By John Frank Adams

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Or shorter: Any rectilinear figure can be squared. 42-45, which will find their sequels in Book II. Prop. 42. To construct, in a given rectilineal angle, a parallelogram equal to a given triangle. The construction is easy enough, compare Fig. ABC and the angle 0 are given and E is the midpoint of BC. 17 40 4. 18 The "given angle" will be a right angle in Euclid's subsequent applications. So we might as well specialize it to this case in the next propositions. ) We replace "parallelogram" by "rectangle" and "given angle" by "right angle" in Euclid's Props.

6. If in a triangle two angles are equal to one another, the sides which subtend the equal angles will also be equal to one another. 5, Euclid first constructs two auxiliary triangles BFC and CGB [Fig. 3]: Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less; and let the straight lines FC, GB be joined. CGB: 24 4. AGB, and especially BG = CF and LBFC = LCGB. BFC by SAS. Now Euclid concludes: Therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG.

9, supplied by E. Hartmann, the point F will be in the southern hemisphere; hence (x' > ~. It should, however, be clear that any Greek mathematician would reply to this objection that he was dealing with plane, not spherical, geometry. ), who wrote about spherical geometry, knew the phenomenon. It seems that nobody noted the error before the end of the nineteenth century, when non-Euclidean geometries and order-relations in geometry came to the attention of mathematicians. The likely reason for Euclid's neglect of questions about the ordering of points on a line (or betweenness) may be that he regarded it as a part of logic-or just took it for granted.