James Whitbread Lee Glaisher

“ It 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. 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. In fact, if we put xy = φ (X + Y) , X being a function of x only and Y a function of y only, we can show that we must have X = Aeqx, y = Beqy ”

“ It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation
log (1 + x) = x − 1/2x2 + 1/3x3 − 1/4x4 + ... is true only when the analytical modulus of x is less than unity. 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. ”

“ Hutton’s suggestions are all the more to be regretted as they occur as a history which is the result of a good deal of investigation and which for years was referred to as an authority by many writers. His prejudice against Napier naturally produced retaliation, and Mark Napier in defending his ancestor has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. In connexion with this controversy it should be noticed that the “Admonitio” on the last page of the Descriptio, containing the reference to the new logarithms, does not occur in all the copies. ”

“ The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa ”

“ 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. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. 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. ”

“ The explanation of this property of the base 10 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. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. ”

“ The work is a small quarto containing fifty-seven pages of explanatory matter and a table of ninety pages (see Napier, John) . 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. The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents. ”

“ In 1794 Vega published his Thesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers from Vlacq’s Arithmetica logarithmica of 1628, and Trigonometria artificialis of 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from the Trigonometria, there are given logarithms for every second of the first two degrees, which were the result of an original calculation. Vega devoted great attention to the detection and correction of the errors in Vlacq’s work of 1628. ”

“ The “new invention in Denmark” to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek letters προσθαφαίρεσις) , which had its origin in the solution of spherical triangles. [7] The method consists in the use of the formula
sin a sin b = 1/2 {cos (a − b) − cos (a + b) }, 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 ”

“ In the preface Robert Napier says that he has been assured from undoubted authority that the new invention is much thought of by the ablest mathematicians, and that nothing would delight them more than the publication of the mode of construction of the canon. ”

“ Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.
The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio. ”

“ John Kepler, who has been already quoted in connexion with Craig’s visit to Tycho Brahe, received the invention of logarithms almost as enthusiastically as Briggs. His first mention of the subject occurs in a letter to Schikhart dated the 11th of March 1618, in which he writes-“Extitit Scotus Baro, cujus nomen mihi excidit, qui praeclari quid praestitit, necessitate omni multiplicationum et divisionum in meras additiones et subtractiones commutata, nec sinibus utitur ”

“ The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849) . In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. ”

“ Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.
The function log x as x increases from 0 towards ∞ steadily increases from −∞ towards +∞. It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e. that x−m log x tends to zero as x tends to ∞ for every positive value of m however small. ”

“ These extracts contain all the original statements made by Napier, Robert Napier and Briggs which have reference to the origin of decimal logarithms. It will be seen that they are all in perfect agreement. Briggs pointed out in his lectures at Gresham College that it would be more convenient that 0 should stand for the logarithm of the whole sine as in the Descriptio, but that the logarithm of the tenth part of the whole sine should be 10,000,000,000. ”

“ There is a short “preface to the reader” by Briggs, and a description of a triangular diagram invented by Wright for finding the proportional parts. The table is printed to one figure less than in the Descriptio. Edward Wright died, as has been mentioned, in 1615, and his son, Samuel Wright, in the preface states that his father “gave much commendation of this work (and often in my hearing) as of very great use to mariners” ”

“ The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences. A full account of this method as applied to the calculation of the Tables du Cadastre is given by Lefort in vol. ”

“ In the years 1791–1807 Francis Maseres published at London, in six volumes quarto “Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton’s historical introduction to his new edition of Sherwin’s mathematical tables ...,” which contains reprints of Napier’s Descriptio of 1614, Kepler’s writings on logarithms (1624–1625) , &c. In 1889 a translation of Napier’s Constructio of 1619 was published by Walter Rae Macdonald. ”

“ The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a301,030 to 1, and so on ”

“ To destroy the first 6 multiply by 1 − .0006 giving 1.000063361744, and multiplying successively by 1 − .00006 and 1 − .000003, we obtain 1.000000357932, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1 + .1nx. ”

“ N = (e−1) L (Napier) , L = (1.0001) 1/10N (Byrgius) , viz. Napier gives logarithms to base e−1, Byrgius gives antilogarithms to base (1.0001) 1/10.
There is indirect evidence that Napier was occupied with logarithms as early as 1594, for in a letter to P. Crügerus from Kepler, dated September 9, 1624 (Frisch’s Kepler, vi. 47) , there occurs the sentence: “Nihil autem supra Neperianam rationem esse puto: etsi quidem Scotus quidam literis ad Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici.” ”