Gottfried Wilhelm Leibniz's (1646–1716) first great mathematical achievement was the ‘arithmetic quadrature’ of the circle through his alternating series: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

He communicated the result to his mathematical mentor Christiaan Huygens (1629–95), who thought it ‘very beautiful and very pleasing’. Isaac Newton (1642–1726) welcomed Leibniz’s work as ‘very elegant’. Leibniz himself wrote that there was no simpler or more beautiful way of expressing the area of a circle using rational numbers.

In a short note concerning the beauty of theorems, Leibniz wrote:

‘Theorems are not intelligible except by their signs or characters. […] The beauty of theorems consists in the beautiful arrangement of their characters.’

To illustrate ‘beautiful arrangement of characters’, Leibniz gave the example of a theorem concerning Berthet’s curve (shown in red in 1st attached image). The detail of its definition is not important here, but it is defined with reference to an arc AC centred at B.

Leibniz's result was a way of finding the tangent to the curve at E: take the tangent to the arc at its intersection with BE (i.e., at D), and find the point F such that FD ∶ DE ∶∶ EB ∶ BD. Then EF is the desired tangent (see 1st attached image).

Why is there a ‘beautiful arrangement of characters’? Because the proportion FD ∶ DE ∶∶ EB ∶ BD is easily remembered via a mnemonic: one can draw the path FD ⋅ DE ⋅ EB ⋅ BD without raising one's pen (2nd attached image).

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#geometry #Leibniz #Huygens #Newton #MathematicalBeauty #aesthetics #HistMath

An enduring locus of mathematical beauty in the seventeenth century concerned curves like the cycloid and the catenary.

A cycloid is the path followed by a point on the circumference of a circle rolling along a straight line (see attached image).

Christopher Wren (1632–1723) proved that the arc length of the cycloid is four times the diameter of its generating circle.

Christiaan Huygens (1629–95) thought Wren's work ‘really beautiful’. Blaise Pascal (1623–62) also called it ‘beautiful’ (even though he also seemed to repudiate any true notion of mathematical beauty in his ‘Pensées’.

Huygens proved that an inverted cycloid was the ‘tautochrone’: the curve along which a body starting from rest and freely accelerated by uniform gravity reaches the lowest point in the same time, independently of its starting point.

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#cycloid #tautochrone #Huygens #HistMath #MathematicalBeauty #Pascal

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Nicht zuletzt wegen der Lichtverschmutzung im RheinNeckarGebiet ist es oft schwierig, Saturn am Himmel zu entdecken, und Titan lässt sich nur mit einem Teleskop finden. Um so erstaunlicher, dass von Darmstadt aus die kleine Sonde Huygens erfolgreich auf dem Saturnmond abgesetzt wurde. Wirf heute Abend einen Blick in den Himmel und staune über das Universum!

#Lichtverschmutzung #RheinNeckarGebiet #Titan #Darmstadt #Huygens #Universum
#Buerstadt #Kalender2026

Corre l'anno 2005. Sotto cieli densi di metano e azoto, la sonda Huygens tocca finalmente il suolo di Titano, la luna più grande di Saturno.

La discesa, durata due ore e mezza, apre una finestra su un mondo che apparentemente non ci appartiene: rocce composte da acqua e idrocarburi, illuminate da una luce spettrale color arancio. Cosa ha a che fare con noi?

La superficie, morbida come sabbia bagnata, ha accolto il lander che si è infilato appena sotto la crosta. Le batterie hanno resistito per novanta minuti, più che abbastanza per raccontarci un paesaggio che somiglia a un sogno congelato.

A –179 °C, Titano racchiude un ambiente chimico - questa è la cosa notevole - che potrebbe ricordare la Terra prima della vita. Ben più che semplice scienza, questo è un invito a immaginare. Forse, in quelle rocce e in quell’aria densa, si nasconde un’eco del nostro stesso passato. Credo sia così: a volte si deve andare lontano, per capire davvero la propria storia.

Crediti immagine : ESA, NASA, JPL, U. Arizona, Huygens Lander

#astrocaffe #huygens #terra #titano #NASA