VENI VIDI DA VINCI

Collections Newsletter Guild About Developers Patron Sign In

Library/Codex Arundel/ff. 220v-221r

Prev262 / 495Next

Codex Arundel — ff. 220v-221r

100%

Reset zoom

British Library, London — Arundel MS 263

0220 v

Qui lo 'ntero è 4 del quale el 1/2 è 2, e questo 2 viene poi a essere e' 2/3 di 3/4.

3/4 = 1/2 : 2/3

Un 1/2 di che numero sarà 2/3? Di' che sarà 2/3 di 3/4. Ora per la prova dirai che lo 'ntero che qui si manegia sia 12, del quale il mezo è 6, e' 2/3 sono 8, e' 3/4 sono 9. Adunque è ben detto che 6 sia e' 2/3 di 9.

8/9 = 2/3 : 3/4

2/3 di qual numero fia e' 3/4? Dirai che sarà e' 3/4 di 8/9. Qui lo 'ntero è 9, e' 2/3 di 9 è 6/9, e' 3/4 di 8/9 si è pure 6/9.

15/16 = 3/4 : 4/5

Qui lo 'ntero si è 16, e' sua 3/4 fia 12/16, il qual 12/16 fia e'4/5 di 15/16.

24/25 = 4/5 : 5/6

Qui lo 'ntero è 25, e' 4/5 di 25 si è 20, il quale 20 è e' 5/6 di 24/25.

35/36 = 5/6 : 6/7

Qui lo 'ntero è 36, del quale e' 5/6 di 36 si è 30/36, e esso 30/36 viene a essere e' 6/7 di 35/36.

27/20 = 3/5 : 4/9

Qui lo 'ntero è 20; e' 3/5 di 20 son 12/20, el quale 12/20 viene a essere e' 4/9 di 27/20, cioè d'uno e 7/20

0221 r

Risolvi il 5 inel rotto resultato, e così il 6, e troverai 5 essere 15/3, e il 6 2/3 essere 20/3 che, come vedi, 15/3 sono e' 3/4 di 20/3, onde la regola è buona.

5/1 : 3/4

7/8 : 8/9 = 63/64

20

6 2/3

3

1

2/3 : 3/4

8

2 2/3

3

9

Tu hai a multiplicare il 3 ch'è sotto il 2, e 'l 3 ch'è sopra il 4, che fa 9, e esso 9 è quello che si mette sotto il resultante, il quale sarà poi 8/9; e 'l 2 ch'è sopra il 3 che dice 2/3, dirà poi 6/9. Adunque e' 2/3 di 3/4 sono 6/8/9

5/1 : 1/3 = 15/1

3/4 : 4/5

15

3 3/4

4

16

Qui e' 3/4 sono 12/16; direno adunque di che numero esso 12/16 sarà e' 4/5, che verrà a essere di 15/16 apunto.

Here the whole is 4, of which the 1/2 is 2, and this 2 then comes to be the 2/3 of 3/4.

3/4 = 1/2 : 2/3

Of what number will 1/2 be 2/3? Say that it will be 2/3 of 3/4. Now for the proof you will say that the whole here handled is 12, of which the half is 6, the 2/3 are 8, and the 3/4 are 9. Therefore it is rightly said that 6 is the 2/3 of 9.

8/9 = 2/3 : 3/4

Of what number will 2/3 be the 3/4? You will say that it will be the 3/4 of 8/9. Here the whole is 9, the 2/3 of 9 is 6/9, and the 3/4 of 8/9 is likewise 6/9.

15/16 = 3/4 : 4/5

Here the whole is 16, its 3/4 will be 12/16, which 12/16 will be the 4/5 of 15/16.

24/25 = 4/5 : 5/6

Here the whole is 25, the 4/5 of 25 is 20, which 20 is the 5/6 of 24/25.

35/36 = 5/6 : 6/7

Here the whole is 36, of which the 5/6 of 36 is 30/36, and this 30/36 comes to be the 6/7 of 35/36.

27/20 = 3/5 : 4/9

Here the whole is 20; the 3/5 of 20 are 12/20, which 12/20 comes to be the 4/9 of 27/20, that is of one and 7/20.

Resolve the 5 into the resulting fraction, and likewise the 6, and you will find 5 to be 15/3, and the 6 2/3 to be 20/3, so that, as you see, 15/3 are the 3/4 of 20/3, whence the rule is good.

5/1 : 3/4

7/8 : 8/9 = 63/64

20

6 2/3

3

1

2/3 : 3/4

8

2 2/3

3

9

You must multiply the 3 that is under the 2, and the 3 that is above the 4, which makes 9, and this 9 is what is put beneath the result, which will then be 8/9; and the 2 that is above the 3 which reads 2/3 will then read 6/9. Therefore the 2/3 of 3/4 are 6/8/9.

5/1 : 1/3 = 15/1

3/4 : 4/5

15

3 3/4

4

16

Here the 3/4 are 12/16; we will therefore say of what number this 12/16 will be the 4/5, which will come to be of 15/16 exactly.

Studies in fractions and proportional means

Rules linking successive fractions such as 2/3 of 3/4, checked with whole numbers

Arithmetic & Proportion

Both pages are filled with worked exercises on fractions and their proportional relations, showing how one fraction is the given fraction of another (for example, that 1/2 is the 2/3 of 3/4, and 2/3 of 3/4 gives 6/9). Leonardo verifies each rule by choosing a convenient whole number so the fractions become integers, and works a ladder of cases from 3/4 : 4/5 up through 5/6 : 6/7. A second column reduces whole numbers to improper fractions and cross-multiplies numerators and denominators to confirm that the rule holds.

On this page

One half as the two-thirds of three-quarters

Taking the whole as 12, its half is 6, its 2/3 are 8, its 3/4 are 9, so that 6 is rightly the 2/3 of 9. The ladder continues: 3/4 : 4/5 gives 15/16, 4/5 : 5/6 gives 24/25, 5/6 : 6/7 gives 35/36.

Cross-multiplying to find a proportional term

Multiply the 3 under the 2 and the 3 over the 4 to get 9, set under the result to give 8/9, and the 2 over the 3 becomes 6/9, so the 2/3 of 3/4 are 6/9. Whole numbers such as 5 and 6 2/3 are first resolved into 15/3 and 20/3 to check the rule.

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