Henry Billingsley, a London haberdasher, substituted his own explanation of multiplication for Euclid's. Henry said, in effect, two multiplied by three was two added to itself three times. Yet this, as the Chinese pointed out centuries ago, equals eight, not six! Thus, the British Empire infected arithmetical education both in England and throughout its settlements and colonies. Yet the folk of New England fought back! After the American War of Independence, (from the British), the flood of England's infected mathematics books slowed to a trickle. Once New England became America, common sense emerged. American books no longer contained Billingsley’s invented and broken multiplication definition. Algebraically, the Americans said a × b was a taken/repeated b times, or a added to itself b – 1 times. So, here are some examples of arithmetical explanations disinfected with common sense.
1818 An Elementary Treatise on Arithmetic, John Farrar (1)
The first principles, as well as the more difficult parts of Mathematics, have, it is thought, been more fully and clearly explained by the French elementary writers, than by the English. ... When the numbers to be added are equal to each other, addition takes the name of multiplication, because in this case the sum is composed of one of the numbers repeated as many times as there are numbers to be added. Reciprocally, if we wish to repeat a number several times, we may do it, by adding the number to itself as many times, wanting one, as it is to be repeated. For instance, by the following addition,
the number 16 is repeated four times, and added to itself three times.
1848 The American equater: or, Arithmetic simplified, Conley Plotts (2)
We have seen in Addition, that it is frequently necessary to increase the same quantity by repeating it, or adding it to itself a certain number of times. Thus, when a number is added to itself once, we say we have two times that number; if twice, we have three times that number; and, if added to itself, three times, we have four times that number.
1856 A Treatise on Arithmetic, theoretical and practical, Elias Loomis (3)
When a number is added to itself once, it is said to be doubled; when it is added to itself twice, it is said to be tripled, etc. In general, the operation of adding a number to itself a certain number of times is called multiplication.
The number which is added to itself is called the multiplicand, and the number which denotes how many times the multiplicand is to be taken is called the multiplier. The multiplier and multiplicand together are called factors. The result of the operation, instead of being called the sum, as in addition, is called the product.
1871 The Art of Teaching School. A Manual of Suggestions, Josiah. R. Sypher (4)
The operations in Multiplication proceed on exactly the same principles as those in Addition; instead of adding one number to another, it is now required to add a number to itself a given number of times; the multiplicand is to be added to itself as many times, less one, as there are units in the multiplier.
1 An Elementary Treatise on Arithmetic, Taken from the [French] Arithmetic of S. F. Lacroix,, by John Farrar, pp. iii, 14-15, Hilliard & Metcalf, University Press, Cambridge, New England, USA, 1818.
2 The American equater: or, Arithmetic simplified, Conley Plotts, p. 35, William Ryan, Philadelphia, USA, 1848.
3 A Treatise on Arithmetic, theoretical and practical, Elias Loomis, p. 44, Harper & Brothers, New York, 1856.
4 The Art of Teaching School. A Manual of Suggestions, Josiah. R. Sypher, p. 17, Stoddart & Co., Bancroft & Co., Philadelphia, USA, 1871.
NOTE: John Farrar’s translation was from Traité élémentaire d'arithmétique, à l'usage de l'École centrale des Quatre-Nations, Par Sylvestre François Lacroix, Paris, 1797.