# To Thomas Jefferson from William Lambert, 6 July 1825

City of Washington, July 6th 1825.

Sir,

The form of the Earth having been ascertained by various experiments in Europe and South America, to be that of an Oblate Spheroid; and it being now admitted, that the ratio of the equatorial diameter to the polar axis is as 320 to 319, the measure of a degree in any latitude, according to that ratio, claims our attention; and the following calculations connected with this subject, are submitted to your consideration.—

The French mathematicians who were sent to the province of Quito, in Peru, during the last century, determined the measure of a degree at or near the Equator, to be 56,750 toises, answering to 363,006.4 American feet. On the principles of Conic sections, the measure of a degree in lat. 30° will be 56,883.68 toises, = 363,861.5 feet, and in lat. 60° 57,151.38 toises, = 365,573.9 feet. supposing these measures to be correct, it is required to find the ratio of the Earth’s diameters.—

Let e, represent the Equatorial diameter, p, the polar axis of the Earth, M, the measure of a degree in the greatest latitude, L, the greatest latitude, = 60° m, the measure of a degree in the least latitude, l, 30° the least latitude, to find e and p

e/p √ | (M ⅓, sine L, + m ⅓, sine l.)(M ⅓ sine L, - m ⅓, sine l.) |

(m ⅓ cosine l, + M ⅓, cosine L)(m ⅓ cosine l, - M ⅓ cosine L.) |

The process of calculation will be given in full, according to the formula.—

M ⅓ log. | 1.8543251. | ||||

L, 60° sine, | 9.9375306. | ||||

1.7918557 | log. of | 61.92353. | |||

m ⅓, log | 1.8536454 | ||||

l, 30° sine | 9.6989700 | ||||

1.5526154. | log. of. | 35.69565. | |||

Sum, | 97.61918 | log. | 1.9895351. | ||

diff: | 26.22788. | log. | 1.4187631. | ||

2 √ | 3.4082982 | ||||

e, = 50.599836 | log. | 1.7041491. | |||

M ⅓, log. | 1.8543251. | ||||

L, cosine | 9.6989700. | ||||

1.5532951 | log. of | 35.75157. | |||

m ⅓, log. | 1.8536454. | ||||

l, cosine, | 9.9375306. | ||||

1.7911760. | log. of | 61.82669. | |||

Sum, | 97.57826, | log. | 1.9893532. | ||

diff. | 26.07512, | log. | 1.4162264. | ||

2 √ | 3.4055796 | ||||

p. = 50.441709 | log. | 1.7027898. | |||

As, 50.599836 | ar. co. log | 8.2958509. | |||

To 50.441709 | log. | 1.7027898. | |||

So 320 | log. | 2.5051500. | |||

To 319 | log. | 2.5037907. |

Hence, the measure of a degree in the latitudes of 30° and 60° according to the ratio of 320 to 319, as herein stated, is found to be correct.

Let the measure of a degree at the Equator be given, = 363,006.4 American feet, to find the measure of a degree in the latitudes of . 30° and 60° N. respectively, according to the ratio of 320 to 319.

p, polar axis, ratio, 319. ar. co. log. | 7.4962093. |

e, equat. diam. ratio, 320 log. | 2.5051500. |

l, latitude 30°0′ tangent, | 9.7614394. |

Arch, (A) 30°4′39″77dec tangent, | 9.7627987. |

Arch (A), triple cosine, ar. comp. | 0.1884300. |

l, 30°0′ triple cosine, | 9.8125918. |

feet | |

measure at the equator, 363,006.4 log. | 5.5599143. |

measure in lat. 30°0′ 363,861.5 log. | 5.5609361. |

p, polar axis, ratio, 319. ar. co. log. | 7.4962093. |

e, equat. diam. ratio, 320 log. | 2.5051500. |

l, latitude 60°0′0′′ tangent, | 10.2385606. |

Arch, (A) 60°4′39″33dec tangent, | 10.2399199. |

Arch (A), triple cosine, ar. comp. | 0.9061509. |

l, , lad 60° triple cosine, | 9.0969100. |

feet | |

measure at the equator, 363,006.4 log. | 5.5599143. |

measure in lat. 60° 365,573.9 log. | 5.5629752. |

This may be done for any other latitude from the equator to either of the poles: to find the measure at the pole, or 90° from the equator, the following process is given—

e, - p, = 0.0013593, × 3, = | 0.0040779. | ||

feet. | |||

measure at the equator, | 363,006.4. | log. | 5.5599143. |

measure at the pole, | 366,431.0. | log. | 5.5639922. |

difference, | 3424.6. |

It is left at your option to accept these calculations for your own use, or to present them to the university near Charlottesville: in either case, they are transmitted as a tribute of the high respect and esteem with which I have the honor to be,

William Lambert

DLC: Papers of Thomas Jefferson.