A prerequisite for good measurement was made with the creation of a reference plane, now it’s time to determine distance. All uptime units rely on an agreed-upon abstract unit of distance from which most other measurement units can be extrapolated from.
The scale of the ruler at the start of the chapter is 1:1, or true scale. It is a paper ruler and the suggestion would be to copy it as soon as possible, and duplicate it as it will be a center of your universe.
The ruler contains two dominant systems of distance measurements, inch, and centimeter. Please, use the centimeter as the inch reference included just to bring some reference to the possibility that the downtime place that you find yourself in has some notion of what inch is and has some of their own recipes and guides. All distance measurements that will be mentioned here will be in centimeters. Centimeter allows for decimal places while inches are usually represented as fractions, and calculation with fractions is a pain in the ass for someone not used to them or mathematically inclined.
Distance measurement is important as it will allow you to calculate your position in the world, and it will allow you to show how to calculate the size of the world. It will enable you to build accurate clocks, mirrors, and telescopes, gauge distances, and the height of buildings, and in general perform instances of engineering miracles.
Now, related units to centimeters are Liter, a unit for volume. 1 Liter is equal to 1 cubic decimeter or 10x10x10 cm cube. That is, to determine a liter build a box whose internal dimensions are 10cm by 10cm by 10cm and fill it with distilled water. 1 liter is also 1 kilogram or 1000 grams.
Rangefinder
and how we perceive distance
A rangefinder is a useful gadget to have as it will enable you to gauge how far something is. Humans do it instinctively, we have two eyes that are spaced apart and each eye sees a slightly different image that when combined and focused right makes one image. The position of our eyes determines the distance of an object. The closer the object the closer our eyes are to each other. We as humans can with some practice and training measure accurately objects that are close to us, to some 10 meters. As objects get closer our eyes converge on the object and muscles that do that send information about their state back to the brain which based on learned knowledge and experience determines the approximate distance of an object. If an object is farther than we use the mentioned distance clues.
This is not the only clue that our eyes get and brain processes to determine how far something is, for example, the Parallax illusion is that far objects move slower than closer objects. We have all noticed this effect when we drive along a scenic route and see hills in the distance and closer objects just whizz by.
If you can devise a very synchronized way of observing the sky from opposite places in the world and recording the position of an object in it, a star, galaxy, or a planet, you can determine the distance of that celestial object. The best way to do it actually is to record the position of a distant sun at a certain date and repeat the measurement on the opposite side of our trip around the sun so you get a bigger difference in your position relative to a stationary star and can then due to some trigonometry. The angle at which the star is compared to the earth is measured and solved using
D = Radius/Tan(angle)
To complete this measurement you will, in addition to angle, only need to know the distance of the earth from the sun which in uptime is standardized and is called the astronomical unit. The astronomical Unit is 149.5 million kilometers. The formula can be applied to any kind of measurement and is the basis for a rangefinder device. Also, a nice thing to know is that earth itself has a radius of 6,357 to 6,378 km as it is not a perfect sphere.
Don’t worry, just give this info to someone who knows a little about mathematics. but to do so you will need to venture to the east if you are not there already. To the Chinese and Indian cultures which were the only ones at that time that had any inkling of the dark art of mathematics.
The rangefinder device consists of a mirror with the bottom or upper half of silverback striped making it transparent.
This mirror is fixed to a piece of wood at a 45-degree angle to the forward direction of viewing. Another mirror is placed some known fixed distance away in line with the angled mirror. This second mirror is on a swivel that enables it to turn 360 degrees around its axis but its zero point is again angled at 45 degrees.
The rotating mirror really only needs to go from 0 to 90 degrees. The light should bounce from the second mirror back to the first one and then to the eye of the person looking. The device functions so that you look at an object in the distance through the half mirrored glass. While looking at the object whose distance you are measuring, turn the second mirror to reflect the image back to the first mirror and align the top with the bottom part, making the object complete. Read the difference in angles and do the calculations.
r = d × cotan(θ)
We just reordered the formula to fit our new situation. Here we are not measuring the distance to stars but at something like a ship. Fundamentals of Mathematics will have their own chapter.
For devices that measure small distances, a calibration can be made by simply making a person walk away from you with a measuring tape. Complete this process for different distances with your homemade rangefinder and mark the distances on the dial below the rotating mirror. This way you can avoid all of this calculation nonsense. This way you could circumvent the need to understand the cotangent or cotan() function. Cotangent is calculated by r/d = cotan(θ).