Units of Measure, Historical Progression • Historical Units, “pre-science” measurement • Human, plants, and animals as standards • No tape measure … so use what’s readily available • Introduction of Metric System • Eliminate inconsistencies, odd multiples, inconvenient units • Towards a global system, international acceptance • Deployment of “Universal” natural constants • Speed of Light available to anybody, anywhere … including E.T. • Accuracy, Precision, Significant figures • Definitions and Applications
Units of Measure • Mass • Historical definitions & difficulties • Mass versus Weight, Planetary influence • Length • Historical definitions & difficulties • Modern Definition • Time • Historical Definitions & difficulties • Modern definitions • Combination Measurements • Area and volume from lengths (2500 sqft home, 1 acre-foot) • Density (grams/cm^3) from mass and volume • Speed (miles/hour) from distance and time
First “standard” Easy access Consistent weight ? What variety to use? Inconveniently small Persists to this day Pharmacy Ammunition 7000 gr. ≡ 1 pound Definition of Mass, “grains”
Mass • “Grain” originated from a Barley grain (now defined as 64.8 milligrams), and still used in pharmacy ( 5 grains of Aspirin) and firearms (M-16 rifle bullet weighs 55 grains) • “Carat” is a unit of mass still used for gems (equals 200 milligrams). The word is based on Greekkeration (fruit of the carob). Carob seeds were once used as weights on scales due to a reputation for having uniform weight. • “Avoirdupois pound” invented by London merchants in 1303 equals 7000 grains. 16 ounce/pound, 1 ounce = 28.3495 grams • “Troy Pound” and “Troy Ounces” (12 per pound) still used in precious metals trade. One Troy Ounce = 31.1035 grams • “Stone” is a unit of mass formerly used in parts of the United Kingdom. It is equal to 14 poundsavoirdupois, and 8 stone make a hundredweight in the Imperial system • “Apothecary” or Pharmacy system of units, mostly phased out. Still used for shotgun shells. One dram= 3.888 grams
Grain used in pharmacy, but being phased out • Standard dose for aspirin is 5 grains, 325mg
Winchester 30-30 • Introduced 1895 for model 1894 Winchester and still one of the most popular cartridges after 115 years • Why “30-30” designation? • 30 caliber (0.300 inch) • 30 grains of gunpowder, 170 grain lead bullet • Similar description 45-70 Springfield & 30-40 US Krag • The “grain” has survived as a unit of measure
Which pound was it ?… • Avoirdupois Pound, from England • 16 ounces per pound, still in use … mostly USA • 1 pound ≡ 7,000 grains of Barley (originally) • Troy Pound, from Troyes in France • 12 Troy ounces per Troy pound • 1 Troy Pound ≡ 5,760 grains of Barley • At least the grains were the same kind! • Tower Pound • Tower Pound = 7200 grains of Wheat … not Barley • 7200 grains of wheat ≈ 5400 grains of Barley
What’s in a pound? • Merchant’s Pound • 9600 wheat grains • 16 Tower ounces • 15 Troy ounces • London Pound • 7200 Troy (wheat) grains • Pennyweight = 22.5 Troy grains • Wool Pound • Wool Pound = 6,992 grains (probably Barley) • Roman Libra (Latin pound), = 327 grams • 12 uncia (ounces) per libra … Libra gives abbreviation “lb”
Apothecary (Pharmacy) Mass“Grain” a constant value in old systemsPharmacy pound unlike other “pound” definitions
Other definitions related to Pounds • Shotgun Gauges, based on weight of lead • 12 gauge = 12 lead balls per pound • 20 gauge = 20 lead balls per pound • Money defined by weights of silver • 1 pound of silver = 1 pound sterling • Today 1 troy ounce of Ag worth ≈ $10 • 12 troy ounce = 1 troy pound ≈ $120 per pound for silver in today’s bullion market. British Pound Sterling worth about $2, so Silver has appreciated 60x
Mass was a mess! • >12 variations on the pound • Other units of measure add to confusion • Drams, pennyweights, Stones, tons … • International trade a problem • A “pound” is not always the same • Standardization difficult • Same names for different metrics (ounce) • A better method was needed …. • And helped give rise to the “metric system”
Fahrenheit scale Arbitrary values Centigrade scale Based on water Kelvin scale Zero not arbitrary Same increments as centigrade = Celsius Definitions of Temperature
Defining Temperature • Temperature scales are arbitrary • Fahrenheit Scale from year 1724 • 0oF based on a mixture of ice, water, salt • Alternatives include history of coldest climate at 0 degrees • 100oF based on temperature inside horse’s mouth • Alternatives include use of armpit=96oF,“too hot to work”=100oF • Water freeze at 32oF, boils at 212oF • Handy size degree, scales divisible by 4, avoids fractions • Celsius (Centigrade) scale based on water • Established in 1744, the work of Anders Celsius • 0oC defined as freezing point of water • 100oC defined as boiling point of water
Arbitrary scales = problematic • Fahrenheit, Celsius require negative values • Inconvenient complexity from minus temperatures • Temperature relationships arbitrary, ratios don’t make sense • 100oF (or 100oC) not “twice as hot” as 50o • 10oF is not “infinitely hotter” than 0oF • Needed: a new scale: linear, proportional … with a “true zero” • Solution to define “absolute zero” as absence of all motion • Kelvin scale resolves the zero problem • Absolute zero translates to water freezing at 273oK, boils at 373oK • Twice the Kelvin temperature IS “twice as hot” • Just add 273 to degrees centigrade to get degrees Kelvin • For Fahrenheit Abs zero = -460oF, “Rankine” scale (rarely used)
Kelvin Scale (oC + 273)utilizes “true zero”, temperatures proportional
Temperature Summary • Fahrenheit named for GermanphysicistGabriel Fahrenheit (1686–1736), proposed in 1724. Several versions of it’s creation include using the mouth of a horse to establish 100 degrees. In this scale, freezing point of water is 32 degrees, and the boiling point is 212 degrees. It’s use continues partly due to smaller increments than Centigrade, which helps avoid fractions. • Centigrade or Celsius scale is based on properties of water, 0 is defined as freezing point and 100 used for the onset of boiling. • Kelvin scale provides an “absolute” value with zero defined as absence of temperature. Absolute zero is -273 degrees referred to water freezing at 0 degrees centigrade, but it’s more logical to define 0 centigrade as 273 Kelvin, avoiding negative numbers and providing linearity (avoids mistake of thinking 100 degrees centigrade is “twice as hot” as 50 degrees). • Temperature definitions largely resolved, have achieved close to absolute zero with boiling point of liquid helium
Temperature Conversions • Zero NOT same temp. in both scales • 0oF ≠ 0oC, Δ =32oF • Celsius degree is larger = 5oC/9oF • 5/9 is “exact” ratio, no sig-figs involved • Fahrenheit = (Celsius * 9oF/5oC) + 32oF • Convert size of degrees first, then add zero offset • (100oC * 9oF/5oC) + 32oF = 180 + 32 = 212oF • Celsius = (Fahrenheit - 32)oF * 5oC/9oF • Get rid of offset first, then use ratio of scales • (212–32)oF * 5oC/9oF = 180*5/9 = 100oC • 98.6oF body temp = (98.6-32)*5/9 = 37.0oC
Kelvin Temperature Conversions • Celsius to Kelvin • “True Zero” = - 273oC • Therefore, Celsius + 273 ≡ Kelvin • Common Examples • Melting Ice = 0oC + 273 = 273oK • “Room Temperature” = 25oC + 273 = 298oK • Boiling Water = 100oC + 273 = 373oK
Arbitrary units Earth cycles reference 12 or 24 hour basis, but 24 is less confusing and avoids AM/PM notation Depends on location “Zulu” time = “GMT” Greenwich Mean Time based on one longitude. “UTC” = universal coord. Time, broadcast by NIST, used by “atomic” clocks. Definitions of Time
Time … is also arbitrary • “Year” is defined as earth’s circumnavigation time around the sun, NOT an integer multiple of any other measurement. All planets are different. • “Day” defined by earth’s Rotation on its own axis. There are 365.25 days per year, leading to “leap years” every 4th year to keep seasons as they are. • “Hour” was defined by the ancient Egyptians as either 1/12 of daytime or 1/12 of nighttime, hence both day and night hours varied with the seasons (must have been confusing), but the sum was always 24 hours per day. • “Minutes” per hour multiple of 60 comes from the Babylonians who used factors of 60 in their counting system • “Seconds” originally referred to Babylonian subdivision of a minute called the “second minute”, later simplified to the “second”… also a factor of 60 • “Leap Seconds” accommodate the Earth's rotation slowing down due to tidal action.. The length of a solar day increased from an exactly defined 86,400 seconds per day (60 sec/min*60min/hr*24hr/day=86,400sec/day) in 1820 to 86,400.002 seconds per day in 1999, The difference amounts to nearly one second per year (365 days * 0.002 sec/day = 0.73 sec) which is handled by inserting a leap second into reference clocks about once a year.
Time • Time is a fundamental measurement • The definition should not be changing • Time denominated by arbitrary definitions • Based on planet where we live, and changing! • NOT the same for other planets • Time also is unidirectional • Unlike other physical measurements • Need to stabilize/freeze the definition • GPS requires nanosecond accuracy
Length • “Foot” originally referred to somebody’s appendage, probably a King • “Inch” defined as 1/12 of “Foot” … why 12? • “Rod” was used in surveying, 16.5 feet long (11 cubits), with a piece of Jasper mineral at each end to prevent wear of the bar. • “Chain” also used for surveying, 66 feet long with 100 links, so land measurements employed both units (e.g. width of property = 2 chains + 4 links) • “Yard” defined as 3 Feet (why 3?) • “Fathom” is distance between fingertips of outstretched arms (now re-defined as exactly 6 feet) … and still in nautical use. • “Furlong” agricultural measurement, based on distance a horse could pull a plough before needing a rest. Still used by racetracks, 660 feet or 220 yards • “Mile” English parlaiment of Elizabeth I defined the mile as eight furlongs or 1,760 yards; 5,280 feet; or 63,360 inches. • “League” originally defined by length of an hour's walk, now considered equal to about three miles. • Train tracks use standard separation of wagon wheels, originally from Roman 2-horse Chariots established by breadth of 2 horse’s behinds (4 ft 8-1/2 inch)
280 Cubits High 440 cubits wide Current unit conversions: Height = 146.5 meters Width = 230.4 meters Mass = 5.9E6 tons Stone Blocks = 2.3E6 Volume = 2.5E6 meter^3 (Egyptian cubit ≈ 20.6 in) Giza Pyramid SpecificationsCompleted 2551 BC, tallest structure for 3800 years
Noah’s Ark Specifications Genesis 6:15 God said to Noah, "The end of all flesh has come before me, for the earth is filled with violence through them. Behold, I will destroy them with the earth.14 Make a ship of gopher wood. You shall make rooms in the ship, and shall seal it inside and outside with pitch.15 This is how you shall make it. The length of the ship will be three hundred cubits, its breadth fifty cubits, and its height thirty cubits. Might just be the first recorded “engineering specification”, two sentences used to define a large scale construction project This simple spec. would never be accepted in today’s world !
What’s a Cubit? Cubit is the name for any one of many units of measure used by various ancient peoples. The natural cubit is based on the distance between thumb and another finger to the elbow on an average person (a more common definition measures from elbow to end of longest finger). In ancient Egypt, the Cubit was approximately 20 inches, based on pyramid specifications. It was employed consistently (for example, to measure cords and textiles) through the Middle-Ages up to the Early Modern Times. The natural cubit measures 24 digits or 6 palms. This is about 45 cm or 18 inches (1.50 ft.) Early engineering projects depended only on human anatomy (no Home Depot for tape measures). Major projects relied on basically simple dimensions based on the people who built them. No permits, no inspections, no safety rules?
How big was the Ark?(by today’s measurements) Dr. Max D. Younce says by his calculations from Genesis 6:15 that Noah’s ark was 450 feet long, 75 feet wide, and 45 feet deep. He goes on to say that it’s equivalent to "522 standard stock cars or 8 freight trains of 65 cars each."
Length Dimension also in trouble • Competing arbitrary definitions • Foot subject to change ?, whose foot? • Unnecessarily complex relationships • Acre = 160 square rods, 43,560 square feet • 1 square rod = 272.25 sq feet • Unnatural multiples • 5280 feet per mile • 12 inches per foot, 3 feet per yard • Need something better … more universal • “meter” invented to solve the problem
Commodities, Currency Historical definitions abound, many still in use Dozen = 12, Baker’s Dozen = 13 Bushel of corn = 9.3 gallons Bale of hay = 45 pounds (small square variety) Cord of wood = 128 cubic feet (4’ x 4’ x 8’) simple to use Barrel of Oil = 42 gallons of crude Acre feet = 43,560 cubic feet (agricultural water, intuitive) Currency Quid = pound sterling, 20 shillings/pound, 12 pence/shilling, UK went metric in 1971, got rid of confusing system Some quantities are just names, no numbers School of fish, Gagle of geese, Pride of Lions
“Pieces of 8” coinage Peso is the name of the currency in many Latin American countries because of Spanish influence 500 years ago. The peso was a silver coin minted in Spain. The Americas had vast deposits of silver which was used to mint millions of pesos over several centuries. The old silver peso was worth 8 reales ("royals"), just as the U.S. Dollar is worth 100 cents. Pesos were often physically cut into eight "bits", or sometimes four quarters, to make smaller change. People began to call the silver peso coins "pieces of eight." A peso cut into quarters came to be called "two bits" a term still used in the United States to mean a quarter of a dollar, or twenty five cents.
Early measurements a mess! • Length based on King’s foot • What happens when we change Kings? (save the foot!) • The King’s foot might change with age … • Mass depends on bits of something natural • Inconsistent by location, time, plant variety, humidity … • Nonsensical multiples abound over time • 4 quarts/gallon, 32 ounces/quart, barrels, cords … • 12 inch/foot, 3 feet/yard, 5280 ft/mile, leagues, fathom … • 7000 grains/pound, 14 pounds/stone • 20 schillings per currency “pound”, reams of paper, … • France attacked the problem • Defined new measurements (no plants or people) • Based values using powers of 10, became the “metric” system
SI system of units • Employ a Decimal System, of powers of 10 • Defined kilometer, meter, centimeter, millimeter, nanometer • Replacing feet, fathoms, knots, cubits, furlongs, etc. • Volume defined as 1 liter = 10 x 10 x 10 cm = 1000cm^3 • Kilogram, gram, metric ton (1000 Kg) • Replacing pounds, stones, grains, ounces, drams • Related to water (1 liter = 1000 cm^3 = 1 kilogram) • Second, millisecond, microsecond • Preserved historical units, impractical to change all clocks • Tied old units to more precise standards
Basic CGS metric schemePreceded SI / ISO system of units (cm vs meter)1 cm^3 = 1 milliliter = 1 gram H2O
Where did the meter come from? • Objective was to dispense with artifacts • Use something “universal” to measure • Why not the earth itself? • Everybody has access • Easy to understand • Conceptually simple • Reproducible result
Original Meter Definition • Early measurement systems were problematic, so a new definition of length was desired NOT to be tied to human or animal-centric artifacts. • In 1791, the French Academy of Sciences adopted a meridional definition to establish a universally accepted definition of the metre. The Bureau des Longitudes commissioned an expedition, from 1792 to 1799, to measure length of the meridian between Dunkerque and Barcelona. This portion of the meridian, which also passes through Paris, was to serve as the basis for the length of the quarter meridian, connecting the North Pole with the Equator. • in 1793, France adopted provisional results from the expedition, but it was later determined that the first definition was short by 1/5 millimetre from miscalculation of flattening of Earth. The length of this quarter-circumference divided by 10,000,000 became the standard meter. This measurement was transferred to scribed lines on a metal bar kept in Paris.
Historical International Prototype Metre bar, made of an alloy of platinum and iridium, which was the standard from 1889 to 1960
Problems with Artifact based Standards • Only ONE “master” or “gold” standard (meter bar in Paris). • Copies made of master become “secondary” or “silver” standards (sent to USA, England, Germany, etc.) • “Silver” standards (our National Secondary standards) used to create “Reference standards” for science and industry, and “Working Standards” are made from those. • Errors are cumulative as copies are made from copies, and defects propagate through the system. Consider a 10th generation Xerox copy … quality degradation obvious. • Users must have access to the standards for calibration, risk of damage to the standard and measurement error. • Complex schemes evolve, groups of objects whose average value is “the standard”, etc. Disk Drive example, NBS used 20 pieces of media to define a “gold average”.
A thought experiment … • Assume we establish radio & TV communication with intelligent life from another galaxy. After sharing pictures of our families and technical data, the questions begin. • When asked “how tall are you”, the response of 6 feet 2 inches is met with an obvious question: “What is a foot?” Same issue with measurement in meters. • Advice for our friends in the other world to come to Paris for a copy of meter bar is not too practical. • A more universal solution is required ….
Light to the rescue! • In 1893, the standard metre was first measured with an interferometer by Albert A. Michelson, the inventor of the device and an advocate of using wavelength of light as a standard of distance. • By 1925, interferometry was in regular use by standards organizations, although the International metal bar Metre remained the primary standard until 1960. • In 1960 the metre was redefined as equal to 1,650,763.73 wavelengths of the orange-redemission line of the krypton-86 atom in a vacuum. The original international prototype of the metre metal bar is still kept in Paris under the conditions specified in 1889.
“Universal” ISO dimensions • The Meter was redefined again in 1983 as the path length traveled by light during a time of 1/(299,792,458) second. This fixes speed of light at exactly 299,792,458 meters per second. Definitions based on light are more precise and reproducible because properties of light are believed to be universally constant. … E.T. would agree. • Length and time are related by the speed of light, a universal constant, so time can be similarly defined. • The “Second” is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. This definition eliminates any relationship to time variations of planetary motion.