Editing Tycho Brahe (section)

===Observational astronomy===
[[File:Fotothek df tg 0005915 Astronomie ^ Messinstrument.jpg|thumb|Brahe's illustration of his [[sextant]], from his [[star catalogue]] ''Astronomiae instauratae mechanica'', 1602]]
Tycho's view of science was driven by his passion for accurate observations, and the quest for improved instruments of measurement drove his life's work. Tycho was the last major astronomer to work without the aid of a [[telescope]], soon to be turned skyward by [[Galileo Galilei]] and others. Given the limitations of the naked eye for making accurate observations, he devoted many of his efforts to improving the accuracy of the existing types of instrument{{snd}}the [[sextant]] and the quadrant. He designed larger versions of these instruments, which allowed him to achieve much higher accuracy. Because of the accuracy of his instruments, he quickly realized the influence of wind and the movement of buildings, and instead opted to mount his instruments underground directly on the bedrock.{{sfn|Christianson|2000|p=83}}

Tycho's observations of [[star|stellar]] and [[planet]]ary positions were noteworthy both for their accuracy and quantity.{{sfn|Swerdlow|1996|pp=207{{ndash}}210}} With an accuracy approaching one arcminute, his celestial positions were much more accurate than those of any predecessor or contemporary{{snd}}about five times as accurate as the observations of Wilhelm of Hesse.{{sfn|Høg|2009}} {{harvcoltxt|Rawlins|1993|p=§B2}} asserts of Tycho's Star Catalog D, "In it, Tycho achieved, on a mass scale, a precision far beyond that of earlier catalogers. Cat D represents an unprecedented confluence of skills: instrumental, observational, and computational, all of which combined to enable Tycho to place most of his hundreds of recorded stars to an accuracy of ordermag 1'!"

He aspired to a level of accuracy in his estimated positions of celestial bodies of being consistently within an [[minute of arc|arcminute]] of their real celestial locations, and also claimed to have achieved this level. But, in fact, many of the stellar positions in his star catalogues were less accurate than that. The median errors for the stellar positions in his final published catalog were about 1.5', indicating that only half of the entries were more accurate than that, with an overall mean error in each coordinate of around 2'.{{sfn|Rawlins|1993|p=12}} 

Although the stellar observations as recorded in his observational logs were more accurate, varying from 32.3" to 48.8" for different instruments,{{sfn|Wesley|1978|pp=42{{ndash}}53, table 4}} systematic errors of as much as 3' were introduced into some of the stellar positions Tycho published in his star catalog{{snd}}due, for instance, to his application of an erroneous ancient value of parallax and his neglect of polestar refraction.{{sfn|Rawlins|1993|p=20, n. 70}} Incorrect transcription in the final published star catalogue, by scribes in Tycho's employ, was the source of even larger errors, sometimes by many degrees.{{refn|1=Victor Thoren{{sfn|Taton|Wilson|1989}} says: "[the accuracy of the 777 star catalogue C] falls below the standards Tycho maintained for his other activities&nbsp;... the catalogue left the best qualified appraiser of it (Tycho's eminent biographer J. L. E. Dreyer) manifestly disappointed. Some 6% of its final 777 positions have errors in one or both co-ordinates that can only have arisen from 'handling' problems of one kind or another. And while the brightest stars were generally placed with the minute-of-arc accuracy Tycho expected to achieve in every aspect of his work, the fainter stars (for which the slits on his sights had to be widened, and the sharpness of their alignment reduced) were considerably less well located." (ii) Michael Hoskin{{sfn|Hoskin|1997|p=101}} concurs with Thoren's finding "Yet although the places of the brightest of the non-reference stars [in the 777 star catalogue] are mostly correct to around the minute of arc that was his standard, the fainter stars are less accurately located, and there are many errors." (iii) The greatest max errors are given by Dennis Rawlins.<ref>{{harvcoltxt|Rawlins|1993}}, p. 42</ref> They are in descending order a 238° scribal error in the right ascension of star D723; a 36° scribal error in the right ascension of D811; a 23° latitude error in all 188 southern stars by virtue of a scribal error; a 20° scribal error in longitude of D429; and a 13.5° error in the latitude of D811.|group=note}}

Celestial objects observed near the horizon and above appear with a greater [[altitude]] than the real one, due to atmospheric [[refraction]], and one of Tycho's most important innovations was that he worked out and published the very first tables for the systematic correction of this possible source of error. But, as advanced as they were, they attributed no refraction whatever above 45° altitude for solar refraction, and none for starlight above 20° altitude.{{sfn|Taton|Wilson|1989|pp=14{{ndash}}15}}

To perform the huge number of multiplications needed to produce much of his astronomical data, Tycho relied heavily on the then-new technique of ''[[prosthaphaeresis]]'', an algorithm for approximating products based on [[List of trigonometric identities|trigonometric identities]] that predated logarithms.{{sfn|Thoren|1988}}