Oughtred gives two processes of long division. The first is identical with the modern process, except that the divisor is written below every remainder, each digit of the divisor being crossed out as soon as it has been used in the partial multiplication. The second method of long division is one of the several types of the old “scratch method.” This antiquated process held its place by the side of the modern method in all editions of the Clavis. The author divides 467023 by 357|0926425, giving the following instructions: “Take as many of the first Figures of the Divisor as are necessary, for the first Divisor, and then in every following particular Division drop one of the Figures of the Divisor towards the Left Hand, till you have got a competent Quotient.” He does not explain abbreviated division as thoroughly as abbreviated multiplication. 17 3̸0̸3̸ 2̸8̸0̸3̸ 1̸0̸9̸9̸3̸0̸ 3̣5̣7̣|0̣9̣2̣6425) 4̸6̸7̸0̸2̸3̸ (1307|80 3̸5̸7̸0̸9̸3̸ 1̸0̸7̸1̸2̸7̸ 2̸5̸0̸0̸ 2̸8̸6̸ Oughtred does not examine the degree of reliability or accuracy of his processes of abbreviated multiplication and division. Here as in other places he gives in condensed statement the mode of procedure, without further discussion. He does not attempt to establish the rules for the addition, subtraction, multiplication, and division of positive and negative numbers. “If the Signs are both alike, the Product will be affirmative, if unlike, negative”; then he proceeds to applications. This attitude is superior to that of many writers of the eighteenth and nineteenth centuries, on pedagogical as well as logical grounds: pedagogically, because the beginner in the study of algebra is not in a position to appreciate an abstract train of thought, as every teacher well knows, and derives better intellectual exercise from the applications of the rules to problems; logically, because the rule of signs in multiplication does not admit of rigorous proof, unless some other assumption is first made which is no less arbitrary than the rule itself. It is well known that the proofs of the rule of signs given by eighteenth-century writers are invalid. Somewhere they involve some surreptitious assumption. This criticism applies even to the proof given by Laplace, which tacitly assumes the distributive law in multiplication.