ARITHMETIC

, the art and science of numbers; or, that part of mathematics which confiders their powers and properties, and teaches how to compute or calculate truly, and with ease and expedition. It is by some authors also defined the science of discrete quantity. Arithmetic consists chiefly in the four principal rules or operations of Addition, Subtraction, Multiplication, and Division; to which may perhaps be added involution and evolution, or raising of powers and extraction of roots. But besides these, for the facilitating and expediting of computations, mercantile, astronomical, &c, many other useful rules have been contrived, which are applications of the former, such as, the rules of proportion, progression, alligation, false position, fellowship, interest, barter, rebate, equation of payments, reduction, tare and tret, &c. Besides the doctrine of the curious and abstract properties of numbers.

Very little is known of the origin and invention of arithmetic. In fact it must have commenced with mankind, or as soon as they began to hold any sort of com- | merce together; and must have undergone continual improvements, as occasion was given by the extension of commerce, and by the discovery and cultivation of other sciences. It is therefore very probable that the art has been greatly indebted to the Phœnicians or Tyrians; and indeed Proclus, in his commentary on the first book of Euclid, says, that the Phœnicians, by reason of their traffic and commerce, were accounted the first inventors of Arithmetic. From Asia the art passed into Egypt, whither it was carried by Abraham, according to the opinion of Josephus. Here it was greatly cultivated and improved; insomuch that a considerable part of the Egyptian philosophy and theology seems to have turned altogether upon numbers. Hence those wonders related by them about unity, trinity, with the numbers 4, 7, 9, &c. In effect, Kircher, in his Oedip. Ægypt. shews, that the Egyptians explained every thing by numbers; Pythagoras himself affirming, that the nature of numbers pervades the whole universe; and that the knowledge of numbers is the knowledge of the deity.

From Egypt arithmetic was transmitted to the Greeks, by means of Pythagoras and other travellers; amongst whom it was greatly cultivated and improved, as appears by the writings of Euclid, Archimedes, and others: with these improvements it passed to the Romans, and from them it has descended to us.

The nature of the arithmetic however that is now in use, is very different from that above alluded to; this art having undergone a total alteration by the introduction of the Arabic notation, about 800 years since, into Europe: so that nothing now remains of use from the Greeks, but the theory and abstract properties of numbers, which have no dependence on the peculiar nature of any particular scale or mode of notation. That used by the Hebrews, Greeks, and Romans, was chiefly by means of the letters of their alphabets. The Greeks, particularly, had two different methods; the first of these was much the same with the Roman notation, which is sufficiently well known, being still in common use with us, to denote dates, chapters and sections of books, &c. Afterwards they had a better method, in which the first nine letters of their alphabet represented the first numbers, from one to nine, and the next nine letters represented any number of tens, from one to nine, that is, 10, 20, 30, &c, to 90. Any number of hundreds they expressed by other letters, supplying what they wanted with some other marks or characters: and in this order they went on, using the same letters again, with some different marks, to express thousands, tens of thousands, hundreds of thousands, &c: In which it is evident that they approached very near to the more perfect decuple scale of progression used by the Arabians, and who acknowledge that they had received it from the Indians. Archimedes also invented another peculiar scale and notation of his own, which he employed in his Arenarius, to compute the number of the sands. In the 2d century of christianity lived Cl. Ptolemy, who, it is supposed, invented the sexagesimal division of numbers, with its peculiar notation and operations: a mode of computation still used in astronomy &c, for the subdivisions of the degrees of circles. Those notations however were ill adapted to the practical operations of arithmetic: and hence it is that the art ad- vanced but very little in this part; for, setting aside Euclid, who has given many plain and useful properties of numbers in his Elements, and Archimedes, in his Arenarius, they mostly consist in dry and tedious distinctions and divisions of numbers; as appears from the treatises of Nicomachus, supposed to be written in the 3d century of Rome, and published at Paris in 1538; as also that of Boethius, written at Rome in the 6th century of Christ. A compendium of the ancient arithmetic, written in Greek, by Psellus, in the 9th century, was published in Latin by Xylander, in 1556. A similar work was written soon after in Greek by Jodocus Willichius; and a more ample work of the same kind was written by Jordanus, in the year 1200, and published with a comment by Faber Stapulensis in 1480.

Since the introduction of the Indian notation into Europe, about the 10th century, arithmetic has greatly changed its form, the whole algorithm, or practical operations with numbers, being quite altered, as the notation required; and the authors of arithmetic have gradually become more and more numerous. This method was brought into Spain by the Moors or Saracens; whither the learned men from all parts of Europe repaired, to learn the arts and sciences of them. This, Dr. Wallis proves, began about the year 1000; particularly that a monk, called Gilbert, afterwards pope, by the name of Sylvester II, who died in the year 1003, brought this art from Spain into France, long before the date of his death: and that it was known in Britain before the year 1150, where it was brought into common use before 1250, as appears by the treatise of arithmetic of Johannes de Sacro Bosco, or Halifax, who died about 1256. Since that time, the principal writers on this art have been, Barlaam, Lucas de Burgo, Tonstall, Aventinus, Purbach, Cardan, Scheubelius, Tartalia, Faber, Stifelius, Recorde, Ramus, Maurolycus, Hemischius, Peletarius, Stevinus, Xylander, Kersey, Snellius, Tacquet, Clavius, Metius, Gemma Frisius, Buteo, Ursinus, Romanus, Napier, Ceulen, Wingate, Kepler, Briggs, Ulacq, Oughtred, Cruger, Van Schooten, Wallis, Dee, Newton, Morland, Moore, Jeake, Ward, Hatton, Malcolm, &c, &c; the particular inventions or excellencies of whom, will be noticed under the articles of the several species or kinds of arithmetic here following, which may be included under these heads, viz, theoretical, practical, instrumental, logarithmical, numerous, specious, universal, common or decadal, fractional, radical or of surds, decimal, duodecimal, sexagesimal, dynamical or binary, tetractycal, political, &c.

Theoretical Arithmetic, is the science of the properties, relations, &c, of numbers, abstractedly considered; with the reasons and demonstrations of the several rules. Such is that contained in the 7th, 8th, and 9th books of Euclid's Elements; the Logisties of Barlaam the monk, published in Latin by J. Chambers, in 1600; the Summa Arithmetica of Lucas de Burgo, printed 1494, who gives the several divisions of numbers from Nicomachus, and their properties from Euclid, with the algorithm, both in integers, fractions, extraction of roots, &c; Malcolm's New System of Arithmetic, theoretical and practical, in 1730, in which the subject is very completely treated, in all its branches, &c. |

Practical Arithmetic, is the art or practice of numbering or computing; that is, from certain numbers given, to find others which shall have any proposed relation to the former. As, having the two numbers 4 and 6 given; to find their sum, which is 10; or their difference, which is 2; or their product, 24; or their quotient, 1 1/2; or a third proportional to them, which is 9; &c.—Lucas de Burgo's works contain the whole practice of arithmetic, then used, as well as the theory. Tunstall gave a neat practical treatise of Arithmetic in 1526; as did Stifelius, in 1544, both on the practical and other parts. Tartalea gave an entire body of practical arithmetic, which was printed at Venice in 1556, consisting of two parts; the former, the application of arithmetic to civil uses; the latter, the grounds of Algebra. And most of the authors in the list before enumerated, joined the practice of arithmetic with the theory.

Binary or Dyadic Arithmetic, is that in which only two figures are used, viz 1 and 0. See Binary. —Leibnitz and De Lagny both invented an arithmetic of this sort, about the same time: and Dangicourt, in the Miscel. Berol. gives a specimen of the use of it in arithmetical progressions; where he shews, that the laws of progression may be more easily discovered by it, than by any other method where more characters are used.

Common or Vulgar Arithmetic, is that which is concerning integers and vulgar fractions.

Decimal or Decadal Arithmetic, is that which is performed by a series of ten characters or figures, the progression being ten-fold, or from 1 to 10's, 100's, &c; which includes both integers and decimal fractions, in the common scale of numbers; and the characters used are the ten Arabic or Indian figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This method of arithmetic was not known to the Greeks and Romans; but was borrowed from the Moors while they possessed a great part of Spain, and who acknowledge that it came to them from the Indians. It is probable that this method took its origin from the ten fingers of the hands, which were used in computations before arithmetic was brought into an art. The Eastern missionaries assure us, that to this day the Indians are very expert at computing on their fingers, without any use of pen and ink. And it is asserted, that the Peruvians, who perform all computations by the different arrangements of grains of maize outdo any European, both for certainty and dispatch, with all his rules.

Duodecimal Arithmetic, is that which proceeds from 12 to 12, or by a continual subdivision according to 12. This is greatly used by most artificers, in calculating the quantity of their work; as Bricklayers, Carpenters, Painters, Tilers, &c.

Fractional Arithmetic, or of fractions, is that which treats of fractions, both vulgar and decimal.

Harmonical Arithmetic, is so much of the doctrine of numbers, as relates to the making the comparisons, reductions, &c of musical intervals.

Arithmetic of Infinites, is the method of summing up a series of numbers, of which the numbers of terms is infinite. This method was first invented by Dr. Wallis, as appears by his treatise on that subject; where he shews its uses in geometry, in finding the areas of superficies, the contents of solids, &c. But the method of fluxions, which is a kind of universal arithmetie of infinites, performs all these more easily; as well as a great many other things, which the former will not reach.

Instrumental Arithmetic, is that in which the common rules are performed by instruments, or some sort of tangible or palpable substance. Such are the methods of computing by the ten fingers and the grains of maize, by the East Indians and Peruvians, above-mentioned; by the Abacus-or Shwanpan of the Chinese; the several sorts of scales and sliding rules; Napier's bones or rods; the arithmetical machine of Pascal, and others; Sir Samuel Morland's instrument, described in 1666; that of Leibnitz, described in the Miscell. Berol.; that of Polenus, published in the Venetian Miscellany, 1709; and that of Dr. Saunderson, of Cambridge, described in the introduction to his algebra.

Integral Arithmetic, or of integers, is that which respects integers, or whole numbers.

Literal or Algebra Arithmetic, is that which is performed by letters, which represent any numbers indefinitely.

Logarithmical Arithmetic is performed by the tables of logarithms. These were invented by baron Napier; and the best treatise on the subject, is Briggs's Arithmetica Logarithmica, 1624.

Logistical Arithmetic. See Logistical.

Numerous or Numeral Arithmetic, is that which teaches the calculus of numbers, or of abstract quantities; and is performed by the common numeral or Arabic characters.

Political Arithmetic, is the application of arithmetic to political subjects; such as, the strength and revenues of nations, the number of people, births, burials, &c. See Political Arithmetic. To this head may also be referred the doctrine of Chances, Gaming, &c.

Arithmetic of Radicals, Rationals, and Irrationals. See Radical, &c.

Sexagesimal or Sexagenary Arithmetic, is that which proceeds by sixties; or the doctrine of sexagesimal fractions: a method which, it is supposed, was invented by Ptolemy, in the 2d century; at least they were used by him. In this notation, the integral numbers from 1 to 59 were expressed in the common way, by the alphabetical letters: then sixty was called a sexagena prima, and marked with a dash to the character, thus I′; twice sixty, or 120, thus II′; and so on to 59 times 60, or 3540, which is LIX′. Again, 60 times 60, or 3600, was called sexagena secunda, and marked with two dashes, thus I″; twice 3600, thus II″; and ten times 3600, thus X″; &c. And in this way the notation was continued to any length. But when a number less than sixty was to be joined with any of the sexagesimal integers, their proper expression was annexed without the dash: thus 4 times 60 and 25, is IV′XXV; the sum of twice 60 and 10 times 3600 and 15, is X″II′XV. So near did the inventor of this method approach to the Arabic notation: instead of the sexagesimal progression, he had only to substitute decimal; and to make the signs of numbers, from 1 to 9, simple characters, and to introduce another character, which should signify nothing by itself, but serving only to fill up places. —The sexagenæ integrorum were soon laid aside, in or- | dinary calculations, after the introduction of the Arabic notation; but the sexagesimal fractions continued till the invention of decimals, and indeed are still used in the subdivisions of the degrees of circular arcs and angles.

Sam. Reyher has invented a kind of sexagenal rods, in imitation of Napier's bones, by means of which the sexagesimal arithmetic is easily performed.

Specious Arithmetic, is that which gives the calculus of quantities as designed by the letters of the alphabet: a method which was more generally introduced into algebra, by Vieta; being the same as literal arithmetic, or algebra.—Dr. Wallis has joined the numeral with the literal calculus; by which means he has demonstrated the rules for fractions, proportions, extraction of roots, &c; of which a compendium is given by himself, under the title of Elementa Arithmeticæ, in the year 1698.

Tabular Arithmetic, is that in which the operations of multiplication, division, &c, are performed by means of tables calculated for that purpose: such as those of Herwart, in 1610; and my tables of powers and products, published by order of the Commissioners of Longitude, in 1781.

Tetractic Arithmetic, is that in which only the four characters 0, 1, 2, 3 are used. A treatise of this kind of arithmetic is extant, by Ethard or Echard Weigel. But both this, and binary arithmetic, are little better than curiosities, especially with regard to practice; as all numbers are much more compendiously and conveniently expressed by the common decuple scale.

Vulgar, or Common Arithmetic, is that which is conversant about integers and vulgar fractions.

Universal Arithmetic, is the name given by Newton to the science of algebra; of which he left at Cambridge an excellent treatise, being the text-book drawn up for the use of his lectures, while he was professor of Mathematics in that University.

previous entry · index · next entry

ABCDEFGHKLMNOPQRSTWXYZABCEGLMN

Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

This text has been generated using commercial OCR software, and there are still many problems; it is slowly getting better over time. Please don't reuse the content (e.g. do not post to wikipedia) without asking liam at holoweb dot net first (mention the colour of your socks in the mail), because I am still working on fixing errors. Thanks!

previous entry · index · next entry

ARIES
ARISTARCHUS
ARISTOTELIAN
ARISTOTELIANS
ARISTOTLE
* ARITHMETIC
ARITHMETICAL
ARTILLERY
ASCENDANT
ASCENDING
ASCENSION