## Sentence Examples

**In****addition****to****the****various****works****of****Brewster****already****noticed**,**the****following****may****be****mentioned**: -**Notes****and****Introduction****to****Carlyle'****s****translation****of****Legendre'****s****Elements****of****Geometry**(1824);**Treatise****on****Optics**(1831);**Letters****on****Natural****Magic**,**addressed****to****Sir****Walter****Scott**(1831);**The****Martyrs****of****Science**,**or****the****Lives****of****Galileo**,**Tycho****Brake**,**and****Kepler**(1841);**More****Worlds****than****One**(1854).**Under****the****general****heading**"**Analysis**"**occur****the****subheadings**"**Foundations****of****Analysis**,"**with****the****topics****theory****of****functions****of****real****variables**,**series****and****other****infinite****processes**,**principles****and****elements****of****the****differential****and****of****the****integral****calculus**,**definite****integrals**,**and****calculus****of****variations**; "**Theory****of****Functions****of****Complex****Variables**,"**with****the****topics****functions****of****one****variable****and****of****several****variables**; "**Algebraic****Functions****and****their****Integrals**,"**with****the****topics****algebraic****functions****of****one****and****of****several****variables**,**elliptic****functions****and****single****theta****functions**,**Abelian****integrals**; "**Other****Special****Functions**,"**with****the****topics****Euler'**,**s****Legendre'**,**s****Bessel'****s****and****automorphic****functions**; "**Differential****Equations**,"**with****the****topics****existence****theorems**,**methods****of****solution**,**general****theory**; "**Differential****Forms****and****Differential****Invariants**,"**with****the****topics****differential****forms**,**including****Pfaffians**,**transformation****of****differential****forms**,**including****tangential**(**or****contact**)**transformations**,**differential****invariants**; "**Analytical****Methods****connected****with****Physical****Subjects**,"**with****the****topics****harmonic****analysis**,**Fourier'****s****series**,**the****differential****equations****of****applied****mathematics**,**Dirichlet'****s****problem**; "**Difference****Equations****and****Functional****Equations**,"**with****the****topics****recurring****series**,**solution****of****equations****of****finite****differences****and****functional****equations**.**The****director**,**Schmalfuss**,**encouraged****him****in****his****mathematical****studies****by****lending****him****books**(**among****them****Leonhard****Euler'****s****works****and****Adrien****Marie****Legendre'****s****Theory****of****Numbers**),**which****Riemann****read**,**mastered****and****returned****within****a****few****days**.**His****first****published****writings****consist****of****articles****forming****part****of****the****Traite****de****mecanique**(1774)**of****the****Abbe****Marie**,**who****was****his****professor**;**Legendre'****s****name**,**however**,**is****not****mentioned**.**This****is****the****subject****with****which****Legendre'****s****name****will****always****be****most****closely****connected**,**and****his****researches****upon****it****extend****over****a****period****of****more****than****forty****years**.**Legendre'****s****researches****connected****with****the**"**gamma****function**"**are****of****importance**,**and****are****well****known**;**the****subject****was****also****treated****by****K**.- -
**Legendre'****s****Theorie****des****nombres****and****Gauss'****s****Disquisitiones****arithmeticae**(1801)**are****still****standard****works****upon****this****subject**. **The****symbol**(**alp**)**which****is****known****as****Legendre'****s****symbol**,**and****denotes****the****positive****or****negative****unit****which****is****the****remainder****when****au**(-1)**s****is****divided****by****a****prime****number****p**,**does****not****appear****in****this****memoir**,**but****was****first****used****in****the****Essai****sur****la****theorie****des****nombres**.**Legendre'****s****formula****x**: (**log****x**-**I**.**Legendre'****s****second****memoir****was****communicated****to****the****Academie****in**1784,**and****relates****to****the****conditions****of****equilibrium****of****a****mass****of****rotating****fluid****in****the****form****of****a****figure****of****revolution****which****does****not****deviate****much****from****a****sphere**.- (
**See****Todhunter'****s****History****of****the****Mathematical****Theories****of****Attraction****and****the****Figure****of****the****Earth**(1873),**the****twentieth**,**twenty-second**,**twenty-fourth**,**and****twenty-fifth****chapters****of****which****contain****a****full****and****complete****account****of****Legendre'****s****four****memoirs**. **The****best****known****of****these**,**which****is****called****Legendre'****s****theorem**,**is****usually****given****in****treatises****on****spherical****trigonometry**;**by****means****of****it****a****small****spherical****triangle****may****be****treated****as****a****plane****triangle**,**certain****corrections****being****applied****to****the****angles**.**Legendre'****s****theorem****is****a****fundamental****one****in****geodesy**,**and****his****contributions****to****the****subject****are****of****the****greatest****importance**.**In**1806**appeared****Legendre'****s****Nouvelles****Methodes****pour****la****determination****des****orbites****des****cometes**,**which****is****memorable****as****containing****the****first****published****suggestion****of****the****method****of****least****squares**(**see****Probability**).- -
**Legendre'****s****name****is****most****widely****known****on****account****of****his****Elements****de****geometrie**,**the****most****successful****of****the****numerous****attempts****that****have****been****made****to****supersede****Euclid****as****a****text-book****on****geometry**. **Legendre'****s****proof****is****similar****in****principle****to****Lambert'**,**s****but****much****simpler**.**It****will****thus****be****seen****that****Legendre'****s****works****have****placed****him****in****the****very****foremost****rank****in****the****widely****distinct****subjects****of****elliptic****functions**,**theory****of****numbers**,**attractions**,**and****geodesy**,**and****have****given****him****a****conspicuous****position****in****connexion****with****the****integral****calculus****and****other****branches****of****mathematics**.**He****received**£50**for****a****translation****of****Legendre'****s****Geometry**;**and****an****introduction**,**explaining****the****theory****of****proportion**,**is****said****by****De****Morgan****to****show****that****he****could****have****gained****distinction****as****an****expounder****of****mathematical****principles**.