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(Postulate)]"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250805","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250806","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250807","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250808","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250809","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250810","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250811","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250812","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250813","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250814","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250815","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250816","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250817","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250818","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250819","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250820","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250821","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250822","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250823","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250824","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250825","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250826","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250827","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250828","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250829","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250830","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250831","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250832","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250833","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250834","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250835","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250836","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250837","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250838","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250839","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250840","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250841","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250842","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250843","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250844","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250845","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250846","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250847","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250848","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250849","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250850","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250851","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250852","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250853","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250854","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250855","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250856","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250857"],"ranges":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255162","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255167","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255170","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255173","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255175","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255180","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255183","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255186","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255189","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255191","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255196","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255199","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255202","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255204","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255208","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255211","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255214","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255217","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255219","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255221","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255226","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255232","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255238","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255244","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255249","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255252","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255255","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255258","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255261","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255267","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255276","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255281","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255292","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255294","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255297","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255299","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255301"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255154","@type":"sc:Range","label":"Die Gerade Linie, Das Dreyeck Und Das Viereck."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250805","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250806"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255162","@type":"sc:Range","label":"Lehrsatz 1. Alle rechte Winkel sind einander gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250807"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255167","@type":"sc:Range","label":"Lehrsatz 2. Jede grade Linie CD, welche mit einer andern AB zusammentrifft, bildet mit dieser Linie zwey Nebenwinkel ACD, BCD, * deren Summe zwey rechten Winkeln gleich ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250808"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255170","@type":"sc:Range","label":"Lehrsatz 3. Zwey grade Linien, welche zwey Punkte A, F mit einander gemein haben, fallen in ihrer ganzen Ausdehnung zusammen, und bilden nur eine einzige grade Linie."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250809","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250810"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255175","@type":"sc:Range","label":"Lehrsatz 5. Wenn zwey grade Linien AB, DE einander schneiden, so sind die Winkel, welche am Durchschnittspunkt einander gegen\u00fcberstehen, und die man Scheitelwinkel nennt, einander gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250811"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255180","@type":"sc:Range","label":"Lehrsatz 6. Zwey Dreyecke decken sich, wenn ein Winkel die beyden ihn einschliessenden Seiten in beyden Dreyecken gleich sind."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250812"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255183","@type":"sc:Range","label":"Lehrsatz 7. Zwey Dreyecke decken sich, wenn eine Seite und die beyden Winkel, welche an ihr liegen, in beyden Dreyecken gleich sind."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250813"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255186","@type":"sc:Range","label":"Lehrsatz 8. In einem Dreyecke ist jede Seite kleiner als die Summe der beyden andern Seiten."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250814","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250815","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250816"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255191","@type":"sc:Range","label":"Lehrsatz 10. Wenn zwey Seiten AB, AC eines Dreyecks ABC zweyen Seiten DE, DF eines andern Dreyecks DEF gleich sind, und der Winkel BAC den die erstern einschliessen, ist gr\u00f6\u00dfer als der Winkel EDF, den die andern einschliessen, so muss die dritte Seite BC, im ersten Dreyeck, gr\u00f6sser seyn als die dritte Seite EF im andern Dreyeck, [und ist umgekehrt BC>EF, so muss auch der Winkel BAC>EDF seyn.]"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250817"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255196","@type":"sc:Range","label":"Lehrsatz 11. Zwey Dreyecke die untereinander gleichseitig sind, sind auch untereinander gleichwinklig, * und decken sich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250818","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250819"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255199","@type":"sc:Range","label":"Lehrsatz 12. In jedem gleichschenkligen Dreyeck sind die Winkel an der Grundlinie *, welche den gleichen Seiten gegen\u00fcberstehn, gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250820"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255204","@type":"sc:Range","label":"Lehrsatz 14. Von zwey Seiten eines Dreyecks ist stets die die gr\u00f6ssere, welche einem gr\u00f6sseren Winkel gegen\u00fcbersteht. - Umgekehrt ist von zwey Winkeln eines Dreyecks stets der der gr\u00f6ssere, welcher einer gr\u00f6ssern Seite gegen\u00fcber steht."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250821"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255208","@type":"sc:Range","label":"Lehrsatz 15. Von einem Punkte A ausserhalb einer graden Linie DE, l\u00e4sst sich nach dieser Linie nur eine einzige senkrechte Linie ziehn."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250822","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250823","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250824","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250825","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250826"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255211","@type":"sc:Range","label":"Lehrsatz 16. Man denke sich von einem Punkte A nach einer graden Linie DE die senkrechte Linie AB, und mehrere schiefaufstehende grade Linien AE, AC, AD etc. gezogen, so ist:\r\n1) die senkrechte AB unter allen diesen Linien die k\u00fcrzeste.\r\n2) Unter den schiefstehenden Linien sind je zwey, z.B. AC AE, welche auf entgegengesetzten Seiten derselben...."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250827"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255214","@type":"sc:Range","label":"Lehrsatz 17. Es sey EF eine auf der graden Linie AB, in deren Mitte C, aufstehendes Perpendikel, so ist 1) jeder Punkt in diesem Perpendikel von den beyden Endpunkten der geraden Linie AB gleich weit entfernt; 2) jeder Punkt ausserhalb des Perpendikels hingegen von diesen Endpunkten ungleich weit entfernt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250828","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250829"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255217","@type":"sc:Range","label":"[Lehrsatz 18.] Zwey Dreyecke DEF, NBL decken sich, wenn in ihnen zwey Winkel und irgend eine Seite gleich sind."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250830","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250831"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255221","@type":"sc:Range","label":"[Lehrsatz 20.] Zwey schiefwinklige Dreyecke decken sich, wenn in ihnen zwey Seiten und einer der Winkel, welcher diesen Seiten gegen\u00fcbersteht, gleich sind, und dabey die zweyten gegen\u00fcberstehenden Winkel beyde spitz, oder beyde stumpf sind."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250832","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250833","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250834","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250835"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255232","@type":"sc:Range","label":"Lehrsatz 22. Wenn auf der geraden Linie AB, eine andere BD senkrecht und eine zweyte AC schief aufsteht, so dass der spitze Winkel BAC nach der Seite jenes Perpendikels zu liegt, so m\u00fcssen BD, AC genugsam verl\u00e4ngert [an der Seite der AB, auf welcher der spitze Winkel liegt] zusammen treffen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250836"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255238","@type":"sc:Range","label":"Lehrsatz 23. Wenn zwey grade Linien AC, BD mit einer dritten AB zwey innere Winkel* CAB, ABD bilden, deren Summe zwey rechte Winkeln gleich ist, so sind sie parallel."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250837","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250838"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255244","@type":"sc:Range","label":"Lehrsatz 24. Wenn zwey grade Linien AI, BD, mit einer dritten AB zwey innere Winkel BAI, ABD bilden, deren Summe kleiner als zwey rechte Winkel ist, so treffen sie genugsam verl\u00e4ngert zusammen, und zwar an der Seite der AB, an welcher die beyden innern Winkel, die kleiner als zwey rechte sind, liegen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250839","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250840"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255249","@type":"sc:Range","label":"Lehrsatz 25. Wenn zwey Parallellinien AB, CD von einer graden Linie EF geschnitten werden, so ist die Summe der beyden innern Winkel AGO, GOC zwey rechten Winkeln gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250841"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255252","@type":"sc:Range","label":"Lehrsatz 26. Zwey grade Linien AB, CD, welche mit einer dritten EF parallel sind, sind untereinander selbst parallel."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250842"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255255","@type":"sc:Range","label":"Lehrsatz 27. Zwey Parallellinien stehn \u00fcberall gleich weit von einander ab, d.h. Perpendikel, die von Punkten in der einen auf die andre gef\u00e4llt werden, sind \u00fcberall gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250843","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250844"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255258","@type":"sc:Range","label":"[Lehrsatz 28.] Zwey grade Linien in einer Ebne, welche nicht parallel sind, stehn \u00fcberall ungleich weit von einander ab, und zwar wird ihr Abstand nach der Seite zu, wo sie einander durchschneiden, immer kleiner, nach der entgegengesetzten immer gr\u00f6sser."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250845"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255267","@type":"sc:Range","label":"Lehrsatz 30. Wenn man die Seite CA eines Dreyecks verl\u00e4ngert, so ist der von der Verl\u00e4ngerung AD und der andern nicht verl\u00e4ngerten Seite AB eingeschlossne \u00e4ussere Winkel am Dreyeck BAD der Summe der beyden innern ihm entgegenstehenden Winkeln B und C gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250846"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255276","@type":"sc:Range","label":"Lehrsatz 31. Die drey Winkel eines Dreyecks sind zusammengenommen zwey rechten Winkeln gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250847","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250848"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255281","@type":"sc:Range","label":"Lehrsatz 32. Die Summe aller innern Winkel eines gradelinigen Vielecks betr\u00e4gt so vielmal zwey rechte Winkel, als das Vieleck Seiten, weniger zwey, hat."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250849","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250850"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255292","@type":"sc:Range","label":"[Lehrsatz 33. ] Wenn man auf den Schenkeln eines Winkels zwey Perpendikel BD, CE, errichtet, so durchschneiden sich diese unter einen Winkel G, welcher dem Winkel A gleich ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250851","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250852"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255294","@type":"sc:Range","label":"Lehrsatz 34. Jedes Parallelogramm wird 1) durch eine Diagonale in zwey sich deckende Dreyecke gehteilt; und 2) sind die sich gegen\u00fcberstehenden Seiten und Winkel derselben einander gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250853","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250854"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255297","@type":"sc:Range","label":"Lehrsatz 35. Umgekehrt ist jedes Viereck, worin die gegen\u00fcberstehenden Seiten [oder die gen\u00fcberstehenden Winkel] einander gleich sind, ein Parallelogramm."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250855","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250856","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250857"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255301","@type":"sc:Range","label":"Lehrsatz 37. Die beyen Diagonalen AC, BD eines Parallelogramms theilen einander wechselseitig in zwey gleiche Theile."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250858","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250859","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250860","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250861","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250862","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250863","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250864","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250865","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250866","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250867","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250868","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250869","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250870","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250871","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250872","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250873","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250874","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250875","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250876","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250877","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250878","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250879","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250880","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250881","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250882","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250883","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250884","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250885","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250886","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250887","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250888","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250889","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250890","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250891","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250892","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250893","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250894","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250895","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250896","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250897","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250898","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250899","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250900","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250901","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250902","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250903","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250904","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250905","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250906","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250907","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250908","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250909","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250910","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250911","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250912","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250913","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250914","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250915","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250916","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250917","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250918","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250919","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250920","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250921","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250922","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250923","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250924","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250925","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250926","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250927","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250928","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250929","https://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Buch. Der Kreis."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250858","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250859","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250860","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250861","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250862","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250863","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250864","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250865","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250866","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250867"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255310","@type":"sc:Range","label":"Erkl\u00e4rungen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250868","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250869"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255313","@type":"sc:Range","label":"[Lehrsatz 1.] 1) Zwey Kreislinien, welche mit gleichem Halbmesser beschrieben sind, decken sich, und schliessen Kreisscheiben von gleicher Gr\u00f6sse ein.\r\n2) Sind umgekehrt zwey Kreisscheiben gleich, so decken sie sich, und haben gleiche Kreislinien und gleiche Halbmesser."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250870"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255316","@type":"sc:Range","label":"Lehrsatz 2. Jeder Durchmesser, z.B. AB, theilt die Kreisscheibe und die Kreislinie in zwey sich deckende Theile."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250871"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255321","@type":"sc:Range","label":"[Lehrsatz 4.] Jede Sehne ED liegt ganz innerhalb, ihre Verl\u00e4ngerung ganz ausserhalb des Kreises."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250872"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255329","@type":"sc:Range","label":"Lehrsatz 5. Jede Sehne die nicht durch den Mittelpunkt geht, ist kleiner als der Durchmesser."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250873","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250874"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255335","@type":"sc:Range","label":"Lehrsatz 7. In einerley Kreise, oder in zwey gleichen Kreisen, geh\u00f6ren zu gleichen Bogen, gleiche Sehnen, und umgekehrt zu gleichen Sehnen, gleiche Bogen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250875"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255346","@type":"sc:Range","label":"Lehrsatz 8. So lange von Bogen die insgesammt kleiner als der Halbkreis sind die Rede ist, geh\u00f6rt in einerley Kreis oder in gleichen Kreisen, zum gr\u00f6ssern Bogen eine gr\u00f6ssere Sehne, [ein gr\u00f6sserer Winkel am Mittelpunkte und ein gr\u00f6sserer Kreisabschnitt,] und umgekehrt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250876","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250877"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255352","@type":"sc:Range","label":"Lehrsatz 9. Ein Halbmesser, welcher senkrecht auf eine Sehe steht, theilt die Sehne und ihren Bogen beyde in zwey gleiche Theile."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250878","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250879"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255358","@type":"sc:Range","label":"Lehrsatz 10. Durch drey gegebne Punkte A, B, C, welche nicht in grader Linie liegen, l\u00e4sst sich stets eine Kreislinie, und zwar nur eine einzige Kreislinie ziehen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250880"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255360","@type":"sc:Range","label":"Lehrsatz 11. 1) Gleiche Sehnen sind vom Mittelpunkt gleich weit entfernt, [und umgekehrt sind alle Sehnen, die gleich weit vom Mittelpunkte  abstehn, gleich.]\r\n2) Von zwey ungleichen Sehnen ist die Kleinere weiter als die Gr\u00f6ssere vom Mittelpunkte entfernt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250881","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250882","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250883","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250884","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250885","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250886"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255363","@type":"sc:Range","label":"Lehrsatz 12. Jede grade Linie welche auf einem Halbmesser in dessen Endpunkte senkrecht steht, ber\u00fchrt den Kreis, und durch jeden Punkt der Kreislinie ist nur eine einzige Tangente m\u00f6glich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250887","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250888"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255369","@type":"sc:Range","label":"Lehrsatz 13. Wenn zwey Parallellinien AB, DE, beyde den Kreis durchschneiden, so sind die Kreisbogen MN, PQ welche zwischen ihnen liegen, gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250889","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250890","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250891","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250892","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250893","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250894"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255380","@type":"sc:Range","label":"[Lehrsatz 14.] Sind umgekehrt zwey Bogen MN, PQ eines Kreises gleich, so sind 1) die Sehnen, welche durch die \u00fcbereinstimmenden Endpunkte beyder gehn, MP, NQ, parallel. - 2) Die Sehnen welche durch die verkehrt liegenden Endpunkte beyder gezogen sind, MQ, NP, durchschneiden sich so, dass die \u00e4hnlich liegenden Theile in beyden gleich sind. - 3) Auch wenn die Sehnen der beyden gleichen Bogen MN, PQ selbst, sich ausserhalb des Kreises schneiden, so sind die abgeschnittnen St\u00fccke in der Verl\u00e4ngerung ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250895","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250896","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250897"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255400","@type":"sc:Range","label":"[Lehrsatz 15.] Nimmt man ausserhalb oder innerhalb eines Kreises einen Punkt A und zieht durch ihn und den Mittelpunkt B eine grade Linie, welche die Kreislinie in den Punkten I und H durchschneidet; so hat 1) unter allen Punkten der Kreislinie der Punkt I, der in der Linie AH mit dem Punkte A zu einerley Seite des Mittelpunkts B liegt, den kleinsten; dagegen der Punkt H, der mit A auf entgegengesetzter Seite des Mittelpunktes B liegt, den gr\u00f6ssten Abstand vom Punkte A ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250898","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250899","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250900","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250901"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255403","@type":"sc:Range","label":"[Lehrsatz 16.] 1) Zwey Kreise deren Mittelpunkte A, B, um die Summe oder um den Unterschied ihrer Halbmesser \u03b1, \u03b2 von einander entfernt sind, ber\u00fchren sich und zwar im ersten Fall \u00e4usserlich im zweyten innerlich.\r\n2) Ist die Summe ihrer Halbmesser gr\u00f6sser oder der Unterschied derselben kleiner, so haben sie keinen Punkt miteinander gemein, und liegen im ersten Fall der eine ganz ausserhalb, im zweyten der eine ganz innerhalb des andern."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250902","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250903"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255487","@type":"sc:Range","label":"[Lehrsatz 17.] Wenn zwey um die Mittelpunkte C und O beschriebne Kreise sich innerlich oder \u00e4usserlich ber\u00fchren, so liegen, im ersten Fall die beyden \u00fcbereinstimmenden Endpunkte E, G, im zweyten die verkehrt liegenden Endpunkte E, G zweyer paralleler Durchmesser DE, FG, mit dem Ber\u00fchrungspunkte I in grader Linie."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250904"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255489","@type":"sc:Range","label":"[Lehrsatz 18.] Zwey Kreise k\u00f6nnen sich nur dann durchschneiden, wenn sowohl die Summe ihrer Halbmesser \u03b1, \u03b2 gr\u00f6sser, als auch der Unterschied derselben kleiner als der Abstand ihrer Mittelpunkte A, B ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250905"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255491","@type":"sc:Range","label":"[Lehrsatz 19.] Zwey Kreise die sich durchschneiden treffen sich stets in zwei Punkten, welche zu den entgegengesetzten Seiten der graden Linie durch beyde Mittelpunkte liegen. Umgekehrt durchschneiden sich alle Kreise, welche zwey Punkte gemein haben, oder die in einem Punkte ausserhalb der Linie durch ihre Mittelpunkte zusammen treffen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250906"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255497","@type":"sc:Range","label":"Lehrsatz 20. Wenn zwey Kreise sich schneiden, so wird ihre gemeinschaftliche Sehne von der graden Linie, die durch die beyden Mittelpunkte O, B geht, senkrecht durchschnitten und halbirt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250907","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250908"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255500","@type":"sc:Range","label":"Lehrsatz 21. Wenn in einerley Kreis oder in zwey gleichen Kreisen, zwey Winkel am Mittelpunkte ABC, DCE sich zu einander wie zwey ganze Zahlen verhalten; so m\u00fcssen auch die beyden Bogen welche von ihnen umspannt werden AB, DE sich wie dieselben Zahlen, und folglich wie jene Winkel verhalten."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250909","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250910","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250911","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250912","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250913","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250914","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250915"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255503","@type":"sc:Range","label":"Lehrsatz 22. Wie auch zwey Winkel ACB, ACD sich zu einander verhalten m\u00f6gen, immer verhalten sich auf dieselbe Art zwey Kreisb\u00f6gen AB, AD, welche um ihren Scheitelpunkt mit gleichem Halbmesser beschrieben und von ihren Schenkeln umspannt werden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250916","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250917","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250918","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250919","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250920","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250921"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255510","@type":"sc:Range","label":"Lehrsatz 23. Jeder einem Kreise eingeschriebene Winkel (d.h. jeder Winkel am Umfange) hat zu seinem Maasse den halben Bogen worauf er steht."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250922","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250923","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250924","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250925","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250926"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255519","@type":"sc:Range","label":"Lehrsatz 24. 1. Jeder Winkel BIE, den eine Tangente IB mit einer Sehne IE macht, welche die Tangente im Ber\u00fchrungspunkte durchschneidet, hat zu seinem Maasse die H\u00e4lfte des Bogens INE der von beyden Linien eingeschlossen wird. [2. Umgekehrt muss jede grade Linie IB, welche durch den Endpunkt einer Sehne IE gegen diese Sehne unter einem Winkel gezogen wird, den die H\u00e4lfte des von ihnen eingeschlossnen Bogens INE misst, den Kreis im Punkte I ber\u00fchren.]"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250927","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250928","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250929","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250930"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255522","@type":"sc:Range","label":"[Lehrsatz 25.] 1. Der Winkel AOD, unter welchem zwey Sehnen AB, DE eines Kreises sich durchschneiden, hat zu seinem Maass die halbe Summe der Bogen, welche seine Schenkel umspannen (AD+BE) : 2.\r\n2. Der Winkel, unter welchem zwey verl\u00e4ngerte Sehnen AO', BO', oder eine verl\u00e4ngerte Sehne..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250931","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250932","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250933","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250934"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255527","@type":"sc:Range","label":"[Lehrsatz 26.] Die Spitzen aller gleichen Winkel, welche \u00fcber derselbe graden Linie BC stehn, liegen insgesammt in einer Kreislinie, die durch die Endpunkte dieser graden Linie B, C geht."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250935","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250936"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255545","@type":"sc:Range","label":"Lehrsatz 27. 1. In jedem Viereck, welches in einem Kreise eingeschrieben ist, sind die gegen\u00fcberstehenden Winkel zusammengenommen zwey rechten Winkeln gleich.\r\n[2. Umgekehrt l\u00e4sst sich um ein Viereck, worin die Summe der gegen\u00fcberstehenden Winkel zwey rechten Winkeln gleich ist, allemal ein Kreis umschreiben.]"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250937","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250938"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255558","@type":"sc:Range","label":"[Lehrsatz 28.] Wenn man auf der Sehne eines Kreisabschnitts AEB ein Perpendikel ED errichtet, und nimmt auf der andern Seite des Perpendikels im Bogen einen Punkt G, und in der Sehneoder deren Verl\u00e4ngerung einen Punkt F, wovon jener von E, dieser von D, eben so weit als diese letztern Punkte selbst vom Punkte B abstehn, so sind G und F vom andern Endpunkte A der Sehne gleich weit entfernt; oder wenn EG = EB und DF = DB ist, so ist immer AG = AF."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250939"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255562","@type":"sc:Range","label":"[Lehrsatz 29.] Wenn eine grade Linie DE die Sehne eines Kreisabschnitts ABD unter einem Winkel AFD, welcher den Winkeln im Abschnitt gleich ist, durchschneidet, so steht der Durchmesser welcher durch den Punkt A geht, auf der durchschneidenden Linie DE senkrecht, und die Durchschnittspunkte D, E sind vom Punkte A gleich weit entfernt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250940","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250941","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250942","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250943","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250944","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250945","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250946"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255571","@type":"sc:Range","label":"[Lehrsatz 30.] Tr\u00e4gt man auf die Verl\u00e4ngerung einer Sehne AB, den Halbmesser des Kreises von B nach O auf, so schneiden die Sehne und der Durchmesser des Kreises, der verl\u00e4ngert durch O geht, von der Kreislinie zwey Bogen AD, BE' ab, wovon jener das Dreyfache dieses ist, oder Bog. AD = 3.Bog.BE'."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250948","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250949","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250950","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250951","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250952","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250953","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250954","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250955","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250956","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250957","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250958","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250959","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250960","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250961","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250962","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250963","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250964","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250965","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250966","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250967","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250968","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250969","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250970","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250971","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250972","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250973","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250974","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250975","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250976","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250977","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250978","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250979","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250980","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250981","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250982","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250983","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250984","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250985","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250986","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250987","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250988","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250989"],"ranges":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255573","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255576","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255579","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255581","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256930","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256933","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3263636","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256936","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256938","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256942","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256945","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256948","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256953","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256960","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256967","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256980","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256992","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257002","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257007","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257010"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3250947","@type":"sc:Range","label":"Anhang. Aufgaben welche zu den beyden ersten B\u00fcchern geh\u00f6ren."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250949"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255573","@type":"sc:Range","label":"Aufgabe 1. Eine gegebne grade Linie AB in zwey gleiche Theile zu theilen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250950"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255576","@type":"sc:Range","label":"Aufgabe 2. Auf eine grade Linie, durch einen in ihr gegebnen Punkt A, eine grade Linie senkrecht zu ziehn: oder an einem gegebnen Punkt A einer graden Linie BC einen rechten Winkel zu bilden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250951","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250952"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255579","@type":"sc:Range","label":"Aufgabe 3. Von einem ausserhalb einer graden Linie gegebnen Punkte A, ein Perpendikel auf diese Linie zu f\u00e4llen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250953"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3255581","@type":"sc:Range","label":"Aufgabe 4. 1. An einem Punkte A der graden Linie AB einen Winkel zu bilden, welcher einem gegebnen Winkel K gleich ist.\r\n[2. Durch einen Punkt R ausserhalb AB, nach dieser Linie eine grade Linie so zu ziehn, dass sie die AB unter einem gegebnen Winkel K durchschneide."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250954"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256930","@type":"sc:Range","label":"Aufgabe 5. Einen gegebnen Kreisbogen oder einen gegebnen Winkel in zwey gleiche Theile zu theilen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250955","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250956"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256933","@type":"sc:Range","label":"Aufgabe 6. Durch einen Punkt A, der ausserhalb einer graden Linie BC gegeben ist, mit dieser graden Linie eine Parallellinie zu ziehn."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250957"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3263636","@type":"sc:Range","label":"Aufgabe 7. Aus zwey gegebnen Winkeln A und B eines Dreyecks, den dritten Winkel zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250958","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250959"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256942","@type":"sc:Range","label":"Aufgabe 10. Wenn zwey Seiten A und B der der Seite B gegen\u00fcberstehende Winkel C eines Dreyecks gegeben sind, das Dreyeck zu beschreiben."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250960"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256945","@type":"sc:Range","label":"Aufgabe 11. Wenn ein Winkel C und zwey ihn einschliessende Seiten A und B eines Parallelogramms gegeben sind, das Parallelogramm zu beschreiben."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250961","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250962"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256948","@type":"sc:Range","label":"Aufgabe 12. Den Mittelpunkt eines gegebnen Kreises, oder eines gegebnen Kreisbogens zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250963","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250964"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256953","@type":"sc:Range","label":"[Aufgabe 13.] Um einen gegebnen Mittelpunkt einen Kreis zu beschreiben, der eine gegebne grade Linie oder einen gegebnen Kreis ber\u00fchrt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250965"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256960","@type":"sc:Range","label":"Aufgabe 14. Durch einen gegebnen Punkt A eine Tangente an einen gegebnen Kreis zu ziehn."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250966","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250967","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250968"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256967","@type":"sc:Range","label":"[Aufgabe 15.] In einem gegebnen Kreise eine Sehne einzutragen, welche einer gegebnen Linie MN (kleiner als der Durchmesser) gleich ist, und 1) durch einen gegebnen Punkt P geht, oder 2) einer gegebnen graden Linie Q parallel l\u00e4uft."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250969","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250970"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256980","@type":"sc:Range","label":"Aufgabe 16. Ueber eine gegebne grade Linie AB einen Kreisabschnitt zu beschreiben, welcher einen gegebnen Winkel C fasst, (d.h. wo jeder in diesen Kreisabschnitt eingeschriebne Winkel, dem Winkel C gleich ist.*)"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250971","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250972"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3256992","@type":"sc:Range","label":"[Aufgabe 17.] Ein Dreyeck, welches mit einem gegebnen Dreyeck PQR gleichwinklig ist, 1) in einem gegebnen Kreis einzuschreiben, und 2) um einen gegebnen Kreis zu umschreiben."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250973","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250974","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250975","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250976"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257002","@type":"sc:Range","label":"Aufgabe 18. Einen Kreis, 1) in ein gegebnes Dreyeck ABC einzuschreiben; 2) um ein gegebnes Dreyeck zu umschreiben."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250977","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250978","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250979","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250980","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250981","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250982","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250983","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250984","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250985","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250986"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257007","@type":"sc:Range","label":"Aufgabe 19. Das Verh\u00e4ltnis zweyer grader Linien AB, CD, welche gegeben sind, in Zahlen auszudr\u00fccken, oder das Zahlverh\u00e4ltnis dieser Linien zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250987","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250988","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250989"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257010","@type":"sc:Range","label":"Aufgabe 20. Wenn zwey Winkel A, B gegeben sind, ihr gemeinschaftliches Maass, und daraus ihr Zahlverh\u00e4ltniss zu 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Buch. Der Inhalt Gradeliniger Figuren."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250992","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250993","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250994","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250995","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250996","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250997","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250998","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3250999","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251000","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251001"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257015","@type":"sc:Range","label":"Erkl\u00e4rungen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251002","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251003"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257021","@type":"sc:Range","label":"Lehrsatz 1. Parallelogramme von gleicher Grundlinie und gleicher H\u00f6he haben gleichen Inhalt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251004"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257028","@type":"sc:Range","label":"Lehrsatz 2. Der Inhalt eines Dreyecks ABC, ist halb so gross als der Inhalt eines Parallelogramms, welches mit dem Dreyecke gleiche Grundlinie und gleiche H\u00f6he hat."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251005","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251006","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251007"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257031","@type":"sc:Range","label":"Lehrsatz 3. Die Fl\u00e4chenr\u00e4ume zweyer Rechtecke von gleicher H\u00f6he, verhalten sich zu einander wie ihre Grundlinien."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251008","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251009","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251010","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251011","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251012","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251013","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251014","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251015","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251016","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251017","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251018","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251019","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251020"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257036","@type":"sc:Range","label":"Lehrsatz 4. Die Fl\u00e4chenr\u00e4ume zweyer Rechtecke ABCD, AEGF verhalten sich zu einander wie die Produkte aus der Grundlinie eines jeden in dessen H\u00f6he, oder es ist allgemein ABCD : AEGF = AB x AD : AE x AF."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251021","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251022"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257041","@type":"sc:Range","label":"Lehrsatz 5. Der Fl\u00e4chenraum eines Parallelogramms wird durch das Produkt aus der Grundlinie in die H\u00f6he gemessen; der Fl\u00e4chenraum eines Dreyecks durch das halbe Produkt aus der Grundlinie in die H\u00f6he."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251023","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251024","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251025"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257050","@type":"sc:Range","label":"Lehrsatz 6. Der Inhalt eines Trapezoid ABCD wird durch das Produkt aus der H\u00f6he EF in die halbe Summe der parallelen Grundlinien desselben, AB, CD* gemessen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251026","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251027","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251028","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251029","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251030","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251031"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257419","@type":"sc:Range","label":"[Lehrsatz 7.] 1. Jede grade Linie, welche, wie DE, durch ein Dreyeck ABC mit einer Seite desselben, z.B. mit BC, parallel gezogen ist, theilt die beyden andern Seiten des Dreyecks in proportionale Theile, so dass sich verh\u00e4lt AD : DB : AB = AE : EC : AC.\r\n2. Sind umgekehrt zwey Seiten AB, Ac eines Dreyecks in den Punkten ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251032","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251033","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251034","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251035","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251036"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257422","@type":"sc:Range","label":"[Lehrsatz 8.] Zwey grade Linien FH, GI, welche man durch einen Punkt E in der Diagonale eines Parallelogramms ABCD, mit den Seiten parellel zieht, theilen 1) die Fl\u00e4chen in vier kleinere Parallelogramme, welche unter sich, und mit dem Gegebnen, gleichwinklig und proportional sind, und 2) die Seiten in zwey proportionale Abschnitte.                                 3) Die parallelen Seiten der Parallelogramme um die Diagonale, GF, HI, AC, stehn in gleichem Verh\u00e4ltniss, und die Erg\u00e4nzungen dieser ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251037","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251038","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251039"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257426","@type":"sc:Range","label":"Lehrsatz 9. Ein Quadrat aus einer zweytheiligen Linie AC ist den Quadraten \u00fcber den beyden Abschnitten AB, BC, und zwey Rechtecken, welche aus den beyden Abschnitten beschrieben sind, zusammengenommen gleich; oder es ist AC\u00b2 d.h. (AB+BC)\u00b2 = AB\u00b2+BC\u00b2+2ABxBC."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251040","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251041","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251042","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251043","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251044"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257750","@type":"sc:Range","label":"Lehrsatz 11. Ein Rechteck aus der Summe und dem Unterschiede zweyer Linien AB, BC beschrieben, ist dem Unterschiede der Quadrate aus beyden Linien gleich, oder (AB+BC)x(AB-BC)=AB\u00b2-BC\u00b2."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251045","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251046","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251047","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251048","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251049","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251050"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257748","@type":"sc:Range","label":"Lehrsatz 12. Das Quadrat der Hypotenuse BC eines rechtwinkligen Dreyecks ABC ist gleich den Quadraten \u00fcber den beyden Katheten zusammen genommen, oder B\u00b2=AB\u00b2+AC\u00b2."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251051","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251052","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251053","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251054","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251055","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251056","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251057","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251058","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251059","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251060"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257764","@type":"sc:Range","label":"[Lehrsatz 13.] In jedem schiefwinkligen Dreyeck ist das Quadrat einer Seite BC, welche einem spitzen Winkel A gegen\u00fcber steht, kleiner, dagegen das Quadrat einer Seite bc, welche einem stumpfen Winkel a gegen\u00fcber steht, gr\u00f6sser als die Summe der Quadrate der beyden andern Seiten. Und zwar, wenn man von einem der Endpunkte dieser Seite, z.B. aus B, auf die gegen\u00fcberstehende Seite AC, oder deren Verl\u00e4ngerung, ein Perpendikel f\u00e4llt; so ist das doppelte Rechteck aus AC und dem Abschnitte dieser Seite ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251061"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257769","@type":"sc:Range","label":"[Lehrsatz 14.] Ein Dreyeck ABC ist bey A rechtwinklig, spitzwinklig oder stumpfwinklig, je nachdem das Quadrat der Seite BC, welche diesem Winkel gegen\u00fcbersteht, den Quadraten der beyden Seiten AB, AC, welche den Winkel A einschliessen, zusammengenommen gleich ist, oder kleiner, oder gr\u00f6sser ist, als diese beyden Quadrate."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251062","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251063","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251064","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251065","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251066","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251067"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257781","@type":"sc:Range","label":"[Lehrsatz 15. ] Wenn man \u00fcber zwey Seiten AB, AC eines Dreyecks ABC zwey Parallelogramme ABDE, ACFG, unter beliebigen Winkeln, und von beliebiger Gr\u00f6sse und Lage beschreibt, die Seiten derselben, welche den Seiten des Dreyecks gegen\u00fcberstehn, bis zu ihrem Durchschnittspunkte H verl\u00e4ngert, und durch diesen und die Spitze des Dreyecks die grade Linie HAK zieht: so ist, wenn diese Linien die Grundlinie BC selbst schneidet, die Summe; wenn sie hingegen die Verl\u00e4ngerung der Grundlinie..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251068","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251069","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251070"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257783","@type":"sc:Range","label":"[Lehrsatz 16.]  1. Ein Perpendikel, welches aus einem der Eckpunkte eines Dreyecks, z.B aus A, auf die gegen\u00fcberstehende Seite BC, oder deren Verl\u00e4ngerung gef\u00e4llt wird, schneidet diese so, dass der Unterschied der Quadrate aus beyden Abschnitten BD, DC, dem Unterschiede der Quadrate aus den beyden andern Seiten des Dreyecks, AB, AC, dem Unterschiede der Quadrate aus den beyden andern Seiten des Dreyecks, AB, AC, gleich ist (BD\u00b2-DC\u00b2=AB\u00b2-AC\u00b2;) und zwar liegt..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251071","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251072","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251073","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251074","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251075","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251076"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257786","@type":"sc:Range","label":"[Lehrsatz 17.] 1. Wenn man in einem Dreyeck ABC von einem der Winkelpunkte, z.B. von A, eine grade Linie AO nach dem Punkte O in der Mitte der gegen\u00fcberstehenden Seite BC zeiht, so ist allemal AB\u00b2+AC\u00b2=2AO\u00b2+2OC\u00b2."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251077"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257789","@type":"sc:Range","label":"Lehrsatz 18. In jedem Parallelogramm ABCD ist die Summe der Quadrate aus allen Seiten, den Quadraten der beyden Diagonalen AC, BD zusammen gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251078","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251079"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257802","@type":"sc:Range","label":"[Lehrsatz 19.] In jedem Trapez ABCD \u00fcbertrifft die Summe der Quadrate aller Seiten, die beyden Quadrate der Diagonalen AC, BD zusammen genommen; und zwar um ein Quadrat, welches man erh\u00e4lt, wenn man \u00fcber zwey aneinander liegende Seiten des Trapezes  ein Parallelogramm ABCE errichtet, den Abstand der beyden Eckpunkte D des Trapezes und E des Parallelogramms, die beyden nicht gemein sind, nimmt, und \u00fcber diesen Abstand DE als Seite, ein Quadrat beschreibt ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251080","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251081","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251082","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251083","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251084","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251085","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251086","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251087","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251088","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251089","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251090","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251091","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251092","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251093","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251094","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251095","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251096","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251097","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251098","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251099","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251100","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251101","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251102","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251103","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251104","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251105","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251106","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251107","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251108","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251109","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251110","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251111","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251112","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251113","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251114","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251115","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251116","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251117","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251118","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251119","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251120"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257805","@type":"sc:Range","label":"[Lehrsatz 20.] Wenn man aus einem der Winkelpunkte eines Dreyecks ABC, z.B. aus A, nach irgend einem Punkte G in der gegen\u00fcberstehenden Seite BC, oder in deren Verl\u00e4ngerung, eine grade Linie AG zieht, so ist allemal [...]"},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251121","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251122","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251123"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257808","@type":"sc:Range","label":"[Lehrsatz 21.] Zieht man aus der Spitze A eines gleichschenkligen Dreyecks ABC nach irgend einem Punkte G in der Grundlinie, oder nach einem Punkte g in deren Verl\u00e4ngerung, eine grade Linie, so ist stets der Unterschied der Quadrate aus dieser Linie und aus einem der gleichen Schenkel, gleich dem Rechteck aus den Abschnitten auf der Grundlinie, BG, CG, oder auf der verl\u00e4ngerten Grundlinie, Gb, Cg."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251124","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251125","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251126","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251127","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251128","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251129","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251130","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251131","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251132","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251133","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251134","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251135","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251136","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251137","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251138"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257811","@type":"sc:Range","label":"[Lehrsatz 22.] 1) Wenn mehrere Sehnen insgesammt durch einen Punkt O im Kreise gehn, so sind die Rechtecke aus den beyden St\u00fccken, welche auf jeder Sehne durch diesen Punkt abgeschnitten werden, sowohl untereinander, als auch mit dem Quadrate der halben Sehne gleich, welche mit dem Punkte O gleich weit vom Mittelpunkte absteht, z.B. AOxOB=DOxOE=OF\u00b2. ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251139","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251140","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251141"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257814","@type":"sc:Range","label":"[Lehrsatz 23.] Wenn zwey Sehnen sich unter rechten Winkeln durchschneiden, so sind stets die Quadrate aus den vier Abschnitten zusammengenommen dem Quadrat des Durchmessers gleich."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251142","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251143","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251144","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251145","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251146","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251147","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251148"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257816","@type":"sc:Range","label":"[Lehrsatz 24.] Wenn man auf dem Durchmesser AB eines Kreises, oder auf dessen Verl\u00e4ngerung, in einem willk\u00fcrlichen Punkte D oder d ein Perpendikel errichtet, und zieht von dem einen Endpunkte des Durchmessers z.B. von A aus, grade Linien, welche die Kreislinie in Punkten F, das Perpendikel in Punkten G oder g durchschneiden, so sind die Rechtecke aus je zwey Abschnitten AF und AG gleich dem Rechtecke aus dem Durchmesser AB und dem Abschnitt AD desselben, der an dem Punkte A anliegt, und jene Rechtecke ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251149","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251150","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251151","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251152","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251153","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251154","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251155","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251156","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251157","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251158","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251159","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251160","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251161","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251162"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257819","@type":"sc:Range","label":"[Lehrsatz 25.] Wenn man durch einen beliebigen Punkt M einer Sehne AH, oder durch irgend einen Punkt m in deren Verl\u00e4ngerung, eine grade Linie MD oder md parallel mit der Tangente AK, welche durch den einen Endpunkt A dieser Sehne geht, zieht, so werden auf jeder andern graden Linie, welche durch den Punkt A geht, und daher den Kreis in einem zweyten Punkte F, die Linie MD oder md hingegen in Punkten G oder g durchschneidet*, zwey St\u00fccke AF und AG oder Ag so abgeschnitten, dass ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251163","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251164","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251165","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251166","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251167","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251168","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251169","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251170","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251171"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257821","@type":"sc:Range","label":"[Lehrsatz 26.] 1) Wenn ein Kreis, mithin auch dessen Mittelpunkt C, und irgend ein zweyter Punkt A gegeben sind, gleich viel ob A innerhalb, oder ausserhalb, oder auf der Kreislinie liegt, und wenn grade Linien, vom Punkte A aus gezogen, sich mit den Sehnen BD dieses Kreises je zwey so in Punkten e durchschneiden, 1) dass entweder Ae\u00b2=BexDe, oder wenn S einen gegebnen Fl\u00e4chenraum bedeutet, Ae\u00b2=BexDe+S ist: und irgend einer dieser Punkte e liegt ausserhalb des Kreises, auf der Verl\u00e4ngerung einer Sehne ..."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251173","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251174","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251175","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251176","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251177","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251178","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251179","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251180","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251181","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251182","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251183","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251184","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251185","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251186","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251187","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251188","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251189","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251190","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251191","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251192","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251193","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251194","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251195","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251196","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251197","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251198","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251199","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251200","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251201","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251202","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251203","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251204","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251205","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251206","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251207","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251208","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251209","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251210","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251211","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251212","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251213","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251214","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251215","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251216","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251217","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251218"],"ranges":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257824","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257827","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257830","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257833","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257836","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257838","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257841","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259191","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259194","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259197","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259200","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259203","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259206","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259209","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259211","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259214","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259217","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259220","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259223","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259226","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259229","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259232"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3251172","@type":"sc:Range","label":"Aufgaben welche zum dritten Buche geh\u00f6ren."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251173","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251174"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257824","@type":"sc:Range","label":"Aufgabe 1. Ein gegebnes Vieleck in ein Dreyeck von gleichem Inhalt zu verwandlen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251175"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257827","@type":"sc:Range","label":"Aufgabe 2. Ein Parallelogramm unter einem gegebnen Winkel zu bilden, welches mit einem gegebnen Parallelogramm, oder Trapezoid, oder Dreyeck, gleichen Inhalt hat."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251176","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251177","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251178","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251179"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257830","@type":"sc:Range","label":"Aufgabe 3. Ueber eine gegebne grade Linie MN ein Parallelogramm zu beschreiben, welches mit einem gegebnen Parallelogramm ABCD gleichen Inhalt hat, und unter gleichen Winkeln enthalten ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251180","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251181","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251182","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251183"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257833","@type":"sc:Range","label":"Aufgabe 4. 1) Eine gegebne gerade Linie AB in beliebig viel gleiche Theile zu theilen.\r\n2) Eine gegebne grade Linie AB, entweder mehrerern gegebnen Linien P, Q, R, oder einer gegebnen eingetheilten Linie MN, proportional einzutheilen. "},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251184","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251185","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251186","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251187"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257836","@type":"sc:Range","label":"Aufgabe 5. 1) Eine gegebne grade Linie BC, nach einem gegebnen Zahlverh\u00e4ltnisse m:n. einzutheilen.\r\n2) Auf der Verl\u00e4ngerung einer gegebnen graden Linie BC so einen Punkt g zu bestimmen, dass die beyden Abschnitte Bg, Cg, in dem gegebnen Zahlverh\u00e4ltnisse m:n stehn."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251188"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257838","@type":"sc:Range","label":"Aufgabe 6. Wenn ein Winkel A, und ein Punkt B gegeben sind, durch diesen Punkt eine grade Linie so zu ziehn, dass durch die Schenkel des Winkels auf ihr zwey St\u00fccke BC, BD abgeschnitten werden, die in einem gegebnen Verh\u00e4ltniss stehn, sich nemlich wie die gegebnen Linien P:Q verhalten."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251189","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251190","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251191"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3257841","@type":"sc:Range","label":"Aufgabe 7. Zu drey gegebnen Linien P, Q, R, die vierte Proportionallinie zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251192","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251193"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259191","@type":"sc:Range","label":"Aufgabe 8. Ein gegebnes Parallelogramm, oder ein gegebnes Dreyeck oder ein gegebnes Trapezoid, in eine Figur von einer dieser drey Gattungen zu verwandeln, welche mit der gegebnen gleichen Inhalt hat, und entweder \u00fcber einer gegebnen Grundlinie MN steht, oder eine gegebne H\u00f6he hat."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251194","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251195","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251196"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259194","@type":"sc:Range","label":"Aufgabe 9. Zu zwey gegebnen Linien P, Q, als \u00e4ussere Glieder einer Proportion, zwey andre Linien Y, X, deren Unterschied oder deren Summe einer gegebnen Linie N gleich ist, als mittlere Glieder der Proportion zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251197"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259197","@type":"sc:Range","label":"Aufgabe 10. Wenn zwey Rechtecke, aus den Linien A, B und P, Q, gegeben sind, eine Linie zu finden, zu welcher sich eine Seite des einen Rechtecks, wie der Inhalt beyder Rechtecke zu einander verh\u00e4lt."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251198"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259200","@type":"sc:Range","label":"Aufgabe 11. Zwey Linien darzustellen, deren Verh\u00e4ltniss aus den gegebnen Verh\u00e4ltnissen dreyer Paar Linien A, B, C und P, Q, R zusammengesetzt ist, oder die sich wie die Produkte dreyer Linien, zu den Produkten dreyer andrer Linien* verhalten."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251199","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251200","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251201"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259203","@type":"sc:Range","label":"Aufgabe 12. Zu zwey gegebnen Linien P, Q die dritte Proportionallinie zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251202","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251203","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251204","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251205"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259206","@type":"sc:Range","label":"Aufgabe 13. 1) Zwischen zwey gegebnen graden Linien P und Q eine mittlere Proportionallinie zu finden, und 2) zwischen einer graden Linie AB und einem gegebnen Abschnitt derselben AD, eine mittlere Proportionallinie darzustellen."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251206","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251207"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259209","@type":"sc:Range","label":"Aufgabe 14. Eine gegebne gradelinige Figur in ein Quadrat zu verwandeln."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251208"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259211","@type":"sc:Range","label":"Aufgabe 15. Ein Quadrat zu bilden, welches der Summe zweyer oder mehrerer gegebner Quadrate gleich ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251209","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251210"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259214","@type":"sc:Range","label":"Aufgabe 16. Ein Quadrat zu bilden, welches dem Unterschiede zweyer oder mehrerer gegebnen Quadrate gleich ist."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251211"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259217","@type":"sc:Range","label":"Aufgabe 17. Ein Quadrat zu beschreiben, zu welchem ein gegebnes Quadrat M\u00b2, in dem Verh\u00e4ltnisse zweyer gegebnen Linien P, Q steht."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251212"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259220","@type":"sc:Range","label":"Aufgabe 18. Eine gegebne grade Linie AB so zu verl\u00e4ngern, dass das Rechteck aus der verl\u00e4ngerten Linie AF, und der Verl\u00e4ngerung BF, dem Rechteck aus zwey gegebnen Linien P und Q gleich, oder AFxBF=PxQ sey."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251213"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259223","@type":"sc:Range","label":"Aufgabe 19. Eine gegebne grade Linie BH, so in zwey Theile BF, FH einzutheilen, dass das Quadrat des einen Theils, gleich sey dem Rechteck aus dem andern Theile und einer gegebnen Linie P, oder BF\u00b2=PxFH."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251214"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259226","@type":"sc:Range","label":"Aufgabe 20. Eine grade Linie AB, in welcher ein Punkt E gegeben ist, aufs neue so in einem Punkte D einzutheilen, dass AD\u00b2=EDxDB sey."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251215"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259229","@type":"sc:Range","label":"Aufgabe 21. Von einem gegebnen Punkte A ausserhalb eines Kreises eine grade Linie so zu ziehn, dass sie von der Kreislinie, zweyen gegebnen Linien P, Q proportional eingetheilt werde."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251216","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251217","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251218"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3259232","@type":"sc:Range","label":"Aufgabe 22. Das Verh\u00e4ltniss zwischen der Seite AB und der Diagonale AC eines Quadrats zu finden."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251220","https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251221"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3251219","@type":"sc:Range","label":"Erkl\u00e4rung der Marginalien."},{"canvases":["https://digital.ub.uni-paderborn.de/i3f/v20/3250002/canvas/3251223"],"@id":"https://digital.ub.uni-paderborn.de/i3f/v20/3250002/range/3251222","@type":"sc:Range","label":"Einige 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