LOGACEDIC VEBSE. 406 LOGARITHMS. This seems to be a modification of the verse used by Sappho: Sedri nv i jSpa* || Xopi | res || koKKiko- | (mi Te — 7 Mowrat 17. Priapaan (a Ghjconic and a Pkerecratic) : - > Co TrrjKTiSa luuia qua? cu l_ /■V I>i8 -> KUfld ponte 1 — d Ion go ludere In English these uicties are rarely found ex- cept in imitations of classic lyric measures. See Versification. liOGARITHMIC CURVE (from Gk. X670S, lofjo.i, ratio, pro])ortion, word + ipiSfiSs, arith- mos, number). A curve of a single branch, hav- ing for its equation y = log a; : log a, or x = a^. In particular, if a = e, then 1/ = log x, the ordi- nate being the logarithm of the abscissa, and the curve representing the Xaperian or hypcrl)olic logarithm. The curve y — e" is sometimes called the cxpcmential curve, and is evidently the sam^ as the logarithmic except as to its orientation, the two being symmetric with resjiect to the bisector of the angle XOY. The logarithmic curve was first studied by Gregory in his Cleornetriw Purs Universalis ( KifiS) , and" was considered under the names 'Io;iiirithwira' and •lopisfica' by Huygens in his De Cnvsa flrariliitis (IG'JO). LOGARITHMIC SERIES. See Seeies. LOGARITHMIC SPIRAL, or Logistic Spiral. A spiral having fur its polar equation T^(ien. Since it possesses the property that tan X = r : r' — I : wi = :i constant, it cuts the radius vector under a constant angle, and is therefore also called the equiangular spiral. It was first brought to the attention of mathemati- cians by Descartes, in his correspondence begin- ning in 1C37. Its properties were first carefully studied by .Jakob Bernoulli in the Acta Erudi- iorum (KiOl), an<l have since then been exten- sively investigated. For bibliography, consult: Brocard, Xotes de bibliograjihie des courbes ge- omitricjiies (Bar-le-Duc. 1897) : Partie coniplc- mentaire (Bar-le-Duc, 1899). See Spiral. LOGARITHMS. A tabular system of num- bers, by which nuiltiplication can be performed by addition, division by subtraction, involution by a single multiplication, and evolution by a single division. The logarithm of a number is the exponent of the base which produces the number. If a^ =zb, x is said to be the logarithm of 6 to the base a. Any finite positive nimiber greater than 1 may be taken for a base, and the logarithms of all positive numbers with respect to the base may be tabulated. Tlie base 10, how- ever, has been found the most convenient, and the system of 'common logarithms' constructed ujion it is universally used for arithmetical computa- tions. The following brief table ■will ser'e for purposes of illustration: 10' = 0.001 io» =1 10'-8"' = 700 103.3OIO ^ n.ooa IQo-e.ao ^ 5 10' = lOflO 10' = 0.01 10' = 10 103-«.4I = 3.500 105«"i = 0.03 10I-8.B1 = 70 103-e»o3 ^ 4900 10-1 = 0.1 10' = 100 10* = 10000 Here 3 means —3, and 3.3010 mean^ — 3 + 0.3010. According to the definition, 3 is the logarithm of 0.001, written log 0.001=3; log 0.002 = 3.3010 ... log 10.000 = 4. The integral part of the logarithm is called the charuclerislic and the decimal part the uiantissii. The charac- teristics of the logarithms of positive numbers less than 1 are negative, of numbers not less than 10 positive, and of numbers between 1 and 10 they are zero. The mantis.sas arc always taken as positive and are generally incommensurable. A table of mantissas expressed to si.x decimal places is sulUciently accurate for all ordinary purposes. In the common system, it is unneces- sary to tabulate the characteristic, since its value is always one less than the number of places to the left of the decimal point. For example, log G59.34 = 2.819109. Another advantage of the common system is that the mantissas of the loga- rithms of numbers having the same sc<|uenee of figures are equal. For example, log 0.59.34 = 2.819109 and log 05934 = 4.819109. These two- properties belong to the common system only, and to them it ow-es it's superiority over others for the purpo.ses of numerical calculations. The logarithm being an exponent, it must obey the laws of exponents, and from these are de- rived the fundamental principles of logarithmic calculation: (1) The logarithm of a product is equal to the sinu of the logarithms of its fac- tors. (2) The logaritiini of the quotient of two numbers is equal to the logarithm of the dividend less the logarithm of the divisor. (3) The loga- rithm of a number affected by an exponent is equal to the exponent times the logarithm of the number. E.g. to multiplv 50 bv 70, log (70.50) — log 70 + log 5 4- log 10 = 1.S451 + 0.0990 + 1=: 3.5441 from the above table. But the num- ber corresponding to the log 3.5441, called the antilogarithm is 3500, .-. 70 , .50 = 3500. Also, to extract the square root of 4900, log y'4900 or log 49002 = lU log 4900 — % . 3.6902 = 1.8451 from the table. But the antilog 1.8451 is 70, .-. 1/4900 = 70. In finding the logarithm of a quotient, especially when the divisor con- tains several factors, it is easier to add the com- plement logarithm or cologarithm of the divisor than to subtract its logarithm. The cologarithm of a number is defined as the logarithm of its reciprocal. The colog »=:log — =: — log n, which justifies its use as explained. John Xapier (1614), a Scotchman, is usually regarded as the inventor of logarithms, although Biirgi (1552-1C32) had probably as early as 1007 computed a table of antilogarithms, but he did not fully understand the importance of the in- vention, and failed to make it public until 1020, when Napier's logarithms were known throughout Europe. Biirgi's Arithmelischc und Gcometrischr Proyress-Tabulen was published in 1620 in Prague, and contains logarithms of ordinary num- bers, while Xapier's ilirifici Lognrithmorum Ca- nonis De'scriptio contains logarithms of trigono- metric functions. (See Triooxometry.) Tables of the numerical values of the trigonometric func- tions had attained a high degree of accuracy at this time, hut their usefulness depended upon ■ abridged methods of calculation, and the search after^such methods led to the discovery of loga- rithms. Napier observed that if in a circle with
