## Sentence Examples

**The****names****of****nearly****all****Napier'****s****classfellows****can****be****traced****as****becoming****determinantes****in**1566**and****masters****of****arts****in**1568;**but****his****own****name****does****not****appear****in****the****lists**.**The****full****title****of****this****first****work****of****Napier'****s****is****given****below**.'- 2
**Napier'****s****Plaine****Discovery****is****a****serious****and****laborious****work**,**to****which****he****had****devoted****years****of****care****and****thought**. **These****rods**,**which****were****commonly****called**"**Napier'****s****bones**,"**will****be****described****further****on**.**Il****mourut****l'an**1616,**et****fut****enterre****hors****la****Porte****Occidentale****d'Edinbourg**,**dans****l'Eglise****de****Sainct****Cudbert**.'"**There****can****be****no****doubt****that****Napier'****s****devotion****to****mathematics****was****not****due****to****old****age****and****the****gout**,**and****that****he****died****in**1617**and****not****in**1616;**still****these****sentences****were****written****within****eighteen****years****of****Napier'****s****death**,**and****their****author****seems****to****have****had****some****special****sources****of****information**.**It****has****been****stated****that****Napier'****s****mathematical****pursuits****led****him****to****dissipate****his****means**.**Briggs****was****greatly****excited****by****Napier'****s****invention****and****visited****him****at****Merchiston****in**1615,**staying****with****him****a****whole****month**;**he****repeated****his****visit****in**1616**and**,**as****he****states**, "**would****have****been****glad****to****make****him****a****third****visit****if****it****had****pleased****God****to****spare****him****so****long**."- (
**See****Logarithm**.)**Napier'****s****Descriptio****of**1614**contains****no****explanation****of****the****manner****in****which****he****had****calculated****his****table**. **Macdonald****at****Edinburgh****in**1889,**and****that****there****is****appended****to****this****edition****a****complete****catalogue****of****all****Napier'****s****writings**,**and****their****various****editions****and****translations**,**English****and****foreign**,**all****the****works****being****carefully****collated**,**and****references****being****added****to****the****various****public****libraries****in****which****they****are****to****be****found**.**Napier'****s****priority****in****the****publication****of****the****logarithms****is****unquestioned****and****only****one****other****contemporary****mathematician****seems****to****have****conceived****the****idea****on****which****they****depend**.**In****that****article****it****is****mentioned****that****a****Scotsman****in**1594**in****a****letter****to****Tycho****Brahe****held****out****some****hope****of****logarithms**;**it****is****likely****that****the****person****referred****to****is****John****Craig**,**son****of****Thomas****Craig**,**who****has****been****mentioned****as****one****of****the****colleagues****of****John****Napier'****s****father****as****justice-depute**.**Of****this****singular****contract**,**which****is****signed**, "**Robert****Logane****of****Restalrige**"**and**"**Jhone****Neper**,**Fear****of****Merchiston**,"**and****is****dated****July**1594,**a****facsimile****is****given****in****Mark****Napier'****s****Memoirs**.**After****Napier'****s****death****his****manuscripts****and****notes****came****into****the****possession****of****his****second****son****by****his****second****marriage**,**Robert**,**who****edited****the****Constructio**;**and****Colonel****Milliken****Napier**,**Robert'****s****lineal****male****representative**,**was****still****in****the****possession****of****many****of****these****private****papers****at****the****close****of****the**18th**century**.**The****transcripts****are****entirely****in****the****handwriting****of****Robert****Napier****himself**,**and****the****two****notes****that****have****been****quoted****prove****that****they****were****made****from****Napier'****s****own****papers**.**The****title**,**which****is****written****on****the****first****leaf**,**and****is****also****in****Robert****Napier'****s****writing**,**runs****thus**: "**The****Baron****of****Merchiston****his****booke****of****Arithmeticke****and****Algebra**.**It**-**is****worth****while****to****notice****that****this****reference****occurs****in****a****chapter**"**De****Multiplicationis****et****Partitionis****compendiis****miscellaneis**,"**which**,**supposing****the****treatise****to****have****been****written****in****Napier'****s****younger****days**,**may****have****been****his****earliest****production****on****a****subject****over****which****his****subsequent****labours****were****to****exert****so****enormous****an****influence**.**Among****the****Merchiston****papers****is****a****thin****quarto****volume****in****Robert****Napier'****s****writing****containing****a****digest****of****the****principles****of****alchemy**;**it****is****addressed****to****his****son**,**and****on****the****first****leaf****there****are****directions****that****it****is****to****remain****in****his****charter-chest****and****be****kept****secret****except****from****a****few**.**The****principle****of**"**Napier'****s****bones**"**may****be****easily****explained****by****imagir**:**ing****ten****rectangular****slips****of****cardboard**,**each****divided****into****nine****squares**.**The****figures****as****written****down****are**12510 6255 14595 1534560**Napier'****s****rods****or****bones****consist****of****ten****oblong****pieces****of****wood****or****other****material****with****square****ends**.**In****multiplications****or****divisions****of****any****length****it****is****generally****convenient****to****begin****by****forming****a****table****of****the****first****nine****multiples****of****the****multiplicand****or****divisor**,**and****Napier'****s****bones****at****best****merely****provide****such****a****table**,**and****in****an****incomplete****form**,**for****the****additions****of****the****two****figures****in****the****same****parallelogram****have****to****be****performed****each****time****the****rods****are****used**.**Nothing****shows****more****clearly****the****rude****state****of****arithmetical****knowledge****at****the****beginning****of****the**17th**century****than****the****universal****satisfaction****with****which****Napier'****s****invention****was****welcomed****by****all****classes****and****regarded****as****a****real****aid****to****calculation**.**Besides****the****logarithms****and****the****calculating****rods****or****bones**,**Napier'****s****name****is****attached****to****certain****rules****and****formulae****in****spherical****trigonometry**.- "
**Napier'****s****rules****of****circular****parts**,"**which****include****the****complete****system****of****formulae****for****the****solution****of****right-angled****triangles**,**may****be****enunciated****as****follows**. - "
**Napier'****s****analogies**"**are****the****four****formulae**-**tan**(**A**+**B**)**cos**=**cos****l**(**a**-**b**)**cotzC**, tan2(**A**-**B**) -**sin**(**a**-**b**)cot2C;**a**(**a**+**b**)**sine**(**a**+**b**)**tan****a**(**a**+**b**) =**cos**(**A**-**B**) tan1c, tan2(**a**-**b**) =**s****in**(**A**-**B**) tan2c. **Only****one****of****the****four****analogies****is****actually****given****by****Napier**,**the****other****three****being****added****by****Briggs****in****the****remarks****which****are****appended****to****Napier'****s****results**.**He****has****been****sometimes****erroneously****called**"**Peer****of****Merchiston**,"**and****in****the**1645**edition****of****the****Flamm****Discovery****he****is****so****styled**(**see****Mark****Napier'****s****Memoirs**,**pp**. 9**and**173,**and****Libri****qui****supersunt**,**p**.**xciv**.).**The****bibliography****of****Napier'****s****workattached****to****W**.**Napier'****s****three****mathematical****works****are****reprinted****by****N**.**The****Boers**,**however**,**strongly****resented****the****contention****of****the****British****that****they****could****not****shake****off****British****nationality****though****beyond****the****bounds****of****any****recognized****British****possession**,**nor****were****they****prepared****to****see****their****only****port****garrisoned****by****British****troops**,**and****they****rejected****Napier'****s****overtures**.**Among****the****English****histories****of****Florence**,**Napier'****s****Florentine****History**(6**vols**.,**London**,1846**-**1847)**and****A**.**Trollope'****s****History****of****the****Commonwealth****of****Florence**(4**vols**.,**London**, 1865)**are****not****without****value****although****out****of****date**.**The****first****announcement****of****the****invention****was****made****in****Napier'****s****Mirifici****Logarithmorum****Canonis****Descriptio**..**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**=10 7**e****L**/Ip7,**so****that**(**ignoring****the****factors****re**,**the****effect****of****which****is****to****render****sines****and****logarithms****integral****to**7**figures**),**the****base****is****C**".**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**1**and****L****is****expressed****by****L**=**r****o**7**loge**10 7 - 10 7 /**or****e****t**=**I**07e**-L**/Ia7**Napier'****s****work**(**which****will****henceforth****in****this****article****be****referred****to****as****the****Descriptio**)**immediately****on****its****appearance****in**1614**attracted****the****attention****of****perhaps****the****two****most****eminent****English****mathematicians****then****living**-**Edward****Wright****and****Henry****Briggs**.**Inl**,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****life****been****spared**.**It****was****published**,**probably****privately**,**in**1617,**after****Napier'****s****death**,'**and****there****is****no****author'****s****name**,**place****or****date**.**The****first****calculation****or****publication****of****Briggian****or****common****logarithms****of****trigonometrical****functions****was****made****in**1620**by****Edmund****Gunter**,**who****was****Briggs'****s****colleague****as****professor****of**1**It****was****certainly****published****after****Napier'****s****death**,**as****Briggs****mentions****his**"**librum****posthumum**."**It****now****remains****to****refer****in****more****detail****to****the****invention****itself****and****to****examine****the****claims****of****Napier****and****Briggs****to****the****capital****improvement****involved****in****the****change****from****Napier'****s****original****logarithms****to****logarithms****to****the****base****ro**.**Rejecting****therefore**,**those****which****he****had****prepared****already**,**Briggs****began**,**at****Napier'****s****advice**,**to****consider****seriously****the****question****of****the****calculation****of****new****tables**.**There****seems**,**however**,**no****ground****whatever****for****supposing****that****Briggs****meant****to****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**.**When****the****Descriptio****was****published****Briggs****was****fiftyseven****years****of****age**,**and****the****remaining****seventeen****years****of****his****life****were****devoted****with****steady****enthusiasm****to****extend****the****utility****of****Napier'****s****great****invention**.**As****regards****priority****of****publication**,**Napier****has****the****advantage****by****six****years**,**and****even****fully****accepting****Bramer'****s****statement**,**there****are****grounds****for****believing****that****Napier'****s****work****dates****from****a****still****earlier****period**.- 560;
**de****Morgan'****s****article****on**"**Tables**"**in****the****English****Cyclopaedia**;**Mark****Napier'****s****Memoirs****of****John****Napier****of****Merchiston**(1834),**p**. 39 2,**and****Cantor'****s****Geschichte****der****Mathematik**,**ii**. **Dr****Craig****was****John****Craig**,**the****third****son****of****Thomas****Craig**,**who****was****one****of****the****colleagues****of****Sir****Archibald****Napier**,**John****Napier'****s****father**,**in****the****office****of****justice-depute**.**We****may****infer****therefore****that****as****early****as**1594**Napier****had****communicated****to****some****one**,**probably****John****Craig**,**his****hope****of****being****able****to****effect****a****simplification****in****the****processes****of****arithmetic**.**Everything****tends****to****show****that****the****invention****of****logarithms**2**See****Mark****Napier'****s****Memoirs****of****John****Napier****of****Merchiston**(1834),**p**. 362.**An****account****has****now****been****given****of****Napier'****s****invention****and****its****publication**,**the****transition****to****decimal****logarithms**,**the****calculation****of****the****tables****by****Briggs**,**Vlacq****and****Gunter**,**as****well****as****of****the****claims****of****Byrgius****and****the****method****of****prosthaphaeresis**.**To****Napier'****s****Descriptio****in****order****to****describe****its****reception****on****the****continent**,**and****to****mention****the****other****logarithmic****tables****which****were****published****while****Briggs****was****occupied****with****his****calculations**.**He****says****that**,**although****Napier'****s****book****had****been****published****five****years**,**he****first****saw****it****at****Prague****two****years****before**;**he****was****then****unable****to****read****it**,**but****last****year****he****had****met****with****a****little****work****by****Benjamin****Ursinus**4**containing****the****substance****of****the****method**,**and****he****at****once****recognized****the****importance****of****what****had****been****effected**.**This****letter****was****written****two****years****after****Napier'****s****death**(**of****which****Kepler****was****unaware**),**and****in****the****same****year****as****that****in****which****the****Constructio****was****published**.**In****the****same****year**(1620)**Napier'****s****Descriptio**(1614)**and****Constructio**(1619)**were****reprinted****by****Bartholomew****Vincent****at****Lyons****and****issued****together**.5**Napier****calculated****no****logarithms****of****numbers**,**and**,**as****already****stated**,**the****logarithms****invented****by****him****were****not****to****base****e**.