The Etymology of Geometry Terms

The Etymology of Geometry Terms Theres an anecdote about how the philosopher-mathematician Pythagoras overcame a students natural dislike of geometry. The student was poor, so Pythagoras offered to pay him an obol for each theorem he learned. Eager for the money, the student agreed and applied himself. Soon, however, he became so intrigued, he begged Pythagoras to go faster, and even offered to pay his teacher. In the end, Pythagoras recouped his losses. Etymology provides a safety net of demystification. When all the words you hear are new and confusing, or when those around you put old words to strange purposes, a grounding in etymology may help. Take the word line. You put your  ruler to paper and draw a line against the straight edge. If youre an actor, you learn your lines line after line of text in a script. Clear. Obvious. Simple. But then you hit Geometry. Suddenly your common sense is challenged by technical definitions*, and line, which comes from the Latin word linea (a linen thread), loses all practical meaning, becoming, instead, an intangible, dimension-less concept that goes off at both ends to eternity. You hear about parallel lines that by definition never meet each other except they do in some warped reality dreamt up by Albert Einstein. The concept you have always known as the  line has been renamed line segment. After a few days, it comes as something of a relief to run into an intuitively obvious circle, whose definition as a set of points equidistant from a central point still fits your previous experience. That circle** (coming possibly from a Greek verb meaning to hoop around or from a diminutive of the circular Roman circus, circulus) is marked with what you would have, in pre-geometry days, called a line across part of it. This line is called a chord. The word chord comes from the Greek word (chordà ª) for a piece of animal gut used as a string in a lyre. They still use (not necessarily cat) gut for violin strings. After circles, youll probably study equiangular or equilateral triangles. Knowing the etymology, you can break those words up into component parts: equi (equal), angular, angle, lateral (of a side/sided), and tri (3). A three-sided object with all sides equal. It is possible that youll see triangle referred to as trigon. Again, tri means 3, and gon derives from the Greek word for corner or angle, gà ´nia. However, youre far more likely to see the word trigonometry trigon the Greek word for measure. Geo-metry is the measure of Gaia (Geo), the Earth. If youre studying geometry, you probably already know you must memorize theorems, axioms, and definitions corresponding with names. Names of Shapes cylinderdodecagonheptagonhexagonoctagonparallelogrampolygonprismpyramidquadrilateralrectanglespheresquare andtrapezoid. While the theorems and axioms are pretty much geometry-specific, the names of shapes and their properties have further applications in science and life. Beehives and snowflakes are both dependent on the hexagon. If you hang a picture, you want to make sure its top is parallel to the ceiling. Shapes in geometry are usually based on the angles involved, so the two root words (gon and angle [from the Latin angulus which means the same thing as the Greek gà ´nia]) are combined with words that refer to number (like triangle, above) and equality (like equiangular, above). Although there are apparent exceptions to the rule, generally, the numbers used in combination with the  angle (from the Latin) and gon (from the Greek) are in the same language. Since hexa is Greek for six, youre unlikely to see hexangle. Youre far more likely to see the combined form hexa gon, or hexagon. Another Greek word used in combination with the numbers or with the prefix poly- (many) is hedron, which means a foundation, base, or sitting place. A polyhedron is a many-sided three-dimensional figure. Construct one from cardboard or straws, if you like, and demonstrate its etymology, by making it sit on each of its many bases. Even if it doesnt help to know that a tangent, the line (or is that line segment?) that touches at only one point (depending on the function), comes from the Latin tangere (to touch) or the oddly shaped quadrilateral known as a trapezoid got its name from looking like a table, and even if it doesnt save a lot of time to memorize the Greek and Latin numbers, instead of just the names of shapes if and when you run into them, the etymologies will come back to add color to your world, and to help you with trivia, aptitude tests and word puzzles. And if you ever do run into the terms on a geometry exam, even if panic sets in, youll be able to count through in your head to figure out whether its a regular pentagon or heptagon that you would inscribe with a traditional five-pointed star. * Heres one possible definition, from McGraw-Hill Dictionary of Mathematics: line: The set of points (x1, . . ., xn) in Euclidean space.... The same source defines line segment as A connected piece of a line. ** For the etymology of circle, see Lingwhizt and the possibility of an ancient Indo-European word for millstone, another round flat object.

Napoleons Empire

Napoleon's Empire The borders of France and the states ruled by France grew during the wars of the French Revolution and the Napoleonic Wars. On May 12th, 1804 these conquests received a new name: the Empire, ruled by a hereditary Bonaparte Emperor. The first – and in the end only – emperor was Napoleon, and at times he ruled vast swathes of the European continent: by 1810 it was easier to list the regions he didn’t dominate: Portugal, Sicily, Sardinian, Montenegro, and the British, Russian and  Ã¢â‚¬â€¹Ottoman Empires. However, while it’s easy to think of the Napoleonic Empire as one monolith, there was considerable variation within the states. The Make-Up of the Empire The empire was divided into a three-tier system. Pays Rà ©unis: this was land governed by the administration in Paris, and included the France of the natural frontiers (i.e. the Alps, the Rhine and the Pyrenees), plus states now subsumed into this government: Holland, Piedmont, Parma, the Papal States, Tuscany, the Illyrian Provinces and a lot more of Italy. Including France, this totaled 130 departments in 1811 – the peak of the empire – with forty-four million people. Pays Conquis: a set of conquered, although supposedly independent, countries which were ruled by people approved by Napoleon (largely his relatives or military commanders), designed to buffer France from attack. The nature of these states ebbed and flowed with the wars, but included the Confederation of the Rhine, Spain, Naples, the Duchy of Warsaw and parts of Italy. As Napoleon developed his empire, these came under greater control. Pays Allià ©s: The third level was fully independent states who were bought, often unwillingly, under Napoleon’s control. During the Napoleonic Wars Prussia, Austria and Russia were both enemies and unhappy allies. The Pays Rà ©unis and Pays Conquis formed the Grand Empire; in 1811, this totaled 80 million people. In addition, Napoleon redrew central Europe, and another empire ceased: the Holy Roman Empire was disbanded on August 6th, 1806, never to return. Nature of the Empire The treatment of states in the empire varied depending on how long they remained part of it, and whether they were in the Pays Rà ©unis or Pays Conquis. It’s worth pointing out that some historians reject the idea of time as a factor, and focus on regions in which pre-napoleon events inclined them to be more receptive to Napoleon’s changes. States in the Pays Rà ©unis before the Napoleonic era were fully departmentalized and saw the benefits of the revolution, with the end of ‘feudalism’ (such as it existed), plus land redistribution. States in both the Pays Rà ©unis and Pays Conquis received the Napoleonic legal Code, the Concordat, tax demands, and administration based on the French system. Napoleon also created ‘dotations’. These were areas of land seized from conquered enemies where the entire revenue was given to Napoleon’s subordinates, conceivably forever if the heirs stayed loyal. In practice they were a huge drain on the loca l economies: the Duchy of Warsaw lost 20% of revenue in dotations. Variation remained in outlying areas, and in some privileges survived through the era, unaltered by Napoleon. His introduction of his own system was less ideologically driven and more practical, and he would pragmatically accept survivals which the revolutionaries would have cut out. His driving force was to keep control. Nevertheless, we can see the early republics being transformed slowly into more centralized states as Napoleon’s reign developed and he envisioned more of a European empire. One factor in this was the success and failure of the men Napoleon had placed in charge of conquered lands – his family and officers – because they varied greatly in their loyalty, sometimes proving more interested in their new land than aiding their patron despite in most cases owing everything to him. Most of Napoleon’s clan appointments were poor local leaders, and an exasperated Napoleon sought more control. Some of Napoleon’s appointees were genuinely interested in effecting liberal reforms and being loved by their new states: Beauharnais created a stable, loyal and balanced government in Italy and was very popular. However, Napoleon prevented him from doing more, and often clashed with his other rulers: Murat and Joseph ‘failed’ with the constitution and Continental System in Naples. Louis in Holland rejected much of his brother’s demands and was ousted from power by an angry Napoleon. Spain, under the ineffectual Joseph, couldn’t really have gone more wrong. Napoleon’s Motives In public, Napoleon was able to promote his empire by stating laudatory aims. These included safeguarding the revolution against Europe’s monarchies and spreading freedom throughout oppressed nations. In practice, Napoleon was driven by other motives, although their competing nature is still debated by historians. It’s less likely that Napoleon began his career with a plan to rule Europe in a universal monarchy – a sort of Napoleon dominated empire which covered the whole continent – and more likely he evolved into wanting this as the opportunities of war brought him greater and greater success, feeding his ego and expanding his aims. However, a hunger for glory and a hunger for power – whatever power that may be - seem to have been his over-riding concerns for much of his career. Napoleon’s Demands on Empire As parts of the empire, the conquered states were expected to assist in furthering Napoleon’s aims. The cost of the new warfare, with greater armies, meant more expense than ever before, and Napoleon used the empire to for funds and troops: success funded more attempts at success. Food, equipment, goods, soldiers, and tax were all drained out by Napoleon, much of it in the form of heavy, often annual, tribute payments. Napoleon had another demand on his empire: thrones and crowns on which to place and reward his family and followers. While this form of patronage left Napoleon in control of the empire by keeping leaders tightly bound to him – although putting close supporters in power didn’t always work, such as in Spain and Sweden – it also let him keep his allies happy. Large estates were carved out of the empire both to reward and to encourage the recipients to fight to keep the empire. However, all these appointments were told to think of Napoleon and France first, and their new homes second. The Briefest of Empires The empire was created militarily and had to be enforced militarily. It survived the failures of Napoleon’s appointments only as long as Napoleon was winning to support it. Once Napoleon failed, it was swiftly able to eject him and many of the puppet leaders, although the administrations often remained intact. Historians have debated whether the empire could have lasted and whether Napoleon’s conquests if allowed to last, would have created a unified Europe still dreamt of by many. Some historians have concluded that Napoleon’s empire was a form of continental colonialism that could not have lasted. But in the aftermath, as Europe adapted, a lot of the structures Napoleon put in place survived. Of course, historians debate exactly what and how much, but new, modern administrations could be found all over Europe. The empire created, in part, more bureaucratic states, better access to the administration for the bourgeoisie, legal codes, limits on the aristocracy an d church, better tax models for the state, religious toleration and secular control in church land and roles.

Napoleons Empire

Napoleon's Empire The borders of France and the states ruled by France grew during the wars of the French Revolution and the Napoleonic Wars. On May 12th, 1804 these conquests received a new name: the Empire, ruled by a hereditary Bonaparte Emperor. The first – and in the end only – emperor was Napoleon, and at times he ruled vast swathes of the European continent: by 1810 it was easier to list the regions he didn’t dominate: Portugal, Sicily, Sardinian, Montenegro, and the British, Russian and  Ã¢â‚¬â€¹Ottoman Empires. However, while it’s easy to think of the Napoleonic Empire as one monolith, there was considerable variation within the states. The Make-Up of the Empire The empire was divided into a three-tier system. Pays Rà ©unis: this was land governed by the administration in Paris, and included the France of the natural frontiers (i.e. the Alps, the Rhine and the Pyrenees), plus states now subsumed into this government: Holland, Piedmont, Parma, the Papal States, Tuscany, the Illyrian Provinces and a lot more of Italy. Including France, this totaled 130 departments in 1811 – the peak of the empire – with forty-four million people. Pays Conquis: a set of conquered, although supposedly independent, countries which were ruled by people approved by Napoleon (largely his relatives or military commanders), designed to buffer France from attack. The nature of these states ebbed and flowed with the wars, but included the Confederation of the Rhine, Spain, Naples, the Duchy of Warsaw and parts of Italy. As Napoleon developed his empire, these came under greater control. Pays Allià ©s: The third level was fully independent states who were bought, often unwillingly, under Napoleon’s control. During the Napoleonic Wars Prussia, Austria and Russia were both enemies and unhappy allies. The Pays Rà ©unis and Pays Conquis formed the Grand Empire; in 1811, this totaled 80 million people. In addition, Napoleon redrew central Europe, and another empire ceased: the Holy Roman Empire was disbanded on August 6th, 1806, never to return. Nature of the Empire The treatment of states in the empire varied depending on how long they remained part of it, and whether they were in the Pays Rà ©unis or Pays Conquis. It’s worth pointing out that some historians reject the idea of time as a factor, and focus on regions in which pre-napoleon events inclined them to be more receptive to Napoleon’s changes. States in the Pays Rà ©unis before the Napoleonic era were fully departmentalized and saw the benefits of the revolution, with the end of ‘feudalism’ (such as it existed), plus land redistribution. States in both the Pays Rà ©unis and Pays Conquis received the Napoleonic legal Code, the Concordat, tax demands, and administration based on the French system. Napoleon also created ‘dotations’. These were areas of land seized from conquered enemies where the entire revenue was given to Napoleon’s subordinates, conceivably forever if the heirs stayed loyal. In practice they were a huge drain on the loca l economies: the Duchy of Warsaw lost 20% of revenue in dotations. Variation remained in outlying areas, and in some privileges survived through the era, unaltered by Napoleon. His introduction of his own system was less ideologically driven and more practical, and he would pragmatically accept survivals which the revolutionaries would have cut out. His driving force was to keep control. Nevertheless, we can see the early republics being transformed slowly into more centralized states as Napoleon’s reign developed and he envisioned more of a European empire. One factor in this was the success and failure of the men Napoleon had placed in charge of conquered lands – his family and officers – because they varied greatly in their loyalty, sometimes proving more interested in their new land than aiding their patron despite in most cases owing everything to him. Most of Napoleon’s clan appointments were poor local leaders, and an exasperated Napoleon sought more control. Some of Napoleon’s appointees were genuinely interested in effecting liberal reforms and being loved by their new states: Beauharnais created a stable, loyal and balanced government in Italy and was very popular. However, Napoleon prevented him from doing more, and often clashed with his other rulers: Murat and Joseph ‘failed’ with the constitution and Continental System in Naples. Louis in Holland rejected much of his brother’s demands and was ousted from power by an angry Napoleon. Spain, under the ineffectual Joseph, couldn’t really have gone more wrong. Napoleon’s Motives In public, Napoleon was able to promote his empire by stating laudatory aims. These included safeguarding the revolution against Europe’s monarchies and spreading freedom throughout oppressed nations. In practice, Napoleon was driven by other motives, although their competing nature is still debated by historians. It’s less likely that Napoleon began his career with a plan to rule Europe in a universal monarchy – a sort of Napoleon dominated empire which covered the whole continent – and more likely he evolved into wanting this as the opportunities of war brought him greater and greater success, feeding his ego and expanding his aims. However, a hunger for glory and a hunger for power – whatever power that may be - seem to have been his over-riding concerns for much of his career. Napoleon’s Demands on Empire As parts of the empire, the conquered states were expected to assist in furthering Napoleon’s aims. The cost of the new warfare, with greater armies, meant more expense than ever before, and Napoleon used the empire to for funds and troops: success funded more attempts at success. Food, equipment, goods, soldiers, and tax were all drained out by Napoleon, much of it in the form of heavy, often annual, tribute payments. Napoleon had another demand on his empire: thrones and crowns on which to place and reward his family and followers. While this form of patronage left Napoleon in control of the empire by keeping leaders tightly bound to him – although putting close supporters in power didn’t always work, such as in Spain and Sweden – it also let him keep his allies happy. Large estates were carved out of the empire both to reward and to encourage the recipients to fight to keep the empire. However, all these appointments were told to think of Napoleon and France first, and their new homes second. The Briefest of Empires The empire was created militarily and had to be enforced militarily. It survived the failures of Napoleon’s appointments only as long as Napoleon was winning to support it. Once Napoleon failed, it was swiftly able to eject him and many of the puppet leaders, although the administrations often remained intact. Historians have debated whether the empire could have lasted and whether Napoleon’s conquests if allowed to last, would have created a unified Europe still dreamt of by many. Some historians have concluded that Napoleon’s empire was a form of continental colonialism that could not have lasted. But in the aftermath, as Europe adapted, a lot of the structures Napoleon put in place survived. Of course, historians debate exactly what and how much, but new, modern administrations could be found all over Europe. The empire created, in part, more bureaucratic states, better access to the administration for the bourgeoisie, legal codes, limits on the aristocracy an d church, better tax models for the state, religious toleration and secular control in church land and roles.

Napoleons Empire

Napoleon's Empire The borders of France and the states ruled by France grew during the wars of the French Revolution and the Napoleonic Wars. On May 12th, 1804 these conquests received a new name: the Empire, ruled by a hereditary Bonaparte Emperor. The first – and in the end only – emperor was Napoleon, and at times he ruled vast swathes of the European continent: by 1810 it was easier to list the regions he didn’t dominate: Portugal, Sicily, Sardinian, Montenegro, and the British, Russian and  Ã¢â‚¬â€¹Ottoman Empires. However, while it’s easy to think of the Napoleonic Empire as one monolith, there was considerable variation within the states. The Make-Up of the Empire The empire was divided into a three-tier system. Pays Rà ©unis: this was land governed by the administration in Paris, and included the France of the natural frontiers (i.e. the Alps, the Rhine and the Pyrenees), plus states now subsumed into this government: Holland, Piedmont, Parma, the Papal States, Tuscany, the Illyrian Provinces and a lot more of Italy. Including France, this totaled 130 departments in 1811 – the peak of the empire – with forty-four million people. Pays Conquis: a set of conquered, although supposedly independent, countries which were ruled by people approved by Napoleon (largely his relatives or military commanders), designed to buffer France from attack. The nature of these states ebbed and flowed with the wars, but included the Confederation of the Rhine, Spain, Naples, the Duchy of Warsaw and parts of Italy. As Napoleon developed his empire, these came under greater control. Pays Allià ©s: The third level was fully independent states who were bought, often unwillingly, under Napoleon’s control. During the Napoleonic Wars Prussia, Austria and Russia were both enemies and unhappy allies. The Pays Rà ©unis and Pays Conquis formed the Grand Empire; in 1811, this totaled 80 million people. In addition, Napoleon redrew central Europe, and another empire ceased: the Holy Roman Empire was disbanded on August 6th, 1806, never to return. Nature of the Empire The treatment of states in the empire varied depending on how long they remained part of it, and whether they were in the Pays Rà ©unis or Pays Conquis. It’s worth pointing out that some historians reject the idea of time as a factor, and focus on regions in which pre-napoleon events inclined them to be more receptive to Napoleon’s changes. States in the Pays Rà ©unis before the Napoleonic era were fully departmentalized and saw the benefits of the revolution, with the end of ‘feudalism’ (such as it existed), plus land redistribution. States in both the Pays Rà ©unis and Pays Conquis received the Napoleonic legal Code, the Concordat, tax demands, and administration based on the French system. Napoleon also created ‘dotations’. These were areas of land seized from conquered enemies where the entire revenue was given to Napoleon’s subordinates, conceivably forever if the heirs stayed loyal. In practice they were a huge drain on the loca l economies: the Duchy of Warsaw lost 20% of revenue in dotations. Variation remained in outlying areas, and in some privileges survived through the era, unaltered by Napoleon. His introduction of his own system was less ideologically driven and more practical, and he would pragmatically accept survivals which the revolutionaries would have cut out. His driving force was to keep control. Nevertheless, we can see the early republics being transformed slowly into more centralized states as Napoleon’s reign developed and he envisioned more of a European empire. One factor in this was the success and failure of the men Napoleon had placed in charge of conquered lands – his family and officers – because they varied greatly in their loyalty, sometimes proving more interested in their new land than aiding their patron despite in most cases owing everything to him. Most of Napoleon’s clan appointments were poor local leaders, and an exasperated Napoleon sought more control. Some of Napoleon’s appointees were genuinely interested in effecting liberal reforms and being loved by their new states: Beauharnais created a stable, loyal and balanced government in Italy and was very popular. However, Napoleon prevented him from doing more, and often clashed with his other rulers: Murat and Joseph ‘failed’ with the constitution and Continental System in Naples. Louis in Holland rejected much of his brother’s demands and was ousted from power by an angry Napoleon. Spain, under the ineffectual Joseph, couldn’t really have gone more wrong. Napoleon’s Motives In public, Napoleon was able to promote his empire by stating laudatory aims. These included safeguarding the revolution against Europe’s monarchies and spreading freedom throughout oppressed nations. In practice, Napoleon was driven by other motives, although their competing nature is still debated by historians. It’s less likely that Napoleon began his career with a plan to rule Europe in a universal monarchy – a sort of Napoleon dominated empire which covered the whole continent – and more likely he evolved into wanting this as the opportunities of war brought him greater and greater success, feeding his ego and expanding his aims. However, a hunger for glory and a hunger for power – whatever power that may be - seem to have been his over-riding concerns for much of his career. Napoleon’s Demands on Empire As parts of the empire, the conquered states were expected to assist in furthering Napoleon’s aims. The cost of the new warfare, with greater armies, meant more expense than ever before, and Napoleon used the empire to for funds and troops: success funded more attempts at success. Food, equipment, goods, soldiers, and tax were all drained out by Napoleon, much of it in the form of heavy, often annual, tribute payments. Napoleon had another demand on his empire: thrones and crowns on which to place and reward his family and followers. While this form of patronage left Napoleon in control of the empire by keeping leaders tightly bound to him – although putting close supporters in power didn’t always work, such as in Spain and Sweden – it also let him keep his allies happy. Large estates were carved out of the empire both to reward and to encourage the recipients to fight to keep the empire. However, all these appointments were told to think of Napoleon and France first, and their new homes second. The Briefest of Empires The empire was created militarily and had to be enforced militarily. It survived the failures of Napoleon’s appointments only as long as Napoleon was winning to support it. Once Napoleon failed, it was swiftly able to eject him and many of the puppet leaders, although the administrations often remained intact. Historians have debated whether the empire could have lasted and whether Napoleon’s conquests if allowed to last, would have created a unified Europe still dreamt of by many. Some historians have concluded that Napoleon’s empire was a form of continental colonialism that could not have lasted. But in the aftermath, as Europe adapted, a lot of the structures Napoleon put in place survived. Of course, historians debate exactly what and how much, but new, modern administrations could be found all over Europe. The empire created, in part, more bureaucratic states, better access to the administration for the bourgeoisie, legal codes, limits on the aristocracy an d church, better tax models for the state, religious toleration and secular control in church land and roles.