# Ptolemy's table of chords

**table of chordshis table of chordscalculation of chordschords tabulated by Ptolemysurviving table of Ptolemytabulated the value of the chord functionthose of**

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy.wikipedia

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### Almagest

**cataloghis book on astronomyMagna Syntaxis**

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy.

### Chord (geometry)

**chordchords chord**

A chord of a circle is a line segment whose endpoints are on the circle.

The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees.

### Trigonometric tables

**trigonometric tableGenerating trigonometric tablessine table**

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy.

These were used to construct Ptolemy's table of chords, which was applied to astronomical problems.

### Sexagesimal

**sexagesimal systembase 60base-60**

The fractional parts of chord lengths were expressed in sexagesimal (base 60) numerals.

In particular, his table of chords, which was essentially the only extensive trigonometric table for more than a millennium, has fractional parts of a degree in base 60.

### Ptolemy

**Claudius PtolemyClaudius PtolemaeusPtolemaic**

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy.

### Ptolemy's theorem

**Ptolemaios' theorem**

He used Ptolemy's theorem on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs.

Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

### Canon Sinuum (Bürgi)

**Canon Sinuum**

One such table is the Canon Sinuum created at the end of the 16th century.

* Ptolemy's table of chords

### Versine

**haversinecoversinehavercosine**

In fact, the earliest surviving table of sine (half-chord) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).

### Cyclic quadrilateral

**cycliccyclic quadrilateralsBrahmagupta quadrilateral**

The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle.

### Aristarchus's inequality

To find the chords of arcs of 1° and 1⁄2° he used approximations based on Aristarchus's inequality.

Ptolemy used the first of these inequalities while constructing his table of chords.

### Scale of chords

**line of chords**

* Ptolemy's table of chords

### Greece

**GreekHellenic RepublicGreeks**

### Egypt

**EgyptianEGYArab Republic of Egypt**

### Theoretical astronomy

**mathematical astronomytheoreticalmathematical astronomers**

### Sine

**sine functionsinnatural sines**

It is essentially equivalent to a table of values of the sine function.

### Hipparchus

**HipparchosHipparchus of NicaeaHipparchus of Nicea**

It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of 7 1⁄2° = ⁄24 radians).

### Circle

**circularcircles360 degrees**

A chord of a circle is a line segment whose endpoints are on the circle. The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides.

### Linear interpolation

**linearly interpolatedlinearly interpolatelinearly interpolating**

Thus, it is used for linear interpolation.

### Euclidean geometry

**plane geometryEuclideanEuclidean plane geometry**

Chapter 10 of Book I of the Almagest presents geometric theorems used for computing chords.

### Euclid

**Euclid of AlexandriaEuklidGreek Mathematician**

Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find the chords of 72° and 36°.

### Euclid's Elements

**ElementsEuclid's ''ElementsEuclid**

Ptolemy used geometric reasoning based on Proposition 10 of Book XIII of Euclid's Elements to find the chords of 72° and 36°.

### Pentagon

**{5}pentagonalpentagons**

That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

### Hexagon

**hexagonal{6}regular hexagon**

That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

### Decagon

**10decagonal{10}**

That Proposition states that if an equilateral pentagon is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

### Quadrilateral

**crossed quadrilateraltetragonquadrangle**

The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides.