# What astronomy efforts required multiplication of large numbers around 1600?

I'm reading an article about the history of logarithms and it says:

One problem that was plaguing people at the time, especially astronomers, was arithmetic. Astronomical calculations required the multiplication and division of very large numbers, something that’s pretty hard to do without a calculator

I'd like to know some specific astronomy examples that would require the multiplication of large numbers by then.

• properly multiplying sines and cosines of (astronomy, sextant) readings to get (orbital parameters, accurate position) takes some doing. Jun 25, 2020 at 20:58

This calculation is always performed when for example one transfers between the coordinate systems. Positions of celestial bodies are usually described in ecliptic coordinates, while observations are made in equatorial coordinates with stationary telescopes, and in practical astronomy (navigation) horizontal coordinates must be used. A typical formula used in navigation looks like this: $$\sin h=\sin\phi\sin\delta+\cos\phi\cos\delta\cos t,$$ where $$\phi$$ is latitude, $$\delta$$ is the declination $$t$$ is the hour angle, and $$h$$ the altitude of a celestial body. To find the ship position with the most common method, using a chronometer, one has to use this formula at least twice (for two stars, for example). Without a chronometer, a more complicated calculation is required.
By the way the first log tables (of Napier) tabulated not $$\log x$$ but $$\log\sin x$$, that is they were specially designed for trigonometric calculations in astronomy/geodesy.