Last Modified on Saturday, August 10, 2002.A Chronology of Square Roots
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Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents. About 1750BC
The Babylonians compile tables of square and cube roots. They have a very accurate value for R(2).
About 425BC
Theodorus of Cyrene shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17 are irrational. Theodorus approximates R(3) as 7/4 since 49/16 is almost equal to 3.
About 375BC
Archytas gives a method for approximating R(n). This method was known to the Babylonians.
About 350BC
Plato approximates R(2) as 7/5 since 49/25 is almost 2.
About 325BC
Aristotle gives a proof that R(2) is irrational. This proof was known long before him.
About 300BC
Euclid, in The Elements, says that the line AB is divided in "extreme and mean ratio" (phi) by C if AB:AC = AC:CB. The Elements also deals with the idea of incommensurability, although the proof that R(2) is irrational was probably added in later editions of the work.
About 250BC
In Measurement of the Circle, Archimedes gives an approximation of the value of pi with a method which will allow improved approximations. He declares that 3 10/71 < pi < 3 1/7 and approximates pi as 211875/67441 = 3.14163. He needs to approximate R(3) to make these calculations, so he bounded R(3) by 1351/780 > R(3) > 265/153, which he probably finds by what will later be called Heron's Method.
About 50
Heron begins to compute approximate ratios, and in his work he gives approximate values for the ratio of the area of the pentagon to the area of the square of one side. He also gives a method for approximating the square root of a number (this method was probably known to the Babylonians).
About 120
Ptolemy calculates the side of a regular pentagon in terms of the radius of the circumscribed circle. He also gives a value for R(3) that is correct to 6 decimal places.
About 130
Chung Hing uses pi=R(10) = 3.16....
1040
Ahmad al-Nasawi writes Al-Muqni'fi Al-Hisab Al-Hindi which studies four different number systems. He explains the operations of arithmetic, particularly taking square and cube roots in each system.
About 1343
John of Meurs extracts R(2) as equal to (R(2000000))/1000, which is approximately equal to 1414/1000 or 1.414. Although it is clear that this is done using base ten computations, he uses mixed notations (which is very common during this time period) and expresses his result in sexagesimal (base 60) terms.
About 1375
Narayana gives a rule to calculate approximate values of a square root.
1509
Pacioli writes Divina Proportione (Divine Proportion), which is his name for the golden ratio. The book contains little new on the topic, collecting results from Euclid, Fibonacci and other sources on the golden ratio. He states (without any attempt at a proof or a reference) that the golden ratio cannot be rational. Pacioli also gave a method for the approximation of square roots (using a special case of Newton's method).
1525
Rudolff introduces a symbol which resembles the modern day notation for square roots in his Die Coss, the first German algebra book. He calculates the solutions for polynomials with rational and irrational coefficients and was aware that ax^2 + b = cx has 2 roots.
1544
Stifel publishes Arithmetica Integra which contains binomial coefficients and the notation +, -, and the modern day square root symbol.
1590
Cataldi uses continued fractions in finding square roots.
1613
Cataldi publishes Trattato del Modo Brevissimo di Trovar la Radice Quadra Delli Numeri in which he finds square roots using continued fractions. His methods make precise some ideas which date back to Heron.
1673
Leibniz demonstrates his incomplete calculating machine to the Royal Society. It can multiply, divide and extract roots.
1814
Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.
1879
Fermat gives his method of infinite descent, which can be used to prove the irrationality of R(2).
Last Modified on Saturday, August 10, 2002.