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The method which Napier terms "Rabdologia" consists in the use of certain numerating rods for the performance of multiplications and divisions.

00The second method, which he calls the "Promptuarium Multiplicationis" on account of its being the most expeditious of all for the performance of multiplications, involves the use of a number of lamellae or little plates of metal disposed in a box.

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.

001890, p. 490) that exp(mldl +m2d2+m3d3+...) = exp (Midi +M2d2+M3d3+...), where now the multiplications on the dexter denote successive operations, provided that pp t exp(MiE+M2 2+M3E3+...) +mlH+m2V+m3S3+..., being an undetermined algebraic quantity.

00where the multiplications on the leftand right-hand sides of the equation are symbolic and unsymbolic respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities.

00(iii.) Multiplications, represented by X, are performed from right to left.

00This rule as to using brackets is not always observed, the convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions.

00r we regard the multiplications as taking place from left to right; and similarly in r A product in which multiplications are taken in this order is called a continued product.

00by successive multiplications by Ada.

00In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other.

00In 1617 he published a small work entitled Rabdologia relating to mechanical methods of performing multiplications and divisions, and in the same year he died.

00It is evident that Wittich's prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations.

00Involution is a direct process, consisting of successive multiplications; the other two are inverse processes.

00(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g.

00(vi) Distributive Law, that multiplications and divisions may be distributed over additions and subtractions, e.g.

00As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions.

00The method which Napier terms "Rabdologia" consists in the use of certain numerating rods for the performance of multiplications and divisions.

00The second method, which he calls the "Promptuarium Multiplicationis" on account of its being the most expeditious of all for the performance of multiplications, involves the use of a number of lamellae or little plates of metal disposed in a box.

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.

00Napier deliberately set himself to abbreviate multiplications and divisions - operations of so fundamental a character that it might well have been thought that they were in rerum natura incapable of abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one 1 The title runs as follows: Arithmetica Logarithmica, sive Logarithmorum chiliades triginta....

00In multiplications or divisions of any length it is generally convenient to begin by forming a table of the first nine multiples of the multiplicand or divisor, and Napier's bones at best merely provide such a table, and in an incomplete form, for the additions of the two figures in the same parallelogram have to be performed each time the rods are used.

001890, p. 490) that exp(mldl +m2d2+m3d3+...) = exp (Midi +M2d2+M3d3+...), where now the multiplications on the dexter denote successive operations, provided that pp t exp(MiE+M2 2+M3E3+...) +mlH+m2V+m3S3+..., being an undetermined algebraic quantity.

00where the multiplications on the leftand right-hand sides of the equation are symbolic and unsymbolic respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities.

00(iii.) Multiplications, represented by X, are performed from right to left.

00This rule as to using brackets is not always observed, the convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions.

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.

00r we regard the multiplications as taking place from left to right; and similarly in r A product in which multiplications are taken in this order is called a continued product.

00by successive multiplications by Ada.

00In the course of reducing such expressions as (AB)C, (AB){C(DE)} and the like, where a chain of multiplications has to be performed in a certain order, the multiplications may be all progressive, or all regressive, or partly, one, partly the other.

00In 1617 he published a small work entitled Rabdologia relating to mechanical methods of performing multiplications and divisions, and in the same year he died.

00It is evident that Wittich's prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations.

00If instead of commencing with I and making successive additions of 1 we commence with any number such as 3 and make successive multiplications by 3, we get a series 3, g, 27,.

00Involution is a direct process, consisting of successive multiplications; the other two are inverse processes.

00(v) Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order: e.g.

00(vi) Distributive Law, that multiplications and divisions may be distributed over additions and subtractions, e.g.

00As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions.

00

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