in the demonstration we have the angles ABC and ABE equal to two right angles, and also the angles ABC and ABD equal to two right angles; and then the former two right angles are equal to the latter two right angles by the aid of the eleventh axiom. Many modern editions of Euclid however refer only to the first axiom, as if that alone were sufficient; a similar remark applies to the demonstrations of I. 15, and I. 24. In these cases we have omitted the reference purposely, in order to avoid perplexing a beginner; but when his attention is thus drawn to the circumstance he will see that the first and eleventh axioms are both used.
We may observe that errors, in the references with respect to the eleventh axiom, occur in other places in many modern editions of Euclid, Thus for example in III. 1, at the step "therefore the angle FDB is equal to the angle GDBB," a reference is given to the first axiom instead of to the eleventh.
There seems no objection on Euclid's principles to the following demonstration of his eleventh axiom.
Let AB be at right angles to DAC at the point A, and EF at right angles to HEG at the point E: then shall the angles BAC and FEG be equal.
Take any length AC, and make AD, EH, HG all equal to AC. Now apply HEG to DAC, so that H may be on D, and HG on DC, and B and F on the same side of DC; then G will coincide with C, and E with A. Also EF shall coincide with AB; for if not, suppose, if possible, that it takes a different position as AK. Then the angle DAK is equal to the angle HEF, and the angle CAK to the angle GEF; but the angles HEF and GEF are equal, by hypothesis; therefore the angles DAK and CAK are equaL But the angles DAB and CAB are also equal, by hypothesis; and the angle CAB is greater than the angle CAK; there-
