#the sohcahtoa battle cry??

[ladysqueakinpip]

cinematic parallels

[rottmnt-2k18]

Have we gushed about Donnie using SOHCAHTOA as a battle cry yet!? YES, moar science shouts, Donatello!!! looooool

my favorite was when he shouted horticulture during shadow of evil 

[sanguinarysanguinity]

Navigatio Britannica: Planar Trigonometry

Chapter 4: Of Trigonometry, Sections I - V

The first half of this chapter is Planar Trigonometry, which I learned in high school and have used on-and-off ever since; the second half of this chapter is Spherical Trigonometry, which I know nothing about. Consequently, I’m dividing this chapter into two parts -- before we let John Barrow attempt to teach me spherical trig (wish me luck!), I want to do a fast recap of what he has to say about planar trig...

Section I: Definitions

Everything is defined geometrically, on the unit circle, via a diagram that I have yet to find in this scan. (Also, even if I do find the page, I don’t have much hope that it will have been scanned correctly, since Google’s scanning machine can turn pages but not unfold them.) Happily, Barrow is pretty good about describing his figures in enough detail that I can reconstruct them as I go, which is the only reason I was able to understand anything in this chapter.

Not many surprises here, although I did learn that cosine, cotangent, and cosecant are the sine, tangent, and secant of the complementary angle, and likewise that the tangent of an angle is called the tangent because its physical instantiation lives on a line tangent to the circle. Also, Barrow defines the Verse-Sine, which was new to me: geometrically, it is the part of the radius that isn’t the cosine. (Algebraically, it is 1 - cosine.) Wikipedia says the versine was important to navigation, so I assume it will come up later.

Section II: Geometrical Constructions of the Tables of Sines, Tangents, Secants, &c.

In which we are instructed to build ourselves a unit circle, mark it off in 1-degree intervals, and construct ourselves a... well, it’s gonna look a bit like a ruler, but it’s going to measure 1 to 90 degrees, on several parallel scales: chords, sines, versed-sines, tangents, etc. To make this thing, you use your compass and measure the length of a chord for a 10-degree angle, then mark it on your chords-scale, and label it “10 deg.” Repeat for the other 89 degrees, and ta-dah, you have a chords-scale! Then do it again for sines, versed-sines, tangents, and so on. When we get to actually solving trig problems, how to use this scale is one of the three standard methods that Barrow is going to teach us.

Section III: Arithmetical Constructions of the Tables of Sines, &c.

First off, Barrow reassures us this is going to be easy-peasy, no need to panic -- which is our first cue that panicking will be required before we’re done.

But true to his word, Barrow starts out easy, using similar triangles to prove all the basic trigonometric identities: tan = sin/cos, sec = 1/cos, etc. All well and good, except it’s all done in proportions and nothing is called out by name, only by referring to various line-segments in his nowhere-to-be-found unit circle diagram. All of which makes it difficult to absorb at a glance, but once you finish decoding everything this is basically just SOHCAHTOA.

Then he proves two variants on the standard trigonometric sum/difference formulas (although he expresses them as proportions and via verbal descriptions, talking about the means of equi-different angles and the differences between them): 

cos x = (1/2) (sin y + sin (y + 2x)) / sin (y + x)

sin x = (1/2) (sin (y + 2x) - sin y) / cos (y + x)

You can verify those via the standard trigonometric sum/difference identities if you want. (I did.) But they’re also pretty straightforward geometrically, if you take the time to very carefully reconstruct what his diagram must have been: in the end, it’s all just similar triangles. He then proves several corollaries -- which in hindsight are simple enough (just straightforward algebraic manipulations, multiplying everything by two, or both sides by the denominator), but sadly, I lost MANY HOURS to a rash of typos in them.

Then. 

Oh, then.

All hell breaks loose as he endeavours to prove that a semi-circle has an arc-length of pi. I admit to not following this bit: I haven’t seen Newton’s notation for calculus since I was seventeen, when that one weirdo physics professor used it in lectures, and I didn’t really feel like re-teaching it to myself for this. Nor did I really want to get into re-teaching myself binomial expansions. Also, the type-face on all the fractions in the expansions was super-squinchy to read, and you know what, fuck it, I think it’s well-established that a semi-circle has an arc-length of pi, let’s move on.

The point of establishing that a semi-circle has an arc-length of pi is so that we can calculate the arc-length of one minute (simply divide pi by 10,800 minutes, easy-peasy), which we will then use as an approximation of the sine of one minute. ... Which, okay, I suppose if your angle is small enough and your applications are practical enough you can get away with that? But it makes the mathematician in me cry, I’m just saying. (Even as I admit that you really can get away with it for most purposes: according to my handy-dandy TI-84 Plus, pi/10800 differs from sin(1′) in the ninth significant digit. But Dr. Roberts and Dr. Chrestenson would never have let me get away with that shit, never mind that I also have an engineering degree and thus should be okay with this kind of ruthless practicality. In my soul there is a mathematician and an engineer battling to the death over questions like these, you simply don’t know how much shit like this wounds me.)

Anyway, once I finally got over my fit of vapours...

Now that we have an approximation for the sine of one minute, we can calculate the cosine of one minute via the pythagorean trigonometric identity, and then...

And now I want to cry again, because now we get to build our table of sines (and along with it, our table of cosines), minute by freaking goddamn minute, by using the above equations like so:

2 cos (1′) sin (1′) - sin (0′) = sin (2′)

2 cos (2′) sin (2′) - sin (1′) = sin (3′)

2 cos (3′) sin (3′) - sin (2′) = sin (4′)

...

Continue until I cry blood and the seas boil dry.

(At one point Barrow admits that it’s possible to build this table in 5-minute increments and interpolate the intervening minutes when you need them. While this reduces the task to 1/5th of the original, I still want to hug and rock myself and cry.)

Happily, I don’t need to cry, because Barrow includes these tables in the book? But someone cried blood to make those tables, and John Barrow wants us all to know it.

Section IV: Actual Trigonometry Problems, At Long Last!

A ton of sample problems, all worked three ways:

Geometrically: Basically, use a compass and straight edge and your scale-thingie of chords/sines/secants that you made earlier, and draw a triangle of  the correct proportions. Then just read/measure your answer right off the actual triangle in question, ta-dah! No abstract math required, just pretty pictures!

Arithmetically: What you learned in high-school, using the tables that someone cried blood over but without calculators (although you can use logs if you want to skip ahead to chapter five for them!) God, it looks miserable and grindingly awful, and I admit I don’t have the strength of character to follow any of these calculations through to the end.

Magic, I mean, Gunter’s Scale: The instructions here are amazingly low-key -- use your chart-dividers (what tumblr calls a pointy-leg-man what it likes to make walk on tippy-toes across charts) to measure off an interval on one scale, and then drop that same interval across the appropriate second scale, and voila! You have found your answer!

Of course I wanted to know what this magical tool is!

Apparently it was a slide rule without the slidey parts -- you used your dividers to accomplish what the slidey bit does on a slide rule -- but with some extra scales especially chosen for the convenience of navigators. 

Apparently these things were so common among navigators that they were simply called Gunters, and I WANT ONE SO BAD. Here’s a nifty article about them, complete with pictures, and did I say? I WANT ONE SO BAD. I collect old-school mathematical tools and I WANT ONE SO BAD.

Ahem.

Anyway, Section V is more trig homework, except now we’re no longer dealing with right triangles. I admit it, I skimmed this like fuck.

And ta-dah! That’s Planar Trigonomometry, according to John Barrow in Navigatio Britannica, or, A Complete Guide to Navigation, pub. 1750!

Up next: Section VI - Spherical Trigonometry, what makes William Bush cry. Do I have the fortitude to teach myself spherical trig? PLACE YOUR BETS NOW.