- Day, Jeremiah. 1853. An Introduction to Algebra, being the first part of a Course of Mathematics, adapted to the method of instruction in the American colleges. 5th edn. New Haven: Durrie & Peck. https://books.google.co.jp/books?id=FHtaAAAAcAAJ&pg=PR1
86. Every multiplier is to be considered a number. We sometimes speak of multiplying bya a given weight or measure, a sum of money, &c. But this is abbreviated language. ... Buf if a part of the weight of a body be fixed upon as a unit, a quantity may be multiplied by a number equal to the number of these parts contained in the body. If a diamond is old by weight, a particular price may be agrred upon each grain [メートル法グレーン 1gr = 50mg]. A grain is here the unit; and it is evident that the value of the diamond, is equal to the given price repeated as many times, as there are grains in the whole weight. We say concisely, that the price is multiplied by the weight; meaning that it is multiplied by a number eual to the number of grains in the weight. In a similar manner, any quantity whatever may be supposed to be made up of parts, each being considered a unit, and any number of these may become a multiplier.
つまり、乗数が不名数でなければならないので、a $/gr × b gr というのは、 a $ × b の糖衣構文。
582. ... But how, it may be asked can geometrical quantities be multiplied into each other? One of the factors, in multiplication, is alwyas to be considered as a number. (Art. 86) ...
To explain this, it will be necessary to observe, that whenever one geogmetrical quantity is multiplied into another, some particular extent is to be considered the unit. It is immaterial what this extent is, provided it remains the same, in different parts of the same calculation. ...
585. ... The product of two lines is often spoken of, as being equal to a surface; adn the product of a line and a surface, as equal to a solid. ... But if a line has no breadh, how can the multiplication, that is the repetition, of a line produce a surface? ...
語学的な注。ここの if は仮定ではなく、事実で、その原因や理由が次に必要。
このあとの流れは、おおまかにはニュートンと同じだが、いきなり数にいかない。 とりあえず底辺と同じ流さで幅 1in の「棒」を作ってから、かけ算にしている。