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Library/Paris Manuscript K/189

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Paris Manuscript K — 189

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Bibliothèque de l'Institut de France, Paris

Foglio 88 v - K3 8 v

Al proposito della passata1.

4 di che numero è li 3/4? Poni il 4 sopra il 3 e arai 4/3, cioè uno e 1/3, il quale uno e 1/3 aggiugni al 4, il quale tu vuoi sapere di che numero esso sia li 3/4, e arai fatto una somma di 5 e 1/3 e questo 5 e 1/3 è il numero di che 4 è li sua 3/4. Ora multiplica il 5 per 3, che sta sopra la riga, e arai 15/3; aggiungivi l'uno che sta sopra la riga e arai 16/3, di che 12 è li 3/4. E questi 12/3 vol dire 4 interi, e li 16/3 vol dire 5 e 1/3, di che 4 è li 3/4. Ora fa il semicirculo e dividilo in2 n m, b d col sopplimento a f c son li 3/4 del semicirculo sopra detto; resta il paralello a b c d per la quarta parte di tutto il semicirculo.

Semicerchio diviso in sedici settori, con:

f - a c - b d - n i m

Folio 88v - K3 8v

Concerning the matter of the previous page. 4 is the 3/4 of what number? Put the 4 over the 3 and you will have 4/3, that is one and 1/3; add this one and 1/3 to the 4, of which you wish to know of what number it is the 3/4, and you will have made a sum of 5 and 1/3; and this 5 and 1/3 is the number of which 4 is its 3/4. Now multiply the 5 by the 3 that stands above the line and you will have 15/3; add to it the one that stands above the line and you will have 16/3, of which 12 is the 3/4. And these 12/3 mean 4 whole, and the 16/3 mean 5 and 1/3, of which 4 is the 3/4. Now make the semicircle and divide it into n m, b d, with the supplement a f c, which are the 3/4 of the semicircle said above; there remains the parallel a b c d for the fourth part of the whole semicircle.

Semicircle divided into sixteen sectors, with:

f - a c - b d - n i m

Of What Number Is Four the Three-Quarters?

A worked fraction calculation applied to a sixteen-sector semicircle

Arithmetic & Proportion Geometry

Following from the previous page, Leonardo works a numerical example: to find of what number 4 is the three-quarters, he inverts to 4/3 (one and 1/3), adds it to 4 to reach 5 and 1/3, and verifies through the fractions 16/3 and 12/3. He then returns to geometry, dividing a semicircle into sixteen sectors so that the supplement a f c gives three-quarters of the figure while the parallel strip a b c d remains as its fourth part. A large labelled semicircle divided into many sectors accompanies the text.

On this page

Finding the whole of which 4 is three-quarters

Put 4 over 3 to get 4/3, that is one and 1/3; add it to 4 to make 5 and 1/3, the number of which 4 is the 3/4. Multiplying 5 by 3 gives 15/3, plus the 1 makes 16/3; here 12/3 means 4 whole and 16/3 means 5 and 1/3.

The sixteen-sector semicircle and its fourth part

The semicircle is divided into sixteen sectors so that the supplement a f c forms the three-quarters, while the parallel strip a b c d remains for the fourth part of the whole semicircle. The diagram is labelled f - a c - b d - n i m.

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