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© Veneranda Biblioteca Ambrosiana, Milan

2 3

3 4

s. 2 12

9 8

8 9
(se 1/)

2

3

3 6

4 8

9

8

8

12

Se vuoi partire 2/3 per 3/4, l'ordinaria regola ti dà che tu debbi multiplicare in croce col dire:

8

2 3

3 4 3 vie 3 fa 9, e poni 9 sotto il 3, il

9

quale 9 fia partitore1. Poi per l'altra

linia dirai: 2 vie 4 fa 8, e questo 8 fia il numero che si de' partire2 per 9, ch'è impossibile. Adunque resta nominato 8/9; ma qui non s'intende di che, perché ogni cosa partita diminuisce di sua quantità e questa è cresciuta nel partire, imperò ch'elli è maggiore numero 8/9 che none 2/3.

Adunque la regola non è molto chiara e però per altra via dì:

23

3 4

2 vie 3 fa 6 e pon 6 sopra una riga;

poi dirai: 2 vie 4 fa 8 e poni questo 8 sotto essa riga e dira' 6/8, e qui trovi ch'elli è minore 6/8 che 2/3. E ancora poi dire:

2 3

3 4

2 vie 4 fa 8 e poni 8 sopra una

riga; poi di': 3 vie 4, 12 e poni tal 12 sotto lo 8 e dira' 8/12, che è medesimamente minore che 2/33.

2 3
3 4
s. 2 12
9 8
8 9
(if 1/)

2
3

3 6
4 8

9
8

8
12

If you wish to divide 2/3 by 3/4, the ordinary rule tells you that you must multiply crosswise, saying:

2 3
3 4 — 3 times 3 makes 9, and put 9 under the 3, which 9 is the divisor. Then along the other line you will say: 2 times 4 makes 8, and this 8 is the number that is to be divided by 9, which is impossible. Thus the result is named 8/9; but here it is not understood of what, because everything divided diminishes in its quantity, and this has grown in the dividing, inasmuch as 8/9 is a greater number than 2/3.

Therefore the rule is not very clear, and so by another way say:

2 3
3 4 — 2 times 3 makes 6 and put 6 above a line; then you will say: 2 times 4 makes 8 and put this 8 below that line, and it will read 6/8, and here you find that 6/8 is less than 2/3. And you may also say:

2 3
3 4 — 2 times 4 makes 8 and put 8 above a line; then say: 3 times 4, 12, and put that 12 under the 8, and it will read 8/12, which is likewise less than 2/3.

How to Divide 2/3 by 3/4, and Why the Cross-Rule Misleads

A worked critique of the ordinary cross-multiplication rule for dividing fractions

Arithmetic & Proportion

This sheet works through the division of the fraction 2/3 by 3/4. Leonardo shows that the ordinary cross-multiplication rule yields 8/9, which he rejects because a quotient should diminish the quantity, yet 8/9 is greater than 2/3. He then demonstrates two alternative arrangements giving 6/8 and 8/12, both less than 2/3, and the head of the page carries the cross-diagrams of the calculation.

On this page

The cross-rule gives an impossible 8/9

To divide 2/3 by 3/4 the ordinary rule multiplies crosswise: 3 times 3 makes 9 as divisor, and 2 times 4 makes 8 to be divided by 9, naming the result 8/9. Leonardo calls this impossible, since everything divided diminishes in quantity, yet 8/9 is greater than 2/3, so the rule is not clear.

Two clearer arrangements: 6/8 and 8/12

By another way he sets 2 times 3 (6) over a line and 2 times 4 (8) below it, giving 6/8, which he finds less than 2/3. A further arrangement sets 2 times 4 (8) over a line and 3 times 4 (12) below, giving 8/12, likewise less than 2/3.

Reader's Edition — described with AI from the e-Leo transcription.