In many problems 10-figure accuracy is not required, but in the above such tables have been described for they are the funda mental tables. A large number of tables exhibiting 4, 5, 6-place logarithms have been published. A good 6-place table is Brem iker's Logarithmorum VI. Decimalium Nova Tabula. (Berlin, 1852.) Several editions appeared with title page in German and English (1875). 7-place logarithms are frequently required and there are a considerable number of accurate and well-arranged tables, including those of Bremiker, Bruhns, Dupuis, Lalande, Sang, Schron and Shortrede. Schron's table Siebenstellige Ge meine Logarithmen—(Braunschweig, 186o) is typical. There have been editions in German, English and French. The arrange ment is the best for a 7-place table and the modern editions are accurate. The figures of the logarithms are grouped 3, 4, the first group being printed only once. When a change occurs in the final figure of this leading group in the course of a row it is shown by an asterisk prefixed to all the groups affected in the row. This method of attracting the attention of the computer is very suc cessful. In 1871 Sang published A New Table of Seven-place Logarithms of Numbers from 20,000 to 200,000 (London).
There is a distinct advantage in choosing this range in place of Io,000 to 1 oo,000. For the latter range the differences at the commencement of the table change so rapidly that the propor tional parts are so numerous that they are either very crowded or some of them are omitted; by making the table start from 20,000 the differences are halved in magnitude and there are one-fourth as many on a page. This table, unlike most 7-figure tables, is mainly the result of a new calculation. There are very few 8-place tables; until quite recent times there was only one such table—John Newton's Trigonometric Britannica (London, 1658), where the logarithms of numbers to ioo,000 are given. The usual arrangement of 7-place tables is due to this Newton, viz., the first four figures of the argument are shown in the left hand margin, while the fifth figure is shown at the head of suc cessive columns. The only other 8-place tables have been pub lished since 189o; the Service Geographique de l'Armee (France) published an abridgement of the Tables du Cadastre (the famous French manuscript tables) under the title Tables des Logarithmes a hisit decimales des nombres de z a 120,000 (Paris, 1891), and in the same year Mendizabal Tamborrel published Tables des Logarithmes a 1144 decimales des nombres de z a (Paris). Bauschinger and Peters, as a result of an entirely new calcula tion to 12 places, published Logarithmisch-trigonometrische Taf eln mit acht Dezimalstellen (Leipzig, 191o). It has appeared with English title and preface. The logarithms to 8 decimals of all numbers to 200,000 may be taken from this directly.
It is sometimes necessary to use logarithms to a greater number of figures than 1o, but owing to the great expense of publishing extensive tables to a large number of figures, several methods have been devised by mathematicians which enable a computer, with the help of a comparatively small table, to calculate the logarithm to the required number of figures. For example, Gray in Tables
for the Formation of Logarithms and Anti-Logarithms to twenty four or any less number of places (London, 1876), explains a method by which the logarithm and antilogarithm can be found to any number of places not greater than 24. Similar tables and methods have been published by Biirgen, Steinhauser, Guillemin, Mansion-Namur, Pineto, Andoyer and Ellis. At present there is in progress an extensive table to 20-decimals, the calculations being carried out by Thompson. The first part appeared in to commemorate the Tercentenary of Briggs's publication of Arithmetica Logarithmica under the title Logarithmetica Britan nice, being a Standard Table of Logarithins to Twenty Decimal Places (Cambridge University Press, 1924). This part gives the logarithms of numbers 90,00o to 1 oo,000; two more parts have been published since Logarithmic Trigonometrical Functions.—The original and fundamental tables of the logarithmic trigonometrical func tions are (r) Vlacq's Trigonometria Artificialis (Gouda, 1633), which exhibits log sines and tangents to every ten seconds of the quadrant to 10 decimal places with differences. (2) Briggs's Trigonometria Britannica (London, 1633), which gives the natural sines to 15 places, tangents and secants to io places, log sines to 14 places and tangents to io places at intervals of o•oo1 degree from o° to 45° with interscript differences. In Vlacq's earlier work of 1628 there are given, in addition to the logarithms of numbers, the log sines, tangents and secants for every minute of the quadrant to io places with differences. The majority of the loga rithmic-trigonometrical tables published since 1633 have been calculated from, or are abridged forms of the tables of Briggs and Vlacq. It is to be noted that Vlacq used the sexagesimal division of the degree, while Briggs used the centesimal division. This step of Briggs was important and it is probable that, if Vlacq's table had not been published in the same year, tables published sub sequently might have used the latter division, and thus ensured a saving of work in interpolations, multiplications, etc. The French mathematicians at the end of the 18th century divided the right angle centesimally, but there is no real advantage in doing this. Michael Taylor in Tables of Logarithms (London, 1792) made a big advance by giving log sines and tangents to every second of the quadrant to 7 places. This table was calculated by inter polation from Vlacq's Trigonometria to io places and then cut down to 7, so that the table should be accurate to the last figure. This table is in inconvenient arrangement. Bagay's Nouvelles Tables Astronomiques et Hydrographiques (Paris, 1829) has always been preferred. This also gives a complete logarithmic trigonometrical canon to every second.