See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)  , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics, a word invented by See also:

• JOHN
• JOHN (1167–1216)
• JOHN (1290-c. 1320)
• JOHN (1296-1346)
• JOHN (1371–1419)
• JOHN (1468-1532)
• JOHN (1801-1873)
• JOHN (Heb. llni')
• JOHN (ZAPOLYA) (1487-1540)
• JOHN, 4TH MARQUESS OF TWEEDDALE (c. 1695-1762)
• JOHN, DON (1545-1578)
• JOHN, DON (1629–1679)
• JOHN, GOSPEL OF ST
• JOHN, or HAYS (1513–1571)
• JOHN, THE APOSTLE
• JOHN, THE EPISTLES OF

John See also:

• NAPIER
• NAPIER, JOHN (1550-1617)
• NAPIER, SIR CHARLES (1786-1860)
• NAPIER, SIR CHARLES JAMES (1782-1853)
• NAPIER, SIR WILLIAM FRANCIS PATRICK (1785-1860)

Napier to denote a particular class of See also:

• FUNCTION

function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation ae = m, then a is called the See also:

• BASE

base, and x is said to be the See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

logarithm of m to the base a . This relation between x, a, m, may be expressed also by the equation x=See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log. m . Properties.—The See also:

• PRINCIPAL

principal properties of logarithms are given by the equations loge (mn) =loge m+loge it, log.(m/n) =loge m—log. n, log . Mr =1' loge m, log . ;(m = (1/r) log. m, which may be readily deduced from the See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition of a logarithm . It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

root of a quantity is equal to (I/r)th of the logarithm of the quantity . Logarithms were originally invented for the See also:

• SAKE

sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

division may be replaced by those of addition and subtraction, and the operations of raising to See also:

• POWERS, HIRAM (1805-1873)

powers and extraction of roots by those of multiplication and division . For the purpose of thus simplifying the operations of See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic, the base is taken to be ro, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated . The logarithm is also a function of frequent occurrence in See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis, being regarded as a known and recognized function like See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin x or tan x; but in mathematical investigations the base generally employed is not to, but a certain quantity usually denoted by the See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter e, of value 2.71828 18284 ... . Thus in arithmetical calculations if the base is not expressed it is understood to be to, so that log m denotes logie m; but in See also:

• ANALYTICAL

analytical formulae it is understood to be e . The logarithms to base to of the first twelve See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers to 7 places of decimals are log I =0•0000000 log 5=0.6989700 log 9=0.9542425 log 2 = 0.30I0300 log 6 =0.7781513 log I0 = I.0000000 log 3 =0.4771213 log 7 =0.8450980 log I I = I.0413927 log 4= 0.6020600 log 8 = 0.9030900 log I2 = I.0791812 The meaning of these results is that I=See also:

• I00

I00, 2 =I00•3010300, 3 =100.4771213, 10 = 101, 11= 101.0413927 12 = 101.0791312 The integral See also:

• PART

part of a logarithm is called the See also:

• INDEX

index or characteristic, and the fractional part the mantissa . When the base is 1o, the logarithms of all numbers in which the digits are the same, no See also:

• MATTER

matter where the decimal point may be, have the same mantissa; thus, for example, log 2.5613 =0.4084604, log 25.613 = I.4084604, log 2561300 = 6.4084604, &c .

In the See also:

• CASE
• CASE, JOHN (d. 1600)

case of fractional numbers (i.e. numbers in which the integral part is o) the mantissa is still kept See also:

• POSITIVE (or PORTABLE) ORGAN

positive, so that, for example, log .25613 =7.4084604, log •0025613 =_3.4084604, &c . the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus 1.4084604 stands for—1+ '4084604 . The fact that when the base is to the mantissa of the logarithm is See also:

• INDEPENDENT

independent of the position of the decimal point in the number affords the See also:

• CHIEF (from Fr. chef, head, Lat. caput)

chief See also:

• REASON (Lat. ratio, through French raison)

reason for the choice of to as base . The ex-planation of this See also:

• PROPERTY

property of the base to is evident, for a See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change in the position of the decimal points amounts to multiplication or division by some power of to, 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 to is added to all the logarithms in the table in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes o, &c . Logarithms thus increased are frequently referred to for the sake of distinction as See also:

• TABULAR

tabular logarithms, so that the tabular logarithm =the true logarithm + lo . In tables of logarithms of numbers to base to the mantissa only is in See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the See also:

• RULE

rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure . It follows very simply from the definition of a logarithm that log. b X loge a = I , logo m = log. m X (I /log. b) . The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the See also:

• CONSTANT, BENJAMIN (1845-1902)

constant multiplier 1/log.b, which is called the modulus of the See also:

• SYSTEM

system whose base is b with respect to the system whose base is a . The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or See also:

• COMMON

common system, of which the base is ro; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/logeio, which is called the modulus of the common system of logarithms . The numerical value of this modulus is o•43429 44819 03251 82765 11289 . . ., and the value of its reciprocal, loge 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 94045 68401 79914 .

. The quantity denoted by e is the See also:

• SERIES (a Latin word from serere, to join)

series, I+I+ T + I + I + ... I I.2 1.2.3 1.2.3.4 the numerical value of which is, 2.71828 18284 59045 23536 02874 . The logarithmic Function.—The mathematical function log x or loge x is one of the small See also:

• GROUP

group of transcendental functions, consisting only of the circular functions (See also:

• DIRECT

direct and inverse) sin x, See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos x, &c., arc sin x or sin-1 x,&c., log x and ee which are universally treated I in analysis as known functions . The notation log x is generally employed in See also:

• ENGLISH

English and See also:

• AMERICAN

American See also:

• WORKS

works, but on the' See also:

• CONTINENT (from Lat. continere, "to hold together "; hence " connected," " continuous ")

continent of See also:

• EUROPE

Europe writers usually denote the function by 1x or lg x . The logarithmic function is most naturally introduced into analysis by the equation logy= J tt (x>o) . This equation defines log x for positive values of x; if x10 o the See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula ceases to have any meaning . Thus log x is the integral function of I/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions . A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x . A relation which is of See also:

• HISTORICAL

historical See also:

• INTEREST

interest connects the logarithmic function with the See also:

• QUADRATURE (from Lat. quadratura, a making square)

quadrature of the See also:

• HYPERBOLA

hyperbola, for, by considering the equation of the hyperbola in the See also:

• FORM (Lat. forma)

form xy=const., it is evident that the See also:

• AREA

area included between the arc of a hyperbola, its nearest asymptote, and two ordinates See also:

• DRAWN

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 b/a . The following fundamental properties of log x are readily deducible from the definition (i.) log xy=log x+log y . (ii.) Limit of (x" -1)/h = log x, wken h is indefinitely diminished . Either of these properties might be taken as itself the definition of log x .

There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1+x), viz . log (I+x)=x—zx2+1x3—zx'+ ... ; the series, however, is convergent for real values of x only when x lies between +1 and - I . Other formulae which are deducible from this equation are given in the portion of this See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• RELATING

relating to the calculation of logarithms . The function log x as x increases from o towards so steadily in-creases frokri -so towards +so . It has the important property that it tends to Infinity with x, but more slowly than any power of x, i.e. that x-" log x tends to zero as x tends to so for every positive value of m however small . The exponential function, exp x, may be defined as the inverse of the logarithm: thus x=exp y if y = log x . It is positive for all values of y and increases steadily from o toward so as y increases from -eo towards + so . As y tends towards so , exp y tends towards so more rapidly than any power of y . The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y . d (u.) Tx exp x =exp x . (iii.) exp x = t +x+x2/2 !

+ x3/3 ! + .. . From (i.) and (ii.) it may be deduced that exp x=(1+1+1/2 1 +1/3 ! + ...)r, where the right-See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand See also:

• SIDE
• SIDE (mod. Eski Adalia)

side denotes the positive xth power of the number 1+I+1/2 ! +1/3 ! + ... usually denoted by e . It is customary, therefore, to denote the exponential function by See also:

• EA(nar...), (p.s„,.,.)
• EA(psr...), (p, „...)• (pqr...)

ea, and the result ex = I +x+x2/2 ! +x3/3 ! . • • is known as the exponential theorem . The See also:

• DEFINITIONS

definitions of the logarithmic and exponential functions may be extended to complex values of x . Thus if x=E+in, log x= ( — Ji t where the path of integration in the See also:

• PLANE

plane of the complex variable t is any See also:

• CURVE (Lat. curvus, bent)

curve which does not pass through the origin; but now log x is not a See also:

• UNIFORM

uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have log (E+in) =logs/ (E2+n2)+i(a+2nir), where a is the numerically least See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle whose cosine and sine are (E'+n2) and n/sl (E'+n2), and n denotes any integer . Thus even when the See also:

• ARGUMENT

argument is real log x has an See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite number of values; for putting n=o and taking E positive, in which case a=o, we obtain for log the infinite system of values log E+2n1ri .

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 i_ log(1 +x) =x—1x2+1 9x'—1x4+ .. . 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 . Also exp (E=ef (cos n+i sin ,t) . An alternative method of developing the theory of the exponential function is to start from the definition exp x = I .l-x+x2/2 ! +x3/3 ! + ... , the series on the right-hand being convergent for all values of x and therefore defining an analytical function of x which is uniform and See also:

• REGULAR

regular all over the plane . Invention and See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

Early See also:

• HISTORY

History of Logarithms.—The invention of logarithms has been accorded to John Napier, See also:

• BARON
• BARON (1835-1895)
• BARON, MICHEL (1653—1729)

baron of Merchiston in See also:

• SCOTLAND
• SCOTLAND, CHURCH OF
• SCOTLAND, EPISCOPAL CHURCH OF

Scotland, with a unanimity which is rare with regard to important scientific discoveries: in fact, with the exception o1 the tables of Justus Byrgius, which will be referred to further on, there seems to have been no other mathematician of the See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time whose mind had conceived the principle on which logarithms depend, and no partial anticipations of the See also:

• DISCOVERY

discovery are met with in previous writers . The first announcement of the invention was made in Napier's hlirifici Logarilhmorum Canonis Descriptio . . . (See also:

• EDINBURGH

Edinburgh, 1614) .

The See also:

• WORK

work is a small See also:

• QUARTO

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 See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion of points in a straight See also:

• LINE

line, and the principle upon which they are based is that of the See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

correspondence of a geometrical and an arithmetical series of numbers . The table gives the logarithms of sines for every See also:

• MINUTE
• MINUTE (Lat. minutes, small; minuere, to make less)

minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences of the two logarithms in the same line, are the logarithms of the tangents . Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say,logarithms to the base e where e= 2.7182818 .; the relation between N (a sine) and L its logarithm, as defined in the Canonis Descriptio, being N= 1o7e L/Ip7, so that (ignoring the factors Io7, the effect of which is to render sines and logarithms integral to 7 figures), the base is Napier's logarithms decrease as the sines increase . If 1 denotes the logarithm to base e (that is, the so-called "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between l and L is expressed by L = Io, loge Io7 — Io71 or et = I07e—L/103 Napier's work (which will henceforth in this article be referred to as the Descriptio) immediately on its See also:

• APPEARANCE (from Lat. apparere, to appear)

appearance in 1614 attracted the See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention of perhaps the two most eminent English mathematicians then living—See also:

• EDWARD

Edward See also:

• WRIGHT, CARROLL DAVIDSON (1840-1909)
• WRIGHT, CHAUNCEY (1830-1875)
• WRIGHT, JOSEPH (1734-1797)
• WRIGHT, SILAS (1795-1847)
• WRIGHT, THOMAS (1809–1884)
• WRIGHT, THOMAS (1810—1877)
• WRIGHT, WILLIAM ALDIS (1836– )

Wright and See also:

• HENRY
• HENRY (1129-1195)
• HENRY (c. 1108-1139)
• HENRY (c. 1174–1216)
• HENRY (Fr. Henri; Span. Enrique; Ger. Heinrich; Mid. H. Ger. Heinrich and Heimrich; O.H.G. Haimi- or Heimirih, i.e. " prince, or chief of the house," from O.H.G. heim, the Eng. home, and rih, Goth. reiks; compare Lat. rex " king "—" rich," therefore " mig
• HENRY, EDWARD LAMSON (1841– )
• HENRY, JAMES (1798-1876)
• HENRY, JOSEPH (1797-1878)
• HENRY, MATTHEW (1662-1714)
• HENRY, PATRICK (1736–1799)
• HENRY, PRINCE OF BATTENBERG (1858-1896)
• HENRY, ROBERT (1718-1790)
• HENRY, VICTOR (1850– )
• HENRY, WILLIAM (1795-1836)

Henry See also:

• BRIGGS, CHARLES AUGUSTUS (1841– )
• BRIGGS, HENRY (1556–1630)

Briggs . 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 See also:

• ORIGINAL

original calculation of the logarithmic tables now in use is mainly due . Both Napier and Wright died soon after the publication of the Descriptio, the date of Wright's See also:

• DEATH

death being 1615 and that of Napier 1617, but Briggs lived until 1631 . Edward Wright, who was a See also:

• FELLOW

fellow of See also:

• CAIUS
• CAIUS [Anglice KEES, KEYS, etc.], JOHN (1510-1573)

Caius See also:

• COLLEGE (Collegium)

College, See also:

• CAMBRIDGE
• CAMBRIDGE, EARLS AND DUKES OF
• CAMBRIDGE, RICHARD OWEN (1717-1802)

Cambridge, occupies a conspicuous See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place in the history of See also:

• NAVIGATION (from Lat. navis, ship, and agere, to move)

navigation . In 1599 he published Certaine errors in Navigation detected and corrected, and he was the author of other works; to him also is chiefly due the invention of the method known as See also:

• MERCATOR, GERARDUS [latinized form of GERHARD KREMER] (1512-1594)

Mercator's sailing . He at once saw the value of logarithms as an aid to navigation, and lost no time in preparing a See also:

• TRANSLATION

translation, which he submitted to Napier himself . The See also:

• PREFACE (Med. Lat. prefatia, for classical praefatio, praefari, to speak beforehand)

preface to Wright's edition consists of a translation of the preface to the Descriptio, together with the addition of the following sentences written by Napier himself: " But now some of our countreymen in this See also:

• ISLAND (O.E. ieg =isle, +land')

Island well affected to these studies, and the more publique See also:

• GOOD, JOHN MASON (1764-1827)

good, procured a most learned Mathematician to translate the same into our vulgar English See also:

• TONGUE (O. Eng. lunge)

tongue, who after he had finished it, sent the Coppy of it to me, to See also:

• BEE (Sanskrit blza, AS. ben, Lat. apis)

bee See also:

• SCENE (Fr. scene, Lat. scaena, Gr. oflvil, a tent or booth, a stage or scene)

scene and considered on by myself e . I having most willingly and gladly done the same, finde it to bee most exact and precisely conformable to my minde and the originall .

Therefore it may please you who are inclined to these studies, to receive it from me and the Translator, with as much good will as we recommend it unto you." There is a See also:

• SHORT, FRANCIS JOB (1857– )

short " preface to the reader " by Briggs, and a description of a triangular See also:

• DIAGRAM

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, See also:

• SAMUEL

Samuel Wright, in the preface states that his See also:

• FATHER

father " gave much See also:

• COMMENDATION (from the Lat. commendare, to entrust to the charge of, or to procure a favour for)

commendation of this work (and often in my See also:

• HEARING (formed from the verb " to hear," O. Eng. hyran, heron, &c., a common Teutonic verb; cf. Ger. Koren, Dutch hooren, &c.; the O. Tent. form is seen in Goth. hausjan; the initial h makes any connexion with " ear," Lat. audire, or Gr. &Kobew very doub

hearing) as of very See also:

• GREAT

great use to mariners "; and with respect to the translation he says that " shortly after he had it returned out of Scotland, it pleased See also:

• GOD

God to See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call him away afore he could publish it." The translation was published in 1616 . It was also reissued with a new See also:

• TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title-See also:

• PAGE
• PAGE, THOMAS NELSON (1853- )
• PAGE, WILLIAM (1811-1885)

page in 1618 . Henry Briggs, then See also:

• PROFESSOR (the Latin noun formed from the verb profiteri, to declare publicly, to acknowledge, profess)

professor of See also:

• GEOMETRY

geometry at See also:

• GRESHAM, SIR THOMAS (1519-1579)
• GRESHAM, WALTER QUINTON (1832-1895)

Gresham College, See also:

• LONDON

London, and afterwards Savilian professor of geometry at See also:

• OXFORD
• OXFORD, EARLS OF
• OXFORD, EDWARD DE VERE, 17TH EARL
• OXFORD, JOHN DE VERE, 13TH EARL OF (1443-1513)
• OXFORD, PROVISIONS OF
• OXFORD, ROBERT DE VERE, 9TH EARL OF (1362-1392)
• OXFORD, ROBERT HARLEY, 1ST

Oxford, welcomed the Descriptio with See also:

• ENTHUSIASM

enthusiasm . In a letter to See also:

• ARCH
• ARCH, JOSEPH (1826– )

Arch-See also:

• BISHOP (A.S. bisceop, from Lat. episcopus, Gr. irio-Korros, " overlooker " or " overseer ")
• BISHOP, ISABELLA (1832-1904)
• BISHOP, SIR HENRY ROWLEY (1786-1855)

bishop See also:

• USHER (O. Fr. ussier, uissier, mod. huissier, from Lat. ostiarius, a door-keeper, ostium, doorway, entrance, os, mouth)
• USHER (or USSHER), JAMES (1581-1656)

Usher, dated Gresham See also:

• HOUSE
• HOUSE (O. Eng. hiss, a word common to Teutonic languages, cf. Dut. huis, Ger. Haus; in Gothic it is only found in gudhiss, a temple; it may be ultimately connected with the root of " hide," conceal)

House, See also:

• MARCH
• MARCH (1) (from Fr. marcher, to walk; the earliest sense in French appears to be " to trample," and the origin has usually been found in the Lat. marcus, hammer; Low Lat. marcare, to hammer; hence to beat the road with the regular tread of a soldier: cf.
• MARCH, AUZIAS (c. "1395-1458)
• MARCH, EARLS OF
• MARCH, FRANCIS ANDREW (1825– )

March Io, 1615, he wrote, " Napper, See also:

• LORD
• LORD (O. Eng. hldford, i.e. hldfweard, the warder or keeper of bread, hlIf, loaf; the word is not represented in any other Teutonic language)
• LORD, JOHN (1810-1894)

lord of Markinston, path set my See also:

• HEAD (in 0. Eng. heafad; the word is common to Teutonic languages; cf. Dutch hoofd, Ger. Haupt, generally taken to be in origin connected with Lat. caput, Gr. KerbOvi7)
• HEAD, SIR EDMUND WALKER, BART
• HEAD, SIR FRANCIS BOND, BART

head and hands a work with his new and admirable logarithms . I See also:

• HOPE, ANTHONY
• HOPE, THOMAS (c. 1770-1831)

hope to see him this summer, if it please God, for I never saw See also:

• BOOK

book which pleased me better, or made me more wonder.' I purpose to discourse with him concerning eclipses, for what is there which we may not hope for at his hands," and he also states " that he was wholly taken up and employed about the See also:

• NOBLE, SIR ANDREW (1832- )

noble invention of logarithms lately discovered." Briggs accordingly visited Napier in 1615, and stayed with him a whole See also:

• MONTH
• MONTH (a common Teutonic word, cf. Ger. Mond, Du. maand, Dan. maaned, &c., and cognate with Lat. mensis, Gr. µi7v, &c., in other branches of the Indo-Germanic family; all ultimately from the root seen in the word for the moon in nearly all those languages

month.2 He brought with him some ' Dr See also:

• THOMAS
• THOMAS (c. 1654-1720)
• THOMAS (d. 110o)
• THOMAS, ARTHUR GORING (1850-1892)
• THOMAS, CHARLES LOUIS AMBROISE (1811-1896)
• THOMAS, GEORGE (c. 1756-1802)
• THOMAS, GEORGE HENRY (1816-187o)
• THOMAS, ISAIAH (1749-1831)
• THOMAS, PIERRE (1634-1698)
• THOMAS, SIDNEY GILCHRIST (1850-1885)
• THOMAS, ST
• THOMAS, THEODORE (1835-1905)
• THOMAS, WILLIAM (d. 1554)

Thomas See also:

• SMITH
• SMITH, ADAM (1723–1790)
• SMITH, ALEXANDER (183o-1867)
• SMITH, ANDREW JACKSON (1815-1897)
• SMITH, CHARLES EMORY (1842–1908)
• SMITH, CHARLES FERGUSON (1807–1862)
• SMITH, CHARLOTTE (1749-1806)
• SMITH, COLVIN (1795—1875)
• SMITH, EDMUND KIRBY (1824-1893)
• SMITH, G
• SMITH, GEORGE (1789-1846)
• SMITH, GEORGE (184o-1876)
• SMITH, GEORGE ADAM (1856- )
• SMITH, GERRIT (1797–1874)
• SMITH, GOLDWIN (1823-191o)
• SMITH, HENRY BOYNTON (1815-1877)
• SMITH, HENRY JOHN STEPHEN (1826-1883)
• SMITH, HENRY PRESERVED (1847– )
• SMITH, JAMES (1775–1839)
• SMITH, JOHN (1579-1631)
• SMITH, JOHN RAPHAEL (1752–1812)
• SMITH, JOSEPH, JR
• SMITH, MORGAN LEWIS (1822–1874)
• SMITH, RICHARD BAIRD (1818-1861)
• SMITH, ROBERT (1689-1768)
• SMITH, SIR HENRY GEORGE WAKELYN
• SMITH, SIR THOMAS (1513-1577)
• SMITH, SIR WILLIAM (1813-1893)
• SMITH, SIR WILLIAM SIDNEY (1764-1840)
• SMITH, SYDNEY (1771-1845)
• SMITH, THOMAS SOUTHWOOD (1788-1861)
• SMITH, WILLIAM (1769-1839)
• SMITH, WILLIAM (c. 1730-1819)
• SMITH, WILLIAM (fl. 1596)
• SMITH, WILLIAM FARRAR (1824—1903)
• SMITH, WILLIAM HENRY (1808—1872)
• SMITH, WILLIAM HENRY (1825—1891)
• SMITH, WILLIAM ROBERTSON (1846-'894)

Smith thus describes the ardour with which Briggs studied the Descriptio: Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit, ... " Vitae quorundam eruditissimorum et illustrium virorum (London, 1707) . 2 See also:

• WILLIAM
• WILLIAM (1143-1214)
• WILLIAM (1227-1256)
• WILLIAM (1J33-1584)
• WILLIAM (A.S. Wilhelm, O. Norse Vilhidlmr; O. H. Ger. Willahelm, Willahalm, M. H. Ger. Willehelm, Willehalm, Mod.Ger. Wilhelm; Du. Willem; O. Fr. Villalme, Mod. Fr. Guillaume; from " will," Goth. vilja, and " helm," Goth. hilms, Old Norse hidlmr, meaning
• WILLIAM (c. 1130-C. 1190)
• WILLIAM, 13TH

William See also:

• LILLY, WILLIAM (16oz-1681)

Lilly's See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of the See also:

• MEETING (from " to meet," to come together, assemble, 0. Eng. metals ; cf. Du. moeten, Swed. mota, Goth. gamotjan, &c., derivatives of the Teut. word for a meeting, seen in O. Eng. Wit, moot, an assembly of the people; cf. witanagemot)

meeting of Napier and Briggs at Merchiston is quoted in the article NAPIER . calculations he had made, and suggested to Napier the advantages that would result from the choice of ro as a base, an improvement which he had explained in his lectures at Gresham College, and on which he had written to Napier . Napier said that he had already thought of the change, and pointed out a further improvement, viz., that the characteristics of numbers greater than unity should be positive and not negative, as suggested by Briggs . In .1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit''in 1617 had Napier's See also:

• LIFE

life been spared .

Briggs's Logarithmorum chilias prima, which contains the first published table of decimal or common logarithms, is only a small See also:

• OCTAVO

octavo See also:

• TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

tract of sixteen pages, and gives the logarithms of numbers from unity to l000 to 14 places of decimals . It was published, probably privately, in 1617, after Napier's death,' and there is no author's name, place or date . The date of publication is, however, fixed as 1617 by a letter from See also:

• SIR

Sir Henry See also:

• BOURCHIER, ARTHUR (1864- )
• BOURCHIER, THOMAS (c. 1404-1486)

Bourchier to Usher, dated See also:

• DECEMBER (Lat. decent, ten)

December 6, 1617, containing the passage—" Our See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind friend, Mr Briggs, hath lately published a supplement to the most excellent tables of logarithms, which I presume he has sent to you." Briggs's tract of 1617 is extremely rare, and has generally been ignored or incorrectly described . See also:

• HUTTON, CHARLES (1737-1823)
• HUTTON, JAMES (1726-1797)
• HUTTON, RICHARD HOLT (1826-1897)

Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers . There is a copy in the See also:

• BRITISH

British Museum . Briggs continued to labour assiduously at the calculation of logarithms, and in 1624 published his Arithmetica logarithmica, a See also:

• FOLIO (properly the ablative case of the Lat. folium, leaf, but also frequently an adaptation of the Ital. foglio)

folio work containing the logarithms of the numbers from t to 20,000, and from 90,000 to 1oo,000 (and in some copies to 101,000) to 14 places of decimals . The table occupies 300 pages, and there is an introduction of 88 pages relating to the mode of calculation, and the applications of logarithms . There was thus See also:

• LEFT

left a See also:

• GAP

gap between 20,000 and 90,000, which was filled up by See also:

• ADRIAN
• ADRIAN, SAINT

Adrian Vlacq (or Ulaccus), who published at See also:

• GOUDA (or TER GouwE)

Gouda, in See also:

• HOLLAND
• HOLLAND, CHARLES (1733–1769)
• HOLLAND, COUNTY AND PROVINCE OF
• HOLLAND, HENRY FOX, 1ST BARON (1705–1774)
• HOLLAND, HENRY RICH, 1ST EARL OF (1S9o-,649)
• HOLLAND, HENRY RICHARD VASSALL FOX, 3RD
• HOLLAND, JOSIAH GILBERT (1819-1881)
• HOLLAND, PHILEMON (1552-1637)
• HOLLAND, RICHARD, or RICHARD DE HOLANDE (fl. 1450)
• HOLLAND, SIR HENRY, BART

Holland, in 1628, a table containing the logarithms of the numbers from unity to 1oo,000 to 10 places of decimals . Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work . He designates it, however, only a second edition of Briggs's Arithmetica logarithmica, the title See also:

• RUNNING

running Arithmetica logarithmica sive Logarithmorum Chiliades centum, . . . editio secunda aucta per Adrianum Vlacq, Goudanum . This table of Vlacq's was published, with an English explanation prefixed, at London in 1631 under the title Logarithmicall Arithmetike ..

. London, printed by See also:

• GEORGE
• GEORGE, 8TH MARQUESS OF TWEEDDALE (1787-1876)
• GEORGE, HENRY (1839—1897)
• GEORGE, LAKE
• GEORGE, SAINT (d. 303)

George See also:

• MILLER, HUGH (1802–1856)
• MILLER, JOAQUIN (CINCINNATUS HEINE) (1841- )
• MILLER, JOE (JosEPH or JosIAS( (1684-1738)
• MILLER, SAMUEL FREEMAN (1816-1890)
• MILLER, WILLIAM (1782-1849)
• MILLER, WILLIAM (1795-1861)
• MILLER, WILLIAM (1796–1882)
• MILLER, WILLIAM HALLOWES (18o1–188o)

Miller, 1631 . There are also copies with the title-page and introduction in See also:

• FRENCH
• FRENCH, DANIEL CHESTER (1850– )
• FRENCH, NICHOLAS (1604-1678)

French and in Dutch (Gouda, 1628) . Briggs had himself been engaged in filling up the gap, and in a letter to John See also:

• PELL, JOHN (16so-1685)

Pell, written after the publication of Vlacq's work, and dated See also:

• OCTOBER

October 25, 1628, he says: " My See also:

• DESIRE

desire was to have those chiliades that are wantinge betwixt 20 and 90 calculated and printed, and I had done them all almost by my selfe, and by some frendes whom my rules had sufficiently in-formed, and by agreement the busines was conveniently parted amongst us; but I am eased of that See also:

• CHARGE
• CHARGE (through the Fr. from the Late Lat. carricare, to load in a carrus or wagon; cf. " cargo ")

charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole See also:

• HUNDRED

hundred chiliades and printed them in Latin, Dutche and Frenche, moo bookes in these 3 See also:

• LANGUAGES

languages, and hathe could them almost all . But he hathe cutt off 4 of my figures throughout; and hathe left out my See also:

• DEDICATION (Lat. dedicatio, from dedicare, to proclaim, to announce)

dedication, and to the reader, and two chapters the 12 and 13, in the See also:

• REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest he hafh not varied from me at all." The original calculation of the logarithms of numbers from unity to rot,o0o was thus performed by Briggs and Vlacq between 1615 and 1628 . Vlacq's table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived . It contains of course many errors, which were gradually discovered and corrected in the course of the next two hundred and fifty years . The first calculation or publication of Briggian or common iogarithms of trigonometrical functions was made in 162o by See also:

• EDMUND
• EDMUND, or EADMUND (c. 980-1016)
• EDMUND, SAINT [EDMUND RICE] (d. 1240)

Edmund See also:

• GUNTER, EDMUND (1581-1626)

Gunter, who was Briggs's colleague as professor of ' It was certainly published after Napier's death, as Briggs mentions his " librum posthumum." This See also:

• LIBER

liber posthumus was the Lonstructio referred to later in this article.See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy in Gresham College . The title of Gunter's book, which is very scarce, is See also:

• CANON

Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals . • The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents and secants for every minute of the quadrant to to places; there were obtained by calculating the logarithms of the natural sines, &c. given in the See also:

• THESAURUS (Gr. B17o-avp6s)

Thesaurus mathematicus of Pitiscus (1613) . 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 . This work was published by Vlacq at his own expense at Gouda in 1633, under the title Trigonometria Britannica . It contains log sines (to 14 places) and tangents (to to places), besides natural sines, tangents and secants, at intervals of a hundredth of a degree .

In the same See also:

• YEAR

year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to to places . This work also contains the logarithms of numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628 . Briggs appreciated clearly the advantages of a centesimal, division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a See also:

• REFORMATION
• REFORMATION, THE

reformation in this respect, and but for the appearance of Vlacq's work the decimal division of the degree might have become recognized, as is now the case with the corresponding division of the second . The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works—Arithmetica logarilhmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometric artificialis (Vlacq), 1633—have not been superseded by any subsequent calculations . In the preceding paragraphs an account has been given of the actual announcement of the invention of logarithms and of the calculation of the tables . It now remains to refer in more detail to the invention itself and to examine the claims of Napier and Briggs to the See also:

• CAPITAL (i.e. capital stock or fund)
• CAPITAL (Lat. caput, head)

capital improvement involved in the change from Napier's original logarithms to logarithms to the base to . The Descriptio contained only an explanation of the use of the logarithms without any account of the manner in which the canon was constructed . In an " Admonitio " on the seventh page Napier states that, although in that place the mode of construction should be explained, he proceeds at once to the use of the logarithms, " ut praelibatis prius usu, et rei utilitate, caetera See also:

• AUT

aut magis placeant posthac edenda, aut minus saltem displiceant silentio sepulta." He awaits therefore the See also:

• JUDGMENT

judgment and censure of the learned " priusquam caetera in lucem temere prolata lividorum detrectationi exponantur "; and in an " Admonitio " on the last page of the book he states that he will publish the mode of construction of the canon " si huius inventi usum eruditis gratum fore intellexero." Napier, however, did not live to keep this promise . In 1617 he published a small work entitled Rabdologia relating to See also:

• MECHANICAL

mechanical methods of performing multiplications and divisions, and in the same year he died . The proposed work was published in 1619 by See also:

• ROBERT
• ROBERT (1275—1343)
• ROBERT, HUBERT (1753-1808)
• ROBERT, LOUIS LEOPOLD (1794-1835)

Robert Napier, his second son by his second See also:

• MARRIAGE

marriage, under the title Mirifici logarithmorum canonis constructio . . . . It consists of two pages of preface followed by sixty-seven pages of See also:

• TEXT (Lat. textum, woven fabric, from texere, to weave)

text .

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 . He therefore issues the work to satisfy their desires, although, he states, it is See also:

• MANIFEST (Lat. manifestus, clear, open to view)

manifest that it would have seen the See also:

• LIGHT

light in a far more perfect See also:

• STATE
• STATE, GREAT OFFICERS OF

state if his father could have put the See also:

• FINISHING

finishing touches to it; and he mentions that, in the See also:

• OPINION (Lat. opinio, from opinari, to think)

opinion of the best See also:

• JUDGES, THE BOOK OF

judges, his father possessed, among other most excellent gifts, in the highest degree the power of explaining the most difficult matters by a certain and easy method in the fewest possible words . It is important to See also:

• NOTICE

notice that in the Constructio logarithms are called artificial numbers; and Robert Napier states that the work was composed several years (See also:

• ALIQUOT (a Lat. word meaning " some," " so many ")

aliquot annos) before Napier had invented the name logarithm . The Constructio therefore may have been written a good many years previous to the publication of the Descriptio in 1614 . Passing now to the invention of common or decimal logarithms, that is, to the transition from the logarithms originally invented by Napier to logarithms to the base 10, the first allusion to a change of system occurs in the "Admonitio " on the last page of the Descriptio (1614), the concluding See also:

• PARAGRAPH

paragraph of which is " Verum si huius inventi usum eruditis gratum fore intellexero, dabo fortasse brevi (Deo aspirante) rationem ac methodum aut hunc canonem emendandi, aut emendatiorem de novo condendi, ut ita plurium Logistarum diligentia,limatior tandem et accuratior, quam unius See also:

• OPERA (Italian for " work ")

opera fieri potuit, in lucem prodeat . Nihil in ortu perfectum." In some copies, however, this " Admonitio " is absent . In Wright's translation of 1616 Napier has added the See also:

• SENTENCE (Lat. sententia, a way of thinking, opinion, judgment; vote, sentire, to feel, think)

sentence—" But because the addition and subtraction of these former numbers may seeme somewhat painfull, I intend (if it shall please God) in a second Edition, to set out such Logarithmes as shall make those numbers above written to fall upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000, &c., which are easie to be added or See also:

• ABATED

abated to or from any other number " (p..19); and in the dedication of the Rabdologia (1617) he wrote " See also:

• QUORUM (Lat. for " of whom ")

Quorum quidem Logarithmorum speciem aliam multo praestantiorem nunc etiam invenimus, & creandi methodum, una cum eorum usu (si See also:

• DEUS
• DEUS, JOAO DE (1830-1896)

Deus longiorem vitae & valetudinis usuram concesserit) evulgare statuimus; ipsam autem novi canonis supputationem, ob infirmam corporis nostri valetudinem, viris in hoc studii genere versatis relinquimus: imprimis vero doctissimo viro D . Henrico Briggio Londini publico Geometriae Professori, et amico mihi longe charissimo." Briggs in the short preface to his Logarithmorum chilias (1617) states that the reason why his logarithms are different from those introduced by Napier " sperandum, ejus librum See also:

• POST

post humum, abunde nobis propediem satisfacturum." The " liber posthumus " was the Constructio (1619), in the preface to which Robert Napier states that he has added an appendix relating to another and more excellent See also:

• SPECIES

species of logarithms, referred to by the inventor himself in the Rabdologia, and in which the logarithm of unity is o . He also mentions that he has published some remarks upon the propositions in spherical See also:

• TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure)

trigonometry and upon the new species of logarithms by Henry Briggs, "qui novi hujus Canonis supputandi laborem gravissimum, See also:

• PRO

pro singulari amicitia quae illi cum Patre meo L . M. intercessit, animo libentissimo in se suscepit; creandi methodo, et usuum explanatione Inventori relictis . Nunc autem ipso ex See also:

• LAC

lac vita evocato, totius negotii onus doctissimi Briggii humeris incumbere, et See also:

• SPARTA (Gr. Eirapril or AaKe5atycav)

Sparta haec ornanda illi sorte quadam obtigisse videtur." In the address prefixed to the Arithmetica logarithmica (1625) Briggs bids the reader not to be surprised that these logarithms are different from those published in the Descriptio : " Ego enim, cum meis auditoribus Londini, publice in Collegio Greshamensi horum doctrinam explicarem; animadverti multo futurum commodius, si Logarithmus sinus totius servaretur o (ut in Canone mirifico), Logarithmus autem partis decimae ejusdem sinus totius, nempe sinus 5 graduum, 44, M . 21, s., esset l0000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempos, et vacationem a publico docendi munere licuit, profectus sum Edinburgum; ubi humanissime ab eo acceptus hacsi per integram mensem: Cum autem inter nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et cupivisse dicebat: veruntamen istos, quos jam paraverat edendos curasse, donee alios, si per negotia et valetudinem liceret, magis commodos confecisset .

Istam autem mutationem ita faciendam censebat, ut o esset Logarithmus unitatis, et t0000000000 sinus totius: quod ego longe commodissimum esse non potui non agnoscere . Coepi igitur, ejus hortatu, rejectis illis quos antea paraveram, de horum calculo serio cogitare; et sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo praecipuos. ostendi, idem etiam tertia aestate libentissime facturus, si Deus ilium nobis tamdiu superstitem esse voluisset." There is also a reference to the change of the logarithms on the title-page of the work . 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 o 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 ro,000,000,000 . He wrote also to Napier at once; and as soon as he could he went to Edinburgh to visit him, where, as he was most hospitably received by him, he remained for a whole month . When they conversed about the change of system, Napier said that he had perceived and desired the same thing, but that he had published the tables which he had already pre-pared, so that they might be used until he could construct others more convenient . But he considered that the change ought to be so made that o should be the logarithm of unity and 10,000,000,000 that of the whole sine, which Briggs could not but admit was by far the most convenient of all . Rejecting therefore, those which he had prepared already, Briggs began, at Napier's See also:

• ADVICE (Fr. avis, from Lat. ad, to, and visum, viewed)

advice, to consider seriously the question of the calculation of new tables . In the following summer he went to Edinburgh and showed Napier the principal portion of the logarithms which he published in 1624 . These probably included the logarithms of the first chiliad which he published in 1617 . It has been thought necessary to give in detail the facts relating to the See also:

• CONVERSION (Lat. conversio, from convertere, to turn or ,change)

conversion of the logarithms, as unfortunately See also:

• CHARLES
• CHARLES (1270-1325)
• CHARLES (1421-1461)
• CHARLES (1525-1574)
• CHARLES (c. 1319-1364)
• CHARLES (Fr. Charles; Span. Carlos; Ital. Carlo; Ger. Karl; derived from O.H.G. Charal,latinized as Carolus, meaning originally " man ": cf. Mod.Ger., Kerl," fellow," A.S. ceorl, Mod. Eng. " churl ")
• CHARLES (KARL EITEL ZEPHYRIN LUDWIG; in Rum. CAROL)
• CHARLES [KARL ALEXANDER] (1712-1780)
• CHARLES, ELIZABETH (1828-1896)
• CHARLES, JACQUES ALEXANDRE CESAR (1746-1823)
• CHARLES, THOMAS (1755-1814)

Charles Hutton in his history of logarithms, which was prefixed to the early See also:

• EDITIONS

editions of his Mathematical Tables, and was also published as one of his Mathematical Tracts, has charged Napier with want of candour in not telling the See also:

• WORLD

world of Briggs's See also:

• SHARE (O. Eng. scearu, chiefly in compounds, e.g. land-scearu, a share of land, from sceran to cut; cf. " shear" )

share in the change of system, and he expresses the suspicion that " Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms." According to Hutton's view, the words, " it is to be hoped that his See also:

• POSTHUMOUS

posthumous work " .

. . which occur in the preface to the Chilias, were a modest hint that the share Briggs had had in changing the logarithms should be mentioned, and that, as no attention was paid to it, he himself gave the account which appears in the Arithmetica of 1624 . There seems, however, no ground whatever for supposing that Briggs meant to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express anything beyond his hope that the reason for the alteration would be explained in the posthumous work; and in his own account, written seven years after Napier's death and five years after the appearance of the work itself, he shows no injured feeling whatever, but even goes out of his way to explain that he abandoned his own proposed alteration in favour of Napier's, and, rejecting the tables he had already constructed, began to consider the calculation of new ones . The facts, as stated by Napier and Briggs, are in See also:

• COMPLETE

complete accordance, and the friendship existing between them was perfect and unbroken to the last . Briggs assisted Robert Napier in the editing of the " posthumous work," the Constructio, and in the account he gives of the alteration of the logarithms in the Arithmetica of 1624 he seems to have been more anxious that See also:

• JUSTICE (Lat. justitia)

justice should be done to Napier than to him-self; while on the other hand Napier received Briggs most hospitably and refers to him as " amico mihi longe charissimo." Hutton's suggestions are all the more to be regretted as they occur as a history which is the result of a good See also:

• DEAL

deal of investigation and which for years was referred to as an authority by many writers . His See also:

• PREJUDICE (Lat. praejudicium)

prejudice against Napier naturally produced See also:

• RETALIATION

retaliation, and See also:

• MARK
• MARK, CARL (1858– )
• MARK, GOSPEL OF ST
• MARK, ST

Mark Napier in defending his ancestor has fallen into the opposite extreme of attempting to reduce Briggs to the level of a See also:

• MERE
• MERE, ADALBERT (1838-1909)

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 . It is printed on the back of the last page of the table itself, and so cannot have been torn out from the copies that are without it . As there could have been no reason for omitting it after it had once appeared, we may assume that the copies which do not have it are those which were first issued . It is probable, therefore, that Briggs's copy contained no reference to the change, and it is even possible that the "Admonitio " may have been added after Briggs had communicated with Napier . As See also:

• SPECIAL

special attention has not been drawn to the fact that some copies have the " Admonitio " and some have not, different writers have assumed that Briggs did or did not know of the promise contained in the " Admonitio " according as it was See also:

• PRESENT

present or absent in the copies they had themselves referred to, and this has given rise to some confusion . It may also be remarked that the date frequently assigned to Briggs's first visit to Napier is 1616, and not 1615 as stated above, the reason being that Napier was generally supposed to have died in 1618 until Mark Napier showed that the true date was 1617 . When the Descriplio was published Briggs was fifty-seven years of See also:

• AGE (Fr. age, through late Lat. aetaticum, from aetas)

age, and the remaining seventeen years of his life were devoted with steady enthusiasm to extend the utility of Napier's great invention .

The only other mathematician besides Napier who grasped the See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea on which the use of logarithm depends and applied it to the construction of a table is Justus Byrgius (See also:

• JOBST

Jobst Biirgi), whose work Ariihmetische send geometrische Progress-Tabulen ... was published at See also:

• PRAGUE (Ger. Prag; Bohemian Praha)

Prague in 162o, six years after the publication of the Descriptio of Napier . This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms . It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus 0, 1.0000 0000 I O, I ,000I 0000 20, 1,0002 0001 990, 1,0099 4967 In the arithmetical See also:

• COLUMN (Lat. columna)

column the numbers increase by Io, in the geometrical column each number is derived from its predecessor by multiplication by i•000i . Thus the number lox in the arithmetical column corresponds to Ioe (i.000l)s in the geometrical column; the intermediate numbers being obtained by See also:

• INTERPOLATION (from Lat. interpolare, to alter, or insert something fresh, connected with pollee, a polish)

interpolation . If we See also:

• DIVIDE

divide the numbers in the geometrical column by rob the correspondence is between iox and (1•000l)s, and the table then becomes one of antilogarithms, the base being (1.0001)1/10, viz. for example (I•o0o1)i16•990=x•00994967 . The table extends to 230270 in the arithmetical column, and it is shown that 230270.022 corresponds to 9'9999 9999 or 109 in the geometrical column; this last result showing that (i•000 1)23027.022 = ro . The first contemporary mention of Byrgius's table occurs on page II of the " Praecepta " prefixed to See also:

• KEPLER, JOHANN (1571-1630)

Kepler's Tabulae Radolphinae (1627); his words are: " apices logistici J . Byrgio multis annis ante editionem Neperianam viam praeiverent ad hos ipsissimos logarithmos . Etsi homo cunctator et secretorum suorum custos foetum in partu destituit, non ad usus publicos educavit." Another reference to Byrgius occurs in a work by See also:

• BENJAMIN
• BENJAMIN, JUDAHPHILIP (1811-1884)

Benjamin Bramer, the See also:

• BROTHER

brother-in-See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law and See also:

• PUPIL (Lat. pupillus, orphan, minor, dim. of pupus, boy, allied to puer, from root pm- or peu-, to beget, cf. "pupa," Lat. for " doll," the name given to the stage intervening between the larval and imaginal stages in certain insects)

pupil of Byrgius, who, See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

writing in 163o, says that the latter constructed his table twenty years ago or more.' As regards priority of publication, Napier has the See also:

• ADVANTAGE

advantage by six years, and even fully accepting Bramer's statement, there are grounds for believing that Napier's work See also:

• DATES

dates from a still earlier See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

period . The power of to, which occurs as a See also:

• FACTOR (from Lat. facere, to make or do)

factor in the tables of both Napier and Byrgius, was rendered necessary by the fact that the decimal point was not yet in use . Omitting this factor in ' Frisch's Kepleri opera omnia, ii . 834 .

Frisch thinks Bramer possibly relied on Kepler's statement quoted in the text (" Quibus forte confisus Kepleri verbis Benj . Bramer . . . ") . See also vol. vii. p . 298 . The claims of Byrgius are discussed in Kastner's Geschichte der Mathematik, ii . 375, and iii . 14; See also:

• MONTUCLA, JEAN ETIENNE (1725-1799)

Montucla's Histoire See also:

• DES

des mathematigues, ii . Io; See also:

• DELAMBRE, JEAN BAPTISTE JOSEPH (1749-1822)

Delambre's Histoire de l'astronomie moderne, i . 560; de See also:

• MORGAN
• MORGAN, DANIEL (1736-1802)
• MORGAN, EDWIN DENNISON (1811-1883)
• MORGAN, JOHN HUNT (1825-1864)
• MORGAN, JOHN PIERPONT (1837–1913)
• MORGAN, LEWIS HENRY (1818-1881)
• MORGAN, SYDNEY, LADY (c. 1783-1859)
• MORGAN, THOMAS (d. 1743)

Morgan's article on " Tables " in the English Cyclopaedia ; Mark Napier's See also:

• MEMOIRS

Memoirs of John Napier of Merchiston (1834), p . 391, and Cantor's Geschichte der Mathematik, ii .

(1892), 662 . See also Gieswald, Justus Byrg als Mathematiker and dessen Einleitung in See also:

• SEINE
• SEINE (Lat. Sequana)
• SEINE, or SEAN (O. Fr. seigne, mod. seine, Lat. sagena, Gr. vaytivrl, a draw-net)

seine Logarithmen (See also:

• DANZIG, or DANTSIC (Polish Gdansk)

Danzig, 1856).the case of both tables, the connexion between N a number and L its " logarithm " is N=(e')'' (Napier), L=(1•000l)See also:

• IOO

ioo (Byrgius), viz . Napier gives logarithms to base e 1, Byrgius gives See also:

• ANTI, or CAMPA

anti-logarithms to base (1•000i)I'a . There is indirect See also:

• EVIDENCE (Lat. evidentia, evideri, to appear clearly)

evidence that Napier was occupied with logarithms as early as 1594, for in a letter to P . Crugerus from Kepler, dated See also:

• SEPTEMBER (Lat. septem, seven)

September 9, 1624 (Frisch's Kepler, vi . 47), there occurs' the sentence: " Nihil autem supra Neperianam rationem ease puto: etsi quidem Scotus quidam literis ad Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici." It is here distinctly stated that some Scotsman in the year 1594, in g. letter to Tycho See also:

• BRAHE, PER, COUNT (1602-1680)
• BRAHE, TYCHO (1546-1601)

Brahe, gave him some hope of the logarithms; and as Kepler joined Tycho after his See also:

• EXPULSION (Lat. expulsio, from expellere)

expulsion from the island of Huen, and had been so closely associated with him in his work, he would be likely to be correct in any assertion of this kind . In connexion with Kepler's statement the following See also:

• STORY, JOHN (c. 1510-1571)
• STORY, JOSEPH (1779-1845)
• STORY, ROBERT HERBERT (1835-1907)
• STORY, WILLIAM WETMORE (1819—1895)

story, told by See also:

• ANTHONY, SAINT