#navigatio britannica

[sanguinarysanguinity]

COMPLEMENTARY SPERICAL TRIANGLES BOO-YEAH

I HAVE FIGURED OUT WHAT COMPLEMENTARY SPHERICAL TRIANGLES ARE AND IT ONLY TOOK ME WEEKS OF EFFORT

(In my defense, all the illustrations are missing from this book, so I’m having to do it without diagrams, plus the notion of complementary spherical triangles seems to have died with the eighteenth century -- or anyway, I haven’t been able to find more recent text, one with illustrations, that discussed them.)

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Kitty has decided that I have gotten too frustrated in my attempts to verify this one proof in Navigatio Britannica, and has remedied the situation in the way she knows best: by laying on my notes so I can no longer read them.

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Everyone who has had Experience in teaching, must be fully convinced, that a methodical Manner of delivering any Science, is the only Way of rendering it intelligible, and that if this be neglected, the Ideas acquired from such a confused Manner of Instruction, will at best be but faint and languid. For this Reason I have been very careful to compose this Work in a regular Method, that the Learner as he proceeds may have a true Idea of the Subject, and, without Confusion, become a complete Master of the whole A R T.

John Barrow, Navigatio Britannica, or, A Complete System of Navigation

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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!)

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Navigatio Britannica: February 24 x 2

Yesterday’s casually-dropped wtf fact in John Barrow’s Navigatio Britannica (A Complete System of Navigation) is that circa 1750, Leap Day (which he calls an Intercalary Day) occured between February 23rd and February 24th, with the result that February 24th was forty-eight hours long.

(That discussion on the linked page might not make much sense unless you’ve read the several pages previous: in short, he’s saying that in a Leap Year, the days of the week progress in a steady pattern through February 24th, but after February 24th you have to account for everything being one day off for the rest of the year. That is, the trick you use for determining the day of the week for a given year -- the year’s “Dominical Letter” -- needs to be kludged in Leap Years: you’ll need to use one “Dominical Letter” up through February 24th, and a second one after.)

According to Wikipedia, apparently the February 23rd/24th thing dates back to Julius Caesar: the new year started on March 1, so they wanted to get the Leap Day in before then, but February had a lot of overlapping religious festivals, such that it was difficult to slip an extra day in anywhere. But there was a gap in the festival schedule at February 23rd/24th, so that was a sensible place to stick the extra day, the years that you needed an extra day.

Wikipedia doesn’t seem to know when the switchover to February 29th happened, though: a couple of pages claim “the late Middle Ages” (although without citation), but here we are in 1750, in a highly respected navigation text, confidently talking about how February 24th is double-length some years, don’t forget to account for that when determining what day of the week February 25 should be!

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Navigatio Britannica: Minutes, Seconds, Thirds, Fourths

Here’s the first thing that I learned from John Barrow’s Navigatio Britannica, or, A Complete System of Navigation, published in 1750.

“Seconds,” as in 1/60th of a minute, are named such because it’s the second time you took the sixtieth of something. There’s a whole sequence here, and it runs:

Minutes, short for minuta prima, literally the first time you divide something into sixtieths

Seconds, short for secunda minuta, literally the second time you divide the thing into sixtieths

Thirds, literally the third time you divide the thing into sixtieths (a third is equal to one-sixtieth of a second)

Fourths, literally the fourth time you divide the thing into sixtieths (a fourth is equal to 1/360th of a second)

The “something” you’re dividing up can of course be hours or degrees. (And presumably other things, but hours and degrees are the ones I know about.)

Furthermore, ca. 1750, they were actually using fourths for astronomical calculations! This text says that a mean solar month is “30 days, 10 hours, 29 minutes, 6 seconds, 18 thirds, and 50 fourths.” Which, presuming that they’re rounding to the nearest ten “fourths,” means that they were measuring the motion of the sun down to the thirty-sixth of a second. 

(Or were claiming to, anyway -- I can’t help but wonder if there’s a significant digits issue at play here.)

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Navigatio Britannica: Tides

Chapter 2, “On the Flux and Reflux, or Ebbing and Flowing of the Sea”

Terminology Surprises: 

The text kept referring to Full and Change days: the first was obviously the full moon, but took me yonks to realize that Change referred to the new moon.

The moon’s Southing is the moon’s zenith, or when it passes the meridian. Similarly, the various points along its daily moonrise-to-moonset arc are called out by compass points: a SW moon is the time roughly halfway between zenith and moonset. It took me a long time to get comfortable with conceptualizing the moon’s travel in the sky as a lateral arc (instead of a vertical arc), but thinking about summer quarter-to-fullish moons, they do swing a half circle from fairly-east to definitely-south -- and I presume they continue on to fairly-west (I wouldn’t know, I’m usually asleep around then). I presume “Southing” is a northern hemisphere thing. Moreover, I’m guessing from this diagram that for equatorial latitudes, during the summer the full moon is in the south and the new moon is in the north, and vice versa during the winter. But that’s me guessing.

The main content of the chapter is that the high tides for any location are constant with respect to the moon’s position in the sky: if high tide occurs at a “SW moon” one day in a certain place, then it will occur at SW and NE moons on any other day at that place. Consequently, with a calendar that gives you the dates of the full/new moons and any local data-point for a high tide (just a date and time), you can do some arithmetic to work out the times of the high tides for any other day ever.

I pulled up a moon table and various tide tables and tested this proposition with two localities I have a passing familiarity with: Westport, Grays Harbor, Washington (right on the ocean); and Seattle Washington (markedly inland but still distinctly tidal). It worked pretty well: given today’s high tide, I was able to predict the next month’s worth of high tides for both locations, accurate to within about forty-five minutes. For kicks, I also tried predicting Brest’s tides for August 7th next week given today’s high tide: again, accurate to within about forty-five minutes.

The book also has a handy-dandy universal tide chart: If you know the current phase of the moon, then find the column with the correct high-tide time for that day; that column is the correct tide-table for that location, with days called out by the phases of the moons instead of dates. (Sadly, several columns are lost to a fold in the page.)

[sanguinarysanguinity]

Navigatio Britannica: First Sunday

As might be gathered from the Leap Day post, another feature of yesterday’s reading in Navigatio Britannica was a trick for calculating the first Sunday of any given year.

Divide the Year, its Fourth, and Four by Seven, What’s left, subtract from Seven, the Letter’s given.

(That mnemonic is originally from Mr. Street’s Memorial Verses on the Ecclesiastical or Civil Calendar, as reprinted in Navigatio Britannica.)

Let’s try 2001, for kicks:

Take the year --> 2001

and 1/4 of the year (round up!) --> 501

and four --> 4

Add those three numbers together --> 2506

Find the remainder when divided by 7 --> 0

Subtract that remainder from 7 --> 7

January 7, 2001 was the first Sunday of the year!

Again, with 2019:

Take the year --> 2019

and 1/4 of the year (round up!) --> 505

and four --> 4

Add those three numbers together --> 2528

Find the remainder when divided by 7 --> 1

Subtract that remainder from 7 --> 6

January 6, 2019 was the first Sunday of the year!

The reason John Barrow gives this rule and calculation is because what he calls a calendar (actually, he calls it a Kalendar) is what we’d call a perpetual calendar: instead of having days of the week written in, January 1 is noted as A, January 2 as B, January 3 as C, and so on until January 8 which is A again, and on around the cycle we go. If January 1 turns out to be a Monday, then every day coded A is a Monday for the rest of the year (unless it’s a Leap Year, in which case it shifts over a day at February 24th). And since the calendar is perpetual/universal like that, it helps to be able to discover which day is the first Sunday of the year -- the “Letter” in the rhyme refers to this sequence of A through G. The pattern of first Sundays repeats every 28 years, hence why we divide by both 4 and 7 (the division by 4 is the “take 1/4th” bit.)

BTW, this perpetual calendar also includes phases of the moon (marked by numbers in gold, on a 19 year cycle), and includes another calculation to know which Golden Number you should be using for a given year.

Barrow also defines the Victorian Period, which, given that he was writing in 1750, has nothing to do with Queen Victoria -- instead, it’s the cycle by which the phases of the moon, days of the month, and days of the week all come to coincide again. Unsurprisingly, the Victorian Period, aka the Dionysian Period or the Paschal Cycle, is 19 x 28 = 532 years long.

Barrow also for some reason includes how to calculate the current year of the Roman Empire’s fifteen-year tax cycle? Which seems to me one bit of trivia too far, but I suppose it mattered to him for some reason.

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@educatedinyellow replied to your post: Navigatio Britannica: Geometry

omg, that little fact about the multiplication vocabulary of ‘rectangles’ (which has since been lost) and 'squares’ (which we’ve kept) is so CUTE! I’m sorry to say that geometry and trigonometry were my least favorite math classes, and I’ve retained very little of them. I was much more an algebra person, which translated over into physics, and eventually calculus. I’ve forgotten most of all that, too, but at least I enjoyed it while I was doing it! :)

Isn’t it marvelous? It struck me the same way. I have to say, this book is squarely hitting the sweet spot of “modern enough to be comprehensible” and “long enough ago that some interesting shifts have happened.” I hadn’t expected to spend quite so much time grooving on vocabulary changes during the last two hundred and fifty years, but here we are.

Re algebra vs. geometry, if you’re good enough with both, it’s amazing how much of either topic can be seen through either an algebraic or a geometric lens, depending on how you phrase it. In fact, I have the impression that it was almost all taught through a geometric lens until relatively recently. (As is somewhat implied by calling products rectangles!) So you lucked out being born into this era, where mathematics education is less geometric than it used to be!

And yes, mathematics absolutely does become lost to the mists of memory if you’re not using it. (Every few years I have to reteach myself calculus again!) But the good news is that if you ever once knew it, you can get it back much more easily than you learned it the first time.

[sanguinarysanguinity]

One last post about Chapter 1 of John Barrow’s Navigatio britannica: or, A complete system of navigation (published 1750): 

This chapter, which is about time, also discussed astronomical observations of the sun and the moon, which required me to frequently consult Wikipedia about things like Sidereal and Synodic months -- after all, Wikipedia usually had longer explanations, as well as pictures, as well as being written from a heliocentric viewpoint (which I’m more used to visualizing.)

I wouldn’t go so far as to say that Navigatio Britannica advances a geocentric model of the universe (although Wikipedia says that it was around this time that the Roman Catholic church was backing off on its proscription against heliocentric books), but it is definitely written from the perspective of “what you see when you stand on the surface of the Earth and look up,” which is a default-geocentric perspective. Cue me spending lots of time tracing little circles in the air with my fingers while sighting past them with my eye, trying to figure out what all this probably looks like through a telescope.

Next up: Chapter 2, “Of the Flux and Reflux, or the Flowing and Ebbing of the Sea.” Tides!

[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.

[sanguinarysanguinity]

Navigatio Britannica: Geometry

Chapter 3: Of Geometry

So this starts off with the etymology that the science of geometry means to measure the earth (as in surveying and navigation), and I am hopping mad that I never realized that before.

This chapter is basically just Euclidean geometry, which of course hasn’t changed significantly in positively yonks, so I skimmed scads of it. But it did include some new-to-me terms:

Superfiecies -- surfaces

Right lines -- what in Euclidean geometry we just call a line, it also includes what I would call a line segment.

Scalenous triangle -- all sides unequal (I dimly recall having seen this term once upon a time maybe?)

The base of a right triangle is explicitly defined to be its longest leg, (which seems unnecessarily specific to me, but okay)

Amblygonum (a term that seems to have died with the 18th century) -- an obtuse triangle

Oxygonium (another term that seems to have died with the 18th century) -- an acute triangle

Rhombus -- what I would call a rhombus, but explicitly excluding squares

Rhomboides -- what I would call a parallelogram

Trapezia -- any quadrilateral that is not a parallelogram; this includes both trapezoids (one pair of parallel sides) AND quadrilaterals with zero pairs of parallel sides. (Interestingly, Wikipedia says “trapezium” has different definitions in British and American English -- in Britain, a trapezium has one pair of parallel sides, while in North America, a trapezium has zero pairs of parallel sides. That split makes a lot more sense to me if circa 1750 trapezium included both classes of figures.) Barrow also gives an alternate term for this group of quadrilaterals: tables.

Nowhere in this list of terms does Barrow define a rectangle, btw -- what I would call a rectangle is a Right-Angled Parallelogram. It seemed a curious oversight: do rectangles not matter in 18th century geometry?

The remainder of the chapter covered:

Compass and straight-line constructions

Euclid’s Greatest Hits

Arithmetic and Geometric Sequences

If you learned these topics in school, then you probably know everything in this chapter.

But! The word Rectangle is defined in the section on Geometric Sequences! It means product, as in the answer to a multiplication problem. It is parallel in usage to the term square, as quickly becomes obvious -- there are a number of  lemmas and corollaries about rectangles and squares, where the first is the product of two different numbers and the second is the product of the same number twice. There’s a pleasing symmetry in that.

Next, on to Chapter 4! Trigonometry, hurray!

[sanguinarysanguinity]

Navigatio Britannica: cosine, &c., and versine

Okay, I’ve started the trigonometry chapter over again, slowly this time, with pen and paper in hand to reconstruct diagrams and verify claims as I go, and now that the text has pointed it out, I’ve realised why the cosine, cotangent, and cosecant are all called that: they’re respectively the sine, tangent, and secant of the complementary angle.

(I feel like a dunce for never having put that together, but I note that while my OED defines cosine as “the sine of the complement of an angle,” my other dictionaries define it just as adjacent-over-hypotenuse, and Wikipedia buries the etymology way the hell down at the bottom of the article.)

btw, this text is defining something called the versed-sine, which I’ve never heard of before: it’s defined as -- well, this is all being done pictorially via the unit circle, so it’s the part of the radius that’s not the cosine, i.e., 1 - cosine.

Ah, here we go: versine, “a trigonometric function found in some of the earliest trigonometric tables.” Wikipedia goes on to say that the haversine -- half the versine -- “is of particular importance in the haversine formula of navigation,” so I assume this is going to be relevant later!

[sanguinarysanguinity]

@clove-pinks tagged me to expand on my 2021 reading plans. Here are some things that I’m currently reading or are sitting near to the top of my mental tbr list:

The Rise of Kyoshi, F.C. Yee, which I just finished last night. I’ve asked the library for its sequel, and expect I’ll read it promptly when it arrives.

Finish catching up on The Rivers of London series. I read a decent chunk of it in 2019, but got sidetracked. I just picked up the series again this fall, and  finished The October Man this morning.

The Lost Future of Pepperharrow, Natasha Pullman. I read The Watchmaker of Filigree Street earlier in the pandemic and loved it. I’ve heard not-so-loved-it things about the sequel, but I’m looking forward to it anyway.

Continue my re-read of the Lord Peter Wimsey novels, which I am reading in conjunction with @wimseypod. They got knocked off their podcasting schedule by the pandemic, so my re-read has been in an extended hiatus while I wait for them to return to the novels.

The Wreck of the Maid of Athens: Being the Journal of Emily Woolridge, 1869-1870. The personal journal of the wife of a sailing captain whose brigantine was wrecked near Cape Horn, stranding them all on an island for several months. I found it in a free box on the street; the woodcut illustrations are wonderful.

The Fighting Temeraire, Sam Willis. Because someday I’m going to write fic about William Bush at Trafalgar, see if I don’t. I started reading it this fall, and it begins with the capture of the first Temeraire in 1759, running through the Trafalgar-Temeraire’s break-up in 1838. It’s proving to be a nice introduction England’s wars with the French during that era, without being so high-level as to make me glaze over. (No detail is too much if you give me a tight throughline to follow, but if you try to back out to a high-level accounting of all the theaters, politics, and economics of the wars I will not be able to scrape together enough attention to continue.)

I’d love to get back to Navigatio Britannica, from which I had been teaching myself 18th century navigation, but which unfortunately had to put down when I abruptly stopped having so much free time at the end of 2019. Dunno if 2021 will permit that kind of time and focus, but we’ll see. (I liveblogged my progress-to-date here.)

Alternatively, Heavenly Mathematics: the Forgotten Art of Spherical Trigonometry, Glen van Brummelen. What it says on the tin. Lots of problems and examples to work, which I enjoy, and this has the advantage that I can download it to my ereader, and thus take it on the bus, which I can’t do with Navigatio Britannica. (WHY won’t Hathi Trust let you download a public domain book, WHY??)

The Flight of the Heron, D.K. Broster. A fic author whom I respect started abruptly writing fanfic for it in the last year or so, which got me interested. I’ve been meaning to read it for half a year now, and have started it at least half a dozen times, never getting more than ten pages in before being called away.

I’ve been itching to re-read Captains Courageous, Rudyard Kipling, which was a childhood favourite of mine. Maybe as a read-aloud to @grrlpup after we finish the Enola novels? We’ll see; boaty things can be a bit hard-going for her.

Speaking of, I’m in the middle of my third re-read — @grrlpup’s first read — of the Enola Holmes novels, as a bedtime read-aloud to @grrlpup. We’re having a good time, but I’m seriously considering making the executive decision to replace all instances of ‘g*psy’ with ‘Romany.’

~

tagging anyone who feels like playing!

[sanguinarysanguinity]

Today in my reading of Navigatio Britannica (yes, my kitty eventually woke up and let me get back to my attempts to verify the proofs), I am reasonably certain that I have found two misprints in Chapter 4, Section 3, Proposition II, Corollary I. In line 2, that should say “(DG - BE)/2″, and in line 3, that should be “DG + BE.” That is relatively straightforward in hindsight, but confusing the issue, of course, was that I have no idea what “mn” and “Dv” in the first two lines are meant to convey: that doesn’t seem to be multiplication, just... an announcement? that the following equations are about mn and Dv respectively? Seriously, I don’t think mn and Dv are meant to be part of those two equations at all.)

Anyway, that took me far too long to suss out, and these two errors are not in the errata list, so I am left shaking my fist at a man who has been dead for 245 years.