ARITHMETIC, the art of numbering ; or, that part of mathematics which consi ders the powers and properties of num bers, and teaches how to compute or calculate truly, and with expedition and ease. By some authors it is also defined the science of discreet quantity. It con sists chiefly in the four great rules or operations of Addition, Subtraction, Mul tiplication, and Division. Concerning the origin and invention of ari thm etic we have very little Information : history fixes nei ther the author nor the time. Some knowledge, however, of numbers must have existed in the earliest ages of man kind. This knowledge would be sug gested to them, whenever they opened their eyes, by their own fingers, and by their flocks and herds, and by the variety of objects that surrounded them. At first, indeed, their powers of numeration would be of very limited extent ; and be fore the art of writing was invented, it must have depended on memory, or on such artificial helps as might most easily be obtained. To their ten fingers they would, without doubt, have recourse in the first instance ; and hence they would be naturally led to distribute numbers into periods, each of which consisted of ten units. This practice was common among all nations, the ancient Chinese, and an obscure p&ople mentioned by Aristotle, excepted. But though some kind of computation must have com menced at a very early period, the intro duction of arithmetic as a science,and the improvments it underwent, must, in a great degree, depend upon the introduc tion and establishment of commerce ; and as commerce was gradually extended and improved, and other sciences were dis covered and cultivated, arithmetic would be improved likewise. It is therefore probable, that if it was not of Tyrian in vention,it must have been much indebted to the Plicenicians or Tyrians. Proclus, indeed, in his comtucntary on the first book of Euclid, says that the Phoenicians, by reason of their traffic and commerce, were the first inventors of arithmetic ; and Strabo also informs us, that in his time it was attributed to the Phoenicians. Others have traced the origin of this art to Egypt ; and it has been a general opi nion, sanctioned by the authorities of So crates and Plato, that Theut or Thot was the inventer of numbers; that from hence the Greeks adopted the idea of ascribing to their Mercury, corresponding to the Egyptian Theut or Hermes, the superin tendance of commerce and arithmetic. With the Egyptians we ought also to as sociate the Chaldeans, whose astronomi cal disquisitions and discoveries, in which they took the lead, required a considera ble acquaintance with arithmetic. From Asia it passed into Egypt, as Josephus says, by means of Abraham. Here it was greatly cultivated and improved ; inso much, that a large part of the Egyptian philosophy and theology seems to have turned altogether upon numbers. Kirch er skews, that the Egyptians explained every thing by numbers ; Pythagoras himself affirming, that the nature of num bers 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 Pythagoras and his followers ; and among them it was the subject of particu lar attention, as we perceive in the wri tings of Euclid, Archimedes, and others ; with the improvements derived from them, it passed to the Romans, and from them it came to us. The ancient arith
metic was very different from that of the moderns in various respects, and particu larly in the method of notation. The In dians are at this time very expert in com puting, by means of their fingers,without the use of pen and ink ; and the natives of Peru, by the different arrangements of their grains of maze, surpass the Euro pean, aided by all his rules, with regard both to accuracy and dispatch. The He brews and Greeks, however, at a very early period, and after them also the Ro mans, had recourse to the letters of their alphabet for the representation of num bers. The Greeks, in particular, had two different methods : the first resembled that of the Romans, which is sufficiently known, as it is still used for distinguish ing the chapters and sections of books, dates, &c. They afterwards hada better met.lr:i in which the first nine letters of the aienabet represented the first num bers from 1 to 9, and the next nine let ters represented any number of tens, from 1 to 9, that is, 10, 20, &c. to 90. Any number of hundreds they expressed by other letters, supplying what they wanted by some other marks, or charac ters : and in this order they proceeded, using the same letters again, with differ ent marks to express thousands, tens of thousands, hundreds of thousands, &c. ; thus approaching very near to the more perfect decuple scale of progression used by the Arabians, who acknowledge, as some have said,that they received from the Indians. Archimedes, also, in " Arenarius," used a particular scale and notation of his own. In the second cen tury of the Christian era,Ptol ern) is sup posed to have invented the sexagesim al numeration and notation, and this method is still used by astronomers and others, for the subdivisions of the degrees of cir cles. These several modes of notation, above recited, were so operose and in convenient, that they limited the extent, and restrained the progress of arithmetic, so that it was applicable, with great diffi culty and embarrassment, to the other sciences, which required its assistance. The Greeks, if we except Euclid, who in his elements furnished many plain and useful properties of numbers, and Archi medes in his Arenarius, contributed little to the advancement of this science to wards perfection. From Boethius we learn, that some Pythagoreans had in vented and employed, in their calcula tions, nine particular characters, whilst others used the ordinary signs, namely, the letters of the alphabet. These cha racters he calls apices; and they are said greatly to resemble the ancient Arabic characters, which circumstance suggests a suspicion of their authenticity. Indeed, the MSS. of Boethius, in which these characters, resembling those of the Ara bian arithmetic, are found, not being more ancient than three or four centuries, con firm the opinion that they are the works of a copyist. Upon the whole, this trea tise of Boethius does not warrant our re jecting the commonly received system with regard to the origin of our arithme tic ; hut if we suppose that the Arabians derived their knowledge of it from the Indians, it is more probable that it was one of the inventions which Pythagoras spread among the Indians, than that those persons should have obtained it from the Greeks.