[4] +dazu gehörigen+: belonging to it. [5] +ganz beliebig ... gegebene Punkte+: points given entirely at pleasure = any given points whatever. [6] +der Centriwinkel+: angle at the centre. [7] +der Peripheriewinkel+: angle at the cirumference. [8] +inhaltsgleich+ equal in area; +der Inhalt+: contents. [9] +die Kathete+: leg of a right-angled triangle. [10] +Sei CAB ein ... Dreieck+: let CAB be a triangle. [11] +der ... Quadrate+; see 1 Note 10. [12] +es lässt sich zeigen+: it allowes itself to be shown = it may be shown. A common use of +lassen+. [13] +Hülfslinien+: auxiliary lines, construction lines. [14] see 1 Note 9. [15] +d. i. (das ist)+: that is to say (i. e.) 7. [1] +nach keiner Seite hin+: in no direction. [2] +wie weit ... auch+: however far. [3] +gleichlaufend+: German equivalent for parallel. In many cases German uses a foreign word and also a native word for the same term. [4, 5] +die ... Seiten+; see 1 Note 10. [6] +verhalten sich+: are to each other. [7] +Die ... Senkrechte+; see 1 Note 10. [8] +anliegend+: adjacent. [9] +es kommt nur darauf an+: it is only required. 8. [1] +Um um+: the first um means _in order_ and belongs to the infinitive +zu beschreiben+; the second +um+ means _around_. [2] +halbiere+: bisect. [3] +herumzutragen+: to lay off. [4] +eines Vielecks+; see 1 Note 10. [5] +die Fläche+: surface, area. [6] +körperliche+: solid. [7] +bis ins Unendliche+: to infinity. [8] +der Körper+: body; (in Geometry) solid. [9] +zwei ... Ebenen+; see 1 Note 10. [10] +entsteht+: is formed. [11] +die Grundfläche+: base (surface). [12] +und zwar+: and moreover. 9. [1] +inhaltsgleich+: of equal (cubic) contents = of equal volume. [2] +die Mantelfläche+: curved surface. [3] +dergestalt+: in such a manner. [4] +Ein ... Abschnitt+; see 1 Note 10. [5] +Wie verhalten sich+: in what ratio are ... to each other. [6] +einem ... Denkmale+; see 1 Note 10. 10. [1] +im Wesentlichen+: essentially. [2] +allseitig+: on all sides. [3] +die Einheit+: unit. [4] +das ... Normalmeter+; see 1 Note 10. [5] +Allgemeine Eigenschaften+: universal properties. [6] +starr+: rigid, solid. [7] +geläufig+: familiar. [8] +selbstverständlich+: of course. 11. [1] +das Beharrungsvermögen+: capacity for persisting = inertia. [2] +sich gerade befindet+: just finds itself = just happens to be. [3] +das Kausalgesetz+: law of causation. [4] +ein ... Körper+; see 1 Note 10. [5] +die Festigkeit+: firmness = strength. [6] +reicht ... hin (hinreichen)+: suffices. [7] +der Annahme widersetzt+: opposes the assumption. [8] +infolgedessen (in Folge dessen)+: in consequence of that. [9] +wegschnellen+: to fillip. [10] +Beispiele+ is the object of +bieten+, and +für+ governs +Drehungsbewegungen+. [11] +entsprechend+: corresponding, appropriate. 12. [1] +äussert sich+: manifests itself. [2] see 1 Note 10. [3] +die Unterlage+: support. [4] +übt ... aus (ausüben)+: exerts. [5] +also+; see 2 Note 2.