صور الصفحة
PDF

work Neutom'am'smo peg le dame, a work on optics. Voltaire: called him his cher cygne dc Padoue.' Returning from a journey to Russia, he met Frederick the Great who m'adeThim a countI of Prussia (1740) and court chamberlain (r747). ' Augustus III.‘ of Poland honoured him with the title of councillor. 'In 1754, after seven years’ residence partly in Berlin and partly in' Dresden, he returned to Italy, living at Venice and then at Pisa, where he died on the 3rd of May 1764. Frederick the Great erected to his memory a monument on the Campo Santo at Pisa. He was a man of wide knowledge, a connoisseur in‘art and music,'

and the friend of most of the leading authors of his time. His, chief work on art is the Saggi supra le belle arti (“ Essays on the

Fine Arts ”). Among his other works may be mentioned Poems, Travels in Russia, Essay on Painting, Correspondence.

The best complete edition with biography was published by D. Michelessi (1791-1794). _ \

ALGARVE, or ALGARVEs, an ancient kingdom and province in the extreme S. of Portugal, corresponding with the modern administrative district of Faro, and bounded on the N. by Alemtejo, E. by the Spanish province of Huelva, and S. and W. by the Atlantic Ocean. Pop. (1900) 255,19r; area, 1937 sq. m. The greatest length of the province is about 85 m. from E. to W.; its average breadth is about 22 m. from N. to S. The Serra de Malhao and the Serra de Monchique extend in the form of a crescent across the northern part of the province, and, sweeping to the south-west, terminate in the lofty promontory of Cape St Vincent, the south-west extremity of Europe. This headland is famous as the scene of many sea-fights, notably the defeat inflicted on the Spanish fleet in February 1797 by the British under Admiral Jervis, afterwards Earl St Vincent. Between the mountainous tracts in the north and the southern coast stretches a narrow‘plain, watered by numerous rivers flowing southward from the hills. The coast is fringed for 30 m. from Quarteira to Tavira, with long sandy islands, through which there are six passages, the most important being the Barra Nova, betwaen Faro and Olhao. The navigable estuary of the Guadiana divides Algarve from Huelva, and its tributaries water the western districts. From the Serra de Malhio flow two streams, the Silves and Odelouca, which unite and enter the Atlantic below the town of Silves. In the hilly districts the roads are bad, the soil unsuited for cultivation, and the inhabitants few. Flocks of goats are reared on the mountain-sides. The level country along the southern coast is more fertile, and produces in abundance grapes, figs, oranges, lemons, olives, almonds, aloes, and even plantains and dates. The land is, however, notv well suited for the production of cereals, which are mostly imported from Spain. On the coast the people gain their living in great

measure from the fisheries, tunny and sardines being caught‘

[blocks in formation]

articles.

The name of Algarvc is derived from the Arabic, and signifies a land lying to the West. The title “ king of Algarvc,” held by the kings of Portugal, was first assumed by Alphonso III., who captured Algarvc from the Moors in 12 53.

ALGA'U, or ALLGAU, the name now given to a comparatively small district forming the south-western corner of Bavaria and belonging to the province of Swabia and Neuburg, butt formerly applied to a much larger territory, which extended as far as the Danube on the N., the Inn on the S. and the Lech on the W. The Algau Alps contain several lofty peaks, the highest of which is Madelegabel (868‘; ft.). The district is Celebrated for its cattle, milk, butter and cheese. ‘

[ocr errors]
[graphic]

ALGEBRA (from the Arab. al-jebr wa’l-muqfibala,"tra'nspositi6n and removal [of terms of an equation],‘the name of a treatise“ by Mahommed ben Musa al-Khwarizmi), a branch of mathematics which may be defined as the generalization and extension of arithmetic. ' *"

The subject-matter of algebra will be treated in the following article under three divisionsz—A. Principles of ordinary algebra; B. Special kinds of algebra; C. History. Special phases of the subject are treated under their own headings, e.g. ALGEBRAIC Foams; BINOMIAL; Corrarm'roaur. Amrvsrs; DETERMINANTS; EQUATION; Com-moan FRACTION; FUNCTION; Gaours, THEORY or; LOGARITHM; NUMBER; PROBABILITY; ‘SERIEs.

A. PRINCIPLES or ORDINARY ALGEBRA- - i:

r. The above definition gives only a partial view of the amp of algebra. It may be regarded as based onarithmetic, or as dealing in the first instancewith formal results of the laws of arithmetical number; and in this sense Sir Isaac Newton gzive the title Universal Arithmetic to a work on algebra. Any definition, however, must have reference to the state of development of the subject at the time when the definition is given.

2. The earliest algebra consists in the solution of'equations. The distinction between algebraical and arithmetical reasoning then lies mainly in the fact that the former is in a more condensed form than the latter; an unknown quantity being represented by a special symbol, and other symbols being used as a kind of shorthand for verbal expressions. This form of algebra was extensively studied in ancient Egypt; but, in accordance with the practical tendency of the Egyptian mind, the study consisted largelyid the treatment of particular cases, very few general rules being obtained. ‘

3. For many centuries algebra was confined almost entirely to' the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use, of the minus sign. The knowledge of these laws, however, does not imply the existence of a conception of negative quantities. ’ The development of symbolic algebra by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1 540—1603). ‘ This led to the idea of algebra as generaliZed arithmetic.

4. The principal step'in the modern development of algebra was the recognition of the meaning of negative quantities. This appears to have been due in the first instance to Albert Girard (1595—1632), who extended Vieta’s results in various branches of mathematics. His work, however, was little known at the time, and later was overshadowed by the greater work of Descartes (1596—1650).

5. The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement. This involved not only the geometrical interpretation of negative quantities, but also the idea of continuity; this latter, which is the basis of modern analysis, leading to two separate but allied developments, viz. the theory of the function and the theory of limits. '

6. The great development of all branches of mathematics in the two centuries following Descartes has led to the term algebra being used to cover a great variety of subjects, many of which are really only ramifications of arithmetic, dealt with by algebraical methods, while others, such as the theory of numbers and the general theory of series, are outgrowths of the application of algebra to arithmetic, which involve such special ideas that they must properly be regarded as distinct subjects. Some writers have attempted unification by treating algebra as concerned with functions, and Comte accordingly defined algebra as the calculus of functions, arithmetic being regarded as the calculus of values.

7. These attempts at the unification of algebra, and its separation from other branches of mathematics, have usually been accompanied by an attempt to base it, as a deductive science, on certain fundamental laws or general rules; and this has tended to increase its difficulty. In reality, the variety of algebra corresponds to the variety of phenomena. Neither mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science. While, therefore, the logical development of algebraic reasoning must depend on certain fundamental relations, it is important that in the early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned. '

8. The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter part of the 19th century to an important reaction against the specialization mentioned in the preceding paragraph. This reaction has taken the form of a return to the alliance between algebra and geometry (§5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions. These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, mensuration and analytical geometry.

9. The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity,- while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition. The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical-measurement with the cardinal and the ordinal aspects of number respectively (see ARITHMETIC). Later, the difficulty recurs in an acute form in reference to the continuous variation of a function. Reference to a geometrical interpretation seems at first sight to throw light on the meaning of a differential coelficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself. One of the most recent developments of algebra is the algebraic theory of number, which is devised with the view of removing these difficulties. The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.

10. Two other developments of algebra are of special importance. The theory of sequences and series is sometimes treated as a part of elementary algebra; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory. The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.

11. One of the most difficult questions for the teacher of algebra is the stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced. On the one hand, the modern developments of algebra began with these ideas, and particularly with the idea of a negative number. On the other hand, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its intrinsic difficulty. Moreover, the ideas which are usually formed on these points at an early stage are incomplete; and, if the incompleteness of an idea is not realized, operations in which it is implied are apt to be purely formal and mechanical. What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (§ 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out.

12. In the present article, therefore, the main portions of elementary algebra are treated in one section, without reference to these ideas, which are considered generally in two separate sections. These three sections may therefore be regarded as to a certain extent concurrent. They are preceded by two sections dealing with the introduction to algebra from the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in §§ q and to above.

[graphic]

I. Arithmetical I ntroductiqn to Algebra.

13. Order of Arithmetical Operations—It is important, before beginning the study of algebra, to have a clear idea as to the meanings of the symbols used to denote arithmetical operations.

(i.) Additions and subtractions are performed from left to right. Thus 3 lb+5 lb—7lb+2 lb means that 5 lb is to be added to 3 lb, 7 lb subtracted from the result, and 2 lb added to the new result.

(ii.) The above operation is performed with 1 lb as the unit of counting, and the process would be the same with any other unit; e.g. we should perform the same process to find 35.+5s.—7s.+2s. Hence we can separate the numbers from the common unit, and replace 3 lb+5 lb — 7 lb+2 lb by (3+5—7+2) lb, the additions and subtractions being then performed by means of an addition-table.

(iii.) Multiplications, represented by X, are performed from right to left. Thus 5X3><7X 1 lb means 5 times 3 times 7 times 1 lb; i.e. it means that 1 lb is to be multiplied by 7, the result by 3, and the new result by 5. We may regard this as meaning the same as 5X3X7 lb, since 7 lb itself means 7X1 lb, and the lb is the unit in each case. But it does not mean the same as 5X21 lb, though the two are equal, i.e. give the same result (see 5 23)

This rule as to the meaning of X is important. If it is intended that the first number is to be multiplied by the second, a special sign such as X should be used.

(iv.) The sign + means that the quantity or number preceding it is to be divided by the quantity or number following it.

(v.) The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers. Thus 16/4 does not mean 16+4, but

(vi.) Any compound operation not coming under the above descriptions is to have its meaning made clear by brackets, the use of a pair of brackets indicating that the expression between them is to be treated as a whole. Thus we should not write 8X7+6, but (8X7)+6, or 8X(7+6). The sign X coming immediately before, or immediately after, a bracket may be omitted; cg. 8X(7+6) may be written 8(7+6).

This rule as to using brackets is not always observed, the convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions. The convention is even pushed to such an extent as to make “ 4i+si of 7+5 ” mean “4%+(s§ of 7)+s ”; though it is not clear what “Find the value of 41}+3§ times 7+5 " would then mean. There are grave objections to an arbitrary rule of this kind, the chief being the useless waste of mental energy in remembering it.

(vii.) The only exception that may be made to the above rule is that an expression involving multiplication-dots only, or a simple fraction written with the solidus, may have the brackets omitted for additions or subtractions, provided the figures are so spaced as to prevent misunderstanding. Thus 8+ (7 X6) +3 may be written 8+7.6+3, and 8+§+3 may be written 8+7/6+3.

[ocr errors]

14. Latent Equations.~—~The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the head of inverse operations; 11:. processes which consist in obtaining an answer to the question “ Upon what has a given operation to be performed in order to produce a given result?” or to the question “ What operation of a given kind has to be performed on a given quantity or number in order to produce a given result?"

(i.) In the case of subtraction the second of these two questions is perhaps the simpler. Suppose, for instance, that we wish to know how much will be left out of ms. after spending 35., or how much has been spent out of res. if 3s. is left. In either case we may put the question in two waysz—(a) What must be added to 35. in orderto produce 105., or (b) To what must 3s. be added in order to produce 10s. If the answer to the question is X, we

[ocr errors]

(ii.) In the above case the two different kinds of statement lead to arithmetical formulae of the same kind. In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.

(a) If 24d. is divided into 4 equal portions, how much will each portion be?

Let the answer be X; then _ 24d.=4XX, X=i of 24d.

(b) Into how many equal portions of 6d. each may 24d. be divided P

Let the answer be x; then

24d. =xX6d., x=24d.+6d.

(iii.) Where the direct operation is evolution, for which there is no commutative law, the two inverse operations are diflerent in kind.

(a) What would be the dimensions of a cubical vessel which would exactly hold 12 5 litres; a litre being a cubic decimetre P

Let the answer be X; then

125 c.dm. =X’, X = ‘7 125 c.dm. = {TF5 dm.

(b) To what power must 5 be raised to produce 125 P

Let the answer be x; then
125=5=, x=log5 125.

15. With regard to the above, the following points should be noted.

(1) When what we require to know is a quantity, it is simplest to deal with this quantity as a whole. In (i.), for instance, we want to find the amount by which ros. exceeds 35., not the number of shillings in this amount. It is true that we obtain this result by subtracting 3 from IO by means of a subtractiontable (concrete or ideal); but this table merely gives the generalized results of a number of operations of addition or subtraction performed with concrete units. We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.

Whether this principle may legitimately be extended to the notation adopted in (iii.) (a) of § 14 is a moot point. But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.

(2) The results may be stated briefly as follows, the more usual form being adopted under (iii.) (a):—

(i.) If A=B+X, or=X+B, then X=A—B.
(ii.) (a) If A=m times x, then x= $1 of A.
(b) If A=x times M, then x=A+M.
(iii.) (a) it n=xP, then x=‘\,/n.
(b) If n=a‘, then x=log.I n.

The important thing to notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that “ if equals are taken from equals the remainders are equal "; the fact being that the two statements are merely different ways of expressing the same relation. To say, for instance, that X is equal to A — B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than definitions of the symbols — , i-offlq V, and log...

An apparent difficulty is that we use a single symbol—to denote the result of the two different statements in (i.) (a) and (i.) (b) of § 14. This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be wise not to use the one symbol— to represent the result of both operations until the commutative law for addition has been fully grasped.

[graphic]

16. In the same way, a statement as to the result of an inverse operation is really, by the definition of the operation, a statement as to the result _of a direct operation. If, for instance, we state that A=X— B, this is really a statement that X=A+B. Thus, corresponding to the results under § 15 (2), we have the following:——

(I) Where the inverse operationis performed on the unknown quantity or number :—

(i.) If A=X—B, then X=A+B. (ii.) (a) IfM =5- of x, then X=m times M. (b) If m=X+M, then X=m times M. (iii.) (a) If a =t’/x, then x =11». b) If p =lognx, then x =0". (2) Where the inverse operation is performed with the unknown quantity or number:— gg If B=A—X, then A=B+X. _ (u. (a) If m=A+X, then A=m times X. (b) 1: M =§ of A, then A=x times M. (iii.) (a) lfp=log,n, then n=xP. (b) If a=\7n, then n=a‘.

In each of these cases, however, the reasoning which enables us to replace one statement by another is of a different kind from the reasoning in the corresponding cases of § 1 5. There we proceeded from the direct to the inverse operations; Le. so far as the nature of arithmetical operations is concerned, we launched

_ out on the unknown. In the present section, however, we return

from the inverse operation to the direct; 11¢. we rearrange our statement in its simplest form. The statement, for instance, that 32—x=25, is really a statement that 32 is the sum of x and 25. ‘

17. The five equalities which stand first in the five pairs of equalities in § 1 5 (2) may therefore be taken as the main types of a simple statement of equality. When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical. The five processes of deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a root, (iii.) (b) taking logarithms. It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the original equation.

18. It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers

_ contemplated in §§ 14-17 are supposed to be positive integers.

If, for instance, we are told that 15==i of (x— 2), what is meant is that (I) there is a number u such that x=u+2, (2) there is a number a such that u=4 times 0, and (3) 15=5 times a. From these statements, working backwards, we find successively that v= 5, u=2o, x=22. The deductions follow directly from the definitions, and such mechanical processes as “clearing of fractions ” find no place (§ 2r (ii.)). The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (§ 27 (ii.)).

19. Expressed Equations—The simplest forms of arithmetical equation arise out of abbreyiated solutions of particular problems. In accordance with § 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g. to represent the former by capital letters and the latter by small letters.

As an example, take the following. I buy 2 lb of tea, and have 6s. 8d. left out of 105.; how much per lb did tea cost?

(1) In ordinary language we should say: Since Gs. 8d. was left, the amount spent was ros. — 6s. 8d., 11¢. was 35. 4d. Therefore 2 lb of tea cost 35. 4d. Therefore 1 lb of tea cost rs. 8d.

(2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality. Thus we say:—

[ocr errors]
[merged small][ocr errors]

(4) The stage which is introductory to algebra consists merely in replacing the unit “ cost of '1 lb tea ” by a symbol, which may be a letter or a mark such as the mark of interrogation, the asterisk, &c. If we denote this unit by X, we have

(2 XX)+6s. 8d. = 105.
2><X==10s.—6s. 8d. =35. 4d.
X= Is. 8d.

20. Notation of M unifies—The above is arithmetic. The only thing whichit is necessary to import from algebra is the notation by which we write 2X instead of 2 X Xor 2 . This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number.

It must be remembered that, if a is a number, 30 means 3 times a, not a times 3; the latter must be represented by aX3 or (1.3. I ,

The number by which an algebraical expression is to be multiplied is called its coefficient. Thus in 3a the coefficient of a is 3. But in 3 . 4a the coefficient of 4a is 3, while the coeflicient of a is 3 . 4.

21. Equations with Fractional Cocficients.;As an example of a special form of equation we may take '

‘ ‘ §x+§x= Io. ' (i.) There are two ways of proceeding.

(a) The statement is that (1') there is a number n such that x= 2u,(2) there is a number 1; such that x= 3'0, and (3) u+v= 10. We may therefore conveniently take as our unit, in place of at, a number y such that x=6y.

We then have 3y+2y= IO, whence ' 5y=ro, y=2, x=6y=r2. ‘

(b) We can collect cocjfioients, i.e. combine the separate

[merged small][ocr errors]
[ocr errors]

is the more advanced process, implying some knowledge of the laws of fractional numbers, as well as an application of the associative law (§ 26 (i.)).

(ii.) Perhaps the worst thing we can do, from the point of view of intelligibility,is to “ clear of fractions ” by multiplying both sides by 6. 2x=6o (and similarly if ix+§x+§x= 10, then 3x+2x+x=6o); but the fact, however interesting it may be, is of no importance for our present purpose. In the method (a) above there is indeed a multiplication by 6; but it is a multiplication arising out of subdivision, not out of repetition (see ARITHMETIC), so that the total (viz. ro) is unaltered.

22. Arithmetical and Algebraical Treatment of Equations.—The following will illustrate the passage from arithmetical to algebraical reasoning. “ Coal costs 3s. a ton more this year than last year. If 4 tons last year cost 104s., how much does a ton cost this year?”

, If we write X for the cost per ton this year, we have 4(X—3s.) = 1045. From this we can deduce sucCessively X—3s.=2os., X=29s. But, if we transform the equation into 4X— 125. = I04s., we make an essential alteration. The original statement was with regard to X—3s.‘ as the unit; and from this, by the application of the distributive law (§ 26 (i.)), we have passed to a statement with regard to X as the unit. Thisis an algebraical process.

In the same way, the transition from (x’+4x+4) —4= 21 to x2+4x+4=25, or from (x+2 2=25 to x+2=425, is arithmetical; but the transition from x3+4x+4=25 to (x—l—z)’: 25 is algebraical, since it involves a change of the number we are thinking about.

[ocr errors]
[graphic]

‘ Generally, we may say that algebraic reasoning in reference to equations consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e.1it cannot be said that “If A=B, then E=F " is arithmetic, while “ If C=D, then E=F ” is algebra. Algebraic treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be equivalent to it. The subsequent reasoning is arithmetical.

2 3. Sign of Equality—The various meanings of the sign of equality (=) must be distinguished.

(i.) 4X3 lb=12 lb.

This states that the result of the operation of multiplying 3 11) by 4 is 12 lb. (ii.) 4X3 lb'=3 X4 lb.

This states that the two operations give the same result; Le. that they are equivalent.

(iii.) A's share =5s., or
3 times A's share = 15s.

Either of these is a statement of fact with regard to a particular quantity; it is usually called an equation, but sometimes a conditional equation, the term “ equation ’-’ being then extended to cover (i.) and (ii.). (iv.) xi=x><xXx.

This is a definition of x“; the sign = is in such cases usually replaced by E. (v.) 24d. =2s.

This is usually regarded as being, like (ii.), a statement of equivalence. It is, however, only true if rs. is equivalent to rzd.,

.and the correct statement is then

T12%TX24d.=2s. If the operator Ilia-X is omitted, the statement is really an

equation, giving rs. in terms of rd. or vice versa.
The following statements should be compared:—
X=A‘s share=i of ro=3>< 5=£r5.
X=A's share=§ of 10 =§ o £30=£r5.

In each case, the first sign of equality comes under (iv.) above, the second under (iii.), and the fourth under (i.); but the third sign comes under (i.) in the first case (the statement being that % of £ro=£5) and under (ii.) in the second.

It will be seen from § 22 that the application of algebra to equations consists in the interchange of equivalent expressions, and therefore comes under (i.) and (ii.). We replace 4(x—3), for instance, by 4x—4. 3, because we know that,whatever the value of x may be, the result of subtracting 3 from it and multiplying the remainder by 4 is the same as the result of finding 4: and 4 . 3 separately and subtracting the latter from the former.

A statement such as (i.) or (ii.) is sometimes called an identity.

The two expressions whose equality is stated by an equation or an identity are its‘members.

24. Use of Letters in General Reasoning—It may be assumed

that the use of letters to denote quantities or numbers will first arise in dealing with equations, so that the letter used will in each case represent a definite quantity or number; such general statements as those of §§ i5 and r6 being deferred to a later stage. In addition to these, there are cases in which letters can usefully be employed for general arithmetical reasoning. , (i.) There are statements, such as A+B=B+A, which are particular cases of the laws of arithmetic, but need not be expressed as such. For multiplication, for instance, we have the statement that, if P and Q are two quantities, containing respectively p and q of a particular unit, then pXQ=qXP; or the more abstract statement that PXq=qXp.

(ii.) The general theory of ratio and proportion requires the use of general symbols.

(iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.

(iv.) Variation is generally included in text-books on algebra, but apparently only because the reasoning is general. It is part of the general theory of quantitative relation, and in its acme)an stages is a suitable subject for graphical treatment

31 .

[merged small][ocr errors]

=xz—a'. Such verifications are of value for two reasons. In the first place, they lead to an understanding of What is meant by the use of brackets and by such a statement as 3(7—l—2) =3 . 7+3 . 2. This does not mean (cf. § 23) that the algebraic result of performing the operation 3(7+2) is 3 . 7+3 . 2; it means that if we convert 7+2 into the single number o'and then multiply by 3 we get the same result as if we converted 3.7 and 3.2 into 21 and 6 respectively and added the results. In the second place, particular cases lay the foundation for the general formula.

Exercises in the collection of coefiicients of various letters occurring in a complicated expression are usually performed mechanically, and are probably of very little value.

26. General Arithmetical Theorems.

(i.) The fundamental laws of arithmetic (q.v.) should be constantly borne in mind, though not necessarily stated. The following are some special points.

(a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of. two'

numberslbefore proceeding to the general case. In the case of .

addition, for instance, suppose that we are satisfied that in a+b+c+d+e we may take any two, as b and 0, together (association) and interchange them (commutation). Then we have a+b+c+d+e=a+c+b+d+a Thus any pair of adjoining numbers can be interchanged. so that the numbers can be arranged in any order.

(6) The important form of the distributive law is m(A+B) =mA+mB.- The form (m+n)A = mA+nA follows at once from the fact that A is the unit with which we are dealing.

(c) The fundamental properties of subtraction and of division

are that A113 +B =A and mX'ln of A=A, since in each case the

second operation restores the original quantity with which we started.

(ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of paiqb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions. Graphic methods are useful here (§ 34 (iv.)). The law of relation of successive convergents to a continued fraction involves more advanced methods (see § 42 (iii.) and CONTINUED FRACTION).

(iii.) There are important theorems as to the relative value of fractions; e.g.

[ocr errors][merged small]

supposed to be positive.)

27. Negative QuantitiesandchtionatNumbers.—(i.) Whatare usually called “ negative numbers ” in arithmetic are in reality not negative numbers but negative quantities. If a person has to receive 75. and pay 55., with a net result of +25., the order of the operations is immaterial. If he pays first, he then has —5s. This is sometimes treated as a debt of 55.; an alternative method is to recognize that our zero is really arbitrary, and that in fact we shift it with every operation of addition or subtraction. But when we say "—55." we mean “-(5s.),” not “ (—5)s.”; the idea of (—5) as a number with which we can perform such operations as multiplication comes later (§ 49).

(ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit. We find that fractions follow certain laws

(All the numbers are, of course,

[graphic]

corresponding exactly with those of integral multipliersflind we are therefore able to deal with the fractional numbers as if they Were integers. . ‘28.. Miscellaneous Developments in Arithmetic.—T he followin are matters which. really belong to arithmetic; they are usually

1 placed under algebra, since the general formulae involve the 1156

of letters. -' w . (i.) Arithmetical Progressions such as 2, 5, 8, . .. .—-The formula

for the rth term is easily obtained. The problem of finding the

sum of 1 terms is aided by graphic representation, which shows

‘ that the terms maybe taken in pairs, working from the outside

to the middle; the two cases of an odd number of terms and an even number of terms may be treated separately at first, and then combined by the ordinarymethod, viz. writing the-series backwards. I . '

In this, as in almost all other cases, particular examples should be worked before obtaining a general formula.

(ii.) The law of indices (positive integral indices only) follows at once from the definition of a”, a“, a‘, . . .. as abbreviations of 0.0, 0.0.0, a.a.a.a, . . ., or (by analogy with the definitions of 2, 3, 4, . . . themselves) of 0.0, 0.02, 0.03, . . . successively. The treatment of roots and of logarithms (all being positive integers) belongs to this subject; a= {7 n and p==logan being the inverses of n=aP (cf. §§ 15, 16). The theory may be extended to the cases of p=r and p=o; so that a3 means a.a.a.r, 02 means a.a.r, a1 means 0.1, and a0 means I (there being/then none of the multipliers a).

The terminology is sometimes confused. In n=oP, o is the root or base, p is the index or logarithm, and n is the power or antilogarithm. Thus a, a”, a“, . . . are the first, second, third, . I; . powers of a. 'But 0" is sometimes incorrectly described as “ o to the power p ”; the power being thus confused with the index or logarithm.

(iii.) Scales of Notation lead, by considering, e. g., how to express in the scale of 10 a number whose expression in the scale of 8 is 2222222, to

(iv.) Geometrical Pragressions.—It should be observed that the radix of the scale is exactly the same thing as the root mentioned under (ii.) above; and it is better to use the term “ root ” throughout. Denoting the root by a, and the number 2222222 in this scale by N, we have

[ocr errors][merged small]

metical point of view, surds present a greater difficulty than

negative quantities and fractional numbers. We cannot solve the equation 7s.+X=4s.; but we are accustomed to transactions of lending and borrowing, and we can therefore invent a negative quantity ~35. such that —3s.+3s.=o. We cannot solve the equation 7X=4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit lis. such that 7th= 15., and can thence pass to the idea of fractional numbers. When,_however, we come to the equation x2= 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us. We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as ~15. We may take it to (say) 4 places of decimals; or we may suppose it to be taken to 1000 places. In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods.

When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractionalnumbers), 45 acquires

« السابقةمتابعة »