212.  What is deceptive in this subject, as I have already observed, is that one feels an inclination to believe that what is the best in the whole is also the best possible in each part.  One reasons thus in geometry, when it is a question de maximis et minimis.  If the road from A to B that one proposes to take is the shortest possible, and if this road passes by C, then the road from A to C, part of the first, must also be the shortest possible.  But the inference from quantity to quality is not always[261] right, any more than that which is drawn from equals to similars.  For equals are those whose quantity is the same, and similars are those not differing according to qualities.  The late Herr Sturm, a famous mathematician in Altorf, while in Holland in his youth published there a small book under the title of Euclides Catholicus.  Here he endeavoured to give exact and general rules in subjects not mathematical, being encouraged in the task by the late Herr Erhard Weigel, who had been his tutor.  In this book he transfers to similars what Euclid had said of equals, and he formulates this axiom:  Si similibus addas similia, tota sunt similia.  But so many limitations were necessary to justify this new rule, that it would have been better, in my opinion, to enounce it at the outset with a reservation, by saying, Si similibus similia addas similiter, tota sunt similia.  Moreover, geometricians often require non tantum similia, sed et similiter posita.

213.  This difference between quantity and quality appears also in our case.  The part of the shortest way between two extreme points is also the shortest way between the extreme points of this part; but the part of the best Whole is not of necessity the best that one could have made of this part.  For the part of a beautiful thing is not always beautiful, since it can be extracted from the whole, or marked out within the whole, in an irregular manner.  If goodness and beauty always lay in something absolute and uniform, such as extension, matter, gold, water, and other bodies assumed to be homogeneous or similar, one must say that the part of the good and the beautiful would be beautiful and good like the whole, since it would always have resemblance to the whole:  but this is not the case in things that have mutual relations.  An example taken from geometry will be appropriate to explain my idea.

214.  There is a kind of geometry which Herr Jung of Hamburg, one of the most admirable men of his time, called ‘empiric’.  It makes use of conclusive experiments and proves various propositions of Euclid, but especially those which concern the equality of two figures, by cutting the one in pieces, and putting the pieces together again to make the other.  In this manner, by cutting carefully in parts the squares on the two sides of the right-angled triangle, and arranging these parts carefully, one makes from them the square on the hypotenuse; that is demonstrating