997 



NUMBERS, APPELLATIONS OF. 



NUMBERS, OLD APPELLATIONS OF. 



the seat on which he sits has some such lever or spring beneath it ; 

 none of these machines, however, has come regularly into use. 



A machine called the timbre additioneur combines the principles of 

 both groups just described : namely, those which print consecutive 

 numbers on tickets and pages, and those which register quantities t>n 

 dial-faces. There are stamping dies, levers, inking-tables, wheel-work, 

 and index-hands. With the full apparatus, the machine can number 

 and stamp such documents as bills, letters, and share-certificates, and 

 can record on the dial the number of stampings thus effected ; while, 

 with the stamping and inking apparatus removed, it may be made to 

 count the passengers through a turnstile, or the revolutions of a coach- 

 wheel, or the length of yarn spun by a machine, or that of cloth 

 woven, or the revolutions of a fly-wheel or water-wheel. 



NUMBERS, APPELLATIONS OF. Various names have been given 

 to classes of numbers, each expressive of properties common to all in 

 its class : they are pointed out in the following list : 



The whole scale, 1, 2, 3, &c., is called that of natural numbers ; it 

 is subdivided into the scale of odd numbers, 1, 3, 5, &c., and even 

 numbers, 2, 4, 6, &c. These again are subdivided into oddly odd 

 numbers, 3, 7, 11, &c. ; evenly odd numbers, 1, 5, 9, &c. ; oddly even 

 numbers, 2, 6, 10, &c. ; and evenly even numbers, 4, 8, 12, &c. These 

 latter appellations are not in universal use, though they are very con- 

 venient. Thus with reference to division by two and by four, all 

 numbers have names ; but not with reference to any higher numbers. 

 The expression of a number which divided by m leaves a remainder n 

 (namely, mx + n, where x is a whole number) is so simple, that it is 

 more easily written than described. When is included in the list, it 

 is considered as divisible without remainder by every number. 



The division of numbers into square numbers, 1, 4, 9, 16, &c. ; cube 

 numbers, 1, 8, 27, 64, &c. ; fourth powers, 1, 16, 81, 256, &c., and so 

 on, may be carried to any extent. 



A prime number is any one of the list 1, 2, 3, 5, 7, 11, 13, &c., no 

 one of which is divisible by any number except unity and itself. A 

 camporite number is any one which is not prime. 



A fii/wate number is any one out of the following series, the first 

 excepted, which is only introduced as a basis. 



3 



6 &c. 



&c. 



&c. 



Each number is the sum of the numbers in the preceding row : thus 

 84 is the sum of 1, 6, 21, and 56, and 84 is the fourth number of 

 the fifth order of figurate numbers. The th number in the first 



n + l . n + 1 + 2 . 



order is n 5 , in the second order n <j 5 in the third 



- and so on. 



234 



Polygonal numbers, as their name imports [POLYGON], may be sub- 

 divided into triangular, quadrangular, pentagonal, hexagonal, &c. 

 To find the numbers which bear the name of an M-sided figure, form a 

 series beginning with 1 and consisting of terms increasing in arithmetical 

 progression, with a common difference 2 : and form the sums of 

 terms of these series in the manner described. Thus for decagonal 

 numbers, we have 



1 9 17 25 33 41 &c. 



1 10 27 52 85 126 &c. 



and the decagonal numbers are 1, 10, 27, &c. The mth number of the 

 n-sided order of figures ia 



m 1 

 1 + nm 



25 36 &c. 

 35 51 &c. 

 45 66 &o. 



The following are some of the polygonal numbers : 



Triangular 1 3 6 10 15 21 &c. 



Quadrangular 1 4 9 16 



Pentagonal 1 5 12 22 



t Hexagonal 1.6 15 28 



. 



Pyramidal numbers are formed by summing the polygonal numbers ; 

 thus, to find pentagonally pyramidal numbers, take the pentagonal 

 number*- ^ ^ ^ ^ 51 &c 



1 6 18 40 75 126 &c. 



Numbers were once considered as abundant, perfect, and defective. 

 An abundant number was one in which the sum of all its divisors 

 (unity included, but not itself) exceeds the number : thus 12 is an 

 abundant number, because 1 + 2 + 3 + 4 + 6 is greater than 12. A 

 perfect number was one in which the sum of all the divisors was equal 

 to the number : thus 6 is 1 + 2 + 3, and is a perfect number, as is 28, 

 orl + 2 + 4 + 7 + 14- A defective number was one in which the sum 

 of the divisors is less than the number, as 10, in which 1 + 2 + 5 is less 

 than 10. Whenever 2" -1 is a prime number, then 2 ; -'(2 1) is 



a perfect number; thus 2' 1, or 127, is a prime number, whence 2" 

 (2' 1), or 64 x 127, or 8128, is a perfect number. 



Amicable numbers are those each of which is equal to the sum of 

 all the divisors of the other. Such are 



284 and 220 



17296 and 18416 



9363583 and 9437056 



Other names have been invented descriptive of classes of numbers ; 

 but the preceding are those which most often occur in the past history 

 of mathematics. With the exception of SQUARE, CUBE, PRIME, 

 even, and odd, the preceding appellatives rarely appear in modern 

 works. 



NUMBERS, OLD APPELLATIONS OF. The student of books 

 verging on the middle ages will occasionally meet with some designa- 

 tions of numbers, or rather of the ratios of numbers 'to numbers, 

 which may need explanation in a work of reference. Corresponding 

 terms are found in the Greek writers, particularly in those on music, 

 and they seem to have obtained universal currency in the middle ages 

 by means of the work of Boethius on Arithmetic, which was a general 

 object of study up to the middle of the 16th century. 



These words illustrate a fault which has been avoided in our day by 

 the adoption of the opposite extreme. The ancients often overloaded 

 a subject with terms; the moderns cannot prepare them as fast as 

 they want them. The higher analysis now abounds with objects of 

 thought for which there are no names except complex algebraical 

 symbols ; the old arithmeticians strove to find names for all the 

 varieties of numerical ratio. 



In describing these words, we shall, where we can, use the English 

 form of the Latin adjectives, to avoid overloading our article with 

 Latin words: the adjectives accompany the word numerus, not ratio. 

 The ratio of the greater integer number to the less was one of the 

 following five, multiple, superparticular, superpartient, multiple super- 

 particular, or multiple superpartient. The ratio of the less to the 

 greater was either submultiple, subsuperparticular, subsuperpartient, 

 multiple subsuperparticular, or multiple subsuperpartient. 



The term multiple has been preserved, and its species, duple (double), 

 triple, quadruple, quintuple, &c. Thus 10 to 2 is a multiple ratio, 

 namely quintuple : that of 2 to 10 is submultiple, namely subquiutuple. 



Superparticular ratio (part, that is, aliquot part, over) is when the 

 greater contains the less and a submultiple of the less : its varie- 

 ties are sescuple or sesquialter, sesquitertius, sesquiquartus, &c. 

 Thus the following ratios are superparticular : 15 to 10, which is 

 sesquialter ; 16 to 12, which is sesquitertius ; 15 to 12, sesquiquartus ; 

 and so on. But the ratio of 12 to 15 is subsuperparticular, namely 

 subsesquiquartus. One of these names is still preserved in our 

 language, in the sesquialter stop of an organ. The ratio of 3 to 2 

 (a sesquialter ratio) is that of the length of a pipe to the length of a 

 pipe which sounds the fifth above the note of the first. Accordingly 

 when a stop was made to sound with the ordinary stop, but a fifth 

 above it, the name sesquialter was given to the stop which gave the 

 higher note. 



Superpartient ratio, according to Boethius, is that in which the 

 major term is twice the minor all but an aliquot part. Its varieties 

 are superbipartient (ratio of 5 to 3), supertripartient (ratio of 7 to 4), 

 superquadripartient (of 9 to 5), and so on. Thus the ratio of one and 

 /ow-fifths to one is super-juarfri-partient. According to Boethius, 

 then, the intermediate ratios of one and two-fifths and one and three- 

 fifths to one have no names. Some of his followers extend the name 

 of superpartient to these, and some would invent the adjectives super- 

 biquintus and supertriquintus to signify them ; other used superbi- 

 partiens quintas and supertripartieus quiutas. Multiple superparticular 

 .and multiple superpartient ratios have in the major term a multiple of 

 the minor together with the fraction which gives the remaining 

 adjectives. Thus, the ratio of 7i to 1 is multiple superparticular, 

 being septuplus sesquiquartus ; and 7J to 1 is multiple superpartient, 

 being septuplus supertripartiens. The preposition sub serves as before, 

 prefixed to super, to denote the inverse ratios. The reader may fancy 

 for himself, if he can, the beauty of a treatise in which the ratio of 4 

 to 11 is expressed by duplus subsupertripartiens. Not that the writers 

 of these works were ignorant of more simple phrases ; and we remem- 

 ber one place in which the example of Aristotle is brought forward to 

 show that it would not be wrong to use them. 



Of means, or medieties, Boethius discusses ten, to which Jordanus 

 added an eleventh. The first three bear the names which have 

 descended to us, arithmetic, geometric, and harmonic ; and all are as 

 follow. Let a, b, c, be three numbers, of which a is the greatest and 

 c the least. 



The works of Boethius and his followers consist in dissertations on 

 what would now be called the most obvious properties of the ratios of 

 numbers, enriched with comments of every species, from numerical to 



