And now I shall let you see that the composition of proportion by multiplication, as it is in the fifth definition of the sixth element, is but another way of adding proportions one to another. Let the proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5, the proportion arising is of 8 to 15. Put in order these three quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15, that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3, and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising is of 10 to 12. Put in order these three numbers, 10, 8, 12. The proportion 10 to 12 is compounded of the proportions of 10 to 8, that is of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you know not this. I find not any more clamour against me for saying the proportion of 1 to 2 is double to that of 1 to 4. Your book, you speak of, concerning proportion against _Meibomius_ is like to be very useful when neither of you both do understand what proportion is. You take exceptions, as that I say, that _Euclid_ has but one word for _double_ and _duplicate_; which nevertheless was said very truly, and that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think you have come off handsomely with asking me whether διπλάσιος and διπλασίων be one word. Nor are you now of the mind you were, that a point is not _quantity unconsidered_, but that in an infinite series it may be safely neglected. What is _neglected_ but unconsidered. Nor do you any more stand to it, that the _quotient_ is the _proportion_. And yet were these the main grounds of your _Elenchus_. But you will say, perhaps, I do answer to the defence you have now made in this your _School Discipline_: ’tis true. But ’tis not because you answer never a word to my former objections against these propositions 19, 89; but because you do so shift and wriggle, and throw out ink, that I cannot perceive which way you go, nor need I, especially in your vindication of your _Arithmetica Infinitorum_. Only I must take notice that in the end of it, you have these words, “Well, _Arithmetica Infinitorum_ _is come off clear_” You see the contrary. For sprawling is no defence. It is enough to me that I have clearly demonstrated both before sufficiently, and now again abundantly, that your book of _Arithmetica Infinitorum_ is all nought from the beginning to the end, and that thereby I have effected that your authority shall never hereafter be taken for a prejudice. And, therefore, they that have a desire to know the truth in the questions between us, will henceforth, if they be wise, examine my geometry, by attentive reading me in my own writings, and then examine, whether this writing of yours confute or enervate mine.