ARISUGAWA

arisugawa, the name of one of the royal families of japan, going back to the seventh son of the mikado go-yozei (d. 1638). after the revolution of 1868, when the mikado mutsu-hito was restored, his uncle, prince taruhito arisugawa (1835-1895), became commander-in-chief, and in 1875 president of the senate. after his suppression of the satsuma rebellion he was made a field-marshal, and he was chief of the staff in the war with china (1894-95). his younger brother, prince takehito arisugawa (b. 1862), was from 1879 to 1882 in the british navy, serving in the channel squadron, and studied at the naval college, greenwich. in the chino-japanese war of 1894-95 he was in command of a cruiser, and subsequently became admiral-superintendent at yokosuka. prince arisugawa represented japan in england together with marquis ito at the diamond jubilee (1897), and in 1905 was again received there as the king's guest. arithmetic (gr. [greek: arithmaetikae], sc. [greek: technae], the art of counting, from [greek: arithmos], number), the art of dealing with numerical quantities in their numerical relations. 1. arithmetic is usually divided into _abstract arithmetic_ and _concrete arithmetic_, the former dealing with numbers and the latter with concrete objects. this distinction, however, might be misleading. in stating that the sum of 11d. and 9d. is 1s. 8d. we do not mean that nine pennies when added to eleven pennies produce a shilling and eight pennies. the sum of money corresponding to 11d. may in fact be made up of coins in several different ways, so that the symbol "11d." cannot be taken as denoting any definite concrete objects. the arithmetical fact is that 11 and 9 may be regrouped as 12 and 8, and the statement "11d. + 9d. = 1s. 8d." is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (s 11). 2. the various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view. there are three main purposes, the practical, the educational, and the scientific; i.e. the subject may be studied with a view to technical skill in dealing with the arithmetical problems that arise in actual life, or for the sake of its general influence on mental development, or as an elementary stage in mathematical study. 3. the practical aspect is an important one. the daily activities of the great mass of the adult population, in countries where commodities are sold at definite prices for definite quantities, include calculations which have often to be performed rapidly, on data orally given, and leading in general to results which can only be approximate; and almost every branch of manufacture or commerce has its own range of applications of arithmetic. arithmetic as a school subject has been largely regarded from this point of view. 4. from the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy. this aspect of the matter, however, belongs mainly to the period when arithmetic was studied almost entirely for commercial purposes; and even then accuracy was not found always to harmonize with actuality. the development of physical science has tended to emphasize an exactly opposite aspect, viz. the impossibility, outside a certain limited range of subjects, of ever obtaining absolute accuracy, and the consequent importance of not wasting time in attempting to obtain results beyond a certain degree of approximation. 5. as a branch of mathematics, arithmetic may be treated logically, psychologically, or historically. all these aspects are of importance to the teacher: the logical, in order that he may know the end which he seeks to attain; the psychological, that he may know how best to attain this end; and the historical, for the light that history throws on psychology, the logical arrangement of the subject is not the best for elementary study. the division into abstract and concrete, for instance, is logical, if the former is taken as relating to number and the latter to numerical quantity (s 11). but the result of a rigid application of this principle would be that the calculation of the cost of 3 lb. of tea at 2s. a lb. would be deferred until after the study of logarithms. the psychological treatment recognizes the fact that the concrete precedes the abstract and that the abstract is based on the concrete; and it also recognizes the futility of attempting a strictly continuous development of the subject. on the other hand, logical analysis is necessary if the subject is to be understood. as an illustration, we may take the elementary processes of addition, subtraction, multiplication and division. these are still called in text-books the "four simple rules"; but this name ignores certain essential differences. (i) if we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) if on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz. addition and subtraction. (iii) the inclusion of the four processes under one general head fails to indicate the essential difference between addition and multiplication, as direct processes, on the one hand, and subtraction and division, as inverse processes, on the other (s 59). 6. the present article deals mainly with the principles of the subject, for which a logical arrangement is on the whole the more convenient. it is not suggested that this is the proper order to be adopted by the teacher. i. number 7. _ordinal and cardinal numbers._--one of the primary distinctions in the use of number is between ordinal and cardinal numbers, or rather between the ordinal and the cardinal aspects of number. the usual statement is that _one, two, three_, ... are cardinal numbers, and _first, second, third_, ... are ordinal numbers. this, however, is an incomplete statement; the words one, two, three, ... and the corresponding symbols 1, 2, 3, ... or i, ii, iii, ... are used sometimes as ordinals, i.e. to denote the place of an individual in a series, and sometimes as cardinals, i.e. to denote the total number since the commencement of the series. on the whole, the ordinal use is perhaps the more common. thus "100" on a page of a book does not mean that the page is 100 times the page numbered 1, but merely that it is the page after 99. even in commercial transactions, in dealing with sums of money, the statement of an amount often has reference to the last item added rather than to a total; and geometrical measurements are practically ordinal (s 26). for ordinal purposes we use, as symbols, not only figures, such as 1, 2, 3, ... but also letters, as a, b, c, ... thus the pages of a book may be numbered 1, 2, 3, ... and the chapters i, ii, iii, ... but the sheets are lettered a, b, c, ... figures and letters may even be used in combination; thus 16 may be followed by 16a and 16b, and these by 17, and in such a case the ordinal 100 does not correspond with the total (cardinal) number up to this point. arithmetic is supposed to deal with cardinal, not with ordinal numbers; but it will be found that actual numeration, beyond about three or four, is based on the ordinal aspect of number, and that a scientific treatment of the subject usually requires a return to this fundamental basis. one difference between the treatment of ordinal and of cardinal numbers may be noted. where a number is expressed in terms of various denominations, a cardinal number usually begins with the largest denomination, and an ordinal number with the smallest. thus we speak of one thousand eight hundred and seventy-six, and represent it by mdccclxxvi or 1876; but we should speak of the third day of august 1876, and represent it by 3. 8. 1876. it might appear as if the writing of 1876 was an exception to this rule; but in reality 1876, when used in this way, is partly cardinal and partly ordinal, the first three figures being cardinal and the last ordinal. to make the year completely ordinal, we should have to describe it as the 6th year of the 8th decade of the 8th century of the 2nd millennium; i.e. we should represent the date by 3. 8. 6. 8. 9. 2, the total number of years, months and days completed being 1875. 7. 2. in using an ordinal we direct our attention to a term of a series, while in using a cardinal we direct our attention to the interval between two terms. the total number in the series is the sum of the two cardinal numbers obtained by counting up to any interval from the beginning and from the end respectively; but if we take the ordinal numbers from the beginning and from the end we count one term twice over. hence, if there are 365 days in a year, the 100th day from the beginning is the 266th, not the 265th, from the end. 8. _meaning of names of numbers._--what do we mean by any particular number, e.g. by _seven_, or by _two hundred and fifty-three_? we can define _two_ as _one and one_, and _three_ as _one and one and one_; but we obviously cannot continue this method for ever. for the definition of large numbers we may employ either of two methods, which will be called the _grouping_ method and the _counting_ method. (i) _method of grouping._--the first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition. thus, on the denary system (s 16) we can give independent definitions to the numbers up to ten, and then regard (e.g.) fifty-three as a composite number made up of five tens and three ones. or, on the quinary-binary system, we need only give independent definitions to the numbers up to five; the numbers _six, seven_, ... can then be regarded as _five and one, five and two_, ..., a fresh series being started when we get to _five and five_ or _ten._ the grouping method introduces multiplication into the definition of large numbers; but this, from the teacher's point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary arithmetical processes. (ii) _method of counting._--the second method consists in taking a series of names or symbols for the first few numbers, and then repeating these according to a regular system for successive numbers, so that each number is defined by reference to the number immediately preceding it in the series. thus _two_ still means _one and one_, but _three_ means _two and one_, not _one and one and one._ similarly _two hundred and fifty-three_ does not mean two hundreds, five tens and three ones, but _one_ more than _two hundred and fifty-two_; and the number which is called one hundred is not defined as ten tens, but as one more than ninety-nine. 9. _concrete and abstract numbers._--number is concrete or abstract according as it does or does not relate to particular objects. on the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. if we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. the three, for instance, are regarded as a whole when we name them _three._ if, however, we count these three as one, two, three, then the number of times we count is an abstract number. thus number in the abstract is the number of times that the act of counting is performed in any particular case. this, however, is a description, not a definition, and we still want a definition for "number" in the phrase "number of times." 10. _definition of "number."_--suppose we fix on a certain sequence of names "one," "two," "three," ..., or symbols such as 1, 2, 3, ...; this sequence being always the same. if we take a set of concrete objects, and name them in succession "one," "two," "three," ..., naming each once and once only, we shall not get beyond a certain name, e.g. "six." then, in saying that the number of objects is six, what we mean is that the name of the last object named is six. we therefore only require a definite law for the formation of the successive names or symbols. the symbols 1, 2, ... 9, 10, ..., for instance, are formed according to a definite law; and in giving 253 as the _number_ of a set of objects we mean that if we attach to them the symbols 1, 2, 3, ... in succession, according to this law, the symbol attached to the last object will be 253. if we say that this act of attaching a symbol has been performed 253 times, then 253 is an _abstract_ (or _pure_) _number._ underlying this definition is a certain assumption, viz. that if we take the objects in a different order, the last symbol attached will still be 253. this, in an elementary treatment of the subject, must be regarded as axiomatic; but it is really a simple case of mathematical induction. (see algebra.) if we take two objects a and b, it is obvious that whether we take them as a, b, or as b, a, we shall in each case get the sequence 1, 2. suppose this were true for, say, eight objects, marked 1 to 8. then, if we introduce another object anywhere in the series, all those coming after it will be displaced so that each will have the mark formerly attached to the next following; and the last will therefore be 9 instead of 8. this is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the theorem is true for 8, it is true for 9. but it is true for 2; therefore it is true for 3; therefore for 4, and so on. 11. _numerical quantities._--if the term _number_ is confined to number in the abstract, then number in the concrete may be described as _numerical quantity_. thus l3 denotes l1 taken 3 times. the l1 is termed the _unit._ a numerical quantity, therefore, represents a certain unit, taken a certain number of times. if we take l3 twice, we get l6; and if we take 3s. twice, we get 6s., i.e. 6 times 1s. thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. if we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. but in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced. ii. notation, numeration and number-ideation 12. _terms used._--the representation of numbers by spoken sounds is called _numeration_; their representation by written signs is called _notation_. the systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. this latter presentation may, in the absence of any accepted term, be called _number-ideation_; this word covering not only the perception or recognition of particular numbers, but also the formation of a number-concept. 13. _notation of numbers._--the system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the hindu, sometimes incorrectly called the arabic, system. the essential features which distinguish this from other systems are (1) the limitation of the number of different symbols, only ten being used, however large the number to be represented may be; (2) the use of the _zero_ to indicate the absence of number; and (3) the principle of local value, by which a symbol in effect represents different numbers, according to its position. the symbols denoting a number are called its _digits_. a brief account of the development of the system will be found under numeral. here we are concerned with the principle, the explanation of which is different according as we proceed on the grouping or the counting system. (i) on the grouping system we may in the first instance consider that we have separate symbols for numbers from "one" to "nine," but that when we reach ten objects we put them in a group and denote this group by the symbol used for "one," but printed in a different type or written of a different size or (in teaching) of a different colour. similarly when we get to ten tens we denote them by a new representation of the figure denoting one. thus we may have: ones 1 2 3 4 5 6 7 8 9 tens 1 2 3 4 5 6 7 8 9 hundreds,\ 1 2 3 4 5 6 7 8 9 &c. / &c. &c. on this principle 24 would represent twenty-four, 24 two hundred and forty, and 24 two hundred and four. to prevent confusion the _zero_ or "nought" is introduced, so that the successive figures, beginning from the right, may represent ones, tens, hundreds, ... we then have, e.g., 240 to denote two hundreds and four tens; and we may now adopt a uniform type for all the figures, writing this 240. 1 2 3 .. .. .. .. 8 9 10 1 2 3 .. .. .. .. o o o .. .. .. .. o o o o o o .. .. .. .. (ii) on the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, repeating this series indefinitely. there is as yet no distinction between the first object marked 1 and the second object marked 1. we can, however, attach to the 0's the same symbols, 1, 2, ... 0 in succession, in a separate column, repeating the series indefinitely; then do the same with every 0 of this new series; and so on. any particular object is then defined completely by the combination of the symbols last written down in each series; and this combination of symbols can equally be used to denote the number of objects up to and including the last one (s 10). in writing down a number in excess of 1000 it is (except where the number represents a particular year) usual in england and america to group the figures in sets of three, starting from the right, and to mark off the sets by commas. on the continent of europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal. the zero, called "nought," is of course a different thing from the letter o of the alphabet, but there may be a historical connexion between them (s 79). it is perhaps interesting to note that the latter-day telephone operator calls 1907 "nineteen o seven" instead of "nineteen nought seven." 14. _direction of the number-series._--there is no settled convention as to the direction in which the series of symbols denoting the successive numbers one, two, three, ... is to be written. (i) if the numbers were written down in succession, they would naturally proceed from left to right, thus:--1, 2, 3,... this system, however, would require that in passing to "double figures" the figure denoting tens should be written either above or below the figure denoting ones, e.g. 1 1, 2, ..., 8, 9, 0, 1, 2, ... or 1, 2, ..., 8, 9, 0, 1, 2, ... 1 the placing of the tens-figure to the left of the ones-figure will not seem natural unless the number-series runs either up or down. (ii) in writing down any particular number, the successive powers of ten are written from right to left, e.g. 5,462,198 is (6) (5) (4) (3) (2) (1) (0) 5 4 6 2 1 9 8 the small figures in brackets indicating the successive powers. on the other hand, in writing decimals, the sequence (of negative powers) is from left to right. (iii) in making out lists, schedules, mathematical tables (e.g. a multiplication-table), statistical tables, &c., the numbers are written vertically downwards. in the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals, though, when they represent values of a continuous quantity, they must be regarded as ordinals (ss 26, 93). (iv) in graphic representation measurements are usually made upwards; the adoption of this direction resting on certain deeply rooted ideas (s 23). 200 50 3 --- 253 === this question of direction is of importance in reference to the development of useful number-forms (s 23); and the existence of the two methods mentioned under (iii) and (iv) above produces confusion in comparing numerical tabulation with graphical representation. it is generally accepted that the horizontal direction of increase, where a horizontal direction is necessary, should be from left to right; but uniformity as regards vertical direction could only be attained either by printing mathematical tables upwards or by taking "downwards," instead of "upwards," as the "positive" direction for graphical purposes. the downwards direction will be taken in this article as the normal one for succession of numbers (e.g. in multiplication), and, where the arrangement is horizontal, it is to be understood that this is for convenience of printing. it should be noticed that, in writing the components of a number 253 as 200, 50 and 3, each component beneath the next larger one, we are really adopting the downwards principle, since the figures which make up 253 will on this principle be successively 2, 5 and 3 (s 13 (ii)). 15. _roman numerals._--although the roman numerals are no longer in use for representing cardinal numbers, except in certain special cases (e.g. clock-faces, milestones and chemists' prescriptions), they are still used for ordinals. the system differs completely from the hindu system. there are no single symbols for two, three, &c.; but numbers are represented by combinations of symbols for one, five, ten, fifty, one hundred, five hundred, &c., the numbers which have single symbols, viz. i, v, x, l, c, d, m, proceeding by multiples of five and two alternately. thus 1878 is