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The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement.

20Arithmetical Introduction to Algebra.

10Order of Arithmetical Operations.

10Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings.

10When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known.

00The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.

00na-nun, one; nar, two; and ne', three, or variants of these; all higher arithmetical ideas being expressed by the word kerpn, which means " many."

00Hence if all the energy supplied to the train is utilized at one axle there is the fundamental relation RV (I) Continuing the above arithmetical illustration, if the wheels to the axle of which the torque is applied are 4 ft.

00He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an arithmetical, and the quantities of acids saturating a given base a geometrical, progression.

00He took a passionate delight in the pursuit of knowledge from his very infancy, and is reported to have worked out long arithmetical sums by means of pebbles and biscuit crumbs before he knew the figures.

00The most famous outcome of his inquiries is the law known as Weber's or Fechner's law which may be expressed as follows:- "In order that the intensity of a sensation may increase in arithmetical progression, the stimulus must increase in geometrical progression."

00In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

00Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.

00The 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.

00They are preceded by two sections dealing with the introduction to algebra from4the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in ï¿½ï¿½ 9 and 1 o above.

00- 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.

00- The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the head of inverse operations; i.e.

00(ii.) In the above case the two different kinds of statement lead to arithmetical formulae of the same kind.

00In 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.

00so far as the nature of arithmetical operations is concerned, we launched out on the unknown.

00Expressed Equations.-The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems. In accordance with ï¿½ 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.

00(2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality.

00Arithmetical and Algebraical Treatment of Equations.-The following will illustrate the passage from arithmetical to algebraical reasoning.

00In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is arithmetical; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is algebraical, since it involves a change of the number we are thinking about.

00The subsequent reasoning is arithmetical.

00In addition to these, there are cases in which letters can usefully be employed for general arithmetical reasoning.

00(v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

00"), that the graphic method leads without arithmetical reasoning to the properties of negative values.

00It must, of course, be remembered (ï¿½ 23) that this is a statement of arithmetical equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.

00(iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple.

00(ii.) The solution of equations is effected by transformation, which may be either arithmetical or algebraical.

00The principles of arithmetical transformation follow from those stated in ï¿½ï¿½ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities.

00These transpositions are purely arithmetical.

00These numbers constitute an arithmetical progression of the rth order.

00In algebraical transformations, however, such as (x-a)2 = x 2 - 2ax+a 2, the arithmetical rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation.

00The divisions in ï¿½ï¿½ 50-52 have been supposed to be performed by a process similar to the process of arithmetical division, viz.

00If, moreover, we examine the process of algebraical division as illustrated in ï¿½ 50, we shall find that, just as arithmetical division is really the solution of an equation (ï¿½ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a multinomial really implies the use of undetermined coefficients (ï¿½ 42).

00(v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division.

00It is often convenient, as in ï¿½ 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to arithmetical calculation; i.e.

00(n- I)(r)xr+ï¿½ï¿½ï¿½}; the symbol " being used to indicate that the equality is only formal, not arithmetical.

00The idea of continuity must in the first instance be introduced from the graphical point of view; arithmetical continuity being impossible without a considerable extension of the idea of number (ï¿½ 65).

00The word " sequence," as defined in ï¿½ 58 (i.), includes progressions such as the arithmetical and geometrical progressions, and, generally, the succession of terms of a series.

00The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages.

00Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts.

00It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved.

00The only known type of algebra which does not contain arithmetical elements is substantially due to George Boole.

00The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded.

00Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical.

00Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra.

00His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical.

00It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds.

00The earlier chapters, treating chiefly of the arithmetical foundations of the science, differ but little in their line of argument from the principles laid down by Pietro Aron, Zacconi, and other early writers of the Boeotian school; but in bk.

00At a time when the Cartesian system of vortices universally prevailed, he found it necessary to investigate that hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it.

00Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms R- x 3 3x 3, 3x 3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x-- 3, x- `3, x-- 3, &c.

00" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

00These elements will enable us to convert, by a simple arithmetical operation, any historical date, of which the chronological characters are given according to any era whatever, into the corresponding date in the Christian era.

00They correspond to the two methods of regarding quantity - the arithmetical and the geometrical.

00Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or geometrical methods.

00This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.

00(ii) The very earliest stages of mensuration should be directly associated with simple arithmetical processes.

00arithmetical mensuration, geometrical mensuration, and analytical mensuration.

00The most elementary stage is arithmetical mensuration, which comprises the measurement of the areas of rectangles and parallelepipeds.

00The third step is the arithmetical calculation of the area or volume of the rearranged figure.

00These last two steps may introduce magnitudes which have to be subtracted, and which therefore have to be treated as negative quantities in the arithmetical.

00Chebichev further constructed an instrument for drawing large circles, and an arithmetical machine with continuous motion.

00A slide rule should be used for the arithmetical operations, as it works to the accuracy obtainable in practice.

00Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division.

00Thus in arithmetical calculations if the base is not expressed it is understood to be io, so that log m denotes log n m; but in analytical formulae it is understood to be e.

00This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus 0, 1,0000 0000 10, I,0001 0000 20, 1,0002 000 I 9 90, 1,0099 4967 In the arithmetical column the numbers increase by io, in the geometrical column each number is derived from its predecessor by multiplication by i 0001.

00The table extends to 2 3 0270 in the arithmetical column, and it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in the geometrical column; this last result showing that (1.0001)23027 022 = 10.

00To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.

00The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios.

00Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series.

00This ratio, invariably denoted by 7r, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number.

00Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a " formula."

001 V 2+2A/ From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation.

00quam Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations.

00Deslandres,s who found that the successive differences in the frequencies formed an arithmetical progression.

00The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

00The arithmetical mean of n quantities is the sum of the quantities divided by their number n.

00The harmonic mean of n quantities is the arithmetical mean of their reciprocals.

00The significance of the word "mean," i.e., middle, is seen by considering 3 instead of n quantities; these will be denoted by a, b, c. The arithmetic mean b, is seen to be such that the terms a, b, c are in arithmetical progression,.

00The arithmetico-geometrical mean of two quantities is obtained by first forming the geometrical and arithmetical means, then forming the means of these means, and repeating the process until the numbers become equal.

00The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.

00He generalized Weber's law in the form that sensation generally increases in intensity as the stimulus increases by a constant function of the previous stimulus; or increases in an arithmetical progression as the stimulus increases in a geometrical ratio; or increases by addition of the same amount as the stimulus increases by the same multiple; or increases as the logarithm of the stimulus.

00An opportunity, however, presented itself: being required to work out from data supplied to him the "defilement" of a proposed fortress (an operation then only performed by a long arithmetical process), Monge, substituting for this a geometrical method, obtained the result so quickly that the commandant at first refused to receive it - the time necessary for the work had not been taken; but upon examination the value of the discovery was recognized, and the method was adopted.

00He held that thought has in itself no power of development, and ultimately reduced it to arithmetical computation.

00He nowhere explains the origin of the notions of unity and number, but merely asserts that through their means we can have absolutely exact arithmetical propositions (Works, i.

00The first desideratum here mentioned - the want, namely, of an accurate statement of the relation between the increase of population and food - Malthus doubtless supposed to have been supplied by the celebrated proposition that "population increases in a geometrical, food in an arithmetical ratio."

00It appears that Pascal contemplated publishing a treatise De aleae geometria; but all that actually appeared was a fragment on the arithmetical triangle (Traite du triangle arithmetique, " Properties of the Figurate Numbers"), printed in 1654, but not published till 1665, after his death.

00care being taken to attend to the sign of a, so that when that is negative the arithmetical values of y and a are to be added in order to give that of $.

00and the corresponding axes parallel, then by changing the signs of x x, y, the values x',17; x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables.

00During the later years of his life he resided in London, devoting himself to the construction of machines capable of performing arithmetical and even algebraical calculations.

00That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

00The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant" (iripas ixov), and as such is styled "quantity" and perhaps "number" but the ideal numbers are distinct from arithmetical numbers.

00Xenocrates, however, failing, as it would seem, to grasp the idealism which was the metaphysical foundation of Plato's theory of natural kinds, took for his principles arithmetical unity and plurality, and accordingly identified ideal numbers with arithmetical numbers.

00In thus reverting to the crudities of certain Pythagoreans, he laid himself open to the criticisms of Aristotle, who, in his Metaphysics, recognizing amongst contemporary Platonists three principal groups - (1) those who, like Plato, distinguished mathematical and ideal numbers; (2) those who, like Xenocrates, identified them; and (3) those who, like Speusippus, postulated mathematical numbers only - has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideas are numbers made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations.

00The central gauge is useful for correcting and checking the others, but in such a perfectly simple case as the straight valley above assumed it may be omitted in calculating the results, and if the other four gauges are properly placed, the arithmetical mean of their results will probably not differ widely from the true mean for the valley.

00As an arithmetical calculator he was not only wonderfully expert, but he seems to have occasionally found a positive delight in working out to an enormous number of places of decimals the result of some irksome calculation.

00" He used to carry on," says his elder son, William Edwin Hamilton, " long trains of algebraical and arithmetical calculations in his mind, during which he was unconscious of the earthly necessity of eating; we used to bring in a ` snack ' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards."

00The arithmetical fact is that I i and 9 may be regrouped as 12 and 8, and the statement "IId.+9d.

00is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (§ II).

00the 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.

00Nature of Arithmetical Reasoning

00The 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.

00Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout.

00In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation.

00- In using material objects as a basis for developing the number-concept, it must be remembered that it is only when there are a few objects that their number can be perceived without either counting or the performance of some arithmetical process such as addition.

00The forms seem to result from a general tendency to visualization as an aid to memory; the letter-forms may in the first instance be quite as frequent as the numberforms, but they vanish in early childhood, being of no practical value, while the number-forms continue as an aid to arithmetical work.

00The possibility of replacing them by a standard form, which could be utilized for performing arithmetical operations, is worthy of consideration; some of the difficulties in the way of standardization have already been indicated (§ 14).

00- The application of arithmetical methods to geometrical measurement presents some difficulty.

00Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.

00are the names given to certain numerical processes which have to be performed in order to find the result of certain arithmetical operations.

00The essential portions of these statements, from the arithmetical point of view, may be exhibited in the form of the diagrams A and B A B or more briefly, as in C or C' and D or D':- C C' the general arrangement of the diagram being as shown in E or E':- E E' Multiplication is therefore equivalent to completion of the diagram by entry of the product.

00The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:

00Thus any arithmetical processes which can be applied to the numbers p, q, r, ..

00The numbers (integral or decimal) by which we represent the results of arithmetical operations are often only approximately correct.

00When the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the error.

00Accuracy after Arithmetical Operations.

00The simplest form of arithmetical reasoning consists in the determination of the term in one series corresponding to a given term in another series, when the relation between the two series is given; and it implies, though it does not necessarily involve, the establishment of each series as a whole by determination of its unit.

00When the unit is not determined, the reasoning is algebraical rather than arithmetical.

00More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae.

00They are not suitable for elementary purposes, as the arithmetical relations involved are complicated and difficult to grasp.

00In the second place, this method fixes the attention at once on the larger, and therefore more important, parts of the quantities concerned, and thus prevents arithmetical processes from becoming too abstract in character.

00Text-books on arithmetic usually contain explanations of the chief commercial transactions in which arithmetical calculations arise; it will be sufficient in the present article to deal with interest and discount, and to give some notes on percentages and rates in the £.

00A sample of De Morgan's bibliographical learning is to be found in his account of Arithmetical Books, from the Invention of Printing (1847), and finally in his [[Budget]] of Paradoxes.

00These calculations not only involved difficult mathematical expressions but also dealt with heavy arithmetical calculations.

00stitched fascicle of 16 bifolios (= 32 folios ); this also includes jottings on arithmetical and geometrical topics.

00The arithmetical half of mathematics, which had been gradually growing into algebra, and had decidedly established itself as such in the Ad logisticen speciosam notae priores of Francois Vieta (1540-1603), supplied to some extent the means of generalizing geometry.

00The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an arithmetical series of numbers.

00When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known.

00Nothing shows more clearly the rude state of arithmetical knowledge at the beginning of the 17th century than the universal satisfaction with which Napier's invention was welcomed by all classes and regarded as a real aid to calculation.

00This single instance of the use of the decimal point in the midst of an arithmetical process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely in the course of performing an arithmetical operation.

00The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.

00na-nun, one; nar, two; and ne', three, or variants of these; all higher arithmetical ideas being expressed by the word kerpn, which means " many."

00Hence if all the energy supplied to the train is utilized at one axle there is the fundamental relation RV (I) Continuing the above arithmetical illustration, if the wheels to the axle of which the torque is applied are 4 ft.

00He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an arithmetical, and the quantities of acids saturating a given base a geometrical, progression.

00He took a passionate delight in the pursuit of knowledge from his very infancy, and is reported to have worked out long arithmetical sums by means of pebbles and biscuit crumbs before he knew the figures.

00The most famous outcome of his inquiries is the law known as Weber's or Fechner's law which may be expressed as follows:- "In order that the intensity of a sensation may increase in arithmetical progression, the stimulus must increase in geometrical progression."

00In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

00Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.

00In order to save arithmetical labour it is convenient to be provided with conversion factors for reducing variously expressed results to the standard form.

00It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the laws of arithmetical number; and in this sense Sir Isaac Newton gave the title Universal Arithmetic to a work on algebra.

00The 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.

00The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement.

00The 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.

00The 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).

00The 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.

00They are preceded by two sections dealing with the introduction to algebra from4the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in Ã¯¿½Ã¯¿½ 9 and 1 o above.

00Arithmetical Introduction to Algebra.

00Order of Arithmetical Operations.

00- 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.

00- The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the head of inverse operations; i.e.

00(ii.) In the above case the two different kinds of statement lead to arithmetical formulae of the same kind.

00In 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.

00so far as the nature of arithmetical operations is concerned, we launched out on the unknown.

00Expressed Equations.-The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems. In accordance with Ã¯¿½ 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.

00(2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality.

00Arithmetical and Algebraical Treatment of Equations.-The following will illustrate the passage from arithmetical to algebraical reasoning.

00In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is arithmetical; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is algebraical, since it involves a change of the number we are thinking about.

00The subsequent reasoning is arithmetical.

00In addition to these, there are cases in which letters can usefully be employed for general arithmetical reasoning.

00General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.

00(i.) Arithmetical Progressions such as 2, 5, 8, ...

00(v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (Ã¯¿½Ã¯¿½ 41, 44)Ã¯¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

00"), that the graphic method leads without arithmetical reasoning to the properties of negative values.

00- (i.) An expression such as a.2.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors.

00It must, of course, be remembered (Ã¯¿½ 23) that this is a statement of arithmetical equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result.

00(iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple.

00- (i.) Special arithmetical results may often be used to lead up to algebraical formulae.

00(ii.) The solution of equations is effected by transformation, which may be either arithmetical or algebraical.

00The principles of arithmetical transformation follow from those stated in Ã¯¿½Ã¯¿½ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities.

00These transpositions are purely arithmetical.

00(i.) The sum of the first n terms of an ordinary arithmetical progression (a+b), (a+2b), ...

00These numbers constitute an arithmetical progression of the rth order.

00In algebraical transformations, however, such as (x-a)2 = x 2 - 2ax+a 2, the arithmetical rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation.

00The use of negative coefficients leads to a difference between arithmetical division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a binomial, as illustrated by the following examples: - 2.10+1) 6.100+5.10+ 1(3.10+I 2.10+I) 6.100+I.10 - I (3.10 - I 6.100+3.10 6.100+3.10 2.10+ I - 2.10 - I 2.10 +I - 2.10 - I In (1) the division is both arithmetical and algebraical, while in (2) it is algebraical, the quotient for arithmetical division being 2.10+9.

00- (i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (Ã¯¿½ 26 (i.)) but differ from them in relating, not to arithmetical value, but to algebraic form.

00The divisions in Ã¯¿½Ã¯¿½ 50-52 have been supposed to be performed by a process similar to the process of arithmetical division, viz.

00If, moreover, we examine the process of algebraical division as illustrated in Ã¯¿½ 50, we shall find that, just as arithmetical division is really the solution of an equation (Ã¯¿½ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a multinomial really implies the use of undetermined coefficients (Ã¯¿½ 42).

00(v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to arithmetical evolution (calculation of a surd) that algebraical division bears to arithmetical division.

00It is often convenient, as in Ã¯¿½ 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to arithmetical calculation; i.e.

00(n- I)(r)xr+Ã¯¿½Ã¯¿½Ã¯¿½}; the symbol " being used to indicate that the equality is only formal, not arithmetical.

00The idea of continuity must in the first instance be introduced from the graphical point of view; arithmetical continuity being impossible without a considerable extension of the idea of number (Ã¯¿½ 65).

00The word " sequence," as defined in Ã¯¿½ 58 (i.), includes progressions such as the arithmetical and geometrical progressions, and, generally, the succession of terms of a series.

00The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages.

00Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts.

00Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings.

00It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of arithmetical interpretation, according to the values attributed to the letters involved.

00It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction.

00The only known type of algebra which does not contain arithmetical elements is substantially due to George Boole.

00The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in arithmetical computations, there are no material advances to be recorded.

00Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical.

00Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra.

00His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical.

00It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds.

00The earlier chapters, treating chiefly of the arithmetical foundations of the science, differ but little in their line of argument from the principles laid down by Pietro Aron, Zacconi, and other early writers of the Boeotian school; but in bk.

00To assist his lectures on astronomy he constructed elaborate globes of the terrestrial and celestial spheres, on which the course of the planets was marked; for facilitating arithmetical and perhaps geometrical processes he constructed an abacus with twenty-seven divisions and a thousand counters of horn.

00At a time when the Cartesian system of vortices universally prevailed, he found it necessary to investigate that hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it.

00Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms R- x 3 3x 3, 3x 3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x-- 3, x- `3, x-- 3, &c.

00" Now for the interpolation of the rest, I considered that the denominators I, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated.

00These elements will enable us to convert, by a simple arithmetical operation, any historical date, of which the chronological characters are given according to any era whatever, into the corresponding date in the Christian era.

00They correspond to the two methods of regarding quantity - the arithmetical and the geometrical.

00Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or geometrical methods.

00This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.

00(ii) The very earliest stages of mensuration should be directly associated with simple arithmetical processes.

00arithmetical mensuration, geometrical mensuration, and analytical mensuration.

00The most elementary stage is arithmetical mensuration, which comprises the measurement of the areas of rectangles and parallelepipeds.

00The third step is the arithmetical calculation of the area or volume of the rearranged figure.

00These last two steps may introduce magnitudes which have to be subtracted, and which therefore have to be treated as negative quantities in the arithmetical.

00Chebichev further constructed an instrument for drawing large circles, and an arithmetical machine with continuous motion.

00A slide rule should be used for the arithmetical operations, as it works to the accuracy obtainable in practice.

00Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division.

00Thus in arithmetical calculations if the base is not expressed it is understood to be io, so that log m denotes log n m; but in analytical formulae it is understood to be e.

00This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus 0, 1,0000 0000 10, I,0001 0000 20, 1,0002 000 I 9 90, 1,0099 4967 In the arithmetical column the numbers increase by io, in the geometrical column each number is derived from its predecessor by multiplication by i 0001.

00Thus the number lox in the arithmetical column corresponds to 10 8 (1.0001) x in the geometrical column; the intermediate numbers being obtained by interpolation.

00The table extends to 2 3 0270 in the arithmetical column, and it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in the geometrical column; this last result showing that (1.0001)23027 022 = 10.

00To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.

00The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios.

00Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series.

00This ratio, invariably denoted by 7r, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number.

00Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a " formula."

001 V 2+2A/ From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation.

00quam Archimedes, we may direct our attention either to the infinite series of geometrical operations or to the corresponding infinite series of arithmetical operations.

00(1) Ile /3 c Tou irvpiov, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Hepi On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) `H Ka06Xov lrpa-yµareta, perhaps a work on the general principles of mathematics in which were included Apollonius' criticisms and suggestions for the improvement of Euclid's Elements; (5) ' (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of 7r than the 37 and 3,4-A of Archimedes; (6) an arithmetical work (as to which see Pappus) on a system of expressing large numbers in language closer to that of common life than that of Archimedes' Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus' comm.

00Deslandres,s who found that the successive differences in the frequencies formed an arithmetical progression.

00The three commonest means are the arithmetical, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.

00The arithmetical mean of n quantities is the sum of the quantities divided by their number n.

00The harmonic mean of n quantities is the arithmetical mean of their reciprocals.

00The significance of the word "mean," i.e., middle, is seen by considering 3 instead of n quantities; these will be denoted by a, b, c. The arithmetic mean b, is seen to be such that the terms a, b, c are in arithmetical progression,.

00The arithmetico-geometrical mean of two quantities is obtained by first forming the geometrical and arithmetical means, then forming the means of these means, and repeating the process until the numbers become equal.

00The quadratic mean of n quantities is the square root of the arithmetical mean of their squares.

00He generalized Weber's law in the form that sensation generally increases in intensity as the stimulus increases by a constant function of the previous stimulus; or increases in an arithmetical progression as the stimulus increases in a geometrical ratio; or increases by addition of the same amount as the stimulus increases by the same multiple; or increases as the logarithm of the stimulus.

00An opportunity, however, presented itself: being required to work out from data supplied to him the "defilement" of a proposed fortress (an operation then only performed by a long arithmetical process), Monge, substituting for this a geometrical method, obtained the result so quickly that the commandant at first refused to receive it - the time necessary for the work had not been taken; but upon examination the value of the discovery was recognized, and the method was adopted.

00He held that thought has in itself no power of development, and ultimately reduced it to arithmetical computation.

00He nowhere explains the origin of the notions of unity and number, but merely asserts that through their means we can have absolutely exact arithmetical propositions (Works, i.

00The first desideratum here mentioned - the want, namely, of an accurate statement of the relation between the increase of population and food - Malthus doubtless supposed to have been supplied by the celebrated proposition that "population increases in a geometrical, food in an arithmetical ratio."

00It appears that Pascal contemplated publishing a treatise De aleae geometria; but all that actually appeared was a fragment on the arithmetical triangle (Traite du triangle arithmetique, " Properties of the Figurate Numbers"), printed in 1654, but not published till 1665, after his death.

00care being taken to attend to the sign of a, so that when that is negative the arithmetical values of y and a are to be added in order to give that of $.

00and the corresponding axes parallel, then by changing the signs of x x, y, the values x',17; x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables.

00During the later years of his life he resided in London, devoting himself to the construction of machines capable of performing arithmetical and even algebraical calculations.

00That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

00Statistical inquiries as to the incidence of taxation or of particular taxes, though ideal or even approximate equality of a palpable arithmetical kind is practically unattainable by governments, are not altogether to be put aside.

00The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant" (iripas ixov), and as such is styled "quantity" and perhaps "number" but the ideal numbers are distinct from arithmetical numbers.

00Xenocrates, however, failing, as it would seem, to grasp the idealism which was the metaphysical foundation of Plato's theory of natural kinds, took for his principles arithmetical unity and plurality, and accordingly identified ideal numbers with arithmetical numbers.

00In thus reverting to the crudities of certain Pythagoreans, he laid himself open to the criticisms of Aristotle, who, in his Metaphysics, recognizing amongst contemporary Platonists three principal groups - (1) those who, like Plato, distinguished mathematical and ideal numbers; (2) those who, like Xenocrates, identified them; and (3) those who, like Speusippus, postulated mathematical numbers only - has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideas are numbers made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations.

00The central gauge is useful for correcting and checking the others, but in such a perfectly simple case as the straight valley above assumed it may be omitted in calculating the results, and if the other four gauges are properly placed, the arithmetical mean of their results will probably not differ widely from the true mean for the valley.

00As an arithmetical calculator he was not only wonderfully expert, but he seems to have occasionally found a positive delight in working out to an enormous number of places of decimals the result of some irksome calculation.

00" He used to carry on," says his elder son, William Edwin Hamilton, " long trains of algebraical and arithmetical calculations in his mind, during which he was unconscious of the earthly necessity of eating; we used to bring in a ` snack ' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards."

00The arithmetical fact is that I i and 9 may be regrouped as 12 and 8, and the statement "IId.+9d.

00

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