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multiplication

multiplication

multiplication Sentence Examples

  • (iii.) The multiplication and division of monomials is effected by means of the law of indices.

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  • The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be respectively the required sum and product of the cardinal numbers in question.

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  • I think that there is never any alcoholic fermentation without there being at the same time organization, development and multiplication of globules, or the continued consecutive life of globules already formed."

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  • The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows.

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  • The opportunity thus given for debate naturally stimulated the movement in favour of constitutional government, which received new impulses from the sympathetic attitude of the emperor Alexander II., his grant in 1879 of a constitution to the liberated principality of Bulgaria, and the multiplication of Nihilist outrages which pointed to the necessity of conciliating Liberal opinion in order to present a united front against revolutionary agitation.

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  • 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.

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  • the multiplication by 3 and the reversal of direction.

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  • By the multiplication of the protoplasts in these merismatic areas the substance of the plant is increased.

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  • 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 p X Q = q X P; or the more abstract statement that p X q= q X p.

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  • We see first that any operation with 4a-3b can be regarded as an operation with (+)4a+(-)3b, subject to the conditions (I) that the signs (+) and (-) obey the laws (+)=(+),(+)(-)=(-)(+)= (-), (-) (-)=(+), and (2) that, when processes of multiplication are completed, a quantity is to be added or subtracted according as it has the sign (+) or (-) prefixed.

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  • has been brought about rather by extermination than specialization, and their distinctive facies by the development and multiplication of the surviving types.

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  • The most frequent are the miracle at Cana, the multiplication of the loaves and fishes, the paralytic carrying his bed, the healing of the woman with the issue of blood, the raising of Lazarus, FIG.

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  • The branches of the stem arise by multiplication of the cells 01 the epidermis and cortex at a given spot, giving rise to a protuber ance, at the end of which an apical meristem is established.

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  • Dean's multiplication table of 44, the number of rented apartments, wasn't perfect, but that num­ber times even a reasonable monthly rental lessened any sympathy he might have felt for the woman's financial plight.

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  • Of this we see evidence in the multiplication of Satans in the Book of Enoch.

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  • A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.

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  • = (1 +Plain) (1 = er,nam; and, by multiplication, II (1 +ala+a2a2+...) = II (1-}-biP+b2P 2 +...

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  • Accordingly, the typical form for such a complex number is x+yi, and then with this notation the above-mentioned definition of multiplication is invariably adopted.

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  • 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 fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.

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  • Growing knowledge of Aristotle's works and the multiplication of translations enabled students to tendency.

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  • She can add and subtract with great rapidity up to the sum of one hundred; and she knows the multiplication tables as far as the FIVES.

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  • If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +��� +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.

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  • If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.

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  • If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.

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  • This observation led him to further work, and he succeeded in showing that in vascular organs the presence of cells in inflammatory exudates is not the result of exudation but of multiplication of pre-existing cells.

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  • Such excessive multiplication of the larger taxonomic divisions shows an imperfect sense of proportion, for if the term " class " be allowed its usual zoological value, no student can fail to recognize that the Hexapoda form a single welldefined class, from which few entomologists would wish to exclude even the Apterygogenea.

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  • He forms n equations from f by separate multiplication by x, ...x, I, in succession, and similarly treats 4) with m multipliers I.

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  • (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.

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  • Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) - 4(x-2) This means that, whatever unit A we take, 2(x+ I) A Sand 5 4(x-2) A are equal.

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  • Multiplication may or may not be commutative, and in the same way it may or may not be associative.

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  • = exp,udl where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem.

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  • They differ from other Endopterygota in the multiplication of their Malpighian tubes, and from all other Hexapoda in the union of the first abdominal segment with the thorax.

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  • They protested against the multiplication of slaves from motives of vanity in the houses of the great, against the gladiatorial combats (ultimately abolished by the noble self-devotion of a monk) and against the consignment of slaves to the theatrical profession, which was often a school of corruption.

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  • We will confine ourselves here to algebraic complex numbers - that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra.

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  • Persian influence is also responsible for the vast multiplication of good spirits or angels, Gabriel, Raphael, Michael, &c., who play their part in apocalyptic works, such as the Book of Daniel and the Book of Enoch.

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  • (multiplication symbolic) ?r1! ?2,�..

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  • Augustus set himself against the undue multiplication of manumissions, probably considering the rapid succession of new citizens a source of social instability, and recommended a similar policy to his successor.

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  • The addition and multiplication of two relation-numbers is defined by taking two relations R and S, such that (I) their fields have no Cf.

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  • 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.

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  • This latter process, however, is itself based on a definition of division in terms of multiplication (�� 15, 16).

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  • The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.

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  • The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.

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  • The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz.

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  • Brooks, on the other hand, as stated above, regards the medusa as the older type and looks upon both polyp and medusa, in the Hydromedusae, as derived from a free-swimming or floating actinula, the polyp being thus merely a fixed nutritive stage, possessing secondarily acquired powers of multiplication by budding.

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  • By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...

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  • For instance, if n= 4, E r = e l e 3, Es= e 2 e 3 e 4, we have IErIE8 = (-e2e4) (- e 1) = ele2e4 = l e3, consequently, by the rule of regressive multiplication, eie3�e2e3e4 = e3.

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  • The multiplication of thongs for purposes of flogging is found in the old Roman flagellum, a scourge, which had sometimes three thongs with bone or bronze knots fastened to them.

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  • The formation and gradually increasing thickness of its bark are explained by the continually increasing need of adequate protection to the living cortex, under the strain of the increasing framework which the enormous multiplication of its living protoplasts demands, and the development of which leads to continual rupture of the exterior.

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  • We therefore define algebraical division by means of algebraical multiplication, and say that, if P and M are multinomials, the statement " P/M = Q " means that Q is a multinomial such that MQ (or QM) and P are identical.

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  • The binary products e i e j, however, are expressible as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question.

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  • All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

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  • In 1857 Pasteur decisively proved that fermentation was a physiological process, for he showed that the yeast which produced fermentation was no dead mass, as assumed by Liebig, but consisted of living organisms capable of growth and multiplication.

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  • Instead of regarding these as only ministering to the construction of the bulky portions, the living protoplasts take the first place as the essential portion of the tree, and all the other features are important mainly as ministering to their individual well-being and to their multiplication.

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  • The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication.

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  • The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multiplication.

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  • (v.) When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication.

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  • where x=a/A; thus the successive terms are obtained by successive multiplication.

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  • Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication.

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  • The unknown he terms arithmos, the number, and in solutions he marks it by the final s; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities.

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  • His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots.

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  • His notation is based on that of Vieta, but he introduced the sign X for multiplication, - for continued proportion, :: for proportion, ' and denoted ratio by one dot.

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  • This last character has since been entirely restricted to multiplication, and ratio is now denoted by two dots (:).

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  • To the mere tradition preserved by memory and handed on by speech was then added the written record and its later multiplication by the mechanical arts of printing, by which it acquired permanence and universal distribution.

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  • Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor - n 2, and thus our equations take the form (b 2 v 2 + n2)E+(a2 - b2) ds (b2 2 + n2)n +(a2 - b2 y =0 (7).

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  • The following books on the population of London have been published: John Graunt, Natural and Political Observations on the Bills of Mortality (1661, other editions 1662, 1665, 1676); Essay in Political Arithmetick (1683); Five Essays on Political Arithmetick (1687); Several Essays in Political Arithmetick (1699, 1711, 1751, 1 755); Essay concerning the Multiplication of Mankind (1682, 1683, 1686), all by Sir William Petty; Corbyn Morris, Observations on the past Growth and present State of the City of London (1751); Collection of the Yearly Bills of Mortality from 1657 to 1758 (ed.

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  • We have, in fact, a form of larval multiplication that recalls the development of digenetic Trematodes.

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  • The conditions which favour the vitality, growth and multiplication of the typhoid bacillus are the following: the soil should be pervious; it should be permeated with a sufficiency of decaying - preferably animal - organic matters; it should possess a certain amount of moisture, and be subject to a certain temperature.

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  • The greater the commercial and industrial prosperity of a town, the more rapid was the multiplication of craft gilds, which was a natural result of the ever-increasing division of labour.

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  • The rapid multiplication that takes place in the larval stage of nearly all endoparasitic forms affects the tissues of the "intermediate" host in which they live.

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  • I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this seriesI X m 2 I X m - 2 This rule I therefore applied to the series to be interpolated.

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  • be produced by the continued multiplication of this series mX m - I X 7n-2 X m-3...

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  • Hence multiplication of his values for capacities by 2.54 reduces them to E.S.

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  • This is the Dionysian or Great Paschal Period, and is formed by the multiplication of the numbers 28 and 19, that is, of the solar and lunar cycles, into each other.

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  • The number of species included in this division is enormous, and the multiplication of families is, to some extent, a natural result of increasingly close study.

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  • The commutative law for multiplication is directly illustrated; and subdivisions or groupings of the units lead to such formulae as (a+ a) (b + 0)=.

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  • The multiplication is shown in the adjoining diagram; the factors s ands are kept outside, so that the sum uo,o+3uo,1+ ...

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  • are obtained from the data by summation and multiplication; and the ordinary methods then apply for calculation of its volume.

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  • (g) Manifestation of Jesus as the heaven-descended living Bread and its contradiction in Galilee, vi.: multiplication of the loaves; walking on the waters; and His discourse on the holy Eucharist.

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  • 46-54); multiplication of the loaves (vi.

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  • 4), the day after the multiplication of the loaves.

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  • Of his seven great symbolical, doctrinally interpreted " signs," John shares three, the cure of the ruler's son, the multiplication of the loaves, the walking on the waters, with the Synoptists: yet here the first is transformed almost beyond recognition; and the two others only typify and prepare the eucharistic discourse.

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  • This enlargement of the bishop's parish and multiplication of the chuches under his care led to a change in the functions of the presbyterate.

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  • From this theory, powerful in Great Britain and her colonies, supreme in the United States of America, has resulted an enormous multiplication of sects.

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  • In France, and to a certain extent in Italy, the multiplication of chapels led to their being placed on the north and south side of the aisles, and in some cases, as at Albi in France, to the suppression of the aisles and the instalment of the chapels in their place.

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  • The Greek system, however, adopted this foot as a basis for decimal multiplication, forming --

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  • 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.

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  • The explanation of this property of the base io is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact.

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  • The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.

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  • ., and the value of its reciprocal, log e io (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.302585092994 0 45 68401 799 1 4

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  • This 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.

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  • Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as 'tis said), to save the tedious multiplication and division in astronomical calculations.

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  • 2 The method consists in the use of the formula sin a sin b=2 {cos(a-b)-cos(a+b)l, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only.

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  • It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method.

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  • - Rotifera are unisexual, with the sexes dimorphic. The ovary is, as in many Platyhelminthes, duplex; one part, the germary, being an organ for the production by cell multiplication of the germ-cells or eggs.

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  • The life of the parasites while in the insect is characterized by an alternation of active periods, during which multiplication goes on, with resting-periods, when the Trypanosomes become attached to the epithelial cells of the host.

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  • The parasites are frequently more numerous in the spleen, bone-marrow, kidneys, &c., than elsewhere, and it has been found that multiplication goes on rather more actively in the capillaries of these organs.

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  • All Trypanosomes are capable of binary longitudinal fission, and this appears to be the chief method of multiplication.

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  • The Tenth Census put it beyond doubt that economic and social forces had been at work, reducing the rate of multiplication.

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  • The change, wrote General Walker, which produced this falling off from the traditional rate of increase of about 3% per annum, was that from the simplicity of the early times to comparative luxury; involving a rise in the standard of living, the multiplication of artificial necessities, the extension of a paid domestic service, the introduction of women into factory labor.2 In his opinion the decline in the birth-rate coincidently with the increase of immigration, and chiefly in those regions where immigration was greatest, was no mere coincidence; nor was such immigrant invasion due to a weakening native increase, or economic defence; but the decline of the natives was the effect of the increase of the foreigners, which was a shock to the principle of population among the native element.

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  • Later changes consist in the growth of the shell-valves over the whole area of the mantle-flaps, and in the multiplication of the gill-filaments and their 'consolidation to form gill-plates.

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  • The multiplication of orders is attended with practical difficulties, and the distinctions between the various groups of the Linnean Neuroptera are without doubt less obvious than those between the Coleoptera (beetles) and the Diptera (two-winged flies) for example.

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  • The definite course is not occasioned so much from the ferments which exist in the leaves themselves, but from an arrest of the digestive process which allows the rapid multiplication of the former in the intestines.

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  • The measures by which the government of India chiefly endeavours to reduce the liability of the country to famine are the promotion of railways; the extension of canal and well irrigation; the reclamation of waste lands, with the establishment of fuel and fodder reserves; the introduction of agricultural improvements; the multiplication of industries; emigration; and finally the improvement where necessary of the revenue and rent systems. In times of famine the function of the railways in distributing the grain is just as important as the function of the irrigation-canals in increasing the amount grown.

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  • (the former part of which is lost, the existing fragment beginning in the middle of the 14th proposition) related to a system of multiplication due to Apollonius of Perga.

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  • The insistence on the unique efficacy of the sacrifice of the altar led to the multiplication of masses, and so of altars, which were placed in the transepts or aisles or in chapels, dedicated to the saints whose relics they enshrined.

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  • Multiplication of altars is another medieval characteristic. This also is probably a result of the edict of Pope Felix already mentioned.

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  • It is in the 6th century that we begin to find traces of the multiplication of altars.

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  • and W., with great certainty, without that multiplication of bearings of the points of the compass usual in our maps, which serves as the root of the others."

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  • In common with all enlightened opinion, he complained bitterly of the excessive multiplication of exemptions, of the exaggerated extension of appeals to Rome, of the luxury of the Roman court, of the venality of the cardinals, and of the injury done to the traditional hierarchy by the very extent of the papal power, which was calculated to turn the strongest head.

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  • Delage has distinguished as multiplication those cases in which the new individual arises from a mass of cells which remain a part of the maternal tissues during differentiation, reserving the term reproduction for those cases in which the spore or cell which is the starting-point of the new individual begins by separating from the maternal tissues; but the distinction is inconvenient in practice and does not appear to carry with it any fundamental biological significance.

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  • gemmae, chlamydospores, resting-cells, cysts, &c.; (3) carpospores, produced by a more or less complex formative process, often in special fructifications, and subserving either or both multiplication and persistence, e.g.

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  • Darwin believed that man advanced to his present high condition through such a struggle, consequent on his rapid multiplication.

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  • Variations in the structure and forms of leaves and leafstalks are produced by the increased development of cellular tissue, by the abortion or degeneration of parts, by the multiplication or repetition of parts and by adhesion.

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  • Experience, however, has shown that even when the movements of the ground are alarming the actual range of motion is so small that a satisfactory record can be obtained only by some mechanical (or optical) method of multiplication.

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  • With pendulums of shorter length, say metres, it is necessary to have a multiplication 80 to too fold by a double system of very light levers, in order to render the extremely slight tilting of their support perceptible.

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  • multiplying and dividing by 5 and finding the 13/4 value were also among the fundamental instruments of calculation, and all multiplication proceeded by repetitions of these processes with addition, e.g.

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  • The nerve cells of the higher vertebrata, unlike their blood cells, their connective tissue cells, and even their muscle cells, early, and indeed in embryonic life, lose power of Nervous multiplication.

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  • The antipodal cells aid more or less in the process of nutrition of the developing embryo, and may undergo multiplication, though they ultimately disintegrate, as do also the synergidae.

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  • In all the multicellular plants of this group which have been adequately investigated, vegetative multiplication by means of what are known as hormogonia has been found to occur.

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  • Special provision for vegetative multiplication is not common among Chlorophyceae.

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  • In Caulerpa no other means of multiplication is as yet known.

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  • Multiplication takes place in some cases by the endogenous formation of zoospores, the organism having come to rest; in others by longitudinal division, when the organism is still motile.

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  • Vegetative multiplication occurs only sparingly in Rhodophyceae.

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  • The great bulk of the code was an obstacle to the multiplication of copies of it, whilst the necessity for them was in a great degree superseded by the publication from time to time of synopses and encheiridia of its contents, composed by the most eminent jurists, of which a very full account will be found in the Histoire au droit byzantin, by the advocate Mortreuil, published in Paris in 1846.

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  • von Herder suggested that increase by multiplication with the consequent struggle for existence had played a large part in the organic world, but his theme remained vague and undeveloped.

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  • Pellican's autobiography describes the gradual multiplication of accessible books on the subjects, and he not only studied but translated a vast mass of rabbinical and Talmudic texts, his interest in Jewish literature being mainly philological.

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  • And by comparing one record with another, we can follow the multiplication of " holy places," and sometimes can even see them being shifted from one spot to another, as the centuries pass.

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  • The multiplication of art periodicals, lectures, books, photographs, meetings of societies and gilds, museums, schools of arts and crafts, polytechnics, scholarships, facilities for travel, exhibitions, even those of the Royal Academy, to which objects of applied art are now admitted, not only encourages many persons to become workers and designers in the applied arts, but exposes everything to the plagiarist, who travesties the freshest idea before it has well left the hands of its originator.

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  • chap. 21) of Leonard Digges 1 (originally published by his son Thomas in 1571, and again in 1591): - "Marvellous are the conclusions that may be performed by glasses concave and convex, of circular and parabolic forms, using for multiplication of beams sometime the aid of glasses transparent, which, by fraction, should unite or dissipate the images or figures presented by the reflection of other."

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  • as division from multiplication; but he really meant that it is a deduction from a hypothesis of the law of a cause to particular effects which, being true, verify the hypothesis.

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  • For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are coplanar provided ij = - ji, for it would then assume the form ij(yz' - zy').

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  • He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: - I T = 12 '=' 2 = _ 1, ij= - ji=k, jk= - kj=i, ki= - ik =j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'+jc'+kd') = A+iB+jC+kD, where A=aa' - bb' - cc' - dd', B = ab'+ba'+cd' - dc', C = ac'+ca'+db' - bd', D =ad' +da'+bc' - cb'.

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  • In particular, two species of multiplication (" inner " and " outer ") of directed lines in one plane were given.

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  • The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product.

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  • But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked.

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  • Again a unit point p. may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (b) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar.

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  • As human intelligence and industry come into play the means of livelihood are proportionately extended; population multiplies, and with this multiplication production increases.

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  • The spirit-land of the Ibo on the Lower Niger had its rivers, forests or hills, its towns and roads, as upon earth: 1 the spirits of the Mordvinian mythology, created by Chkai, not only resembled men, they even possessed the faculty of reproduction by multiplication.

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  • Elements of race-consciousness help to shape the outlook on nature or life: and slight differences of linguistic use in the coining of descriptive terms sometimes lead to the multiplication of divine forms. Exacter observation of nature; closer attention to its contrasts of life and death, or light and darkness, or male and 9 Cf.

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  • 1784) and Shubael Stearns, "New Light" evangelists who became Baptists, the spread of Baptist principles and the multiplication of Baptist churches throughout the southern colonies were in great measure due.

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  • were not for its numerous enemies "the multiplication of ostriches would be quite unexampled."

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  • The psalms have already been dealt with, but it may be noted again how the multiplication of saints' festivals, with practically the same special psalms, tends in practice to constant repetition of about one-third of the Psalter, and correspondingly rare recital of the remaining two-thirds, whereas the Proprium de Tempore, could it be adhered to, would provide equal opportunities for every psalm.

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  • As symbols of light and joy a candle is held on each side of the deacon when reading the Gospel at Mass; and the same symbolism underlies the multiplication of lights on festivals, their number varying with the importance of the occasion.

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  • 23 ff.), the multiplication of sacrifices and the minute regulations with regard to the blood and oil.

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  • There is a multiplication of taxes in trade which recalls the old colonial alcabala tax, and it serves to restrict commerce and augment the cost of goods in much the same way, if not to the same degree.

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  • W.) having given up the specializations of func tion impressed on them during evolution and simply carrying out the fundamental functions of nutrition, growth, and multiplication which mark the generalized activities of the bacterial cell, and at the same time rendered as accessible to the environment by isolation and consequent extension of surface, we should doubtless find them exerting changes in the fermentable fluids necessary to their life similar to those exerted by an equal mass of bacteria, and that in proportion to their approximation in size to the latter.

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  • the multiplication of the living organisms in the tissues; this is essential to, and regulates, the supply of toxins.

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  • It is to be noted that there is no fixed relation between toxin production and bacterial multiplication in the body, some of the organisms most active as toxin producers having comparatively little power of invading the tissues.

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  • In the lastmentioned disease even the local multiplication depends upon the presence of other bacteria, as the tetanus bacillus has practically no power of multiplying in the healthy tissues when introduced alone.

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  • The most important are: the Treatise of Taxes and Contributions (1662, 1667 and 1685); Political Arithmetic, presented in MS. to Charles II., but, because it contained matter likely to be offensive to France, kept unpublished till 1691, when it was edited by Petty's son Charles; Quantulumcunque, or a Tract concerning Money (1682); Observations upon the Dublin Bills of, Mortality in 1681 and the State of that City (1683); Essay concerning the Multiplication of Mankind (1686); Political Anatomy of Ireland (1691).

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  • In the 8th century the mayors of the palace are found affixing their personal seals to royal diplomas; and, once the idea was started, the multiplication of seals naturally followed.

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  • The "struggle for existence," the rate of multiplication of animals, and the dependence of their average numbers upon food supply are very clearly demonstrated, and the following conclusion was reached: "Those that prolong their existence can only be the most perfect in health and vigour; ...

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  • By 1888 a new state of chaos had grown up as the result of the multiplication of bodies, and the new Redistribution Act of 1885 paved the way for a further reorganization of local matters by the Local Government Act of 1888, followed by that of 1894.

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  • The difficulty has been increased by the confusion resulting from the multiplication of genera founded on slight characters, and from the description (in consequence of their wide distribution) of identical plants under several different genera.

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  • The excessive multiplication of the title has tended to deprive it of much social value in itself, and under the democratic constitution of Italy it confers neither power nor precedence.

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  • The multiplication of pilgrims after Islam soon made it necessary to clear away the nearest dwellings and enlarge the place of prayer around the Ancient House.

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  • As an illustration, we may take the elementary processes of addition, subtraction, multiplication and division.

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  • (i) If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number.

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  • (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication.

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  • Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

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  • (iii) The inclusion of the four processes under one general head fails to indicate the essential difference between addition and multiplication, as direct processes, on the one hand, and subtraction and division, as inverse processes, on the other (§ 59).

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  • Multiplication and Division 3.3.2 34.

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  • Multiplication of Fractional Numbers 6.3 66.

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  • Multiplication and Division of Surds 8.4 87.

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  • Multiplication Table 11.1.5 99.

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  • Multiplication By A Small Number 11.1.6 103.

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  • Multiplication of a Mixed Number 11.1.8 106.

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  • Multiplication 11.2.2 114.

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  • The first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition.

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  • 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.

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  • Later, a system similar to the Hebrew was adopted, and extended by reproducing the first nine symbols of the series, preceded by accents, to denote multiplication by moo.

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  • The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17).

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  • Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ II), and is taken a certain number of times.

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  • is equivalent to 5 X 12d.; here the multiplication arises out of a subdivision of the original unit is.

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  • Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times.

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  • The result of the multiplication is called the product of the unit by the number of times it is taken.

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  • Diagram of Multiplication.-The process of multiplication is performed in order to obtain such results as the following:- If I boy receives 7 apples, then 3 boys receive 21 apples; or If Is.

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  • The 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.

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  • Successive Multiplication.-In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e.

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  • When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication; and so on indefinitely.

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  • Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units.

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  • Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings.

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  • Division.-From the commutative law for multiplication, which shows that 3 X 4d.

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  • Successive Division may be performed as the converse of successive multiplication.

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  • Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

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  • This gives a rule for testing multiplication, which is found in most text-books.

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  • - Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout.

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  • For multiplication we make use of the formula (§ 60 (i)) (a'= a) (b' (3) = a'b'+a/3 = (a'(3+b'a).

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  • ., the series of indices being the standard series; and we have also seen (§ 44) that multiplication of any two of these numbers corresponds to addition of their indices.

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  • Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

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  • The series is formed by successive multiplication, and any antilogarithm to a larger number of decimal places is formed from it in the same way by multiplication.

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  • For practical purposes the number taken as base is so; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, i.e.

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  • To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects.

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  • The multiplication or division here involves the omission of the unit " boy," and the operation is incomplete.

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  • The conversion of the unit may be represented as multiplication by a factor 4 boy apples, so that the operation is 4 apples X 5 boysboy 4 apples r boy Y 5 X I X I boy = 5 X 4 apples= 20 apples.

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  • In §§ 33 4 2 We Have Dealt With The Parallelism Of The Original Number Series With A Series Consisting Of The Corresponding Multiples Of Some Unit, Whether A Number Or A Numerical Quantity; And The Relations Arising Out Of Multiplication, Division, &C., Have Been Exhibited By Diagrams Comprising Pairs Of Corresponding Terms Of The Two Series.

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  • addition or subtraction of two numbers or quantities, and multiplication by some small numbers - are with a little practice performed more quickly and more accurately from left to right.

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  • For Multiplication And Division We Use A Multiplication Table, Which Is A Multiple Table, Arranged As Explained In § 36, And Giving The Successive Multiples, Up To 9 Times Or Further, Of The Numbers From 1 (Or Better, From O) To 10, 12 Or 20.

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  • These Are Examples Of The Associative Law For Multiplication (§ 58 (Iv)).

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  • Multiplication By A Large Number.

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  • For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corre up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier.

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  • Contracted Multiplication.

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  • For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has i as its numerator.

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  • The method E of § ioi being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 1 01); and the multiplication is performed from the left, two extra figures being kept in.

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  • Other units are derived from these by multiplication or division by powers of io, the names being denoted by prefixes.

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  • The prefixes for multiplication by io, io 2, io 3 and Io 4 are deca-, hecto-, kilo- and myria-, and those for division by io, io 2 and io 3 are deci-, centi- and milli-; the former being derived from Greek, and the latter from Latin.

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  • The dissemination of cheap literature and the multiplication of cheap newspapers could not compensate the nation for the ruin of an important trade.

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  • Meanwhile the chief objection, that of "novelty," was gradually removed by the multiplication of local manifestations, the genuineness of which was proved to the satisfaction of the Roman Congregation of Rights, and in 1765 it was allowed for houses of the Visitation and certain countries.

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  • While strongly discouraging the arbitrary multiplication of public or private fasts, the English Church seems to leave to the discretion of the individual conscience every question as to the manner in which the fasts she formally enjoins, are to be observed.

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  • The new jurisprudence encouraged the multiplication of small properties.

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  • This multiplication of the seed-bearing branches might at first sight be considered advantageous; but in practice the quality of the grain is found to be inferior, as if the force that should have been devoted to the maturation of the grain were, in a measure, diverted and expended in the production of additional branches to the spike.

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  • Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

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  • Multiplication of two Determinants of the same Order.

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  • To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

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  • We have already mentioned that in astronomical practice right ascensions are expressed in time, so that no multiplication by 15 is necessary.

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  • For the investigation of the general scheme of sidereal structure, the multiplication of results of the kind is indispensable.

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  • So long as Mr. Lloyd George was Minister, Dr. Addison was his right-hand man in the strenuous labours of the office, resulting in the enormous multiplication of engines of war, and in the redeeming of many vital industries, fertilizers, tungsten and potash from German control; and when Mr. Lloyd George formed a Government himself in December 1916, he placed him at the head of the department.

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  • The very multiplication of offices, so noticeable at this time, furthered this triumph of feudalism by multiplying the links of personal dependence, and neutralizing more and more the direct action of the central authority.

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  • The festival of the new doctrine, which turned the head of the new pontiff (June 8), the loi de Prairial, or code of legal murder (June 10), which gave the deputies themselves into his hand; and the multiplication of executions at a time when the victory of Fleurus (June 25) showed the uselessness and barbarity of this aggravation of the Reign of Terror provoked against him the victorious coalition of revenge, lassitude and fear.

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  • The successive issues of assignats, and the Financial multiplication of counterfeits made abroad, had so policy depreciated this paper money that an assignat of joo otflic francs was in February 1796 worth only 30 centimes; 01).

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  • Alteration in the symmetrical arrangement as well as in the completeness and regularity of flowers has been traced to suppression or the non-development of parts, degeneration or imperfect formation, cohesion or union of parts of the same whorl, adhesion or union of the parts of different whorls, multiplication of parts, and deduplication (sometimes called chorisis) or splitting of parts.

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  • Multiplication, or an increase of the number of parts, gives rise to changes.

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  • Flowers become double by the multiplication of the parts of the corolline whorl; this arises in general from a metamorphosis of the stamens.

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  • The staminal row is more liable to multiplication of parts than the outer whorls.

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  • plants as Mesembryanthemum, where there is a multiplication of petals in several rows.

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  • Dean's multiplication table of 44, the number of rented apartments, wasn't perfect, but that num­ber times even a reasonable monthly rental lessened any sympathy he might have felt for the woman's financial plight.

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  • Just for explaining a little, a genetic algebra is basically an algebra in which multiplication is not associative.

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  • Recall that in Heisenberg's first paper, he defined a multiplication which was not necessarily commutative.

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  • Then = n (f g ). In other words, the discrete Fourier transform maps convolution to pointwise multiplication.

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  • convolution in the time domain is equivalent to multiplication in the frequency domain, which is equivalent to addition in the cepstrum domain.

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  • homomorphism between both groups because this would require testing the full multiplication table.

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  • Logarithms are ' traditionally ' used for multiplication (by adding the logarithms are ' traditionally ' used for multiplication (by adding the logarithms) and division (by subtracting the logarithms ).

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  • Math: I never learned multiplication tables and cannot do long division, even on paper.

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  • matrix multiplication, you can use the product global function.

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  • On the set of all n × n matrices over a field F it is possible to define a multiplication in many ways.

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  • For example, consider a floating point pipeline performing a multiplication.

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  • This sort of multiplication is given a special name; it is scalar multiplication, because the numbers are sometimes called scalars.

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  • Like viruses, such strategies take advantage of rapid multiplication to explode the message to thousands, to millions.

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  • To do inner product matrix multiplication, you can use the product global function.

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  • multiplication table to him!

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  • multiplication square and comment on them.

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  • multiplication register where each gain stage was defined by model 2 was also simulated using the same approach.

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  • multiplication factor F of the gene is 10.

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  • multiplication facts up to 7 now.

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  • multiplication sum.

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  • The matrix-vector multiplication routine must be supplied by the user.

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  • The basic element by element operators are therefore: element-by-element multiplication: " .* " element-by-element division: " .

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  • This will depend on which comes first - part of the vector multiplication rules; do you remember?

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  • All that is needed is the replacement of new lines for old in the seed multiplication chain.

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  • multiplication of the bacteria.

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