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Antiphon was also a capable mathematician. Antiphon, alongside his companion Bryson of Heraclea , was the first to give an upper and lower bound for the value of pi by inscribing and then circumscribing a polygon around a circle and finally proceeding to calculate the polygons' areas.
The idea originated in the late 5th century BC with Antiphon, although it is not entirely clear how well he understood it. [1] The theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. [2]
Machin's formula [4] (for which the derivation is straightforward) is: = The benefit of the new formula, a variation on the Gregory–Leibniz series ( π / 4 = arctan 1), was that it had a significantly increased rate of convergence, which made it a much more practical method of calculation.
Antiphon was a statesman who took up rhetoric as a profession. He was active in political affairs in Athens, and, as a zealous supporter of the oligarchical party, was largely responsible for the establishment of the Four Hundred in 411 (see Theramenes); upon restoration of the democracy shortly afterwards, he was accused of treason and condemned to death. [1]
– Antiphon: c. 470 BC – 410 BC – Hippocrates: 465 BC – 398 BC – Theodorus: ... Timeline of mathematics; References This page was last edited on 30 March ...
In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized ...
Printed antiphonary (ca. 1700) open to Vespers of Easter Sunday. (Musée de l'Assistance Publique – Hôpitaux de Paris)An antiphonary or antiphonal is one of the liturgical books intended for use in choro (i.e. in the liturgical choir), and originally characterized, as its name implies, by the assignment to it principally of the antiphons used in various parts of the Latin liturgical rites.
In the Gregorian system, psalm tones are based on the mode of the antiphon. Ambrosian chants, including psalm antiphons, do not conform to the Gregorian system of modes. Each Ambrosian psalm antiphon belongs to one of four different series depending on its final pitch. Within each series, there are several possible psalm tones corresponding to ...