaudus, Wingate-1661. Reyherus1662. Strauchins- -1663. Duke, Levera, Schottus-1664. Biermans 1665. Kircher 1666. Branker, Pajottus, Pell

1668. Baker, Philips, Tylkawski

∙1657.

1669

Clavel, Voigt Beverege, Graffenriedt, Hodder, Jamblichus, Kerfey, Zaragofa-1670. Brown, Jackjon, Newton,-1671. Clark, Fontaine-1673. Morland, Severius, Tabing, Taffius—1674. Braffer, Gottignus, Mercator1676. Forbes, Hugerus-1680. Ozanam, Tartaglia- -1687. For daine, Moore- -1690. Hawkins, Mandey, Pickering- ∙1693. Lybourn 1694. Preflet 1696. Chamberlain, Jeake1697. Mofc1700. Ayres, Cocker, Cole, Heinlin, Sturmius, Treu1710. Harris, Lydal, Royer, Ward, Wolfius 1720. Fuller, Gardm, Hatton, Jones, Sharpe

1725. Claircombe

1714. Cunn

-1423. Wells.

1728. Chambers, Hayes, Hill

1730.

Makelm-1731. Leadbetter, Hodgkin-1732. Grey, Wilfon

1735. Kirby, Martin, Shelley, Stephens 1735. Barreme, Wefton -1737. Gore, Stoneboujë- -1740. Fletcher, Weber

Mork -1743. Holanes- 1744. Fiber, Worley-
Chopple, Markham, Pardon-1746. Dilworth, Holliday
Richards, Kevad

1749. Lowe 1751. SmithPotter, Thorpe, Fenning-1755. Martin.

1742:

1745

1747

1753.

Having thus given fome Account of the Antiquity and Progrefs of Arithmetic, we hail now proceed to fay fomething concerning its Ufefulness; which at firft Glance appears to be fo great, as to be ineitimable; for, without it, we could not so much as have a com

pleat

pleat Idea of Number, Weight, and Measure; and, without it, Traffic, &c. muft immediately ceafe.

Plato held it in fuch Eftimation, that he faid, "Take away Arithmetic, which is the Art by which we come to the Knowledge of Weight and Meafure, and all that remains is base and of no "Eftimation." It is an Introduction to all other Arts and Sciences, and, without fome Knowledge firft obtained of this, it would be impoffible to make a Progrefs in any other.

Numbers are so much the Measure of every Thing that is valuable, that Sir Richard Steele thinks * "it is not poffible to demonftrate the Success of any Action, or the Prudence of any Undertaking, without them."

The most ancient Method of Numbering was by the Fingers; to which Solomon is fuppofed to allude, when he fays, Wildem cometh with Length of Days in her right Hand; and Solomon fpeaks greatly in its Praife in another Place, where he breaks out, Thou O Lord, haft difpofed all Things, in Measure, Number, and Weight.

Arithmetic will fhew us the great Advantages that may redound to Science in General, by a happy Notation or Expreffion of our Thoughts; for it is owing to the Arabian Method of Notation (fee Chap. II. Effay 1.) that the most complicated Operations are ma naged with fo much Eafe and Difpatch.

This Art will fhew us the Conduct and Manner of the Mind, when employed in the Exercife of Invention; and the great Advantages derived from an artful Method of claffing our Perceptions. For by confidering Numbers as divided into Parts, (by the Method of Notation) we are able, eafily and readily, to perform that by confidering their refpective Parts, which would perplex and confound the Mind, to confider their Wholes, without confidering their Parts feparately. This manifeftly appears in the Operations of Addition, Subtraction, Multiplication, Divition, &c. for, though it goes beyond the Limits of the Human Mind to find the Sum or Product of two very large Numbers, without confidering their Parts feparately, yet, by finding the Sum or Product of their feveral Parts, we, with very little Trouble to the Mind, easily and readily, find the Sum or Product of their Wholes: Since the Sum or Product of all their Parts must be equal to the Sum or Product of their Wholes.

It is now high Time, and neceffary, to give fome Account of the prefent Performance; for, as Arithmetic has been treated of by fuch a Multitude of Authors, perhaps fome may think there is no Need of this, or any other new Treatife on this Subject; and, therefore, to fend a Book into the World on a Subject that has gone through fo many Hands as this hath, without fome Introductory Account, is but little better than expofing it. I fhall therefore firft obferve, that, fuppofing I could not make any Improvement in Arithmetic, yet, as my Intention is to publish a Courfe of the Ma