Plane Geometry - Bruce E. Shapiro

32 SECTION 7. EUCLID’S ELEMENTSFigure 7.1: Illustration of Euclid’s Parallel postulate. The two angles αand β add to les than 180 degrees, hence the lines h and k meet at a pointS on the same same side of g as α and β.Euclid’s Common Notions1 Things equal to the same thing are also equal to one anotherWe might write this today as: If a = b and b = c then a = c.2 And if equal things are added to equal things then the wholes are equal.We might write this as: if a = c then a + b = c + b.3 And if equal things are subtracted from equal things then the remainders areequal.Which we might write this as: if a = c then a − b = c − b.4 And things coinciding with one another are equal to one another.By this Euclid meant he could imagine picking up the “picture” of a triangle(or some other object) and lay it on top of another; if they were the same thenthey were considered “equal” (or maybe congruent).5 And the whole [is] greater than the partWhich we might write as: if a > 0 and b > 0 then a + b > a and a + b > b.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012