In the Dresdenverse short story “Bombshells,” the only story told in first-person POV by Harry’s apprentice Molly Carpenter, she describes a way to incorporate math (!) into her version of the first spell she learned from him- the tracking spell- and uses it to estimate the distance to her target (Thomas Raith, using a few of his hairs) without having to actually go all the way there. This occurred to me as pretty significant because it’s not something Harry ever showed her how to do (or even figured out for himself) and it’s a vivid demonstration of Molly’s own strengths and intelligence.
Here’s how she did it.
The basic idea is that if the target of your tracking spell is close to you, and you’re moving, it will appear to shift a lot relative to your position. If it’s really far away, it won’t appear to move much at all when you move- it stays pretty much the same direction from you. Compare looking at faraway things as you drive past them to the things right at the side of the road that whoosh by your window. You may have even used some form of this trick in video games like Call of Duty, Skyrim, Arkham City, or anything else that lets you see a pinned location relative to which way you’re facing. For many years (before electronics got decently sophisticated) airplane pilots used bearing changes and a mechanical calculator to fudge estimates of their distance from radio navigation aids.
Jim Butcher didn’t stick an actual equation into the story, and rightly so, because it would have dragged the pace down to nothing and alienated the readership. But for those of us who are obsessive nerds who enjoy that level of detail, it’s surprisingly easy math to do. Despite the implication that Molly’s technique would involve high-school-level trigonometry, you can do it in your head, using only an ordinary magnetic compass and a tracking spell (or its equivalent).
Step 1: Go ahead and put that blood or hair or whatever in your mouth and follow the tingle of your lips (like Molly does) or dangle it from a string or a chain (like Harry Does, if you don’t relish the idea of putting such things in your mouth) and determine the direction of your target. Use the compass to determine the exact number of degrees that is relative to magnetic north. For now, let’s say that the target happens to be directly (0°) north of us.
Step 2: Turn so you’re facing perpendicular to the way the tracking spell points, so the target is directly to your right or to your left. For example, with our hair donor directly north of us, we’d need to face directly east or west. Now, walk a reasonable distance to measure (Molly uses the convenient unit known as a “Molly-pace”) keeping the target exactly off your shoulder. Make sure to go at least far enough to register a slight change in the direction you’re facing according to the compass.
Step 3: Measure the change in your compass bearing. Continuing our example, let’s say we started with our target directly north of us, and walked fifteen paces west. Checking the tracking spell against the compass, our target is now four degrees east (004°) of dead north, and we’re not facing directly west anymore- we’re facing four degrees north of that (274°). We’re now ready to plug in some numbers. Do not fear trigonometry- that’s not what we’re doing. Instead, do this:
Step 4: (Molly-paces x 60) ÷ degrees changed = Molly-paces to the target
To finish our example, we took fifteen paces to travel four degrees. Fifteen times sixty is nine hundred. Divide that by four degrees, and we’ve got a result of two hundred and twenty-five paces to the target… however far that is. If you’re not as tall as Molly, your results may vary.
I’m certain that the math nerds in the crowd started mumbling about cosines and reached for their scientific calculators before this last step. The reason this trick works, however, is not because it’s a 30-60-90 triangle, nor because it approximates an isosceles triangle. What we’ve done is approximate an arc-length of a circle.
As we already know, a circle (a) contains 360 degrees, and (b) has a constant ratio between its circumference and diameter, known as pi, or 3.1415926blahblahblah, which, for the sake of rough simplicity, we will approximate as 3. What Molly’s trying to figure out is the distance (in Molly-paces) from the center of the circle (the target) to the perimeter, a portion of which she’s just paced off. That distance (the radius ) is half of the diameter, so we’re going to use (in rough simplicity, 6) as the total number of Molly-paces it would take to walk around the entire circle, and then solve for
Since we know how much of the circle we’ve walked around (“degrees changed” out of 360), we also know what portion of the circumference we’ve paced off (“Molly-paces” out of 6). Since these are equal portions, all we need to do is simplify:
degrees changed = Molly-paces 360 6r
degrees changed * 6r = Molly-paces * 360
degrees changed * r = Molly-paces * 60
r = Molly-paces * 60/degrees changed
It’s not terrifically precise, but it doesn’t have to be. It was close enough for Molly to locate Thomas in Svartalheim, and now you’re just that much cooler (and/or more dorky) for knowing it. Now, for your homework, go find Mouse. Ten paces off your shoulder gives you two degrees of bearing change.
This wonderfully nerdy guest post was brought to you by Andy Hammond!