## Sentence Examples

**Zeuthen**,**Geschichte****der****Mathematik****im**16.**Zeuthen**,**Die****Lehre****von****den****Kegelschnitten****im****Altertum**(**Copenhagen**, 1886**and**1902).**Zeuthen****that****at****most****eight****of****these****are****real**.**It****was****assumed****by****Plucker****that****the****number****of****real****double****tangents****might****be**28, 16, 8, 4**or****o**,**but****Zeuthen****has****found****that****the****last****case****does****not****exist**.**But****although****we****thus****arrive****by****projection****at****the****notion****of****a****circuit**,**it****is****not****necessary****to****go****out****of****the****plane**,**and****we****may**(**with****Zeuthen**,**using****the****shorter****term****circuit****for****his****complete****branch**)**define****a****circuit****as****any****portion**(**of****a****curve**)**capable****of****description****by****the****continuous****motion****of****a****point**,**it****being****understood****that****a****passage****through****infinity****is****permitted**.**It****was****thus****that****Zeuthen**(**in****the****paper****Nyt****Bydrag**, "**Contribution****to****the****Theory****of****Systems****of****Conics****which****satisfy****four****Conditions**" (**Copenhagen**, 1865),**translated****with****an****addition****in****the****Nouvelles****Annales**)**solved****the****question****of****finding****the****characteristics****of****the****systems****of****conics****which****satisfy****four****conditions****of****contact****with****a****given****curve****or****curves**;**and****this****led****to****the****solution****of****the****further****problem****of****finding****the****number****of****the****conics****which****satisfy****five****conditions****of****contact****with****a****given****curve****or****curves**(**Cayley**,**Comptes****Rendus**,**t**.**He****determines****in****every****case****the****characteristics**(µ,**v**)**of****the****corresponding****systems****of****cubics**(4p), (3**p**,**il**), &**c**.**The****same****problems**,**or****most****of****them**,**and****also****the****elementary****problems****in****regard****to****quartics****are****solved****by****Zeuthen**,**who****in****the****elaborate****memoir**"**Almindelige****Egenskaber**, &**c**.,"**Danish****Academy**,**t**.**The****methods****of****Maillard****and****Zeuthen****are****substantially****identical**;**in****each****case****the****question****considered****is****that****of****finding****the****characteristics****Cu**,**v**)**of****a****system****of****curves****by****consideration****of****the****special****or****degenerate****forms****of****the****curves****included****in****the****system**.**Zeuthen****in****the****case****of****curves****of****any****given****order****establishes****between****the****characteristics****pc**,**v**,**and**18**other****quantities**,**in****all**20**quantities**,**a****set****of**24**equations**(**equivalent****to**2 3**independent****equations**),**involving**(**besides****the**20**quantities**)**other****quantities****relating****to****the****various****forms****of****the****degenerate****curves**,**which****supplementary****terms****he****determines**,**partially****for****curves****of****any****order**,**but****completely****only****for****quartic****curves**.**Zeuthen**, "**Abzahlende****Methoden**,"**Bd**.**Zeuthen**,**Die****Lehre****von****dem****Kegelschnitten****in****Alterthum**(1886).