LOGARITHM LOGGE DI RAFFAELLO. 



515 



arc the exponents of the different powers to which a 

 constant number must be raised, in order to be 

 equal to those numbers ; the principles, therefore, 

 which apply to exponents in general, apply to log- 

 arithms." To constitute a logarithm, it is necessary 

 that the exponent should refer to a system or series. 

 These exponents, therefore, constitute a series of 

 numbers in arithmetical proportion, corresponding to 

 as many others in geometrical proportion. Take, for 

 instance, the series, 10'= 10 ; 10'= 100 ; 10 3 =1000 ; 

 10 4 = 10,000: then we have the logarithm of 10=1 ; 

 logarithm, 100=2 ; logarithm, 1000=3 ; logarithm, 

 10,000=4, &c. Perhaps the definition of a logarithm 

 may be more scientifically expressed thus : Logarithm, 

 is a mathematical term for a number by which the 

 magnitude of a certain numerical ratio is expressed 

 in reference to a fundamental ratio. The value of 

 a ratio becomes known to us by the comparison of 

 two numbers, and is expressed by a number called 

 the quotient of the ratio ; for instance, 12 : 4 is ex- 

 pressed by 3, or 18 : 9 by 2 ; 3 and 2 being called the 

 quotients of the two proportions, 12 : 4 and 18 : 9. If 

 we now imagine a series of proportions, which have 

 all the same value or quotient, as, for instance, 1 to 

 3, 3 to 9, 9 to 27, 27 to 81, &c. (in which 9 and 3, 21 

 and 9, 81 and 27, are in the same ratio as 3 and 1), 

 and if we at the same time adopt the ratio 3 to 1, as 

 the fundamental ratio (or the unit of these ratios), 

 then 9 to 1 is the double of this ratio, 27 to 1 the 

 triple, 81 to 1 the quadruple, and so on. The num- 

 bers 1, 2, 3, 4, which indicate the value of such ratios, 

 in respect to the fundamental ratio, are called logar- 

 ithms. If, therefore, in this case, 1 is the logarithm 

 of 3, 2 must be the logarithm of 9, 3 of 27, 4 of 81, 

 &c. If we adopt, however, the ratio of 4 : 1 as the 

 fundamental one, and hence 1 as the logarithm of 4, 

 then 2 would be the logarithm of 16, 3 of 64, &c. 

 The logarithms of the numbers which lie between, 

 must be fractions, and are to be calculated and put 

 in a table. A table of logarithms, made according 

 to an assumed basis or fundamental ratio, of all 

 numbers, to a certain limit, is called a logarithmic 

 system. The most common, at present, is that of 

 Briggs, in which the fundamental basis is 10 to 1 ; 

 hence 1 is the logarithm of 10, 2 of 100, 3 of 1000, 4 

 of 10,000, &c. It is evident that all logarithms of 

 numbers between 1 and 10, must be more than 0, yet 

 less than 1, i. e. a fraction ; thus the logarithm of 6 

 is 0-7781513. In the same way, the logarithms of 

 the numbers between 10 and 100 must be more than 

 1, but less than 2, &c.; thus the logarithm of 95 

 is=l-9777236. All logarithms of the numbers be- 

 tween 0, 10, 100, 1000, &c., are arranged in tables, 

 the use of which, particularly in calculations with 

 large numbers, is very great. The process is sim- 

 ple and easy. If there are numbers to be multiplied, 

 we only have to add the logarithms ; if the numbers 

 are to be divided, the logarithms "are merely to be 

 subtracted ; if numbers are to be raised to powers, 

 their logarithms are multiplied ; if roots are to be 

 extracted, the logarithms are merely to be divided 

 by the exponent of the root. In a table of logarithms, 

 the integer figure is called the index or characteristic. 

 The decimals are called, by the Germans and Ital- 

 ians, the mantissa. In general, the logarithms of 

 the system in which 1 indicates 10, are called com- 

 mon or Briggs's logarithms. The use of logarithms 

 in trigonometry was discovered by John Napier, 

 (q. v.) a Scottish baron, and made known by him in a 

 work published at Edinburgh, in 1614. Logarith- 

 mic tables are of great value, not only to mathema- 

 ticians, but to all who have to make calculations with 

 large numbers. The best logarithmical tables are 

 those of Vega and of Callet. The former are calcu- 

 lated with ten decimals. Logarithms are of incal- 



culable importance in trigonometry and in astronomy. 

 Vega's edition of Vlacq's tables contains a trigono- 

 metrical table of the common logarithms of the ra- 

 dius or log. sin. tot. =10-0000000, which gives the 

 logarithms of sines, arcs, co-sines, tangents, and co- 

 tangents for each second of the two first and two last 

 degrees, and for each ten seconds of the rest of the 

 quadrant. Under Napier's direction, B. Ursinius 

 first gave the logarithm of the sines of the angles 

 from 10 to 10 seconds, the logarithm of the tangents, 

 which are the differences of the logarithms of each 

 sine and co-sine, together with the natural sine for a 

 radius of 100,000,000 parts. Kepler turned his at- 

 tention particularly upon the invention of Napier, 

 and gave a new theory and new tables. Briggs was 

 also conspicuous in the construction of tables. 

 Mercator shows a new way for calculating the loga- 

 rithms easily and accurately. Newton, Leibnitz, 

 Halley, Euler, L'Huillier, and others, perfected the 

 system much, by applying to it the binomial theorem 

 and differential calculus. The names of Vlacq, Sher- 

 win, Gardiner, Hutton, Taylor, Callet, and others, 

 deserve to be honourably mentioned. The edition o f 

 Vlacq, within a few years, by Vega, is particularly 

 valuable. During the French revolution, when all 

 measures were founded on the decimal division, new 

 tables of the trigonometrical lines and their logarithms 

 became necessary. The director of the bureau du ca. 

 tastre, M. Prony, was ordered, by government, to have 

 tables calculated, which were to be not only extreme- 

 ly accurate, but to exceed all other tables in magni- 

 tude. This colossal work, for which the first 

 mathematicians supplied the formulas and the methods 

 for using the differences in the calculations, was 

 executed, but the depreciation of the paper money 

 prevented its publication. The tables would have 

 occupied 1200 folio pages. (Notices sur les grandes 

 Tables Logarithmiques et Trigonometriques, calcules 

 au Bureau du Catastre a Paris, an IX.) 



LOG AU, FREDERIC, baron of; an epigrammatist, 

 born in Silesia, 1604, and died in 1655. He early 

 showed poetical talents, but, at a later period, his 

 avocations appear to have prevented him from at- 

 tempting any large poems, and his poetical productions 

 were confined to short pieces and epigrams. He pub- 

 lished a selection of 200 epigrams, which were so well 

 received, as to induce him (probably in 1654) to 

 publish a new collection of 3000. A contemporary 

 of Opitz, he followed in the steps of his great prede- 

 cessor, and often expresses himself with as much 

 vigour. Many of his epigrams are original and 

 happy, and are the more striking as this department 

 has been little cultivated by German writers. Lo- 

 gau is particularly original in the gnome, and truly 

 poetical in a form which is now become foreign to 

 poetry. Rainier and Lessing, who edited a collec- 

 tion of his epigrams in 1759, revived his reputation. 

 After Lessing's death, Ramler republished the col- 

 lection, in 1791. Select poems of Logau are contained 

 in W. Miiller's Bibliothek Deutscher Dichter det 17 

 Jahrn, (Library of the German Poets of the seven- 

 teenth Century, volume vi., Leipsic, 1824). 



LOGGE DI RAFFAELLO ; part of the Vatican, 

 and one of those beautiful scenes to be found no- 

 where but in Rome. Leo X. had these logge or 

 arcades built under the direction of the immortal 

 Raphael. There are three stories which enclose a 

 court called il Cortile di S. Damaso. The middle 

 story is the most celebrated. It is formed by thirteen 

 arches, and the vault of each contains four paintings 

 in fresco, representing scenes from the Old Testa- 

 ment, and executed by Giulio Romano, Pierin dal 

 Vaga, Pellegrino da Modena, Policloro,and Maturinn 

 da Caravaggio, and others, after cartoons prepared 

 by the great Raphael himself. The number of these 

 2K 2 



