OF GIOVANNI BATTISTA

BENEDETTI

VENETIAN PATRICIAN,

PHILOSOPHER TO THE MOST SERENE

CARLO EMANUELE, DUKE OF THE ALLOBROGES.

Arithmetical Theorems.

Decorative historiated woodcut initial P depicting a landscape with figures and cattle/sheep.
The ancient mathematical philosophers handed down to posterity many things excellently devised concerning numbers and their effects; yet, since they gave hardly any explanation for them, or certainly a very slight one, the opportunity was provided to me by various problems proposed by the Most Serene Duke of Savoy. There occurred to me several matters to be contemplated regarding those things which were proposed by the ancients, which I have judged not useless to commend to posterity, lest these thoughts of mine should perish, and to provide an opportunity for as many as possible to investigate these abstruse things, which, being involved in problems and theorems, have hardly found anyone to unravel them.

Among the other things sought from me, indeed, was this theorem.

FIRST THEOREM.

The Most Serene Duke of Savoy asked me by what reasoning it could be known scientifically and speculatively (as it is said) that the product of two fractional numbers is smaller than either of the factors. To which I replied, that it must be conceived in the mind and thought that the fractional factors, together with the fractional products, are not of one and the same nature, but rather of a far different one.

For example, given the proposed fractional numbers $a. i.$ and $a. c.$, whose wholes are $a. b.$ and $a. d.$, which are to be conceived as lines: it would indeed be manifest that the product $c. i.$ would be superficial [an area], which would take its name from the superficial product $d. b.$ generated from one of the linear wholes into the other. For if $a. i.$ were established as an eighth part of $a. b.$, and $a. c.$ as a half of $a. d.$, then by multiplying $a. i.$ by $a. c.$, a sixteenth part of $d. b.$ would be produced. Therefore, $d. b.$ would be the relative whole of $c. i.$, and not any whole of the factors. Thus it is no wonder if the product $c. i.$ seems smaller than its factors, since it is compared with a whole of a nature different from the first; for a fraction is named from a whole of the same nature, whether linear, superficial, or corporeal.

But if, for the sake of fuller knowledge, anyone should wish to speculate from the precepts of science...