The logarithmic formulae for these concentration cells indicate that theoretically their electromotive force can be increased to any extent by diminishing without limit the concentration of the more dilute solution, log c i /c 2 then becoming very great.
The definitions of the logarithmic and exponential functions may be extended to complex values of x.
The results give equations of the same logarithmic form as those obtained in a somewhat different manner in the theory of concentration cells described above, and have been verified by experiment.
On the analogy between this case and that of the interface between two solutions, Nernst has arrived at similar logarithmic expressions for the difference of potential, which becomes proportional to log (P 1 /P 2) where P2 is taken to mean the osmotic pressure of the cations in the solution, and P i the osmotic pressure of the cations in the substance of the metal itself.
A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be left unnoticed.
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.
The elementary idea of a differential coefficient is useful in reference to the logarithmic and exponential series.
remarkable for the beauty of its logarithmic curves; Iwaki-san (5230 ft.), known as Tsugaru-Fuji, and said by some to be even more imposing than Fuji itself; and the twin mountains Gassan (6447 ft.) and Haguro-san (5600 ft.).
In his investigations respecting cycloidal lines and various spiral curves, his attention was directed to the loxodromic and logarithmic spirals, in the last of which he took particular interest from its remarkable property of reproducing itself under a variety of conditions.
Like another Archimedes, he requested that the logarithmic spiral should be engraven on his tombstone, with these words, Eadem mutata resurgo.
The logarithmic function is most naturally introduced into analysis by the equation log x= x ?
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.
The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.
The former translated the work into English; the latter was concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms, and it is to him that the original calculation of the logarithmic tables now in use is mainly due.
The title of Gunter's book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.
During the last years of his life Briggs devoted himself to the calculation of logarithmic sines, &c. and at the time of his death in 1631 he had all but completed a logarithmic canon to every hundredth of a degree.
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.
to Napier's Descriptio in order to describe its reception on the continent, and to mention the other logarithmic tables which were published while Briggs was occupied with his calculations.
It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.
The only other logarithmic canon to every second that has been published forms the second volume of Shortrede's Logarithmic Tables (1849).
Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede's Logarithmic tables already referred to, and in Filipowski's Table of antilogarithms (1849).
Such tables can scarcely be said to come under the head of logarithmic tables.
Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him.
The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by I or 2, the prime factors may all be within the range of the logarithmic tables.
The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).
As originally proposed, many of these formulae were cast in exponential form, but the adoption of the logarithmic method of expression throughout the list serves to show more clearly the relationship between the various types.
For simplicity of calculation Rankine chose logarithmic curves for both the inner and outer faces, and they fit very well with the conditions.
Gunter's Line, a logarithmic line, usually laid down upon scales, sectors, &c. It is also called the line of lines and the line of numbers, being only the logarithms graduated upon a ruler, which therefore serves to solve problems instrumentally in the same manner as logarithms do arithmetically.
On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, &c.), and on the other side the corresponding artificial or logarithmic ones.
anamorphosis operator which is similar to the logarithmic transform is the square-root operator.
Understand the concept of damping and the meaning of damped natural frequency, damping factor and logarithmic decrement.
The logarithmic curves which accompany his paper demonstrate that within wide ranges of maximum induction W = 0.01 B 1.6 = 0.5271 1 ' 62 very nearly.
8vo, a collection of logarithmic and trigonometrical tables which has passed through many editions, a very useful one volume stereotype edition having been published in 1840 by Masse.
pp. 66-92, 1873, Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form I t I r n to 20 places.
As it unfolds it follows a pattern very similar to the logarithmic spiral, known as the Golden Spiral, which follows the Fibonacci Sequence.
respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...
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