# Arithmetical Sentence Examples

The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between

**arithmetical**and geometrical measurement.Thus, while

**arithmetical**numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.He 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.He 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.In 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.**Arithmetical**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.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.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.**Arithmetical**and Algebraical Treatment of Equations.-The following will illustrate the passage from**arithmetical**to algebraical reasoning.In 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.AdvertisementThe subsequent reasoning is

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

**arithmetical**reasoning.These transpositions are purely

**arithmetical**.These numbers constitute an

**arithmetical**progression of the rth order.In 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.AdvertisementThe 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.Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of

**arithmetical**problems and the statement of**arithmetical**facts.It 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.The only known type of algebra which does not contain

**arithmetical**elements is substantially due to George Boole.The 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.AdvertisementInvestigation 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.Although 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.His 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.It 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.The 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.AdvertisementAt 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.Now 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.These 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.They correspond to the two methods of regarding quantity - the

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

**arithmetical**or geometrical methods.This 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.The most elementary stage is

**arithmetical**mensuration, which comprises the measurement of the areas of rectangles and parallelepipeds.The third step is the

**arithmetical**calculation of the area or volume of the rearranged figure.These last two steps may introduce magnitudes which have to be subtracted, and which therefore have to be treated as negative quantities in the

**arithmetical**.Chebichev further constructed an instrument for drawing large circles, and an

**arithmetical**machine with continuous motion.A slide rule should be used for the

**arithmetical**operations, as it works to the accuracy obtainable in practice.Logarithms 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.Thus 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.The 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.To 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.The 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.Besides 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.This 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.Archimedes'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."Deslandres,s who found that the successive differences in the frequencies formed an

**arithmetical**progression.The three commonest means are the

**arithmetical**, geometrical, and harmonic; of less importance are the contraharmonical, arithmetico-geometrical, and quadratic.The

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

**arithmetical**mean of their reciprocals.The 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,.The 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.The quadratic mean of n quantities is the square root of the

**arithmetical**mean of their squares.He 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.He held that thought has in itself no power of development, and ultimately reduced it to

**arithmetical**computation.The 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."It 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.During 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.The 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.Xenocrates, 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.In 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.The 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.As 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.The

**arithmetical**fact is that I i and 9 may be regrouped as 12 and 8, and the statement "IId.+9d.The grouping method introduces multiplication into the definition of large numbers; but this, from the teacher's point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary

**arithmetical**processes.Thus

**arithmetical**processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout.In 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.The 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.The 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).Thus any

**arithmetical**processes which can be applied to the numbers p, q, r, ..The numbers (integral or decimal) by which we represent the results of

**arithmetical**operations are often only approximately correct.When 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.The 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.When the unit is not determined, the reasoning is algebraical rather than

**arithmetical**.More 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.They are not suitable for elementary purposes, as the

**arithmetical**relations involved are complicated and difficult to grasp.In 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.Text-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 £.A 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.These calculations not only involved difficult mathematical expressions but also dealt with heavy

**arithmetical**calculations.The

**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.The 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.Nothing 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.This 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.It 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.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).General

**Arithmetical**Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.It 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.To 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.Thus 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.Statistical 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.He introduced the words cosine and cotangent, and he suggested to Henry Briggs, his friend and colleague, the use of the

**arithmetical**complement (see Brigg's Arithmetica Logarithmica, cap. xv.).When 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.The 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.Hence 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.In order to save

**arithmetical**labour it is convenient to be provided with conversion factors for reducing variously expressed results to the standard form.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.Even 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.