Thursday, 31 October 2013

Pesi e carrucole

Leonardo da Vinci - Codice Arundel

Wednesday, 23 October 2013

Robert Grosseteste in the history of science

Recently I have studied some treatises written by Robert Grosseteste.
Here the links to some papers:

1. arXiv:1302.1885 [pdf] Reflection and refraction in Robert Grosseteste's De Lineis, Angulis et Figuris, Amelia Carolina Sparavigna ,  Subjects: History and Philosophy of Physics (physics.hist-ph)
2. arXiv:1301.3037 [pdf] The four elements in Robert Grosseteste's De Impressionibus Elementorum, Amelia Carolina Sparavigna ,  Subjects: History and Philosophy of Physics (physics.hist-ph)
3. arXiv:1212.6336 [pdf] Robert Grosseteste's colours, Amelia Carolina Sparavigna ,  Subjects: History and Philosophy of Physics (physics.hist-ph)
4. arXiv:1212.1007 [pdf] Sound and motion in the De Generatione Sonorum, a treatise by Robert Grosseteste, Amelia Carolina Sparavigna , Subjects: History and Philosophy of Physics (physics.hist-ph)
5. arXiv:1211.5961 [pdf] Translation and discussion of the De Iride, a treatise on optics by Robert Grosseteste, Amelia Carolina Sparavigna , Subjects: History and Philosophy of Physics (physics.hist-ph)

Wednesday, 16 October 2013

Divergence of a Radial Vector Field

Divergence of a Radial Vector Field, Tags: divergence, field, radial, vector ,

adapted from the discussion at

http://www.physicsforums.com/showthread.php?t=56413

Monday, 7 October 2013

Leonardo da Vinci e l'attrito

Studio di Leonardo da Vinci sull'attrito.

Dal codice ARUNDEL

Tuesday, 1 October 2013

Come vincere la gravità

Un artista in via Garibaldi.

Sunday, 11 August 2013

"Moving dunes on the Google Earth" is my paper on arXiv, published 4 January 2013. It shows how using GH time series you can see the motion of dunes. Here an example.

To see a movie, please visit Moving sand dunes ... post

http://physics-sparavigna.blogspot.it/2013/06/moving-sand-dunes-on-google-earth.html

NASA spacecraft detects changes in Martian sand dunes

May 9, 2012 — NASA's Mars Reconnaissance Orbiter has revealed that movement in sand dune fields on the Red Planet occurs on a surprisingly large scale, about the same as in dune fields on Earth.

Tuesday, 30 July 2013

HD 189733: An Eclipsing Exoplanet Seen In X-rays For First Time

Friday, 12 July 2013

Chords in trigonometry

Source: Wikipedia.

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part. The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos $\theta$ , sin $\theta$ ), and then using the Pythagorean theorem to calculate the chord length:

$\mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right).$

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function.

The chord function satisfies many identities analogous to well-known modern ones:

Name	Sine-based	Chord-based
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1 \,$	$\mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \,$
Half-angle	$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \,$	$\mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \,$
Apothem (a)	$c=2 \sqrt{r^2- a^2}$	$c=\sqrt{D ^2-4 a^2}$
Angle (θ)	$c=2 r \sin \left(\frac{\theta }{2}\right)$	$c=D \sin \left(\frac{\theta }{2}\right)$