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© Veneranda Biblioteca Ambrosiana, Milan

Foglio 273 a Recto

Del moto delle dita delle mani.

Li moti delli diti sono principali¹, cioè ascendere e piegare. L'astendere e piegare si varia in modi, cioè alcuna volta si piega tutto d'un pezzo sopra la prima giuntura, alcuna volta si piega o dirizza mezzo sopra la seconda giuntura, e alcuna volta si piega tutto in sé, e nel medesimo tempo si piega in tutte le sue 3 giunture. E se alle 2 prime giunture sarà proibito il piegamento, allora la terza giuntura sarà piegata con maggiore facilità che prima, né mai si potrà piegare per sé sola, essendo libere l'altre giunture e se non si pieghino tutte le 3 giunture. Oltre alli predetti moti ce n'è 4 altri principali, de' quali 2 sono tra su e giù e 2 altri tra qua e là, e ciascun di questi è fatto colla sua semplice corda. Di questi ne seguita infiniti altri moti, fatti sempre con 2 corde, e lasciato una d'esse corde, va ripigliando l'altra. Fu fatto le corde grosse di dentro al dito e le sottili di fori. Fu fatto le corde di dentro a ogni giuntura e di fori no.

Della potenzia che fan le 3 corde dentro alli diti nelle 3 giunture.

Foglio 273 b Recto

Conceptioni.

Dall'alto.

Dare superfizie rette equale a superfizie curve.

Tutte le superfizie curve, alle quali si dà quadrati rettilini equali a loro, son quadrati in [s]e medesimi.

Lungo il margine destro, figura con:

* Se da una superfizie piana di lati curvi si leva li termini curvi, essa resterà superfizie rettilinia. E se la valuta d'essi termini curvi sarà levata dentro a li termini d'essa superfizie curva, allora tal superfizie acquisterà doppia curvità e fia equale al primo triangolo ortogonio¹.

Se da un semicircolo leverai le sue due massime porzioni, el rimanente sarà un triangolo ortogonio.

Figura con:

Sar – d – cb - vt - [Prima].

Figura con:

o c – f – g - n p - (Seconda).

Figura con:

t S n - v - Terza.

ad triangol curvilinio vale dbc triangolo rettilinio. raS, tv vale ad². Ma nella seconda figura è cresciuto il campo tutto³ co porzioni. Ma quadra li campi e trai via la differenzia e tal differenzia, ovvero eccesso quadrato, varrà esse dite porzione. Nella terza dimostrazione si fa la quadratura del triangolo della prima, cioè in essa terza il triangolo Sv è convertito nel quadrilatero rettangolo nSt.

Nella seconda dimostrazione dimin...

In basso, figura:

Quarta.

Entro la figura:

Pieno - campo.

Sheet 273a Recto

On the motion of the fingers of the hands.

The motions of the fingers are the principal ones, namely to extend and to bend. Extending and bending vary in several ways: sometimes the finger bends all in one piece at the first joint; sometimes half of it bends or straightens at the second joint; and sometimes it bends entirely upon itself, bending at all three of its joints at the same time. And if bending is prevented at the first two joints, then the third joint will bend more easily than before; nor can it ever bend by itself alone while the other joints are free, unless all three joints bend together. Besides the aforesaid motions there are 4 other principal ones, of which 2 are between up and down and 2 between here and there, and each of these is made by its own single cord. From these follow infinitely many other motions, always made with 2 cords, and leaving one of these cords it takes up the other again. The thick cords were placed inside the finger and the thin ones outside. The cords were placed inside at every joint, and not outside.

Of the power that the 3 cords make inside the fingers at the 3 joints.

Sheet 273b Recto

Conceptions.

From the top:

To give rectilinear surfaces equal to curved surfaces.

All curved surfaces to which rectilinear squares equal to them are given are squared in themselves.

Along the right margin, figure with:

* If from a plane surface with curved sides the curved boundaries are removed, it will remain a rectilinear surface. And if the value of those curved boundaries is removed within the boundaries of that curved surface, then such a surface will acquire double curvature and will be equal to the first right triangle.

If from a semicircle you remove its two greatest portions, the remainder will be a right triangle.

Figure with:

S a r – d – c b – v t – [First].

Figure with:

o c – f – g – n p – (Second).

Figure with:

t S n – v – Third.

The curvilinear triangle a d is worth the rectilinear triangle d b c. r a S, t v is worth a d. But in the second figure the whole field has grown by portions. But square the fields and take away the difference, and that difference, or squared excess, will be worth those said portions. In the third demonstration the squaring of the triangle of the first is carried out, that is, in this third one the triangle S v is converted into the rectangular quadrilateral n S t.

In the second demonstration dimin...

Below, figure:

Fourth.

Within the figure:

Full – field.

Finger motion and the squaring of curved triangles

Two mounted leaves: the anatomy of the fingers' tendons, and 'Conceptioni' on quadrature

Geometry Anatomy

Two leaves are mounted side by side. The narrow left strip carries a paragraph 'On the motion of the fingers of the hands,' analysing how the joints extend and bend, the extra up-down and side-to-side motions, and how the thick and thin cords (tendons) run inside and outside the finger at each joint. The larger right sheet is headed 'Conceptioni' and returns to quadrature, giving rectilinear surfaces equal to curved ones and showing, through a numbered series of figures, that a curvilinear triangle can be converted into an equal rectilinear one. Further untranscribed prose runs across the right sheet.

Leonardo describes the principal finger motions, extending and bending, which may act on the first, second or all three joints at once; if the first two joints are blocked, the third bends more easily. Beyond these there are four further motions — two up-and-down, two side-to-side — each made by a single cord, from which infinitely many others follow using two cords. The thick cords lie inside the finger and the thin ones outside, with cords set inside at every joint and none outside.

Rectilinear surfaces equal to curved ones

Under the heading 'Conceptioni,' Leonardo states the aim of giving rectilinear surfaces equal to curved surfaces. All curved surfaces to which equal rectilinear squares can be given are squared in themselves.

A semicircle less its two portions is a right triangle

If the curved boundaries are removed from a plane figure of curved sides, it remains rectilinear; if their value is taken within the figure, it gains double curvature and equals the first right triangle. Removing the two greatest portions from a semicircle leaves a right triangle.

Converting a curvilinear triangle into a rectilinear one

Across a numbered series of figures (Prima, Seconda, Terza, Quarta), the curvilinear triangle a d is shown worth the rectilinear triangle d b c. In the third demonstration the triangle S v is converted into the rectangular quadrilateral n S t, squaring the triangle of the first figure.

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

Foglio 273 a Recto

Foglio 273 b Recto

Dall'alto.

Lungo il margine destro, figura con:

Figura con:

Figura con:

Figura con:

In basso, figura:

Entro la figura:

Sheet 273a Recto

Sheet 273b Recto

From the top:

Along the right margin, figure with:

Figure with:

Figure with:

Figure with:

Below, figure:

Within the figure:

Finger motion and the squaring of curved triangles

On this page

The motion of the fingers and their tendons

Rectilinear surfaces equal to curved ones

A semicircle less its two portions is a right triangle

Converting a curvilinear triangle into a rectilinear one