_B._ And what you thought had been true, if a spherical angle were a very angle. For all men that have written of spherical triangles take for the ground of their calculation, as Regiomontanus, Copernicus, and Clavius, that the arch of a spherical angle is the side opposite to the angle. You should have considered also that he makes the angle V P M 12 deg., but sets down no arc to answer it. But that you may find I am in the right, look into the definitions which Clavius hath put down before his treatise of spherical triangles, and amongst them is this; “the arc of a spherical triangle is a part of a great circle intercepted between the two sides drawn from the pole of the said great circle.” _A._ The book is nothing worth; for it is impossible to subtract an arc of a circle out of a spherical angle. And I see besides that he takes the superficies that lieth between the sides L P and L M for an arch, which is the quantity of an angle; and is a line, and cannot be taken out of a superficies. I wonder how any man that pretends to mathematics could be so much mistaken. _B._ It is no great wonder. For Clavius himself striving to maintain that a right angle is greater than the angle made by the diameter and the circumference, fell into the same error. A corner, in vulgar speech, and an angle, in the language of geometry, are not the same thing. But it is easy even for a learned man sometimes to take them for the same, as this author now has done; and proceeding he saith, subtract 8 deg. 38 min. from the angle P V L, and there remains the angle L V M. _A._ That again is false, because impossible. What was it that deceived him now? _B._ The same misunderstanding of the nature of a spherical angle. Which appears farther in this, that when he knew the arc V P was part of a great circle, he thought V M, which he maketh 8 deg. 30 min., were also parts of a great circle; which is manifestly false. For two great circles, because they pass through the centre, do cut each other into halves. But V P is not half a circle. He sure thought himself at Vaygates, and that P M V was equal to P V, although in the same hemisphere. _A._ But how proves he that the arc P M is 8 degrees 30 minutes? _B._ Thus. We have in two triangles, P L M and P V M, two sides and one angle included, to find P M the distance of the magnetical pole from the pole of the earth 8 deg. 30 min. _A._ Is that all? It is very short for a demonstration of two so difficult problems, as the quantity of 8 deg. 30 min.; and of the place of the magnetical pole. But he has proved nothing till he has showed how he found it. And though P M be 8 deg. 30 min., it follows not that M is the magnetical pole. _B._ Nor is it true. For if P M be 8 deg. 30 min., and V M 8 deg. 38 min., the whole arc P M V will be 17 deg. 8 min., which should be 20 deg. Besides, whereas the variations were east and west, the subtracting of them should be also east and west, but they are north and south.