#Hornblower is very harsh about his lack of mathematical understanding

[sanguinarysanguinity]

Having in the foregoing Section, explained the Geometrical Method of Constructing the Tables of Sines, Tangents, Secants, &c. we come now to shew the Arithmetical Method, which we shall endeavour to render as plain and easy as the Nature of the Subject will admit of, that the Learner may proceed the more chearfully, and, with a little Application, became a perfect Master of this noble Branch of the Mathematics, it being the Foundation on which the whole Doctrine of Trigonometry, and, consequently, of Navigation, Astronomy, &c. is entirely grounded.

John Barrow, Navigatio Britannica, or, A Complete System of Navigation, published 1750.

What a pleasingly reassuring sentence! Everything will be fine! Do not worry! Proceed chearfully! 

But chearfully rhymes with fearfully for a reason...

Because immediately after this point Barrow explains how to build trigonometric tables by hand, and in the process he descends into the square roots of rational expressions, binomial expansions of the same, and calculus. Fucking calculus, dude. With ZERO explanation of any of the above. Seriously, none. He was defining “rectangles” (products) a few pages back, and now we’re dealing with fluxions.

SINK OR SWIM TIME.

I’m barely managing to follow all this, and I have advanced degrees in this shit. Of course it serves me right for attempting to skim it, but honestly, I am cracking up SO HARD here at Barrow’s optimism in the preface to this section.

(And we haven’t even gotten to spherical trigonometry yet, oh boy!)