ARITHMETIC.
227
greater: thus IV signifies four, XL forty, XC ninety, CD fourhundred.
Sometimes 13 is used instead of D for 500, and the value isincreased ten times, by annexing 3 to the right hand.
Thus If) signifies 500. Also G'jf) is used for 1000If)f) 5000 ( C]3f) for 10000
1,033 50000 CCC1333 f° r 100000
Sometimes thousands are represented by drawing a line overthe top of the numeral, V being used for five thousand, L forfifty thousand, CC two hundred thousand.
(5.) A new kind of arithmetic, called Sexagesimal, was.in-vented, as is supposed, by Claudius Ptolouurus, about A. D.200. The design of it was to remedy the difficulties of the com-mon method, especially with regard to fractions. In this kind ofarithmetic, every unit was supposed to be divided into 60 parts,and each of these into 60 others, and so on: hence any numberof such parts were called sexagesimal fractions; and to make thecomputation in whole numbers more easy, lie made the progressionin these also sexagesimal. Thus from 1 to 59 was marked in thecommon way: thhn 60 was called a sexagesima prima, or first sexa-gesimal integer, and had one singWlash over it; so ()0 was ex-pressed thus 1'; and so on to 5S) times 60, or 3540, which wasthus expressed LIX.'. He now proceeded to 60 times 6o, whichhe called a sexagesima sec.unda, and was thus expressed I". Inlike manner, twice 60 limes 60, or 7200, was expressed by 11";and so on till he came to 00 times 3600, which was a third sexa-gesimal, and expressed thus, I"'. If any number less than 60 wasjoined with these sexagesimals, it was added in its proper charac-ters, without any dash ; thus I'XV represented 60 and 15, or 75;IV'XXV is four times 60 and 25, or 265 ; XTl'XV, is ten times3600, twice 60 and 15, or 36,135, &e. Sexagesimal fractionswere marked by putting the dash at the foot, or on the left' handof the letter: thus 1„ or'1, denoted^; 1, or "I, -j^, &c.
The numeral characters which we now use, and which af-ford the most perfect method of notation, came into Europe fromthe Arabian', by the way of Spain. The Arabs , however, do notpretend to lie the inventors of them, but acknowledge that theyreceived them from the J ndians.
Dr. Wallis mentions an inscription on a chimney in the parson-age house of lielendon in Northamptonshire where the dale isexpressed by M° 133, instead of 1133. Mr. Lutl’kin furnishesa still earlier instance of their use, in the window of a house,part of which is a Roman wall, near the market-place in Col-chester ; where between two carved lions stands an escutcheonwith the figures 1090. Dr. Wallis is of opinion that these cha-racters must have been used in England at least as long ago as theyear 1050, if not in ordinary affairs, at least in mathematical ones,and in astronomical tables.
The introduction of the Arabian characters in notation did notimmediately put au end to the sexagesimal arithmetic. As thishad been used in all the astronomical tables, it was for their sakesretained for a considerable time. The sexagesimal integers wentfirst out, but the fractions continued till the invention of de-cimals.
(6.) The most ancient treatises extant upon the theory of Arith-metic are the 7th, 8th, and 9th, books of Euclid’s elements,where he treats of proportion and of prime and composite num-bers.
About the same time, Nicomachus the Pythagorean , wrote atreatise of the theory of arithmetic, consisting chiefly of the dis-tinctions and divisions of numbers into classes, as plain, solid,triangular, quadrangular, numbers odd and even, &c. Ilis arith-metic was published at Paris in 1538.
The next remarkable writer on this subject is Boethius , wholived at Rome in the time of Theodoric the Colh. He is sup-posed to have copied most of his work from Nichomacluis. Acompendium of the ancient arithmetic, written in Greek, bvPsellus , in the ninth century from our Saviour, was given us inLatin by Xylander , in 1556.
From this time no remarkable writer on arithmetic appearedtill about the year 1200, when Jordanus of Namur wrote a treatiseon this subject, which was published and demonstrated by JoannesFaber Stapulensis in the 15th century, soon after the invention ofprinting. The same author also wrote upon the new art of com-6
putalion by the Arabic figures, am! called this book “ AlgorismusDemonstratus.” Dr. Wallis says this manuscript is in the Savil-lian library at Oxford, but it hath never yet been printed.
(2.) The first entire body of practical arithmetic, was given byNic. Tartaglia, a Venetian, in 1556, consisting of two books; 1.the application ot arithmetic to civil uses ; 2. the grounds of alge-bra. Something had been done before by' Stitelius, in 1544,where we have several particulars concerning the application ofirrationals, cossics, &c. no where else to be met with.
The theory of arithmetic is joined with the practice, and evenimproved in several parts, by Maurolycus, in ins “ Opuscuia Ma-th nnatica, 1575 by Heneschius, in his “ Arithmetica Pcrfecta,1609,” where the demonstrations are all reduced into the form ofsyllogisms ; and by Tacquet , in his “ Theoria et Praxis Arith-metices, 1704.”
Among tlie innumerable multitude of treatises on practicalarithmetic those of Dr. Hutton and Air. Ronnycastie justly claimtlie preference.
As to the circulating decimals, Dr. Wallis is tlie first who tookmuch notice of them. He is also the author of tlie -Arithmetic ofinfinites, which Iras been very usefully applied to geometry.
Tlie greatest improvement, however, which tlie art of computa-tion ever received, is tire invention of logarithms. The honour ofthis invention is unquestionably due to Lord Napier, of Merchis-I011 in Scotland , about tlie end of the I6U1 or beginning of the17th century. P>y these means arithmetic has advanced to a de-gree of perfection which the ancients could never have imaginedpossible, much less imped to attain; ami we believe it may nowbe reckoned one of those few sciences which have arrived at theirutmost height, and winch is in its nature capable of little furtherimprovement.
Explanation of the Signs.
(7.) Tlie following characters or signs are frequently used inarithmetical operations:
= • , /'equality n. /is equal to x / 4 & 4=8
-j- j i addition jrfi i plus or more 1 1 S-(-2=10
— 6 <6 subtraction yew minus or less 3 « 8—2=6
X f '.5 J multiplication f g J into f 7 8x2=16
—■ /" gA division /■_«> "S by / \ 8=2=4
y/ ft '3 i the square root k -3 1 the square root a i y/4=2
a y/ I the cube root w a f the cube root 1 e 3/8=2
::: J Vproportion J Vis to, as, is to / V2 : 3 :: 4:6
Besides these, x between two numbers denotes their difference.
Notation and Numeration.
(8.) Notation is tlie expressing of any proposed number, citherby words or characters. All numbers are expressible by these tencharacters or figures ;1 2 3456 7 8
One, two, three, four, five, six, seven, eight,
9 °
nine, cypher. Tlie nine first are called significant figures ordigits. When placed .singly, they denote tlie simple numberssubjoined to the characters. The usual method ot notation bythese figures is so contrived, that any character is increased in itsvalue in a tenfold proportion, for every' place it is removed fartherto the left. Thus, in these figures 333, the first 3 (reckoningfrom tlie rigid to the left) is 3 ones; the second is 3 tens, and thethird is 3 hundreds. And although the cypher signilies nothingby itself, yet when put on the right of any of the other figures, itincreased! their value in a tenfold proportion, merely by changingtheir position. Thus, though 2 standing alone, or in the firstplace, represents only two ones, yet when a cypher is written.onthe right of it thus, 20, it represents 2 tens, or twenty; and if an-other cypher be atlixed thus, 200, it will represent two hun-dreds, &c.
Since tlie class or scale of numbers formed by means of these tencharacters lias been universally received by most nations whohave cultivated this science, it is probably the most convenientfor general use. Other scales may lie assumed: thus, if eightwere the scale, 6 limes 3 would be two classes and two uin-s,and the number 18 would then be represented by 22. If 12were the scale, 5 times 9 would be three classes and nine units,and 45 would be represented by 39, &c.
The cypher is of great importance in arithmetic, and seems tobe of the most easy use ui tlie decadary system; the ancient
Greeks
Lv