Measurement assigns numerical values to things that can’t be counted – lengths, weights, times, etc. – by choosing a unit and counting how many units are equal to the thing being measured. The earliest measurements were rough estimates, such as using a human forearm to measure length; one man’s forearm might be longer than another’s, but the difference wasn’t enough to matter in most TL0 societies. With the growth of trade and the emergence of bureaucracy at TL1, more exact measures were needed; using the wrong measurement might cost someone money! Early governments often prescribed what units should be used in their marketplaces, and inspected measuring devices to make sure they were accurate. The balance (LT p. 44) became a symbol of justice very early in history.
Measurement of angles started at TL0 with awareness of how far above the horizon the sun has traveled (which also measures time) and of the four cardinal directions. At TL1, right angles were used in architecture and civil engineering. The ancient Mesopotamians divided the circle into six parts, and subdivided each part into 60 degrees (a total of 360 degrees) and each degree into 60 minutes. At TL4, minutes were further divided into 60 seconds. Instruments for angular measurement were important in surveying and, later, in geometry, astronomy, and navigation.
Level (TL1). An A-shaped frame with a plumb bob hanging from the apex. When it’s placed on a flat surface, the plumb line’s deflection from a marked center point on the crossbar indicates the slope. $15, 4 lbs.
Surveyor’s Cross (TL1). This tall staff (also called a groma) has a horizontally balanced cross on top and a sharp spike at the bottom for planting it firmly in the ground. Adjustable plumb bobs on the cross’ ends keep it level in a plane. Sighting along the plane lets the user mark spots at the same elevation; sighting with the plumb lines marks an alignment. $75, 6 lbs.
Chorobates (TL2). A long (up to 20’), narrow bench with a water trough and plumb bobs hanging from the bottom. Both the water level and the angle of the bobs provide a level reading. Typically used only on large-scale projects, such as road and aqueduct construction. A 10’ chorobates: $540, 145 lbs.
Cross-Staff (TL2). A short (3’-4’) staff with a sliding crossbar. The user points the staff at one point and slides the crossbar until it appears to touch another desired point. The distance along the staff indicates the visual angle between the two points. $45, 4 lbs.
Dioptra (late TL2). A tube or set of sights on a platform whose position can be adjusted by screws. The dioptra can give vertical and horizontal angles from the observation point to an object – but only for stationary objects, due to the adjustment time (at least 2-3 minutes per observation). $120, 5 lbs.
Astrolabe (TL3). Developed in the second century A.D., the astrolabe came into widespread use in the Muslim Near East. It has four parts. The mater is a flat plate 5”-10” in diameter, marked with celestial coordinates for a given latitude, centered on the pole and including the horizon, the meridian, and altitude and azimuth circles. Some astrolabes have interchangeable plates for different latitudes. On top of this is the rete, a metal grid with pointers for different stars. On the back is the alidade, a rotating pointer with a sighting hole used to point it at a particular star. A pin through the center holds the other parts together. The astrolabe doesn’t merely measure angles – it can perform hundreds of computations. Treat it as basic equipment for Astronomy. Small astrolabe: $250, 5 lbs. Model with interchangeable plates: $200, 4 lbs., plus $100, 1 lb. per plate.
Kamal (TL3). Used by Muslim navigators to measure a celestial body’s height above the northern or southern horizon. A square board is held at a distance where it just spans the visible gap between the body and the horizon; the length of a cord attached to it indicates the angle. $25, 1 lb.
Quadrant (TL3). A piece of solid material in the shape of a quarter circle, with degrees marked along the edge. The user sights on a celestial body along one edge; a plumb bob hangs down vertically, indicating the body’s elevation above the horizon in degrees. It can also be used to estimate an object’s height via trigonometry. $35, 3 lbs. (Much larger quadrants are used in TL3 astronomical observatories; see GURPS Low-Tech Companion 1.)
Gunner’s Quadrant (TL4). Invented in 1545 by Tartaglia, this gadget has a long arm attached to the gun barrel and a short arm at right angles to it. A plumb bob indicates the gun’s elevation, from which range can be estimated. $45, 4 lbs.
Primitive units of length are mostly based on the human body. Common examples are the hand (4”, used to measure the height of horses), the foot, and the cubit (the distance from elbow to fingertips, typically 18”). Measuring Rod (TL1). Standardized measures of length came into use in the oldest civilizations, including Egypt, Sumer, and the Indus Valley. Egypt had a standard royal cubit, a 21” granite rod to which other measuring rods could be compared. The Egyptian cubit rod was divided into 28 digits, and was often marked with fractions of a digit, from 1/2 down to 1/16. $5, 0.5 lb. Odometer (TL2). A cart or chariot wheel turns a gear as it rolls; after every mile, a pebble drops into a box, giving a running count of miles traveled. Vitruvius described this mechanism around 15 B.C., but Alexander the Great’s chroniclers gave travel distances accurate to better than 1 mile in 250, which were almost certainly mechanically measured. Chinese inventor Zhang Heng (78-139 A.D.) is credited with a similar device. $100, 10 lbs. Area There’s no direct way to measure area. A rectangle’s area can be found by measuring its length and width, and multiplying them together. Areas of other shapes can be broken up into rectangles. Geometry started out as formulas for the area of fields of different shapes. Land measurement is the task of surveyors. Surveyor’s Kit (TL2). A well-equipped surveyor from Rome to the Renaissance has a surveyor’s cross, a dioptra, two 10’ poles, 120’ of cord (stiffened with wax to retain its length), and 20 posts to mark points on the ground. $245, 40 lbs.
Volume can be measured directly, by filling a standard container with water or sand and pouring it into a larger container repeatedly. It can also be calculated geometrically – especially when the larger container is full and emptying it isn’t convenient. Volume measurement was an outgrowth of large-scale agriculture at TL1.
Measuring Basket or Jug (TL1). A basket, cup, jug, or jar with a standard volume, normally marked on the outside. Subdivisions are estimated, not measured; early containers are opaque, and gradations on the interior surface would be awkward to read. Sizes vary from tiny cups to 35-cubic-foot barrels, or tuns (the origin of the word “ton”). See Containers and Storage (p. 34).
Units of weight originated at TL1, as an outgrowth of trade. The smallest unit of weight is often one grain of the local staple food.
Balance (TL1). The original weighing device, with two pans hanging from opposite ends of a beam that pivots on a central point. One pan holds the thing being weighed; the other holds standardized weights, which are counted when the pans are in equilibrium. Balances come in varied sizes; the ones described here are portable models, with lead weights. They can’t weigh anything heavier than the total of their counterweights! Small balance: $25, 1 lb. (set of lead weights: $10, 5 lbs.). Larger balance: $75, 4 lbs. (set of lead weights: $50, 20 lbs.).
Steelyard (TL2). The type of scale that many people have encountered in a doctor’s office: The person or object being weighed rests on a platform or in a pan, and a relatively small counterweight is slid along a beam until its leverage balances the weight. The counterweight’s position is read as a weight with the help of numbered gradations. This device depends on a good understanding of leverage. A small steelyard was found in the ruins of Pompeii. Steelyard that can weigh up to 300 lbs.: $100, 20 lbs. (weights included).
Time, like length, has a natural starting place for measurements: the apparent movement of heavenly bodies across the sky. Every society knows about the day; most societies use the month and/or the year. Keeping track of time on this scale is done with calendars.
Times shorter than a day become important in TL1 societies, for such purposes as keeping records of how long people have worked. Tracking the sun across the sky offers one way to do this, but a variety of inventions provide more precise measures.
A clock measures time on a continuing basis throughout the day. There are two styles of measuring time. One divides day and night each into the same number of hours, and makes daytime hours longer in summer and nighttime hours longer in winter. The other keeps every hour the same length year-round. Mechanical clocks can be made more accurate with more precise construction. Use the equipment grades on p. B345. Good-quality timepieces are twice as accurate; fine-quality ones are five times as accurate.
Clepsydra (TL1). Invented in Egypt around 1550 B.C., the water clock is a vessel filled with water, which flows out through a small hole in the bottom. In Egyptian clocks, this was drilled through a gem set into a larger opening. Lines at intervals down the inside mark off the hours. Water flows out more slowly as its level falls; a properly designed clepsydra has tapered sides to compensate. The clepsydra measures fixed-length hours. However, it can be built with different scales for different times of year, to give variable hours. Flow speed varies with temperature and humidity; the clepsydra is accurate to the nearest 10 minutes. $500, 15 lbs.
Sundial (TL1). Another Egyptian invention, dating to 1500 B.C., the sundial consists of a vertical projection, or gnomon, that casts a shadow onto a painted or carved surface. The shadow’s position marks the hours. The sundial measures hours of variable length. Sundials are almost perfectly accurate at the latitude for which they’re made. However, they only work during the day – and only if there’s enough sunlight to cast a shadow. $300, 95 lbs.
Portable Sundial (TL2). A more advanced Greek invention for telling time: a sundial small enough to be carried easily. To tell time in a new location, it must be aligned to the noonday sun. $100, 2 lbs.
Regulated Clepsydra (TL2). The Greek inventor Ctesibius devised a clepsydra that avoided the flow rate changing as the tank emptied. A float valve in the tank (like the one in a toilet tank) let in more water as the level sank. Time was measured by the water level in a second tank that received the outflow. Accurate to the nearest 2 minutes. $750, 25 lbs.
Graduated Candle (TL3). First mentioned in Chinese writings of the sixth century A.D.; the form described here dates to the reign of Alfred the Great of England (878). Consists of six 12” candles, each of which takes 4 hours to burn down; each candle is divided into 12 20-minute sections. The burning candle is kept in a case with translucent horn sides. This and later clocks measure hours of fixed length. Candles: $40, 6 lbs. Case: $15, 0.7 lb.
Water-Driven Clock (TL3). The Chinese experimented with mechanical clocks that worked like later European weight-driven clocks, but with a water tank rather than a solid weight. A regulated water flow drove a mill wheel that turned gears. The invention never came into common use.
Weight-Driven Clock (TL3). A clock powered by hanging weights, whose fall turns a shaft within the mechanism. Lacking a pendulum or other regulator, it’s accurate only to the nearest hour, which it signals by ringing a bell. Some models had dials, but these functioned as astronomical displays rather than for keeping time. Each clock was individually made and should be treated as a prototype, but on average: $450, 100 lbs.
Pendulum Clock (TL4). This was invented by Dutch scientist Christiaan Huygens in 1656, based on the discovery that a pendulum always takes the same amount of time to swing back and forth. Such clocks could achieve an accuracy of 1 minute in a day. Wall or mantel clock: $300, 20 lbs. Longcase (“grandfather”) clock: $600, 100 lbs.
Spring-Driven Clock (TL4). In 1660, Robert Hooke and Christiaan Huygens (they had a dispute over priority) invented the balance spring as a way of making the balance wheel reliable; this performed the functions of a pendulum, but was much smaller. It made possible the first reliable pocket watches, accurate to 10 minutes a day. $100, neg.
A timer doesn’t keep running for as long as a clock – usually an hour or less. It can serve as an improvised clock, but its main use is to measure a rapid process (like the later stop watch) or to monitor the time assigned to a task.
Miniature Clepsydra (TL3). Invented by Chinese artificer Li Lan in 450, this variant on the water clock was made of jade and used mercury as its working fluid. It holds enough mercury to run for two Chinese hours (28 minutes 48 seconds). For its time, it’s an ultra-precision instrument, measuring intervals as short as 1/20 of a Chinese hour. $500, 8 lbs.
Sandglass (TL3). The oldest record of this device is a painting by Ambrogio Lorenzetti, dated 1338. A sandglass consists of two glass vessels joined by a narrow neck through which a granular material flows at a steady rate. Despite the name, ordinary sand isn’t suitable; fine marble dust works better. Sandglasses – being relatively unaffected by conditions at sea – were used both in navigation (pp. 50-52) and to time watches. Standard half-hour sandglass: $50, 3 lbs.
Scientific experimenters at TL4 invented devices to measure other aspects of the physical world. These early instruments were often slow and inaccurate, but the very idea of measuring heat or pressure led to a revolution in physics. The thermometer and barometer are experimental prototypes, not commercial products, in this period.
Barometer (TL4). In 1643, Evangelista Torricelli invented the barometer: a column of mercury, the height of which fluctuated with changing air pressure. In 1660, Otto von Guericke used it to predict the weather. This design is a mercury-filled glass tube with height gradations, sealed at the top but open to a pool of mercury at the bottom.
Thermometer (TL4). In 1600, Galileo devised an experimental thermometer that used air as the working fluid. Several other models were developed in the 17th century, culminating with Gabriel Fahrenheit’s alcohol thermometer in 1709. A thermometer can ensure accurate temperature measurement for processes such as distillation (see Distillation, LT pp. 11-12).