ARITHMETIC

ARITHMETIC, Arithmetica, the Art of Numbering;or, that Part of Mathematics which considers the Powers and Properties of Numbers, and teaches how to compute or calculate truly, and with Expedition and Ease. See NUMBER, MATHEMATICS, COMPUTATION, etc.

Some Authors choose to define Arithmetic, the Science of discrete Quantity. See DISCRETE and QUANTITY.



Arithmetic consists chiefly in the four great Rules or Operations of Addition, Subtraction, Multiplication, and Division. See each in its Place, Addition, Subtraction, Multiplication, and Division.

'Tis true, for the facilitating and expediting of Computations, both Mercantile and Astronomical, divers other useful Rules have been contrived; as, the Rule of Proportion, of Alligation, of False Position, Extraction of Square and Cube Roots, Progression, etc.

But these are only Applications of the first four Rules.

See RULE; see also PROPORTION, ALLIGATION, POSITION, EXTRACTION, etc.

We have very little Intelligence about the Origin and Invention of Arithmetic;History neither fixes the Author, nor the Time.—In all Probability, however, it must have taken its Rise from the Introduction of Commerce; and consequently, be of Syrian Invention. See COMMERCE.

From Asia it passed into Egypt, (Josephus says by means of Abraham.) Here it was greatly cultivated and improved;inasmuch, that a great part of their Philosophy and Theology, seem to have turned altogether upon Numbers. Hence those Wonders related by them about Unity, Trinity; the Numbers Seven, Ten, Four, &c. See UNITY, TRINITY, TETRACTYS, etc.

In effect, Kircher, in his Oedipus Aegyptiacus, Tom. II. p. 2, shows, that the Egyptians explained everything by Numbers; Pythagoras himself affirming, that the Nature of Numbers goes through the whole Universe; and that the Knowledge of Numbers is the Knowledge of the Deity.

See PYTHAGOREAN.

From Egypt Arithmetic was transmitted to the Greeks, who handed it forward, with great Improvements, which it had received by the Computations of their Astronomers, to the Romans; from whom it came to us. The ancient Arithmetic, however, fell far short of that of the Moderns: All they did was to consider the various Divisions of Numbers; as appears from the Treatises of Nicomachus, wrote in the third Century of Rome, and that of Boethius, still extant. A Compendium of the ancient Arithmetic, wrote in Greek, by Psellos, in the ninth Century from our Saviour, was given us in Latin by Xylander, in 1556.—A more ample Work of the same Kind, was wrote by Jordanus, in the Year 1200; published with a Commentary by Faber Stapulensis in 1480.

Arithmetic, under its present State, is variously divided, into various Kinds;Theoretical, Practical, Instrumental, Logarithmical, Numerous, Specious, Decimal, Dynamical, Teralycal, Duodecimal, Sexagesimal, Vulgar, Decimal, Finite, Infinite, etc.
Theoretical Arithmetic, is the Science of the Properties, Relations, &c. of Numbers considered abstractedly; with the Reasons and Demonstrations of the several Rules.

See NUMBER.

Euclid furnishes a Theoretical Arithmetic, in the seventh, eighth, and ninth Books of his Elements.—Basilius Monachus has also given a Theory for demonstrating the common Operations, both in Integers and broken Numbers, in his Logistica, published in Latin by Chambers, in 1600.—To which may be added Lucas de Burgo, who, in an Italian Treatise published in 1523, gives the several Divisions of Numbers from Nicomachus, and their Properties from Euclid; with the Algorithm, both in Integers, Fractions, Extractions of Roots, etc.

Practical Arithmetic, is the Art of Computing; that is, from certain Numbers given, of finding certain others whose Relation to the former is known—As, if a Number be required equal to two given Numbers 6 and 8. The first entire Body of Practical Arithmetic, was given by Niccolò Tartaglia, a Venetian, in 1556, consisting of two Books; the former, the Application of Arithmetic to civil Uses; the latter, the Grounds of Algebra. Something had been done before by Stifel, in 1544; where we have several Particulars concerning the Application of Irrationals, Cossics, &c. no where else to be met withall.— We omit other merely practical Authors which have come since, the Number whereof is almost infinite; as Gemma Frisius, Metius, Wingate, &c.

The Theory of Arithmetic is joined with the Practice, and even improved in several Parts, by Maurolycus in his Opuscula Mathematica, 1575; Henricus Henischius in his Arithmetica Perfecta, 1659, where the Demonstrations are all reduced into the Form of Syllogisms; and Zachariah Pasqualigo in his Theoria et Praxis Arithmetices, 1704.

Instrumental Arithmetic, is that where the common Rules are performed by means of Instruments contrived for Ease and Dispatch;such are Napier’s Bones, described under their proper Article; Sir Samuel Morland’s Instrument, the Description whereof was published by himself in 1666; that of Gottfried Wilhelm Leibniz, described in the Miscellanea Berolinensis, and that of Giovanni Poleni, published in the Venetian Miscellany, 1709.—To these may be added,Logarithmical Arithmetic, performed by Tables of Logarithms. See LOGARITHM. The best Piece on this Subject, is Henry Briggs's Arithmetica Logarithmica, 1624.

To this Head may also be added, the universal Arithmetical Tables of Prosthaphaeresis, published in 1610, by Hulsius; whereby Multiplication is easily and accurately performed by Addition, and Division by Subtraction.

The Chinese have little Regard to any Rules in their Calculations;instead of which, they use an Instrument made of a little Plate, a Foot and half long, across which are fitted ten or twelve Iron Wires, on which are strung little round Balls. By drawing these together, and dispersing them again one after another, they count, somewhat after the Manner in which we do by Counters; but with so much Ease and Readiness, that they will keep pace with a Man reading a Book of Accounts, let him make what Expedition he can: And at the End the Operation is found completely done; and they have their Way of Proving it. See COMPTE.

Numerous Arithmetic, is that which gives the Calculus, of Numbers or indeterminate Quantities;and is performed by the common Numeral, or Arabic Characters. See ARABIC and CHARACTER.

Specious Arithmetic, is that which gives the Calculus of Quantities;using Letters of the Alphabet instead of Figures, to denote the Quantities. See SPECIOUS ARITHMETIC.

Specious Arithmetic coincides with what we usually call Algebra.

See ALGEBRA.

Dr. Wallis has joined the Numeral with the literal Calculus;and by means hereof, demonstrated the Rules of Fractions, Proportions, Extractions of Roots, etc. A Compendium of which is given by Dr. Wells, under the Title of Elements of Arithmetic, An. 1698.

ARITHMÉTIQUE