Applications of trigonometry (from En. Of Trig.-pag.84)
Marine sextants like this are used to measure the angle of the
sun or stars with respect tothe horizon. Using trigonometry and an
accurate clock, the position of the ship can thenbe determined from
several such measurements.There are an enormous number of
applications of trigonometry and trigonometricfunctions. For
instance, the technique of triangulation is used in astronomy to
measurethe distance to nearby stars, in geography to measure
distances between landmarks, andin satellite navigation systems.
The sine and cosine functions are fundamental to thetheory of
periodic functions such as those that describe sound and light
waves.Fields which make use of trigonometry or trigonometric
functions include astronomy(especially, for locating the apparent
positions of celestial objects, in which sphericaltrigonometry is
essential) and hence navigation (on the oceans, in aircraft, and in
space),music theory, acoustics, optics, analysis of financial
markets, electronics, probabilitytheory, statistics, biology,
medical imaging (CAT scans and ultrasound), pharmacy,chemistry,
number theory (and hence cryptology), seismology,
meteorology,oceanography, many physical sciences, land surveying
and geodesy, architecture,phonetics, economics, electrical
engineering, mechanical engineering, civil engineering,computer
graphics, cartography, crystallography and game development.
Orientare cu busola pe harta O busola de calitate acul magnetic
oscileaza intr-o cutie cu lichid, ceea ce ii ofera acestuia o mai
mare stabilitate. Timpul de imobilizare a acului magnetic ar trebui
sa fie de maxim 7 secunde. Corpul busolei fiind transparent este
posibila stabilirea unei directii de deplasare fara orientarea
(nordarea) prealabila a hartii, operatie obligatorie in cazul
utilizarii tuturor celorlalte tipuri de busole existente (cu
oglinda in cutie metalica de exemplu). Principalele parti
componente ale busolei de orientare sunt: placa de baza
dreptunghiulara, transparenta si alungita ce are incrustate pe ea
celelalte piese ale busolei; cadranul divizat mobil care se roteste
o data cu cutia cu lichid. Aceasta are pe partea inferioara mai
multe linii paralele intre ele, pe numele lor linii directoare
sageata directoare o gasim pe cutia cu lichid in dreptul diviziunii
0 sau Nord, N acul magnetic are varful nordic colorat (rosu sau
negru), si prezinta o portiune fluorescenta lupa pentru marirea
diferitelor elemente de pe hartaDEPLASAREA PE O DIRECTIE FOLOSIND
BUSOLAStabilirea directiei de deplasare pe harta se face urmand
pasii:1. Trasam o linie pe harta intre pozitia A in care ne aflam
si locul B care reprezinta destinatia noastra.2. Fixam marginea
stanga a placii de baza a busolei de-a lungul liniei trasate,
astfel incat sageata directiei de pe placa sa fie indreptate spre
punctul vizat3. Rotim cadranul mobil divizat in asa fel incat
liniile directoare de pe partea inferioara a cutiei cu lichid sa
fie paralele cu liniile de nord magnetic ale hartii4. Dupa acesti
pasi, directia este deja stabilita. Ridicam busola de pe harta si
ne rotim, tinand-o orizontal, pana ce varful nordic al acului
magnetic ajunge in dreptul diviziunii 0 a cadranului, respectiv a
literei N (nord), si paralel cu liniile directoare. Directia de
deplasare este indicata de corpul busolei, respectiv de sageata
directiei de pe placa de baza.How to Make a CompassWith a few
household items, you can make a compass. Do this project with your
child and make learning science fun.
Things You'll Need Needle Magnet Cork Glass of
waterInstructions1. 1Get a one inch sewing needle. The needle can
be slightly larger or a little smaller, but should be around that
size. Once you have the needle in hand, rub it against a magnet. Be
sure to keep rubbing it in the same direction. 2Find a piece of
cork. Cork from a wine bottle will work great. Make sure the piece
of cork is dry and clean. 3Stick the needle threw the piece of
cork. On this step be careful because the needle is sharp and you
could injure yourself. You shouldn't let a child attempt this step.
Stick the needle in until the cork is centralized on the needle.
4Place the needle and cork in a glass of water. Make sure there is
enough water in the glass so that the piece of cork is able to
float. 5Put the glass of water on a stable surface. Once
stabilized, you should notice the needle start to spin until it
reaches a point. When the needle stops, it will be pointing north.
Now that you have a working compass, you are able to move the glass
about and it will continue to point in the same direction.History
of Cartography
WORLD MAP OF PIRRUS DE NOHA, CA. 1414History topic: The history
of cartography
This article describes how map making has played an important
role in the development of mathematics. It is hardly surprising
that cartography should be considered as a mathematical discipline
in early times since cartography measures positions of places
(mathematics was the science of measurement) and represents a the
surface of a sphere on a two dimensional map.Of course what
constitutes a map is hard to say, especially when one goes back to
the very earliest times. In around 6200 BC in Catal Hyk in Anatolia
a wall painting was made depicting the positions of the streets and
houses of the town together with surrounding features such as the
volcano close to the town. The wall painting was discovered in 1963
near the modern Ankara in Turkey. Whether it is a map or a stylised
painting is a matter of debate.
Early attempts at maps were severely limited by lack of
knowledge of anything other than very local features. In Egypt
geometry was used from very early times to help measure land. The
annual flooding by the Nile meant that without such measurements it
was impossible to reconstruct the boundaries that had existed
before the flood. Such measurements, however, seem only to have
been of local use and there is no evidence that the Egyptians
integrated the measurements into maps of large areas. The oldest
extant example of a local Egyptian map is the Turin papyrus which
dates from around 1300 BC.Early world maps reflect the religious
beliefs of the form of the world. For example maps have been
discovered on Babylonian clay tablets dating from around 600 BC.
One such map shows Babylon and the surrounding area in a stylised
form with Babylon represented by a rectangle and the Euphrates
river by vertical lines. The area shown is depicted as circular
surrounded by water which fits the religious image of the world in
which the Babylonians believed.
The earliest ancient Greek who is said to have constructed a map
of the world is Anaximander, who was born in 610 BC in Miletus (now
in Turkey), and died in 546 BC. He is said to have studied under
Thales but sadly no details of his map have survived. Of course,
although only a very limited portion of the Earth was known to
these ancient Greeks, the shape of the Earth was always going to be
of fundamental importance in world maps. Pythagoras, in the
6thcentury BC, is believed to be the first to put forward a belief
in a spherical Earth while Parmenides certainly argued in favour of
this in the following century. Around 350 BC Aristotle put forward
six arguments to prove that the Earth was spherical and from that
time on scholars generally accepted that indeed it was a
sphere.Eratosthenes, around 250 BC, made major contributions to
cartography. He measured the circumference of the Earth with great
accuracy. He sketched, quite precisely, the route of the Nile to
Khartoum, showing the two Ethiopian tributaries. He made another
important contribution in using a grid to locate positions of
places on the Earth. He was not the first to use such a grid for
Dicaearchus, a follower of Aristotle, had devised one about 50
years earlier. Today we use latitude and longitude to determine
such coordinates and Eratosthenes' grid was of a similar nature.
Note, of course, that the use of such positional grids are an early
form of Cartesian geometry. Following Dicaearchus, Eratosthenes
chose a line through Rhodes and the Pillars of Hercules (present
day Gibraltar) to form one of the principal lines of his grid. This
line is, to a quite high degree of accuracy, 36 north and
Eratosthenes chose it since it divided the world as he knew it into
two fairly equal parts and defined the longest east-west extent
known. He chose a defining line for the north-south lines of his
grid through Rhodes and drew seven parallel lines to each of his
defining lines to form a rectangular grid.
Hipparchus was critical of the grid defined by Eratosthenes,
saying reasonably enough that it was chosen arbitrarily. He
suggested that a grid should be chosen with astronomical
significance so that, for example, points on the same line would
all have the same length of longest day. Although Hipparchus never
constructed a map as far as we know, he did make astronomical
observations to describe eleven parallels given by his astronomical
definition. Although no copies of the work by Eratosthenes and
Hipparchus survives, we know of it through theGeographical
Sketchesof Strabo which was completed in about 23 A.D. Although
Strabo gives a critical account of earlier contributions to
cartography, he devotes only a small discussion to the problem of
projecting a sphere onto a plane. He states clearly that his work
is not aimed at mathematicians, rather at statesmen who need to
know about the customs of the people and the natural resources of
the land.The final ancient Greek contribution we consider was the
most important and, unlike that of Strabo, was written by a noted
mathematician. In about 140 A.D. Ptolemy wrote his major workGuide
to Geography,in eight books, which attempted to map the known world
giving coordinates of the major places in terms of what are
essentially latitude and longitude. The first volume gives the
basic principles of cartography and considers the problem of map
projection, that is mapping the sphere onto the plane. He gave two
examples of projections, and also described the construction of
globes. Right at the beginning Ptolemy identifies two distinct
types of cartography, the first being [1]:-... an imitation through
drawing of the entire known part of the world together with the
things which are, broadly speaking, connected with it.The second
type is [1]:-... an independent discipline[which]sets out the
individual localities.Now the main part ofGeographyconsisted of
maps but Ptolemy knew that although a scribe could copy a text
fairly accurately, there was little chance that maps could be
successfully copied. He therefore ensured that the work contained
the data and the information necessary for someone to redraw the
maps. He followed previous cartographers in dividing the circle of
the equator into 360 and took the equator as the basis for the
north-south coordinate system. Thus the line of latitude through
Rhodes and the Pillars of Hercules (present day Gibraltar) was 36
and this line divided the world as Ptolemy knew it fairly equally
into two. The problem of defining lines of longitude is more
difficult. It required the choice of an arbitrary zero but it also
required a knowledge of the circumference of the Earth in order to
have degrees correspond correctly to distance. Ptolemy chose the
Fortune Islands (which we believe are the Canary Islands) as
longitude zero since it was the most western point known to him. He
then marked off where the lines of longitude crossed the parallel
of Rhodes, taking 400 stadia per degree.
Had Ptolemy taken the value of the circumference of the Earth
worked out by Eratosthenes then his coordinates would have been
very accurate. However he used the later value computed by
Posidonius around 100 BC which, although computer using the correct
mathematical theory, is highly inaccurate. Therefore instead of the
Mediterranean covering 42 as it should, it covers 62 in Ptolemy's
coordinates. Books 2 to 7 ofGeographycontain the coordinates of
about 8000 places, but although he knew the correct mathematical
theory to compute such coordinates accurately from astronomical
observations, there were only a handful of places for which such
information existed. It is not surprising that the maps given by
Ptolemy were quite inaccurate in many places for he could not be
expected to do more than use the available data and, for anything
outside the Roman Empire, this was of very poor quality with even
some parts of the Roman Empire severely distorted.Ptolemy used data
from most of his predecessors, particularly Marinus of Tyre. He
used information from reports of travellers who gave such
information as "after ten days travel north we reached ...". In
order to estimate distances from such data, Ptolemy had to estimate
the difficulty of the terrain, how straight the route the
travellers taken had been, and many other unknowns. Given the way
that he gathered the data it is certainly not surprising that the
maps were inaccurate but they represented a considerable advance on
all previous maps and it would be many centuries before more
accurate world maps would be drawn.Little progress was made in
cartography over the next centuries. That the Romans made few
contributions is slightly strange given their skills at road
building which required accurate surveying measurements. Also the
very precise military strategies which their commanders used would
seem to give them the motivation to create maps to help their
military campaigns. Perhaps it was the mathematical nature of a map
which prevented the non-mathematical Romans from advancing the
subject. In China, however, there is evidence that mathematics had
been used an a major way in surveying and cartography. In [12] Liu
Hui's 3rdcentury work theSea Island mathematical manualis studied.
The book gives a good insight into the history of surveying in
China and its links with cartography. The main driving force in
China to survey and draw maps was often for military reasons but
also for problems such as water conservancy.Once Christianity
spread across Europe those of learning were Churchmen and the truth
about the world, they argued, was contained in the Bible and not to
be found by scientific investigation. Where Bible quotations
appeared to contradict pre-Christian scientific discoveries, then
good science was dismissed as pagan foolishness. Biblical
quotations convinced some that the Earth was a circle, certainly
not a sphere, while for others quotations such as "the four corners
of the Earth" in Isaiah proved that the Earth was rectangular.In
the Arabic/Persian/Muslim world, progress was made in cartography,
however, and in fact far more progress than was realised for a long
time, for it is only in recent years that the full significance of
these contributions has been realised. Ptolemy'sGeographywas
translated into Arabic in the 9thcentury and soon improvements were
being made using data obtained from the explorations being carried
out. Al-Khwarizmi wrote a major work on cartography which gave the
latitudes and longitudes for 2402 localities as a basis for his
world map. The book, which was based on Ptolemy'sGeography,lists
with latitudes and longitudes, cities, mountains, seas, islands,
geographical regions, and rivers. The manuscript does include maps
which are more accurate than those of Ptolemy, in particular it is
clear that where more local knowledge was available to al-Khwarizmi
such as in the regions of Islam, Africa and the Far East then his
work is considerably more accurate than that of Ptolemy, but for
Europe al-Khwarizmi seems on the whole to have used Ptolemy's
data.The major work by Sezgin, see [8], [9], and [10], has done
much to demonstrate that the medieval Islamic geographers had an
important influence on the development of geography in Europe up to
1800. In [8] he presents a reconstruction of al-Khwarizmi's map of
the world which he believes used a stereographic projection of the
terrestrial hemisphere, with pole on the terrestrial equator.
Sezgin also argues that Ptolemy'sGeographymay not have included a
world map, and that some later world maps are based, at least in
part, on Islamic sources.The next important Islamic scholar we
should mention is al-Biruni who wrote hisCartographyin around 995.
In it he discussed map projections due to other scientists, then
gives his own interesting mapping of a hemisphere onto a plane. A
detailed description of this projection is given in [17]. Al-Biruni
wrote a textbook on the general solution of spherical triangles
around 1000 then, some time after 1010, he applied these methods on
spherical triangles to geographical problems. He introduced
techniques to measure the Earth and distances on it using
triangulation. He computed very accurate values for the differences
in longitude and latitude between Ghazni in Afghanistan and Mecca.
He found the radius of the earth to be 6339.6 km, a value not
obtained in the West until the 16thcentury. HisMasudic
canoncontains a table giving the coordinates of six hundred places,
some of which were measured by al-Biruni himself, some being taken
al-Khwarizmi's work referred to above.At a time when Christian
Europe was producing religious representations of the world rather
than scientific maps, another type of map, or perhaps more
accurately chart, for the use of sailors began to appear. These
were calledportolanmaps (from the Italian word for a sailing
manual) and were produced by sailors using a magnetic compass. The
earliest examples we know about date from the beginning of the
14thcentury, and were Italian or Catalan portolan maps. The
earliest portolan maps covered the Mediterranean and Black Sea and
showed wind directions and such information useful to sailors. The
coast lines shown on these maps are by far the most accurate to
have been produced up to that time. The Catalan World Map produced
in 1375 was the work of Abraham Cresques from Palma in Majorca. He
was a skilled cartographer and instrument maker and the map was
commissioned by Charles V of France. The western part of his map
was partly based on portolan maps while the eastern part was based
on Ptolemy's data.
The 15thcentury saw cartography revolutionised in Europe. The
first steps involved the translation of Ptolemy'sGeographyinto
Latin which was begun by Emmanuel Chrysoloras and completed in 1410
by Jacobus Angelus. The main motivation to improve cartography came
with the discoveries of new lands made by the Portuguese explorers
of the 15thcentury. Brother Mauro, a monk from Murano near Venice,
had an excellent reputation in cartography by the middle of the
15thcentury. In 1457 he was commissioned by the King of Portugal to
produce a new world map containing details of the new lands
discovered by the Portuguese explorers, and charts drawn by these
explorers were sent to him. Producing a map which did not follow
Ptolemy clearly worried Mauro who wrote (see for example [5]):-I do
not think it derogatory to Ptolemy if I do not follow his
'Cosmografia', because, to have observed his meridians or parallels
or degrees, it would be necessary in respect to the setting out of
the known parts of this circumference, to leave out many provinces
not mentioned by Ptolemy. But principally in latitude, that is from
south to north, he has much 'terra incognita', because in his time
it was unknown.Brother Mauro added the new discoveries to his maps
but he made no improvements in the science of cartography. Despite
1300 years passing since Ptolemy's time, Mauro is still not able to
give a good approximation for the circumference of the Earth
writing:-I have found various opinions regarding this
circumference, but it not possible to verify them ... they are not
of much authenticity, since they have not been tested.The means to
make maps widely available also happened in the 15thcentury with
the invention of the printing press around the middle to the
century. The first printed version of Ptolemy'sGeographyappeared in
1475 being the Latin translation referred to above. This edition
only contained the text and not maps. The date of the first edition
to contain maps is still disputed but may be the one printed in
Rome in 1478 which contained 27 maps. Many printed editions with
maps followed in quick succession, and newly discovered lands were
soon included. New maps were added to various editions to include
more accurate and detailed information about Europe, the first
being in the Florence edition of 1480 which contained new maps of
France, Italy, Spain and Palestine based on recent knowledge. The
first to show the New World was a new edition of the 1475 Rome
edition, which appeared in 1508 with 34 maps. The edition which
many consider to be the first modern atlas (although the term
'atlas' was not used until Gerardus Mercator coined it around 1578)
was published in Strasburg in 1513 with 27 maps of the ancient
world and 20 new maps based on recent knowledge produced by Martin
Waldseemller. He made a clear distinction between the two parts
(see [7] where the following quotation is given):-We have confined
the Geography of Ptolemy to the first part of the work, in order
that its antiquity may remain intact and separate.Waldseemller's
map of the world was the first to cover 360 of longitude and to
show the complete coast of Africa. Another first for Waldseemller
occurred in an earlier work in 1507 in which he proposed the naming
of America (see [16] where the following quotation is given):-Since
another fourth part[of the world]has been discovered by Americus
Vesputius, I do not see why anyone should object to its being
called after Americus the discoverer, a man of natural wisdom, Land
of Americus or America, since both Europe and Asia have derived
their names from women.Waldseemller also made important
contributions to the science of cartography. He wrote on surveying
and perspective and produced a booklet on how to use globes.Arabic
science continued to flourish, now along side European science, and
mathematical geography saw important developments with Sulayman
al-Mahri'sTuhfat al-fuhul fi tamhid al-usuland the commentary on
itKitab sarhwritten in the early sixteenth-century. Al-Mahri used
astronomical observations of the height of stars to determine the
difference in latitude between two places. Trigonometric methods
allowed differences in longitude to be calculated. He also
developed a good understanding of how to compensate for the errors
caused by short-cuts in his mathematical calculations and also for
errors caused by inaccurate data.It was the 16thcentury which saw
the first major mathematical improvements in cartography in Europe
although Regiomontanus had led the way towards the end of the
15thcentury. He set up a new press in Nuremburg in 1472 and
announced his intention to publish maps and books including
Ptolemy'sGeography.With an interest in trigonometry, mathematical
instruments, astronomy, and geography, Regiomontanus was in a good
position to give a lead. He set up a workshop in Nuremburg to make
mathematical instruments, and published works giving details of the
use of the instruments. He realised that accurate coordinates of
places were required to draw accurate maps and that the greatest
problem was in computing the longitude. He proposed the method of
lunar distances to determine longitude which was an important
proposal. Johann Werner was a follower of Regiomontanus from
Nuremburg. Werner's most famous work on geography isIn Hoc Opere
Haec Cotinentur Moua Translatio Primi Libri Geographicae
Cl'Ptolomaeiwritten in 1514. This book contains a description of an
instrument with an angular scale on a staff from which degrees
could be read off. It also gives a method to determine longitude
based on eclipses of the Moon and makes a study of map projections.
This work by Werner strongly influenced Gerardus Mercator.Albrecht
Drer visited Regiomontanus' workshops in Nuremburg when he was
young lad. He was fascinated with the ideas of projecting a sphere
and also of what the Earth would look like if viewed from the
heavens. He employed his ideas of perspective on maps, and in
particular he collaborated with Johann Stabius in the construction
of globes in 1515. Apianus, a noted mathematician, began his
publishing career with producing a world mapTypus orbis
universaliswhich he based on an earlier 1520 world map by Martin
Waldseemller. His 1524 publicationCosmographia seu descriptio totis
orbiswas a work based largely on Ptolemy which provided an
introduction to astronomy, geography, cartography, surveying,
navigation, weather and climate, the shape of the earth, map
projections, and mathematical instruments.Gemma Frisius was another
mathematician who made significant contributions to cartography. He
produced his own version of Apianus'sCosmographiaa few years after
the original edition. In 1530 he publishedOn the Principles of
Astronomy and Cosmography, with Instruction for the Use of Globes,
and Information on the World and on Islands and Other Places
Recently Discoveredwhich made major contributions to cartography.
In particular he described how longitude could be calculated using
a clock to determine the difference in local and absolute times,
being the first to make such a proposal. In 1533 Gemma Frisius
publishedLibellus de locurumwhich described the theory of
trigonometric surveying and in particular contains the first
proposal to use triangulation as a method of accurately locating
places. This provided an accurate means of surveying using
relatively few observations. Positions of places were fixed as the
point of intersection of two lines and, as Frisius pointed out,
only one accurate measurement of actual distance was required to
fix the scale. Not only did Frisius propose an efficient
theoretical method for surveying which was needed to produce
accurate maps, but he also produced the instruments with which to
undertake the surveys and he published accurate maps using the data
gathered from such surveys.
Following Gemma Frisius, major contributions were made by
Gerardus Mercator who studied under Frisius. Mercator made many new
maps and globes, but his greatest contribution to cartography must
be theMercator projection. He realised that sailors incorrectly
assumed that following a particular compass course would have them
travel in a straight line. A ship sailing towards the same point of
the compass would follow a curve called a loxodrome (also called a
rhumb line or spherical helix), a curve which Pedro Nunes, a
mathematician greatly admired by Mercator, had studied shortly
before. A new globe which Gerardus Mercator produced in 1541 was
the first to have rhumb lines shown on it. This work was an
important stage in his developing the idea of the Mercator
projection which he first used in 1569 for a wall map of the world
on 18 separate sheets. The 'Mercator projection' has the property
that lines of longitude, lines of latitude and rhomb lines all
appear as straight lines on the map. In this projection the
meridians are vertical and parallels having increased spacing in
proportion to the secant of the latitude. Edward Wright published
mathematical tables to be used in calculating Mercator's projection
in 1599, see [20] for details.
Of course the Mercator projection has the property that
distances near the poles are greatly distorted so it was not easy
to use the map to measure distances. Gerardus Mercator gave
instructions on the map so that for two places if one knew any two
of the following four pieces of data:difference on
longitudes,difference in latitudes,direction between them,distance
between them,then his formula allowed one to find the other two. It
is interesting to realise that on a map of the world drawn with the
Mercator projection, Greenland (whose area is about 2 million km2)
appears to be larger than Africa (whose area is about 30 million
km2). As a world map the Mercator projection then has considerable
disadvantages (as necessarily do all projections) but for sea
charts it is undoubtedly the best projection and was eventually
adopted by all sailors.Abraham Ortel, known by his Latinised name
of Ortelius, was born in Antwerp on 4 April 1527. He studied Greek,
Latin and mathematics and, strongly influenced by Gerardus
Mercator, went on to open a map making business. He published
theTheatrum orbis terrarumin 1570 which consisted of 70 maps
presented in a uniform style using the most up-to-date data. It was
the most popular atlas of its time, and it is important in the
history of cartography partly because Ortelius quotes 87
authorities for the data on which his maps are based. It appeared a
few years before the atlas of Mercator began publication and many
argue that Mercator delayed in order to let his younger friend have
priority. This, however, seems unlikely and it is much more
probable that Mercator's work was delayed, for by the 1570s he was
an old man with health problems.
By the 17thand 18thcenturies scientific advances had paved the
way for further improvements in cartography. Not only were new
methods being developed, but there were also arguments to produce a
different type of map. For example Thomas Burnet inThe theory of
the earth(London, 1684) wrote:-I do not doubt but that it would be
of very good use to have natural maps of the Earth . . . as well as
civil. . . . Our common maps I call civil, which note the
distinction of countries and of cities, and represent the
artificial Earth as inhabited and cultivated: But natural maps
leave out all that, and represent the Earth as it would be if there
were not an Inhabitant upon it, nor ever had been - the skeleton of
the Earth, as I may so say, with the site of all its parts.
Methinks also a Prince should have such a draught of his country
and dominions, to see how the ground lies in the several parts of
them, which highest, which lowest; what respect they have to one
another, and to the sea; how the rivers flow, and why; how the
mountains lie, how heaths, and how the marches. Such a map or
survey would be useful both in time of war and peace, and many good
observations might be made by it, not only as to natural history
and philosophy, but also in order to the perfect improvement of a
country.Progress in cartography now became dependent on having the
means of accurately determining the position of places in the
world. Calculating latitude was easy, and had long been achieved
with a sextant, but the problem of accurately calculating the
longitude proved a great challenge. The story of attempts at
solving this problem are given in our two essays Longitude and the
Acadmie Royale and English attack on the Longitude Problem and it
is to these essays that we refer the reader for information on many
later developments in cartography. The Low Countries had dominated
developments in cartography through the 16thand early
17thcenturies. However after this the centre of activity moved to
France where a national survey based on a mathematical approach to
trigonometric surveying led the way.There is another problem with
longitude, other than methods to calculate it, namely that a zero
needs to be set arbitrarily. At first, as is to be expected,
several different places were chosen as the zero such as Paris,
Cadiz, Naples, Pulkova, Stockholm and London. International
agreement was needed to set cartographic standards and the
International Meridian Conference held in Washington D.C. USA in
1884 had delegates from 26 countries. They standardised the
Greenwich Meridian as the zero for longitude and, after some delay,
all countries adopted this and the equator as the basic reference
lines.There is, of course, another decision to be taken in order to
standardise maps, namely how the map is oriented. It is fairly
logical to have either north or south at the top, but which is
chosen is a completely arbitrary decision. Early Christian maps had
north at the top while early Arabic/Muslim maps had south at the
top. Without any international agreement, it has become standard
practice to have north at the top of a map. Other collaborative
international projects have been less successful. In 1891 there was
an International Geographical Congress in Bern which established
theInternational Map of the World.Standards were set and a symbol
convention was chosen. The scale was to be 1:1000000 and several
nations agreed to cooperate to produce a world map to this
standard. Some, but not all, of the proposed maps have been
produced but the project has never been completed.Article by:J J
O'ConnorandE F Robertson
Suppose you walk past a barber's shop one day, and see a sign
that says:Do you shave yourself?If not, please come in and I'll
shave you!I shave anyone who does not shave himself,and noone
else.So the question is: Who shaves the barber?Bertrand Russell
(1872-1970), Barber paradox
Hodometrul cu el se msurau distanele mari n antichitate