## Units of Measurement – Introduction

### Definitions

**Measurement** means the **act of measuring** or **the size** of something.

*To Measure means to ascertain the dimensions, capacity, or amount (quantity) of something.*

**A unit of measurement** is a definite **magnitude of a physical quantity**, defined and adopted **by convention and/or by law**, that is **used as a standard for measurement** of the same physical quantity. **Any other value of the physical quantity** can be expressed as a simple **multiple of the unit of measurement.**

**People have always found it necessary to measure time, distance, area, volume and weight, and have devised units that measure these quantities.** For **time**, there is an **absolute standard in the motions of the heavens**, but for the other quantities the units have had to be chosen arbitrarily.

Official view is that **only recently** have we **succeeded in creating system of measurement accepted all over the world** as the standard system for use in science and trade: **The International System of Units (SI)**. However some researchers suggest that** in ancient times** people were commonly using units of measure similar in value and closely related to each other.

# Ancient Metrology

All ancient cultures used units of measures. The earliest known uniform systems of weights and measures seem to have all been created sometime in the 4th and 3rd millennia BC among the ancient peoples of **Mesopotamia, Egypt and the Indus Valley.** Indeed, people who have not previously been regarded as civilized in the literal sense, manifestly utilised this sophisticated measurement system to extraordinary degrees of accuracy.

Similarity of certain units used by ancient architects around the world raises these questions:

*Was there a system as a whole, which all civilizations used as reference, that predated them all?*

*Were ancient people able to arrive to the identical systems of measure because they used nature to define such units?**Was there re any direct cultural contacts between the disparate peoples who used the identical system?*

The modern practice of **dividing a circle into 360 degrees, of 60 minutes each, began with the Sumerians.**

### Units of Time

**Observing movement of the Sun and the stars** suggested that **Earth is spinning around its own axis and that the Sun is moving against the background of constellations. **This suggested there **are 2 cycles: earth axis spin cycle and earth around the sun orbital cycle.**

In ancient times it was easy to observe the **Sun in order to establish units** of time so it is good assumption that such units would be **“solar units of time”.** We use them today and call them **“solar days” or simply days** (as opposed to stellar background based **“sidereal days”**).

**After these cycles were noticed (discovered) the next step would be to quantify them** (describe them using specific units of measurement).

Numbers like **360, 72, 30, 12 and multiples thereof were intentionally plotted in ancient myths.** It was as if the storyteller were trying to convey a secret code. Here’s what the figures signify in the precession cycle:

- 360 degrees = 12 X 30 degrees, or one full circuit through the zodiac constellations
- 72 years = the time it takes for the stars to shift 1 degree
- 30 degrees = one astrological age (a different zodiac constellation rises with the Sun every 2,160 years)
- 12 = the total number of zodiac signs or astrological ages. 12 times 2,160 = 25,920 years, or one full precession cycle

- In Babylonia, the ancient scribe Berossus wrote that
**mythical kings ruled before the Great Flood for a total 432,000 years.** - In India, the
**Rigvida contains exactly 432,000 syllables.**And although the calculation has created some confusion of late, the**Vedic Kali Yuga**(representing the current world age) is said to be comprised of**432,000 years.** - On the other side of the globe,
**Mayan calendar units parrot the precessional figures.**

For example: 1 tun (an astronomical year) = 360 days;

6 tuns = 2,160 days;

1 katun = 7200 days,

6 katuns = 43,200.

**The standard Mayan base of 20 (ours is 10) is arrived at by dividing 43,200 by 2,160.**

Today we use decimal system (multiples of 10) however when it comes to **measuring angles, we use ancient convention:** a circle is not divided into 100 (or 1000 parts), instead it divided in 360 units (called degrees) and each unit is further divided into 60 equal pats called minutes and each one of these is further divided into 60 equal units called seconds.

**A full circle has 360 degrees, 21,600 (360×60) minutes and 1,296,000 (360x60x60) seconds.**

Ancient divided the whole sky into **12 equal parts** called constellations.

*Note: *As the sun passes through the twelve zodiac signs, the four signs that govern the four cardinal events in the sun’s journey are the most significant. Of supreme importance is the sign under which the sun crosses the celestial equator on the spring equinox. Astrological ages are named after this sign. For example, today we are somewhere at the end of the age of Pisces, because Pisces is the sign behind the sun when it crosses its midway point in the spring. Due to a slight imbalance in the earth’s wobble, these four signs change roughly every 2,200 years, in a gradual process called the precession of the equinoxes. It takes an entire 26,000 years for all twelve signs of the zodiac to pass behind the place where the sun crosses the celestial equator during the spring equinox. Every 72 years we slip backwards 1 degree of the zodiac, meaning that soon we will be entering the age of Aquarius.

Before the present age of Pisces was the age of Aries from about 2400BC to 200BC, and before that was the age of Taurus from 4600BC to 2400BC. During that period, the spring equinox was in Taurus, the summer solstice in Leo, the winter solstice in Aquarius, and the fall equinox in Scorpio. Although Scorpio is today represented by the Scorpion, that part of the sky used to be represented by another constellation, the Eagle or Phoenix. The symbols that represent these signs – the Lion, Eagle, Bull and Man – are often found in religious and mythological texts that were developed during the age of Taurus.

Age of Taurus, 4600BC – 2400BC

Fall Equinox: Scorpio Summer Solstice: Leo

Winter Solstice: Aquarius Spring Equinox: Taurus

There are several references to these four animals in the Old Testament (e.g. Ezekiel), which were later copied into the New Testament book of Revelation.

The first living creature was like a lion, the second like a bull, the third living creature had a human face, and the fourth living creature was like a flying eagle. (Revelation 4:7)

These four symbols, which represented the four seasons and the four elements (fire, earth, water, air), were later assigned to four specific apostles whose names were given to the four books of the gospels.

Matthew = Human, Mark = Lion, Luke = Ox, John = Eagle. Source: http://www.goodreads.com/reader/10021-jesus-potter-harry-christ

**Zodiac (circle division): 12 equal parts (or 4×3): 4 quarters each into 3 and further each third into 10 (4, 3, 10)****Each Constellation had 30 degrees (360/12)****30 degrees = 1,800 minutes = 108,000 seconds.**

Dividing the daily cycle into equal parts established units of time.

**Ancient divisions were not decimal but based on 24 and 60:**

1/24 of the Earth spin cycle was unit we call now 1 hour, Each hour is divided into 60 equal units (called minutes) and each minute is divided into 60 equal units (called seconds).

The finest unit of time in ancient times was one second:

1/(24x60x60) = 1/86400 part of one spin cycle (1 day). **In other words 1 rotation cycle of the Earth (one day) has 24 hours, 1,440 minutes and 86,400 seconds.**

Each day was divided into 24 parts called hours, each hour into 60 minutes and each minute in to 60 seconds (today we divide seconds using decimal system; 1/10, 1/100, 1/1000 (millisecond).

**Each day has 24 hours = 1440 minutes (24×60) = 86400 seconds (24x60x60).**

The Earth turns 15 degrees per hour.

**Notes:**

*The average (typical) resting heart rate in a healthy adult is 60–80 bits per minute 1-1.333 bits per second**In 1 second light travels 299,792.458 km, in 1.000692286 milliseconds – 300 km in a vacuum*

**Ancients established value for the earth axis spin cycle (called day) and used this as measuring unit for the longer cycle of earth orbiting the sun** (or the sun returning to the same constellation on the sky.) Earth axial speed is 360 deg/axial cycle (day). Earth orbital speed is 360 deg/solar cycle (year).

**If we choose time units based on a solar day (86,400 seconds), sidereal day will be 365/366 x 86,400 seconds = 86,163.929 seconds. It means it will be shorter than solar day by 236.1 seconds = 3.9345 minutes = 3 min 56.1 seconds (rounded to 4 min)**

**Ancient Units of length**

**The Egyptian cubit**, the Indus Valley units of length referred to above and the **Mesopotamian cubit** were used in the 3rd millennium BC and are the **earliest known units used by ancient peoples to measure length.** The measures of length used in ancient India included the dhanus (bow), the krosa (cry, or cow-call) and the yojana (stage).

**The common cubit** was the length of the forearm from the elbow to the tip of the middle finger. It was divided into the span of the hand (one-half cubit), the palm or width of the hand (one sixth), and the digit or width of the middle finger (one twenty-fourth) and the span or the length between the tip of little finger to the tip of the thumb.

**The Sacred Cubit (aka Royal Cubit),** which was a **standard cubit enhanced by an extra span**—**thus 7 spans or 28 digits long**—was used in **constructing buildings and monuments and in surveying in ancient Egypt.** **The inch, foot, and yard evolved from these units through a complicated transformation not yet fully understood. Some believe they evolved from cubic measures; others believe they were simple proportions or multiples of the cubit.**

*The Egyptian Royal Cubit rod, from the Turin collection, has an official length of 20.618 inches. Its refined value, under the sexagesimal geodetic system, was calculated mathematically to be 20.61818182 inches.*

*Note: **Royal Cubit consists of 28 units, digits, which is the same as 7 palms of 4 digits. The names of divisions of royal cubit may suggest anatomical origin, however the divisions indicate astronomical origin of the cubit (7 days per week, 28 days lunar calendar, 4 weeks per lunar month)…*

**Interesting Relationship between ancient units of lengths**

From measurements of the King’s chamber and other dimensions in the Great Pyramid by John Greaves, Sir Isaac Newton realized that the King’s Chamber was 10 x 20 Royal Cubits (or Thoth Cubits) so that the Royal Cubit is determined as equal to 1.719 (1.72) feet.

Therefore, we can take the following as “ideal values” in metric system of 3 fundamental ancient units of length:

**1MY = 1 foot + 1RC = 2.72 feet = 0.829m
1 Royal Cubit = 1.72 feet = 0.632 MY = 0.524m
1 Remen = 0.7071 Royal Cubit = 0.3715 m **

Notice the relationship between all 3 units can be well approximated as follows:

**1 Megalithic Yard = sqrt(5) x 1 Remen = 1 Royal Cubit x sqrt(5/2) = 1.5811388 RC**

Let’s notice the relationship between all 3 units can be well approximated as follows:

1 Megalithic Yard = sqrt(5) x 1 Remen = 1 Royal Cubit x sqrt(5/2) = 1.5811388 RC

1 Royal Cubit = sqrt(2) x 1 Remen = 0.525 m

1 Megalithic Yard = sqrt(5) x 1 Remen = 0.83 m

It may be seen that, from the basic square side of the Remen, the length of the Royal Cubit can be derived by multiplying the Remen by the square root of 2; similarly, the Megalithic yard can be derived by multiplying the Remen by the square root of 5.

Another geometric illustration of the relationship between Remen, Royal Cubit and Megalithic Yard, where1M.Y. is circumference of the circle (0.823m) inscribed in 1/3 R.C. square (with 99.3 % accuracy):

If Megalithic Yard was defined as equal to the circumference of the circle inscribed in ½ of Royal Cubit square:

1 MY=1/2 x RC x “Pi” so 1 MY = 1.570795 RC

For RC=1 we get and 1MY=1.570795 and 1 Remen = 1/sqrt(2)=0.7071

*NOTE:*

**For the Great Pyramid:**

Height = **280 Royal Cubits**

Base Side = 440 Royal Cubits = **280 MY**

In whichever case, **the Greeks and Romans inherited the foot from the Egyptians.**

The **Roman foot (~296 mm) was divided into both 12 unciae (inches) (~24.7 mm) and 16 digits (~18.5 mm).**

The Romans also introduced the **mille passus (1000 paces) or double steps,** the pace being equal to five Roman feet (~1480 mm).

The **Roman mile of 5000 feet (1480 m)** was introduced into England during the occupation. Queen Elizabeth I (reigned from 1558 to 1603) changed, by statute, the mile to **5280 feet (~1609 m) or 8 furlongs, a furlong being 40 rod (unit)s (~201 m) of 5.5 yards (~5.03 m)each.**

**The introduction of the yard (0.9144 m) as a unit of length came later, but its origin is not definitely known. Some believe the origin was the double cubit, others believe that it originated from cubic measure.** Whatever its origin, the** early yard was divided by the binary method into 2, 4, 8, and 16 parts called the half-yard, span, finger, and nail.** The association of the yard with the “gird” or circumference of a person’s waist or with the distance from the tip of the nose to the end of the thumb of King Henry I (reigned 1100–1135) are probably standardizing actions, since several yards were in use in Britain. There were also Rods, Poles and Perches for measurements of length. The following table lists the equivalents:

Length

12 lines = 1 inch

12 inches = 1 foot

3 feet = 1 yard

1760 yards = 1 mile

36 inches = 1 yard

440 yards = quarter mile

880 yards = half mile

100 links = 1 chain

10 chains = 1 furlong

8 furlongs = 1 mile

4 inches = 1 hand

22 yards = 1 chain

5.5 yards = 1 rod, pole or perch

4 poles = 1 chain

40 poles = 1 furlong

**Ancient Metrology**

Interest in ancient metrology was triggered by research into the various Megalith building cultures and the Great Pyramid of Giza.

In 1637 John Greaves, professor of geometry at Gresham College, made his first of several studies in Egypt and Italy, making **numerous measurements of buildings and monuments, including the Great Pyramid.** These activities **fuelled many centuries of interest in metrology of the ancient cultures by the likes of Isaac Newton and the French Academy.**

**The first known description and practical use of a physical pendulum is by Galileo Galilei, however, Flinders Petrie, a disciple of Charles Piazzi Smyth, is of the opinion that it was used earlier by the ancient Egyptians.** Writing in an article in Nature, 1933 Petrie says:

*If we take the natural standard of one day divided by 105, the pendulum would be 29.157 inches (0.7405878 m) at lat 30 degrees. Now this is exactly the basis of Egyptian land measures, most precisely known through the diagonal of that squared, being the Egyptian double cubit. The value for this cubit is 20.617 inches, while the best examples in stone are 20.620±0.005inches. *

No explanation is offered as to why no Egyptian pendulums have been found, despite the extremely rich archaeological material from this culture, nor to the question as to why none of the rich historic material from Egypt mentions this, or indeed why a divisor of 105 would have been chosen or measured.

### Royal Cubit (Sacred Cubit)

**Uniformity of royal cubits.**

It is difficult to imagine how a supposedly anatomical measure could turn up in different nations with distinct subdivisions yet have a suspiciously similar length. If they were exaggerating in order to make their own king look the larger than life, why would the lengths be similar? **There is even mention of English, Chinese and Mexican Aztec cubits within the range 518–531 mm (20.4 to 20.9 in).**

Uniformity of royal / architectural cubits

**Civilization Length (mm)**

Mesopotamia 522–532

Persia 520–543

Egypt 524–525

**Mysterious royal cubit origin.** ‘The anatomical length … cannot possibly be as long as the royal cubit of 525 mm.’24 (Unless, of course, it came from a people taller than the Egyptians.) Egyptian royal cubits had seven palms and 28 fingers in a cubit. The Babylonian had 30 divisions. **Both numbers indicate astronomical origin of the cubit (28 or 30 day month with 4 weeks of 7 days).**

**Respect for the royal cubit.** This indicates an important legacy, like a standard handed down from the ‘Gods’. The ‘Gods’ of certain cultures could be early post-flood founders a few generations after Noah. In Egypt, building overseers required the **Royal Egyptian Cubit to be calibrated against a precision standard at regular intervals. Failure to do so was punishable by death.**

http://www.worldwideflood.com/ark/noahs_cubit/cubit_references.htm

**Charles Piazzi Smyth**

**John Taylor**, in his 1859 book ** “The Great Pyramid: Why Was It Built? & Who Built It?”**, claimed that the Great Pyramid was planned and the building supervised by the biblical Noah, and that it was:

**built to make a record of the measure of the Earth.**A paper presented to the Royal Academy on the topic was rejected.

Taylor’s theories were, however, the inspiration for the deeply religious **archeologist Charles Piazzi Smyth** to go to Egypt to study and **measure the pyramid**, subsequently publishing his book * Our Inheritance in the Great Pyramid (1864)*, claiming that the

**measurements he obtained from the Great Pyramid of Giza indicated a unit of length, the pyramid inch, equivalent to 1.001 British inches, that could have been the standard of measurement by the pyramid’s architects.**From this he extrapolated a number of other measurements, including the pyramid pint, the sacred cubit, and the pyramid scale of temperature.

Smyth claimed—and presumably believed—that

**the inch was a God-given measure handed down through the centuries from the time of Israel, and that the architects of the pyramid could only have been directed by the hand of God.**To support this

**Smyth said that, in measuring the pyramid, he found the number of inches in the perimeter of the base equalled 1000 times the number of days in a year, and found a numeric relationship between the height of the pyramid in inches to the distance from Earth to the Sun, measured in statute miles.**

Smyth used this as an argument against the introduction of the metre in Britain, which he considered a product of the minds of atheistic French radicals.

### The Grand Scheme

By the time measurements of Mesopotamia were discovered, by doing various exercises of mathematics on the definitions of the major ancient measurement systems, various people (Jean-Adolphe Decourdemanche in 1909, August Oxé in 1942) came to** the conclusion that the relationship between them was well planned.**

**Livio C. Stecchini** claims in his A History of Measures:

**The relation among the units of length can be explained by the ratio 15:16:17:18 among the four fundamental feet and cubits.** Before I arrived at this discovery, Decourdemanche and Oxé discovered that the cubes of those units are related according to the conventional specific gravities of oil, water, wheat and barley.

Stecchini makes claims that imply that** the Egyptian measures of length, originating from at least the 3rd millennium BC, were directly derived from the circumference of the earth with an amazing accuracy.** According to* “Secrets of the Great Pyramid”* (p. 346), his claim is that the

**Egyptian measurement was equal to 40,075,000 meters, which compared to the International Spheroid of 40,076,596 meters gives an error of 0.004%.**No consideration seems to be made to the question of, on purely technical and procedural grounds,

**how the early Egyptians, in defining their cubit, could have achieved a degree of accuracy that to our current knowledge can only be achieved with very sophisticated equipment and techniques.**

**Alexander Thom**

**Oxford engineering professor Alexander Thom**, doing **statistical analysis of survey data taken from over 250 stone circles in England and Scotland**, came to the **conclusion** that there must have been a **common unit of measure which he called a megalithic yard.** This research was published in the ** Journal of the Royal Statistical Society** (Series A (General), 1955, Vol 118 Part III p275-295) as a paper entitled A Statistical Examination of the Megalithic Sites in Britain.

As Professor Thom observed in his book ** Megalithic Sites in Britain** (1967):

“*It is remarkable that one thousand years before the earliest mathematicians of classical Greece, people in these islands not only had a practical knowledge of geometry and were capable of setting out elaborate geometrical designs but could also set out ellipses based on the Pythagorean triangles*.”

**Robin Heath**

Later, these ideas were further developed as **defence for the Imperial units against the emerging metric system**, and adopted by parts of the anti-metric movement. Robin Heath, in his book **Sun, Moon & Stonehenge,**** connects the megalithic yard (and thus Stonehenge) to the imperial foot**, and manages to **connect a few astronomical phenomena, and the Egyptian Royal Cubit (and thus the Great Pyramid) into one grand equation (MY is an abbreviation for megalithic yard):**

*if the lunar year is represented by 12 MY then 1 ft corresponds precisely to the extra 10.875 days to coincide with the end of the solar or seasonal year. Furthermore, the period between the end of the solar year and 13 lunations – 18.656 days – is represented by another unit of length from antiquity, the ‘Royal Cubit’ of 20.63″ or 1.72 ft.*

This seems to bring pseudoscientific metrology to new heights, especially in view of the conclusion:

*Hence the equally astonishing revelation that 1 MY = 1 ft + 1 RC. Assuming that the MY was the primary unit, then the derivative foot and cubit appear to have formed a logical and essential part of the astronomical and calendrical researches of our Neolithic ancestors. If, however, the foot preceded the MY in time – and here we must remember that 1/1,000th of a degree of arc around the equatorial circumference of the Earth is just 365.244 ft in length! – then knowledge of the roundness of the Earth must have predated use of the MY…i.e. well before 3,000BC. There are no other choices readily apparent!*

**Megalithic Yard**

A measuring unit defined by astronomical and/or geodetic properties of the Earth would have to contain information about the size of the Earth.** Such unit was used by ancient builders of megaliths – it is called the Megalithic Yard (MY).**

The MY turns out to be much more than an abstract unit such as the modern metre, it is a highly scientific measure repeatedly constructed by empirical means. It is based upon observation of three fundamental factors:

- The orbit of the Earth around the sun
- The spin of the Earth on its axis
- The mass of the Earth

**The Megalithic Yard is a unit of measurement, about 2.72 feet (32.4 in or 0.829 m), that some researchers believe was used in the construction of megalithic structures. The proposal was made by Alexander Thom* as a result of his surveys of 600 megalithic sites in England, Scotland, Wales and Brittany.**

C**hristopher Knight and Alan Butler** further develop the work of Smyth’s and Stecchini’s “Grand Scheme” in their* Civilization One* hypothesis, which describes a

**megalithic system of units.**This system is claimed to be

**the source of all standard units used by civilization, and is so named after the Neolithic builders of megaliths.**Knight and Butler contend the

**reconstructed megalithic yard (0.82966m) is a fundamental part of a megalithic system.**Although the megalithic yard is the work of Alexander Thom, Knight and Butler make a novel contribution by speculating on how the MY may have been created by using a pendulum calibrated by observing Venus. It also

**explains the uniformity of the MY across large geographical areas.**The accuracy claimed for this procedure is disputed by astronomers.

Knight and Butler describe a procedure for Neolithic astronomers to make a “

**Venus Pendulum**“,

**using the transit of Venus across the sky to give both time and distance units.**

**Measures of volume and massare derived from the megalithic yard.** It is divided into 40 megalithic inches. Knight and Butler claim that a cube with a side of 4 megalithic inches has a volume equal to one imperial pint and weighs one imperial pound when filled with unpolished grain. They also posit ratio relationships with the imperial acre and square rod.

**A Megalithic Yard is a unit of measurement, about 2.72 feet (0.83 m)**, that some researchers believe was **used in the construction of megalithic structures.** The proposal was made by Alexander Thom as a result of his surveys of 600 megalithic sites in England, Scotland, Wales and Britanny. Thom additionally proposed the **Megalithic Rod of 2.5 MY** and suggested the Megalithic Rod could be divided into one hundred and the Meglithic Yard divided into forty, which he called the **Megalithic Inch of 2.073 centimetres (0.816 in).** Thom applied the statistical lumped variance test of J.R. Broadbent on this quantum and found the results significant while others have challenged his statistical analysis and suggested that Thom’s evidence can be explained in other ways, for instance the average length of a pace.

*Source: Wikipedia*

Michell claims that **all over the world traditional units of measurements are related.**

He goes on to point out the value of the pu that still survives in Indo-China is given in L.D’A. Jackson’s Modern Metrology (available on the net) as **2.7272 miles** with the fraction repeating. Without knowledge of the pu’s existence its former use in Britain was deduced by J. F. Neal, who called it the **Megalithic Mile** because the ratio is similar to that between the foot and the Megalithic Yard. **Since the ratio between the dimensions of the Earth and Moon is 10:2.7272 the following relationships unambiguously exist.**

Earth’s diameter = 7920 miles

Moon’s diameter = 792 megalithic miles

Perimeter of the square containing the circle of the Earth = 31,680 miles

Perimeter of the square containing the circle of the Moon = 3,168 megalithic miles.

Sun’s diameter = 864,000 miles = 316,800 megalithic miles.

**The Imperial System**

**Britain introduced Imperial Units**, based on the **yard, pound, and second**, in the 19th century **to resist the metric system** and to uphold **an alternative comprehensive system.**

In engineering, English units were divided decimally just like metric ones, especially in the United States.

**Both the Imperial units and US customary units derive from earlier English units.** Imperial units were mostly used in the **British Commonwealth** and the former **British Empire.** **US customary units are still the main system of measurement used in the United States despite Congress having legally authorized metric measure on 28 July 1866**. Some steps towards US metrication have been made, particularly the redefinition of basic US units to derive exactly from SI units, so that in **the US the inch is now defined as 0.0254 m (exactly),** and the avoirdupois **pound** is now defined as **453.59237 g (exactly).**

**The basic English unit of length was the yard of three feet, or the fathom of six**. Each

**English foot was divided into 12**and each inch into 3

*inches*,*barleycorns*or 12

*lines*.

**Eventually, one inch was defined as exactly 25.4 mm, which tied the English and metric units together.**In the United States, a

**meter was sometimes defined as exactly 39.37 inches**, which gave

**1 inch = 25.40005 mm**, just enough different to be annoying in geodesy.

**12 such inches made a**, used by the Coast and Geodetic Survey. 5 feet, 6 inches, and 7 lines was written 5′ 6″ 7″‘. The single and double apostrophes have survived into modern times, but not the triple.

*survey foot***The modern feet are descended from the Roman measurement of the same name and approximate value.** The Roman foot, **11.65 modern inches (29.6 cm)**, was usually **divided into 16 inches, not 12,** however (as four palms of four Roman inches, about 3 modern inches, each). **Divisions by powers of 2 are specially useful,** since they are *binary*, and much more adapted to computers than powers of 10. **The English inch was later divided into halves, quarters, eighths, and so on,** because of the utility and extendibility of this system, which completely replaced the use of lines.

**Roman standards were relatively uniform**, an interlude between times of confusion. **Most Roman units of length survived in name or spirit in the English and other systems, even if changing somewhat in absolute value.** For example, the ** stadium**, which was 1/8 of a Roman mile, or 202 yards, became the

**, 1/8 of an English mile, or 220 yards. The**

*furlong***, a forearm’s length, was**

*cubit***1-1/2 Roman feet or 6 palms, and typically used in building.**

**Some ancient cubits seem to have been longer than this, up to about 22 inches.**

*Hands*of 4 inches are still used to measure the height of a horse (at the shoulders).

Note how

**length units were conveniently based on parts of the body used to measure distances.**

**The Roman mile consisted of 1,000 double paces, or 5,000 Roman feet, or 1480 metre, or 1619 yards.**- Distances on Roman roads were measured by odometers attached to carriage axles, as described in Vitruvius, and marked on mile stones.
**The English mile of 5280 feet is 1609 metre**(a “metric mile” is, apparently,**1,500 metre**). As explained in the article “Chaining,” it was defined as**80 chains of 66 ft each**, and this is the reason for the odd number**5280**. Gunter’s chain of 100 links was a successful attempt to create a portable length standard that was not as stretchy as a cord.**The English mile happened to come out a little larger than the Roman mile, to which it was intended to be an approximation.****The nautical mile is 1852 metre, which corresponds to one minute of arc of latitude, approximately.****The ‘geographical’ mile was 7420 metre**, and the**Prussian mile 7532 metre.****These long miles were about five Roman miles.**- The
was another measure of journeys, usually*league***3 English miles.**France had an assortment of leagues: 2,000 toise for the*lieue de poste*, 3 Roman miles for the*lieue de terre*, 4 kilometers for the*lieue kilométrique*, and 3 nautical miles for the*lieue marine*. - The Greeks had the
, and the*stadium*of 580-622 feet*plethron*of 97-100 feet. - The ancient Persian
*parasang*was 3.25 to 3.3 miles, 30 Greek stadia.

Any great accuracy in the size of old units is illusory unless a critical study is made. The standards have, of course, disappeared, and their magnitude can be determined only by remeasuring a distance in modern terms.

*Source: History of units of measure: http://mysite.du.edu/~jcalvert/tech/oldleng.htm*

# The Metric System

**The metric system**, originating in the French Revolution and propagated widely in the 19th century, **has brought a dreary but convenient uniformity to units of measurement.**

**A number of metric systems of units have evolved since the adoption of the original metric system in France in 1791.** The current international standard metric system is** the International System of Units (SI).** An important feature of modern systems is **standardization. Each unit has a universally recognized size.**

**In the establishment of the metric system, the quadrant of the earth was measured as 5 130 738.62 toise, which was set equal to 10 000 000 metre.** This was a bad choice for defining a unit of length, worse than simply making a couple of arbitrary scratches on a bar, since it involved a difficult and tedious procedure, especially when the news arrived in France that **the earth was not a sphere to a sufficient approximation.** At any rate, **a scratched bar was later adopted as standard**, and the earth’s quadrant allowed to be whatever it turned out to be in terms of metres.

A similar mistake was made in **defining the kilogram as the weight of a cubic decimetre (a litre) of water**, since the density of water changes with temperature, and weight can be measured more accurately than a decilitre can be measured. **Thus, both the litre and the kilogram were defined arbitrarily.** The clumsiness was reflected in the slight difference between a cubic centimetre and a millilitre that thereby arose.

The **second of time remained 1/86400 of a mean solar day**, with common **units of time not related decimally, but by factors of 24 and 60.** The decimal calendar was a ludicrous failure. The French also defined a right angle as 100 grads, another superfluity. The units of time and angles were already uniform, and required no work. As has been mentioned above, the French population ignored the metric system until it was forced upon them.

**The circumference of the Earth**

From the 18th century, inspired by the statement of Aristotle that the circumference of the Earth was calculated as 400,000 stadia, it became a belief among members of the French Académie des Sciences that ancient linear measures were all derived directly from the circumference of the Earth. Archaeologist Jean Antoine Letronne, in 1822, tried to show the connection to a supposed pre-Greek measurement of the Earth.

Ronald Zupko claims that since Gunter suggested the concept of division of the earth’s arc into length in the seventeen century, Cassini in 1720 suggested dividing the earth’s circle to 360 degrees, of 60 miles of 1000 fathoms of 6 feet, which again was the inspiration of the metric system, it is not all that unreasonable to suggest that this could have not happened in an earlier time. Indeed, he claims that the error in the Greek foot lies wholy in the range of the geographic measures (since the earth is not spherical), and that the multiples of it follow the sexagesimal division of the earth. Zupko, however, provides no evidence how the Greek would have actually measured the Earth’s actual circumference, if it was the basis of their units of measurement.

*Table: Some milestones in measuring the Earth meridian. stands for the length of the meter calculated as the 40,000,000th part of the meridian (for some important cases. In the results expressed in the form ` ‘ xxxxx stands for the length of one degree meridian arc ( in the text). Ancient estimates have to be taken with large uncertainty:*

*Table source: http://www.roma1.infn.it/~dagos/history/sm/node6.html*

**The Seconds Pendulum**

In August 1790 the French National Assembly entrusted the reform to the Academy of Sciences. The Academy nominated a preliminary commission,which adopted a decimal scale for all measures, weights and coins. The commission presented its report on 27 October 1790. A second commission was charged with choosing the unit of length. The commission was set up on 16 February 1791 and reported to the Academy of Sciences on 19 March 1791. On 26 March 1791 the National Assembly accepted the Academy’s proposals of the decimal system and of a quarter of the meridian as the basis for the new system and the adoption of the consequent immediate unit.

**The guiding ideas of the French scientists** are well expressed in the introduction to the document presented to the Academy:

*The idea to refer all measures to a unit of length taken from nature has appeared to the mathematicians since they learned the existence of such a unit as well as the possibility to establish it: they realized it was the only way to exclude any arbitrariness from the system of measures and to be sure to preserve it unchanged for ever, without any event, except a revolution in the world order, could cast some doubts in it; they felt that such a system did not belong to a single nation and no country could flatter itself by seeing it adopted by all the others.*

Actually, if a unit of measure which has already been in use in a country were adopted, it would be difficult to explain to the others the reasons for this preference that were able to balance that spirit of repugnance, if not philosophical at least very natural, that peoples always feel towards an imitation looking like the admission of a sort of inferiority. As a consequence, there would be as many measures as nations.

Three were the candidates considered by the commission:

- the seconds pendulum;
- a quarter of the meridian;
- a quarter of the equator.

**The latter two units are based on the dimension of Earth.** Indeed, Earth related units had had also quite a long history, though they were not as popular as the seconds pendulum, probably because their intrinsic difficulty to be determined.Mouton had suggested in 1670 the unit that we still use in navigation and call now** nautical mile: the length of one minute of the Earth’s arc along a meridian, equal to 1852m**. In 1720 the astronomer

**Jean Cassini had proposed the radius of Earth**, a

**(he had also indicated the**

*“natural”*unit for a spherical object**). However, neither of these old, French proposals are mentioned in the report of the commission.**

*one ten-millionth part of the radius as the best practical unit*The **quarter of the equator was rejected**, mainly because considered hard to measure and somehow “not democratic”.

*So, we believe we are bound to decide to assume this kind of unit of measure and also to prefer the quarter of the meridian to the quarter of the equator. The operations that are necessary to establish the latter could be carried out only in countries that are too far from ours and, as a consequence, we should have to undertake expenditures as well as to overcome difficulties that would be superior to the advantages that seem to be promised. The inspections, in case somebody would like to carry them out, would be more difficult to be accomplished by any nation, at least until the progress of the civilization reaches the peoples living by the equator, a time that still seems to be unfortunately far away. The regularity of this circle is not more assured than the similarity or regularity of the meridians. The size of the celestial arc, that corresponds to the space that would be measured, is less susceptible to be determined with precision; finally it is possible to state that all peoples belong to one of Earth’s meridians, while only a group of peoples live along the equator.*

As far as **the length of the seconds pendulum is concerned**, during the 18th century** its value was known with sub-millimeter accuracy in several places in France and around the world**, often related to work of rather famous people like Isaac Newton, Mersenne, Giovan Battista Riccioli, Picard, Jean Richer, Gabriel Mouton, Huygens, Jean Cassini, Nicolas Louis de Lacaille, Cassini de Thury and La Condamine. For example, in **1740 Lacaille and Cassini de Thury had measured the length of the seconds pendulum in Paris (48 latitude), obtaining a value of 440.5597 lignes** corresponding **to 0.993828m **(the metric conversion was fixed by the French law of 10 December 1799, that established the meter to be equal to *3 pied and 11.296 lignes;* *Metric equivalent*:** ligne [ line] = 2.25583mm).
Newton himself had estimated the length of the seconds pendulum at several latitudes between 30 and 45 degrees: his value at 45 degrees was 440.428 lignes, i.e. 99.353cm**. A measurement at the equator, made by La Condamine during the Peru expedition gave

**439.15 lignes (99.065cm).**

**Space-time Connection?**

The well known small angle formula that gives the period T of the simple pendulum (i.e. the elementary text book pendulum) as a function of its length L is:

**T= 2*Pi*sqrt(L/g)**

where **g** is the gravitational acceleration, approximately **9.80665 m/s ^{2}** at sea level on Earth

**For L=1 m T=2.007 s, in other words each swing takes 1.0035 seconds**

( more info: http://www.roma1.infn.it/~dagos/history/sm/node18.html)

* Source: http://www.roma1.infn.it/~dagos/history/sm/index.html*

**The International System of Units (abbreviated SI from French), **

**established in 1960,**

**is the**and is generally a

**modern form of the metric system**

**system of units of measurement devised around seven base units and the convenience of the number ten.****The SI,** **is the world’s most widely used system of measurement**, which is **used both in everyday commerce and in science.** The system has been nearly globally adopted with the **United States being the only industrialized nation that does not mainly use the metric system** in its commercial and standards activities. The **United Kingdom has officially partially adopted metrication**, with no intention of replacing customary measures entirely. **Canada has adopted it for all legal purposes but imperial/US units are still in use, particularly in the buildings trade.**

*One common complaint about the metric system is that it doesn’t provide a natural way to divide things into thirds; even dividing things into quarters requires one to go down two levels, instead of just one, in the system of units, since 0.25 is the decimal that represents 1/4. The metric system did not divide the day into 100,000 parts; instead, the hour, minute, and second were retained to allow the day to be divided neatly into quarters and thirds. In response to an instance of the occasionally-heard suggestion that the metric system should have been built on base 12 instead of base 10, it occurred to me that the precedent of an everyday unit, the day, standing in such a relationship to the metric unit, the second, that the day can be exactly divided into 27 parts, each of which consists of an even number of seconds (3200 seconds, or 53 minutes and 20 seconds), one could, for example, use as everyday units a metric pound of 453.6 grams (instead of approximately 453.69 grams) and a metric inch of 2.52 centimeters (instead of 2.54 centimeters). 453.6 grams divides evenly into 81 units of 5.6 grams, and also into 7 units of 64.8 grams – and, for that matter, into 8 units of 56.7 grams. 2.52 centimeters divides evenly into 9 units of 0.28 centimeters, and also into 7 units of 0.36 centimeters – and, for that matter, into 4 units of 0.63 centimeters. However, I can’t really expect that this very wild idea of using this metric pound and metric inch as everyday units, and measuring things out in these pounds and inches, so that they can be evenly divided into thirds, ninths, and sevenths if one uses the regular metric scale, would catch on. Source: http://www.quadibloc.com/other/cnvint.htm*

# The International System of Units (SI)

**All systems of weights and measures, metric and non-metric, are linked through a network of international agreements supporting the International System of Units.** The International System is called the SI, using the first two initials of its French name Système International d’Unités. The key agreement is the **Treaty of the Meter** (Convention du Mètre), **signed in Paris on May 20, 1875**. 48 nations have now signed this treaty, including all the major industrialized countries. The United States is a charter member of this metric club, having signed the original document back in 1875.

The SI is maintained by a small agency in Paris, the **International Bureau of Weights and Measures (BIPM, for Bureau International des Poids et Mesures),** and it is updated every few years by an international conference, the **General Conference on Weights and Measures** (CGPM, for Conférence Générale des Poids et Mesures), attended by representatives of all the industrial countries and international scientific and engineering organizations. As BIPM states on its web site, “**The SI is not static but evolves to match the world’s increasingly demanding requirements for measurement**.”

At the heart of the SI is a **short list of base units defined in an absolute way without referring to any other units.** The base units are consistent with the part of the metric system called the MKS system. In all there are **seven SI base units:**

- m – the meter for distance,
- kg – the kilogram for mass,
- s – the second for time,
- A – the ampere for electric current,
- K – the kelvin for temperature,
- mol – the mole for amount of substance, and
- cd – the candela for intensity of light.

Definitions of 3 fundamental base units:

**meter or metre (m)**– the metric and SI**base unit of distance.**

Originally, the**meter was designed to be one ten-millionth (1/10,000,000) of a quadrant, the distance between the Equator and the North Pole.**In other words, meter was defined as 1/10,000,000 of the distance from the Earth’s equator to the North Pole measured on the circumference through Paris. (The Earth is difficult to measure, and a small error was made in correcting for the flattening caused by the Earth’s rotation. As a result, the meter is too short by a bit less than 0.02%. That’s not bad for a measurement made in the 1790’s.) For a long time,**the meter was precisely defined as the length of an actual object, a bar kept at the International Bureau of Weights and Measures in Paris.**In recent years, however, the SI base units (with one exception) have been redefined in abstract terms so they**can be reproduced to any desired level of accuracy in a well-equipped laboratory.**The 17th General Conference on Weights and Measures**in 1983 defined the meter as that**In other words, “The metre is the length of the path travelled by light in vacuum during a time interval of 1/299, 792, 458 of a second.”*“distance that makes the speed of light in a vacuum equal to exactly 299, 792, 458 meters per second”.***The speed of light in a vacuum, c, is one of the fundamental constants of nature**. Since c defines the meter now, experiments made to measure the speed of light are now interpreted as measurements of the meter instead.

**The meter is equal to approximately 1.093 613 3 yards, 3.280 840 feet, or 39.370 079 inches.**Its name comes from the Latin metrum and the Greek metron, both meaning “measure.” The unit is spelled meter in the U.S. and metre in Britain; there are many other spellings in various languages

**second (s or sec or “)**– a fundamental**unit of time**in all measuring systems and the SI base unit of time. The name simply means that**this unit**is the**second division of the hour, the minute being the first**. The day is divided in 24 hours, each hour divided in 60 minutes, each minute divided in 60 seconds.**The second was defined as 1/86,400 mean solar day until astronomers discovered that the mean solar day is actually not constant**. A second is 1 / (24 × 60 × 60) of the solar day. The definition was then changed to**1/86,400 of the specific mean solar day 1900 January 1.**Since we can’t go back and measure that day any more, this wasn’t a real solution to the problem.**In 1967, scientists agreed to define the second as that**In other words, if we really want to measure a second, we count 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 refers to a caesium atom at rest at a temperature of 0 K.” This definition allows scientists to reconstruct the second anywhere in the world with equal precision.*“period of time which makes the frequency of a certain radiation emitted by atoms of cesium-133 equal to 9, 192, 631, 770 hertz (cycles per second)”.*

**kilogram (kg)**– the base**unit of mass**in the SI and MKS versions of the metric system.

The kilogram is**defined as the mass of the standard kilogram, a platinum-iridium bar in the custody of the International Bureau of Weights and Measures (BIPM) near Paris, France.**Copies of this bar are kept by the standards agencies of all the major industrial nations, including the U.S. National Institute of Standards and Technology (NIST). One kilogram equals exactly 1000 grams, or about**2.204 622 6 pounds.**Original definition:**The mass of one litre of water.**A litre is one thousandth of a cubic metre. Since the redefinition of the metre in 1960,**the kilogram is the only unit which is directly defined in terms of a physical artifact rather than a property of nature.**

Other SI units, called SI **derived units**, are defined algebraically in terms of these fundamental units.

Currently there are 22 SI derived units.They include:

- the radian and steradian for plane and solid angles, respectively;
- the newton for force and the pascal for pressure;
- the joule for energy and the watt for power;
- the degree Celsius for everyday measurement of temperature;
- units for measurement of electricity: the coulomb (charge), volt (potential), farad (capacitance), ohm (resistance), and siemens (conductance);
- units for measurement of magnetism: the weber (flux), tesla (flux density), and henry (inductance);
- the lumen for flux of light and the lux for illuminance;
- the hertz for frequency of regular events and the becquerel for rates of radioactivity and other random events;
- the gray and sievert for radiation dose; and
- the katal, a unit of catalytic activity used in biochemistry.

Future meetings of the CGPM may make additions to this list; the katal was added by the 21st CGPM in 1999.

In addition to the 29 base and derived units, the** SI permits the use of certain additional units,** including:

- the traditional mathematical units for measuring angles (degree, arcminute, and arcsecond);
- the traditional units of civil time (minute, hour, day, and year);
- two metric units commonly used in ordinary life: the liter for volume and the tonne (metric ton) for large masses;
- the logarithmic units bel and neper (and their multiples, such as the decibel); and
- three non-metric scientific units whose values represent important physical constants: the astronomical unit, the atomic mass unit or dalton, and the electronvolt.

**The SI currently accepts the use of certain other metric and non-metric units traditional in various fields.** These units are supposed to be “defined in relation to the SI in every document in which they are used,” and “their use is not encouraged.” These barely-tolerated units might well be prohibited by future meetings of the CGPM. They include:

- the nautical mile and knot, units traditionally used at sea and in meteorology;
- the are and hectare, common metric units of area;
- the bar, a pressure unit, and its commonly-used multiples such as the millibar in meteorology and the kilobar in engineering;
- the angstrom and the barn, units used in physics and astronomy.

**The SI does not allow use of any units other than those listed above and their multiples.** In particular, it does not allow use of any of the English traditional units (the horsepower, for example), nor does it allow the use of any of the algebraically-derived units of the former CGS system, such as the erg, gauss, poise, stokes, or gal. In addition, the SI does not allow use of other traditional scientific and engineering units, such as the torr, curie, calorie, or rem.

*Sources: Wikipedia and Dictionary of Units of Measurement – by Russ Rowlett*

# Global Coordinates

The global coordinate system is constructed to match the surface of the Earth. It consists of **360 divisions** that intersect at the North Pole and the South Pole and parallel lines that circle the Earth and are parallel to the equator. The lines that intersect the poles are called lines of longitude or meridians. Those parallel to the equator are called lines of latitude or parallels. Medians are perpendicular to the equator.

The equator divides the Earth into the Northern and Southern hemispheres. Lines of latitude north of the equator are described as degrees north (N), represented by the symbol ” °.” Lines of latitude south of the equator are described as degrees south (S). The equator is at 0°. The North Pole is at 90° N, and the South Pole is at 90° S. Lines of latitude are equally spaced from each other. Each degree of latitude is approximately 60 nautical miles or 69 statute miles (**111** kilometers) from the next.

The prime meridian divides the Earth into the Eastern and Western hemispheres. The prime meridian is the line of longitude that runs through Greenwich, England. Points on the globe are measured from Greenwich in an eastward or westward direction in units called degrees, from 0§ longitude at Greenwich to 180° east or west. All the lines of longitude converge at the two poles and are equally spaced from each other only at the equator.

Each degree of latitude and longitude is divided into 60 minutes, indicated by an apostrophe, and each minute is divided into 60 seconds, indicated by a quotation mark.

Every location on the Earth’s surface can be described in terms of its latitude and longitude, counting degrees north or south of the equator and east or west of Greenwich. When describing a position, the latitude is listed first. Minutes and seconds can be used to give a more precise location description. For instance, a part of Detroit, Michigan, is located at 42° 30′ N, 83° W.

Time is also measured from Greenwich. The time at Greenwich is referred to as Greenwich Mean Time (GMT). Universal Coordinated Time (UTC), or ZULU. The globe is divided into 24 time zones—12 to the east of Greenwich and 12 to the west of Greenwich. Time retreats by one hour as pilots fly every 15° westward from Greenwich and advances by one hour as pilots fly every 15°eastward until reaching the International Dateline halfway around the globe (or at 180§ longitude). Pilots use GMT to refer to the time of day rather than using local time zones.

Source: http://www.centennialofflight.gov/essay/Dictionary/Global_Coordinates/DI80.htm

### Subject Related Resources and Links

- A History of Measures
- http://www.metrum.org/

- http://greatpyramidmath.weebly.com/

- http://www.unc.edu/~rowlett/units/
- https://standards.nasa.gov/history_metric.pdf
- http://www.sizes.com/units/meter.htm
- http://www.gizapyramid.com/cliveross3.htm
- http://www.roma1.infn.it/~dagos/history/sm/node18.html
- http://science.yourdictionary.com/measurement
- The Measure of All Things
- http://relationary.wordpress.com/
- http://www.holyblasphemy.net/
- http://www.quadibloc.com/science/cal09.htm

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