529 



ARITHMETIC, POLITICAL. 



ARMA'DA. 



530 



The only material addition which has been made to this groundwork 

 of arithmetic is the invention of decimal fractions. This is an exten- 

 sion of the principal of local value, of so simple a character, that it is 

 surprising the Hindoos never adopted it. They write fractions as we 

 do, omitting only the line .which separates the numerator and deno- 

 minator, and they make great use of decimal fractions in approxi- 

 mating to the square roots of numbers, but without any peculiarity of 

 notation. 



The first fractional notation which we find among the Greeks con- 

 sisted in writing the denominator where we now write the exponent. 



3"6'5. 



This 



207 



Thus, retaining our imitation, would be written 2"7 



Ho 



system is principally used by Diophantus ; and in Eutocius we also 

 find a peculiar symbol, something like K, for one half. Ptolemy made 

 a further step, in the application of the method of dividing the circle 

 to all units whatsoever, known by the name of the sexagesimal notation. 

 The degree of the circle is divided into sixty minutes, the minute into 

 sixty seconds, that again into sixty thirds, and so on : Ptolemy divides 

 every unit in the same manner. We have still retained in our division 

 of the circle the ', ", '", &c., used by him. In the notation alluded to 

 (which is that of Ptolemy in the particular point referred to) 



would denote 



27 units, 



27 33' 21" 3'" 

 33 21 3 



60 3600 



and 



216,000 



This sexagesimal notation retained ita ground until the introduction of 

 the Arabic numerals, and with the aid of tables of reduction, was of 

 material use. 



Stifelius and Stevinus (P. 440) used circumflexed digits instead of 

 ', ", Ac., in the sexagesimal system, and an application of the same 

 principle to the decimal system was first made by Albert Girard in or 

 about 1590. This consisted in expressing fractions by tenths, 

 hundredth*, &c., in the following way 



3 4 (0) (1) (2) 



16 |0 j QQ would be written 1634, 



the number in parentheses over a digit being the exponent of the power 

 of ten, which must be used with that digit as a denominator. Here 

 the application uf the principle of local value practically begins ; aud it 

 is clear, from the examples cited by Mr. Peacock, that the cipher was 

 made use of to denote a vacant column. One of those examples is 



941 304 



Decimal fractions were slowly introduced into England in the first 

 half of the 17th century, by Wright, Napier, Wingate, Johnson, 

 Oughtred, &c. From that time the modern form of the Indian 

 arithmetic must be considered as established. The invention of 

 LOGARITHMS (which see) is the principal aid to calculation which has 

 been engrafted upon the system. 



We subjoin a list of names, which the reader may consult on 

 various points connected with the history of arithmetic, either in this 

 work or elsewhere. The figures refer to the century before or after 

 Christ in which the individual is supposed or known to have lived ; 

 and the works cited are in quotation marks. 



B.C. 6 Pythagoras. 4 Euclid, Aristotle, Plato. 3 Archimedes, 

 Apollonius. 1 Vitruviua. 



A.C. 2 -Ptolemy, Diophantus. 3 Nichomachus. 4 Pappus, 

 Theon. 5 Proclus, Eutocius. 6 Boethius. 9 Mahommed Ben 

 Musa. 11 Gerbert. 12 Jordanus, Leonardo Bonacci. 13 Sacro- 

 boaco, Planudes. 15 Lucas di Borgo. 16 Scheubelius, Stifehus, 

 Recorde, Albert Girard. 17 Briggs, Napier, Oughtred, Stevinus, 

 Wright, Bouillaud, Mersenne, Wallis, ' Algebra ; ' Bachet de Meziriac. 

 18 Weidler, ' Historia Astronomisc ; ' Ktestner, ' Geschichte der 

 Mathematik ; ' Montucla, ' Hist, des Mathdmatiques ; ' Delambre, 

 'Hist, de 1' Astronomic Ancienne;' Button, 'Tracts,' 'History of 

 Algebra ; ' Colebrooke, Preface to ' Bija Ganita ;' Chasles, various 

 writings ; Libri, ' History of Mathematics in Italy.' 



We need not of course refer to the work of Dr. Peacock, which we 

 have so often cited. 



ARITHMETIC, POLITICAL. [STATISTICS, INTEREST, ANNUITIES, 



I'MH I.ATION, MORTALITY, &c.] 



ARITHMETIC Oh' SINES. [TRIGONOMETRY.] 



ARITHMETIC, SPKCIOUS. [VIETA.] 



ARITHMETICAL COMPLEMENT is that which a number wants of 

 the next highest decimal denomination. Thus what 7 wants of 10, or 

 3; 32 -.1 100, n 8; 159 of 1000, or 841 ; '017 of 1, or '983 : are the 

 arithmetical complements of these numbers. The best way to find 

 them is to begin from the left, subtract every figure from 9, and the 

 last significant figure from 10, as in the following examples, which 

 include all the cases : 



No. 

 Ar. Co. 



17634 

 82J06 



19-0018 

 80-9982 



1734000 

 8266000 



ARITHMETICAL MEAN", liy the arithmetical mean is meant, 

 that number or fraction which lies between two others, and is equally 

 distant fnjin both. Thus the arithmetical mean between 6 and 14 is 



ARTS A.ND 3CI. D1V. VOL. I. 



10. To find this arithmetical mean, take the half sum of the two 

 numbers. Thus, that of 4 and 17 is 10J. But any numbers are also 

 said to be arithmetical means between two others, when all together 

 form a series of equally increasing or decreasing numbers. Thus, 8, 

 10, 12, are three arithmetical means between 6 and 14. To interpose 

 any number of arithmetical means between two numbers, divide the 

 difference of those two numbers by one more than the number of 

 means required, which gives the difference between the means. Thus, 

 to interpose four arithmetical means between 27 and 102, divide 75 

 (102 27) by 5 (4 + 1) which gives 15. The means are, therefore, 

 27 + 15 or 42, 42 + 15 or 57, 57 + 15 or 72, and 72 + 15 or 87. If 

 the means are fractional, the same process is employed. [AVERAGE.] 



ARITHMETICAL PROGRESSION is a name given somewhat 

 improperly to a series of numbers which increase or decrease by equal 

 steps, such as 1, 2, 3, &c.; 2, 4, 6, &c. ; 14, 2, 24, &c. The difference 

 between any two successive terms, being common to all, is called the 

 common difference. The data which distinguish one arithmetical 

 progression from another, are the first term, the common difference, and 

 the number of terms : from these it is easy to find the last term and 

 the sum of all the terms. To find the last term, multiply the common 

 difference by one less than the number of terms, and add the first 

 term to the product. To find the sum of all the terms ; take 



the number of terms, 



the sum of the first and Lost, 



and multiply the half of either (whichever is most convenient) by the 

 other, or take half the product of the two. Thus, for 100 terms of 

 either of the series 



6 9 12 &c. (A) 



14 2 2J ... &.c. (B) 



To find the last, or 100th, term of (A), multiply 3, the common dif- 

 ference by 99 (1001) and add 3, the first term, which gives 300. 

 Similarly to find the last, or 100th, term of (B), multiply 4 by 99 and 

 add 1, which gives 504- For the sums we have 



(A) (B) 



No. of terms 100 100 



Sum of first and last 303 514 



Multiply half of 100 by 303, aud by 514, which gives 15150 for the 

 sum of (A), and 2575 for that of (B). 



Algebraically, let a, be the first term, x the common difference, 

 and n the number of terms. Let z be the last term, and S the sum. 

 Then 



2 = a + (n 1) x 



H j 



S = 4 > (a + z) = ia + n j: 



2 



from which any three of the letters being given, the other two can be 

 found. 



For the theory of which this article is a part, see SERIES, DIFFER- 

 ENCES, INTEGRATION. 



ARITHMETICAL PROPORTION, the relation which exists 

 between four numbers, of which the first and second have the same 

 difference as the third and fourth. Thus : 



12 81 82 



73 16 12 



24 3J If 24 



are severally in arithmetical proportion, and in every such proportion 

 the sum of the extremes is equal to that of the means. Thus : 



12 + 7 = 3 + 16. 



ARK, a chest or coffer. This term is frequently used by our earliest 

 English and Scottish poets. 



In 1347, in the brewhouse of the priory of Lindisfarue, was an ark 

 for meal (see Raine's ' North Durham," p. 92) ; and among other articles 

 of furniture occurring in an inventory of the household goods belong- 

 ing to Sherborn hospital, taken in 1636, in the boulting-house, is, 

 ' 1 boulting ark.' (Hutch. ' Hist. Durh.' ii. p. 599.) The same word is 

 still in use, in the north of England, for the chest which is employed 

 in containing meal. 



Noah's ark was so named from its supposed resemblance to an ark 

 or chest; by which name it occurs both in the Gothic and Anglo- 

 Saxon versions of the passage in Luke, xvii. 27. Wiclif, in this passage, 

 instead of ark, reads ship. The same term ark is used in our transla- 

 tion of the Old Testament, for the basket or cradle in which the infant 

 Moses was laid when he was put into the Nile. (See Boucher's ' Glos- 

 sary,' by Stevenson.) 



The ark of the tabernacle is of the same derivation. While travelling 

 in the Wilderness the Israelites bore it in the form of a shrine, though 

 it is said to have been fifty-five feet long, eighteen broad, and eighteen 

 high. When the temple was built, the form was preserved, an oblong 

 square. Into this none but priests were admitted, the people being 

 assembled; and the sacrifices performed, in a court in front, while the 

 sacred apartment was in the innermost recess. 



ARMA'DA. This term, which is derived from the Latin word 

 armata, ' armed,' and consequently comes from the same root as the 

 French armie and our army, is used in Spain to denote exclusively a 

 naval armed force, or fleet of war. Flota is used in the same language 

 for a fleet of merchant-men. Annado, which occurs in Shakspere's 



M M 



