# Enclosure: William Lambert’s Table of Logarithms, 4 October 1822

# Enclosure

# William Lambert’s Table of Logarithms

A Table of Logarithms for reducing the Moon’s equatorial horizontal parallax from a sphere to an oblate spheroid, admitting the ratio of the equatorial diameter to the polar axis of the Earth, to be as 320 to 319.

This Table has been constructed on correct principles, for every degree of latitude, (by observation with a quadrant, sextant, &c.) from 0° to 90°: the log. for intermediate minutes and seconds, may, in most instances, be found by taking their proportional part of the first differences annexed; if greater accuracy be required, the equation arising from second differences, with the affirmative sign +, may be applied; but, in no case, will it amount to more than one unit in the last place of the decimal fraction, which would not affect the reduced horizontal parallax 1⁄100 part of a second.

To find the log. for the latitude of the Capitol in Washington,

38.° 52.′ 45.″ North.

Lat. 38.° 0′ | Log. | 9.9994858. | ||

Diff. 231. prop: part for 52.′ 45″ | – 203. | |||

Constant log. for lat. 38.° 52.′ 45″ | 9.9994655. | |||

To find the log. for Greenwich, lat. 51.° 28.′ 40.″ North. | ||||

Lat. 51.° 0.′ | Log | 9.9991800. | ||

Diff: 231. prop. part for 28.′ 40″ | – 110. | |||

Constant log. for lat. of Greenwich, | 9.9991690. | |||

To find the log. for lat. 5.° 20.′ allowing the equation of second diff: | ||||

Lat. 5.° 0.′ | Log. | 9.9999896. | ||

Diff: 45. prop: part for 20′ | – 15. | |||

Second diff.–8 Equation for 20′ | + 1. | |||

Log. correct for Lat. 5.° 20′ | 9.9999882. |

To find the reduced horizontal parallax.

To the log. for the lat. required, add the log. sine of the Moon’s equatorial hor. parallax, the sum, rejecting radius, will be the log. sine of the horizontal parallax, reduced. Or,

The common log. in seconds, of the equat. hor. parallax, may be used instead of the log. sine, without a sensible variance in the result.

Examples.

Reduced the reduced hor: par: for lat. 38.° 52.′ 45.″ when the Moon’s equatorial hor. parallax is 1.° 0.′ 20.″?

Log. for lat. required, | 9.9994655. | ||

° ′ ″ | |||

Moon’s eq: hor. par. | 1.0.20 | log. sine, | 8.2442611. |

〃 hor: par: reduced, | 1.0.15. 547 dec1 | Sine | 8.2437266. |

Or,

1.° 0.′ 20″ = | 3620″ | log. | 3.5587086. |

Log. for lat. | 38.° 52.′ 45″ | 9.9994655. | |

hor: par: reduced, | 1.° 0.′ 15.″ 547. dec. | = 3615.″ 547 dec log. | 3.5581741. |

It will be always easier in practice, and sufficiently exact, to use the common log. in Seconds, of the Moon’s equat. hor. parallax, instead of the log. sine.

Thus may the Moon’s hor. parallax be reduced for any lat. from 0.° to 90°, according to the ratio of 320 to 319, by means of this Table, with little labour, and great accuracy. The reduced latitude, from 0.° to 90.° according to the same ratio, may be found at all times, by the following process—

To the constant log. 9.9972814, add the log. tangent of the latitude by observation, the sum, rejecting radius, will be the log. tangent of the latitude, reduced.

Examples.

Lat. of the Capitol, in Washington, by obsn | 38.° 52.′ 45.″ tangt | 9.9064956. | |

Constant log | 9.9972814. | ||

Lat. of the Capitol, reduced, | 38.° 42.′ 14.″ 51 dec tangt | 9.9037770. | |

Required the reduced lat. of Greenwich, | 51.° 28.′ 40.″ tang. | 10.0990491. | |

Constant log. | 9.9972814. | ||

Lat. reduced, | 51.° 18.′ 10.″ 418 dec | 10.0963305. |

The advantage of a table of this kind, and of the rule to reduce the latitude, will be seen in calculations of the longitude of a place on the earth, from solar eclipses, and occultations of fixed Stars by the Moon, which are known to be among the best methods hitherto devised for the purpose.

William Lambert.

MS (ViU: TJP); entirely in Lambert’s hand.

1. Lambert evidently rewrote these two words to replace a missing number and uncanceled “dec” affected by a chip at bottom of page.

# Index Entries

- astronomy; and lunar calculations search
- astronomy; and solar observations search
- astronomy; astronomical instruments search
- Capitol, U.S.; latitude and longitude measurements at search
- Greenwich Observatory, England; and prime meridian search
- Lambert, William; Table of Logarithms search
- latitude; calculations for U.S. Capitol search
- latitude; logarithms for calculating search
- longitude; W. Lambert on methods of calculating search
- mathematics; logarithms search
- moon; and astronomical calculations search
- quadrants; in astronomy search
- sextants; in astronomy search
- sun; and calculation of longitude search