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Library/Paris Manuscript K/184

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Paris Manuscript K — 184

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Bibliothèque de l'Institut de France, Paris

Foglio 86 r - K3 6 r Tre triangoli sovrapposti, con:

a - d - e b c f

Il triangolo d e f è doppio al triangolo d b c e doppio è il triangolo a b c al medesimo triangolo d b c; e quelle cose che son doppie a una sono equali in fra loro.

Que' triangoli saranno di quantità equali, quando l'altezza del primo supera tanto l'altezza del secondo, quanto la basa del secondo supera la basa del primo.

Come se 'l triangolo a b c fussi doppio in altezza al triangolo d e f e che la basa del triangolo d e f fussi doppia al triangolo1 a b c. Dico essi triangoli essere in fra loro equali. Per provarlo io tirerò dall'angolo d del triangolo d e f le 2 linie d b e d c alli termini della basa b c, e sarà b c per l'ipotesi la metà della basa e f, e fia il triangolo d b c equale per altezza al triangolo d e f, e sia d e f di basa doppia alla basa b c; esso triangolo d e f è doppio al triangolo d [b c], e doppio è il triangolo a b c al triangolo d b c, e quelle c[ose] [che] [son] duple a una sono equali [in] [fra] [loro]2.

Sheet 86r (K3 6r) - Three superimposed triangles

a - d - e b c f. The triangle d e f is double the triangle d b c, and double is the triangle a b c to the same triangle d b c; and those things that are double to one are equal among themselves. Those triangles will be of equal quantity when the height of the first exceeds the height of the second as much as the base of the second exceeds the base of the first. As, if the triangle a b c were double in height to the triangle d e f, and the base of the triangle d e f were double to the triangle a b c: I say these triangles to be equal among themselves. To prove it I shall draw from the angle d of the triangle d e f the 2 lines d b and d c to the ends of the base b c, and b c will be by hypothesis the half of the base e f, and the triangle d b c will be equal in height to the triangle d e f, and d e f is of base double to the base b c; this triangle d e f is double the triangle d (b c), and double is the triangle a b c to the triangle d b c, and those things that are double to one are equal among themselves.

Two triangles double a third are equal in area

A rule for equal areas: height exceeds as much as base falls short

Geometry Arithmetic & Proportion

Because triangle d e f and triangle a b c are each double the triangle d b c, and things double the same are equal, the two are equal to one another. Leonardo then generalises the rule: triangles are equal in area when the first exceeds the second in height by as much as the second exceeds the first in base, proving it by drawing d b and d c from apex d to the ends of base b c. The diagram shows three overlapping triangles sharing an apex region, with a broad base labelled b, c, d, f and an inner apex d.

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Triangles double the same triangle are equal

Triangle d e f is double triangle d b c, and triangle a b c is likewise double d b c; and things double one thing are equal among themselves, so the two are equal. The construction draws d b and d c from apex d to the base b c, with b c being half of base e f.

Equal area when height gain balances base loss

Triangles are of equal quantity when the height of the first exceeds the height of the second by as much as the base of the second exceeds the base of the first. Here triangle a b c is double in height to d e f while d e f is double in base, so their areas match.

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