Math

Instruments and mathematics

The measurement of latitude
(latitude of Monterrey : 25°40')

Determination of the latitude using the Polar star

Marching from the equator to the latitude of Monterrey

Determination of the latitude using the altitude of the Sun at noon

The position of the Sun at noon, the day of an equinox, on the equator

The position of the Sun at noon, the day of an equinox, at Monterrey

The position of the Sun at noon, another day (in fact summer solstice), at Monterrey

The formula is :

Latitude = 90° - (altitude of the Sun - declination)

What happened if the Sun is below the celestial equator ?

The position of the Sun at noon (the winter solstice), at Monterrey

The formula is :

Latitude = 90° - (altitude of the Sun + declination)

The beauty of negative numbers and the law of signs :

If the Sun is below the celestial equator, and if we take as a convention that the declination is negative, then the formula is transformed to become the same as when the sun is over the celestial equator.

Navigation : checking ship position

Measuring with logs and the problem of calculating the longitude

(from Oeuvre de Champlain, t. I, Québec, 1870, p. 1381)

Example 1 : Correction of an estimate, by comparing latitude
How on can find out if a pilot has made an accurate reckoning, and prick the chart.

(From Heidenreich, C.E. Explorations and Mapping of Samuel de Champlain 1603-1632, Toronto : York University, 1976)

Example 2 : How to prick the chart

distance sailed = difference of latitude / cosine of course

Example 3 : Another way of reckning and pricking the chart