The greater angle of a Triangle is opposite to the greater side.
 Let ABC be a Triangle having the ∠ B > the angle C: then shall AC be > AB. For if not, it must be equal or less. It is not equal, for then the angle B would = the angle C. It is not less, for then the angle B would < the angle C. ∴ AC > AB. Q. E. D. |
If we now fold the figure along AD, then AB will fall along AC; and B will fall between A and C if we suppose that AB is the shorter of the two unequal Lines AB and AC. The line DB therefore takes the position DB' within the angle ADC. But the angle AB'D, which = the angle ABC, is exterior to the triangle DCB' and ∴ > the angle C.
Conversely, if the angle ADB < the angle ADC, the line DB will fall within the angle ADC, and ∴ B will fall between A and C; that is, AB will be less than AC. This always happens (see above) if the angle ABC > the angle C, for then the angle BDA < the angle ADC. Theorem. In every Triangle the greater side is opposite to the greater angle, and conversely, the greater angle is opposite to the greater side. |
Now, if you could get some schoolmaster—one who had no bias whatever in favour of Euclid or of Henrici—to teach these two columns (one containing 169, the other 282 words) to two ordinary boys of equal intelligence, or rather of equal stupidity, what result would you expect?
