But let us see how you answer to that which Clavius has objected already. “_They are heterogeneous_,” says he, “_because the angle of contact, how oft soever multiplied, can never exceed a rectilineal angle_.” To answer which, you allege _it is no angle at all; and that therefore, it is no angle at all, because the lines have no inclination one to another_. How can lines that have no inclination one to another, ever come together? But you answer, _at least they have no inclination in the point of contact_. And why have two straight lines inclination before they come to touch, more than a straight line and an arch of a circle? And in the point of contact itself, how can it be that there is less inclination of the two points of a straight line and an arch of a circle, than of the points of two straight lines? But the straight lines, you say, will cut; which is nothing to the question; and yet this also is not so evident, but that it may receive an objection. Suppose two circles, A G B and C F B, to touch in B, and have a common tangent through B. Is not the line C F B G A a crooked line? and is it not cut by the common tangent D B E? What is the quantity of the two angles F B E and G B D, seeing you say neither D B G nor E B F is an angle? It is not, therefore, the cutting of a crooked line, and the touching of it, that distinguisheth an angle simply, from an angle of contact. That which makes them differ, and in kind, is, that the one is the quantity of a _revolution_, and the other, the quantity of _flexion_. In the seventh chapter of the same treatise, you think you prove the angle of contact, if it be an angle, and a rectilineal angle to be (_homogeneous_) of the same kind; when you prove nothing but that you understand not what you say. Those quantities which can be added together, or subtracted one from another, are of the same kind; but an angle of contact may be subtracted from a right angle, and the remainder will be the angle of a semicircle, &c. So you say, but prove it not, unless you think a man must grant you that the superficies contained between the tangent and the arch, which is it you subtract, is the angle of contact; and that the plane of the semicircle is the angle of the semicircle, which is absurd; though, as absurd as it is, you say it directly in your _Elenchus_ , p. 35, l. 14, in these words: “_When Euclid defines a plane angle to be the inclination of two lines, he meaneth not their aggregate, but that which lies between them_.” It is true, he meaneth not the aggregate of the two lines; but that he means that which lies between them, which is nothing else but an indeterminate superficies, is false, or Euclid was as foolish a geometrician as either of you two.