2 4 6|9 1 4 7 2|5 3
-------
7 4 0 7
1 2 3 5
4 9
1 7
-------
8 7 0 8
The second case supposes that “you would have the Product with some places of parts” (decimals), say 4: “Set the place of Unity of the lesser Number under the Fourth place of the Parts of the greater.” The multiplication of 246|914 by 35|27 is now performed thus:
2 4 6|9 1 4 7 2|5 3
---------------
7 4 0 7 4 2 0 0
1 2 3 4 5 7 0 0
4 9 3 8 2 8
1 7 2 8 4 0
---------------
8 7 0 8|6 5 6 8
In the third and fourth cases are considered factors which appear as integers, but are in reality decimals; for instance, the sine of 54° is given in the tables as 80902 when in reality it is .80902.
Of interest as regards the use of the word “parabola” is the following: “The Number found by Division is called the Quotient, or also Parabola, because it arises out of the Application of a plain Number to a given Longitude, that a congruous Latitude may be found.”[26] This is in harmony with etymological dictionaries which speak of a parabola as the application of a given area to a given straight line. The dividend or product is the area; the divisor or factor is the line.
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.
A word should be said on Oughtred’s definition of + and -. He recognizes their double function in algebra by saying (Clavis, 1631, p. 2): “Signum additionis, sive affirmationis, est + plus” and “Signum subductionis, sive negationis est - minus.” They are symbols which indicate the quality of numbers in some instances and operations of addition or subtraction in other instances. In the 1694 edition of the Clavis, thirty-four years after the death of Oughtred, these symbols are defined as signifying operations only, but are actually used to signify the quality of numbers as well. In this respect the 1694 edition marks a recrudescence.