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© Veneranda Biblioteca Ambrosiana, Milan

Non si varierà la proporzione geometrica per crescere equalmente il maggiore e minore extremo di ciascuna sua parte. Sia 6 e 12, cioè dupla. Ora cresci al 12 il suo terzo che è 4, e arai 16; e al 6 cresci il suo terzo che è 2, e arai 8; sì che tu vedi che la medesima proporzione è osservata, perché dupla come prima.

Sì che note secondo geometrice lo aggiugnere e lo scemare s'intende secondo le sue parte, cioè che se sopra l'uno tu poni il suo quarto, ancora all'altro porrai il suo quarto, e così equalmente s'intende secondo le dette parti. E se tu levi all'uno il terzo, leva ancora il terzo dell'altro; ma della aritmetica il crescere e discrescere s'intende secondo la equalità delle differenzie. Crescendo1 equalmente e minore e maggiore extremo di ciascuna proporzione geometrice, sempre sta ferma la medesima proporzione.

Crescendo equalmente l'uno e l'altro stremo aritmetice di qualu[n]che proporzione, sempre diminuisce la proporzione geometrica. Sia la proporzione dupla infra 6 e 12, a li quali extremi aggiugni 2, saran poi 8 e 14 e fien cresciuti equalmente aritmetice; ma la proporzione geometrica, ch'era dupla, è diminuita, perché s'è fatta2, perché fra 8 e 14 è proporzione super tripartiens quartas, cioè l'una è dupla e l'altra è 1 e 3/4, sì che le manca 1/4 alla geometrica3.

Diminuendo4 equalmente aritmetice l'uno e l'altro extremo di ciascuna proporzione, sempre cresce la proporzione geometrica che prima era. Sia 6 e 12 delli cui extremi si levi equalmente aritmetice5, cioè 2 per ciascuno; resterà 4 e 10. Dico la proporzione infra 4 e 10 esser maggiore della dupla che prima era infra 6 e 12.

The geometric proportion will not be altered by increasing equally the greater and lesser extreme of each of its parts. Let it be 6 and 12, that is, double. Now add to 12 its third, which is 4, and you will have 16; and to 6 add its third, which is 2, and you will have 8; so that you see the same proportion is kept, because it is double as before.

So note that in geometric terms adding and subtracting is understood according to the parts, that is, if upon the one you set its quarter, upon the other too you will set its quarter, and so it is understood equally according to the said parts. And if you take a third from one, take also a third from the other; but in arithmetic, increasing and decreasing is understood according to the equality of the differences. By increasing equally both the lesser and greater extreme of each geometric proportion, the same proportion always stays fixed.

By increasing equally the one and the other extreme arithmetically of any proportion, the geometric proportion always diminishes. Let the double proportion be between 6 and 12, to whose extremes add 2, and they will then be 8 and 14, increased equally in arithmetic; but the geometric proportion, which was double, is diminished, because between 8 and 14 there is the proportion super tripartiens quartas, that is, the one is double and the other is 1 and 3/4, so that it lacks 1/4 of the double one.

By diminishing equally in arithmetic the one and the other extreme of each proportion, the geometric proportion that was before always increases. Let it be 6 and 12, from whose extremes one takes away equally in arithmetic, that is, 2 from each; there will remain 4 and 10. I say that the proportion between 4 and 10 is greater than the double that was before between 6 and 12.

Geometric versus arithmetic change of a proportion

How proportional change keeps a ratio, while equal absolute change alters it

Arithmetic & Proportion Geometry

A single sustained argument distinguishes 'geometric' from 'arithmetic' increase of a ratio. Adding to each extreme its own proportional part (a third of 12 and a third of 6) keeps the ratio double, since 16 to 8 is still 2:1, because in geometric terms increase is reckoned by the parts. Adding the same absolute amount to both extremes instead shrinks the ratio (6 and 12 become 8 and 14, a 'super tripartiens quartas'), while subtracting the same amount enlarges it (6 and 12 become 4 and 10, more than double). Faint untranscribed mechanical sketches occupy the lower half of the sheet.

On this page

A ratio kept by proportional increase of both extremes

Increasing the greater and lesser extreme each by its own like part leaves the geometric proportion unchanged: from 6 and 12, adding a third to each (2 and 4) gives 8 and 16, still double. Leonardo generalises that in geometric terms adding or removing is reckoned by the parts, so a quarter set on one extreme calls for a quarter on the other.

Equal absolute change alters the ratio

By contrast, adding the same number to both extremes diminishes the ratio: 6 and 12 plus 2 become 8 and 14, a 'super tripartiens quartas' that falls a quarter short of double. Subtracting equally does the reverse, so 6 and 12 less 2 become 4 and 10, a ratio greater than the original double.

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