Concave burning mirror: angle of incidence and the converging light pyramid

Constructing the angle of incidence between equal opposite angles

From line a d a compass draws the curve d f e; lines a f and b f (parallel to a d) fix the point f, where a semicircle d h g is drawn. Carrying the measure d h onto the semicircle at i g yields the angle of incidence h f i, set between the two equal opposite angles d f h and g f i. The same procedure is then repeated with line a g and c g.

Measuring the sun to calculate the burning mirror's focus

A cancelled note proposes taking the measure of the sun at the summer solstice in June, placing its pyramid at the centre of the world and cutting it at the diameter of the mirror, to see what the parallels a b d f restrict within one braccio a d. Leonardo insists on first obtaining the sun's diameter and distance before making the calculation.

Circles in proportion as the squares of their diameters

A cancelled maxim, written upside down, states that the proportion from circle to circle is as that from square to square built on their diameters. It underpins the numerical estimates of how the light pyramid shrinks.

Calculating the potency of the one-oncia pyramid

Taking n as the base of a pyramid of one oncia, Leonardo tabulates its diminishing power down the pyramid: d 20736, c 82944, b 331776, a 1327104, worked out through columns of multiplication (144, 576, 20736, 41472, 82944, 165888). The smaller circle is reckoned 1/20736 of the larger and 20736 times more potent.

Rays do not intersect at the centre of the concave mirror

If the rays n m had to pass through the centre o they would end only at a b and would not fill the whole concavity c a b d. Since experience shows the contrary, Leonardo concludes that these rays do not intersect at the centre o.

Chord and sagitta of the mirror's concavity

The pyramid a b c is to be cut in the middle of its hypotenuse at o d; the semicircle b f a g c is half its base built on the semidiameter b d (or a d). Because the sought concavity has lips equal to the diameter b c, the line b c is the chord of the required arc, whose sagitta is o d (or m d).

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Nella parte mediana del foglio, a sinistra della figura al centro, scritte non speculari di altra mano:

Al centro del foglio figura con lettere.

Sotto la figura:

Sul lato sinistro della figura:

Nella metà superiore del foglio, in alto a sinistra:

A sinistra della figura posta al centro del foglio, testo cassato:

In basso a destra, altro testo cassato:

Nella metà superiore del foglio, al centro sopra e di fianco ad alcune operazioni matematiche, capovolto, testo cassato:

Lungo il margine superiore, capovolti, figura di tre cerchi con lettera n affiancata dal testo cassato:

Nella metà superiore del foglio a sinistra lungo il margine, capovolti, testo cassato affiancato da una figura, e sopra verso il centro, figura con lettere a b c d e altri diagrammi seguiti da operazioni matematiche (cassate quelle lungo il margine):

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Figura con lettere e numeri.

A sinistra della figura:

A destra della figura:

Nel margine destro del foglio figura con lettere.

A lato della figura:

Nel margine destro, sotto la figura:

Nella metà inferiore del foglio, al centro, figura con lettere.

A sinistra della figura:

In the middle part of the sheet, to the left of the central figure, non-mirror writing in another hand:

At the centre of the sheet, a figure with letters.

Beneath the figure:

On the left side of the figure:

In the upper half of the sheet, at the top left:

To the left of the figure placed at the centre of the sheet, cancelled text:

At the bottom right, another cancelled text:

In the upper half of the sheet, at the centre above and beside some mathematical operations, upside down, cancelled text:

Along the upper margin, upside down, a figure of three circles with the letter n beside the cancelled text:

In the upper half of the sheet on the left along the margin, upside down, cancelled text beside a figure, and above towards the centre a figure with letters a b c d and other diagrams followed by mathematical operations (those along the margin cancelled):

Figure with letters and numbers.

To the left of the figure:

To the right of the figure:

In the right margin of the sheet, a figure with letters.

Beside the figure:

In the right margin, beneath the figure:

In the lower half of the sheet, at the centre, a figure with letters.