All right angles are equal to each other.

If the present petition is considered by us as manifest, and as requiring no demonstration, it is not a petition according to the opinion of Geminus, but an axiom; for it affirms a certain essential accident of right angles, not commanding us to perform any thing according to a simple conception. But neither is it a petition according to the division of Aristotle: for petition, according to his opinion, requires some demonstration. But if we should say it is demonstrable, and enquire after its demonstration, yet according to the opinion of Geminus, it ought not to be placed among petitions. The equality, therefore, of right angles, appears from our common conceptions; for since a right angle has the relation of unity or bound to the infinite increase and decrease of the angles on each side, it is equal with respect to every right angle, since we constitute the first right angle after this manner, by a right line making angles on each side of the right line on which it stands equal to each other; but if it be requisite to produce a linear demonstration of this, let there be two right angles, one _a b c_, the other _d e f_.

I say that they are equal; for if they are not equal, one of them must be greater, suppose the angle at _b_. If then the line _d e_ be adapted to the line _a b_, the line _e f_ shall fall within. Let it fall as _b g_, and let the line _b c_ be produced to _h_; because, then _a b c_ is a right angle, _a b h_ also shall be a right angle, and they shall be mutually equal to each other, from the tenth Definition: the angle _a b h_ therefore, is greater than the angle _a b g_. Let again the line _g b_ be produced to _k_, because, therefore _a b g_ is a right angle, the successive angle _a b k_ shall be a right one, and consequently equal to _a b g_. Hence, the angle _a b h_, shall be less than the angle _a b g_; but it was also greater, which is impossible: but this has been shewn by other expositors, and requires no great consideration. But Pappus very properly admonishes us, that the converse of this Petition is not true; I mean, that every thing equal to a right angle, is a right angle; though if it be rectilinear, it is without doubt a right angle. But a curvilinear angle may also be exhibited equal to one that is right: for let there be conceived two equal right lines, _a b_, and _b c_, making the angle at the point _b_, right;

and on them let the semicircles _a e b_, _b f c_, with a proper centre and interval be described; because, therefore, the semicircles are equal, they shall have a mutual congruence, and the angle _e b a_, is equal to the angle _f b c_, and _a b f_ is common: the whole right angle, therefore, is equal to the lunular, i.e. to _e b f_, and yet the lunular is not a right angle. In the same manner, if the angle _a b c_ should be obtuse or acute, a lunular angle may be shewn equal to it (for this is that genus of curvilinear angles which agrees with such as are rectilinear), only this is to be observed, that in a right and obtuse angle, it is requisite to add the middle angle, which is contained by the line _a b_, and the circumference _b f_; but in an acute angle to take this away: for the right line _c b_, in these cases, cuts the circumference _b e_. The truth of which, will be evident from the following figures:

And hence, it appears, that all right angles are mutually equal to each other, and that not every thing equal to a right angle, is consequently a right angle: for if it be not rectilinear, how can it be called right. But it is also manifest from this Petition, that angular rectitude is allied to equality, in the same manner as acuteness and obtuseness are related to inequality. For rectitude and equality, as also similitude, are of the same co-ordination, (for each exists under bound): but acuteness and obtuseness, as also dissimilitude, are of the same series with inequality. For they are all produced from _bound_ and _infinite_. Hence some, regarding the quantity of angles, say, that a right angle is equal to a right: but others, considering their quality, affirm that one is similar to another. For similitude in qualities is the same as equality in quantities.