the dividend to the divisor in geometrical proportion, so also the remainder after subtraction is the measure of proportion arithmetical. You object in the next place, “that if the proportion of one quantity to another be nothing but the excess or defect, then, wheresoever the excess or defect is the same, there the proportion is the same.” This you say follows in your logic, and from thence, that the proportion of three to two, and five to four is the same. But is not three to two, and five to four, where the excess is the same number, the same proportion arithmetical? And is not arithmetical proportion, proportion? You take here (_ratio_) proportion, which is the _genus_, for that _species_ of it which is called geometrical, because usually this species has the name of proportion simply. Also that the proportion of three to two, is the same with that of nine to six; is it not because the excesses are one and three, the same portions of three and nine, that is to say the same excesses comparatively? I wonder you ask me not what is the _genus_ of arithmetical and geometrical proportions, and what the _difference_? The _genus_ is (_ratio_) proportion, or comparison in magnitude, and the _difference_ is that one comparison is made by the absolute quantity, the other by the comparative quantity, of the excess or defect, if there be any. Can anything be clearer than this? You after come in with _ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon dulness or madness: they are not voluntary faults. But when men adventure voluntarily to talk of that they understand not censoriously and scornfully, I may tell them of it. This difference between the excesses or defects, as they are simply or comparatively reckoned, being thus explained, all the rest of that you say in your objections to this eleventh chapter (saving that art. 5 for _ratio binarii ad quinarium est superari ternario_, as it is in other places, I have put too hastily _ratio binarii ad quinarium est ternarius_), will be understood by every reader to be frivolous, and to proceed from the ignorance of what proportion is. At the twelfth chapter you only note that I say, _that the proportion of inequality is quantity, but the proportion of equality not quantity_, and refer that which you have to say against it to the chapter following; to which place I shall also come in the following lesson. ========== OF THE FAULTS THAT OCCUR IN DEMONSTRATION. TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN THE UNIVERSITY OF OXFORD. LESSON III.