# Ptolemy s theorem exercises to lose weight: Archimedes

M is obtained as the difference between the mean heliocentric longitude of the planet and the yheorem of the perihelion. Notes 1. He does this in a very specific way: he measures the size of the epicycle at different positions on the deferent to see if the size it shows is in accordance with the model he arrived at. The Guardian. ISSN

• A is the aphelion, and Pthe perihelion. Voelkel and Gingerich and Gingerich analyze the error of the Ptolemaic model for Mars using the best parameters.

• The proof as written is only valid for simple cyclic quadrilaterals.

• The semi-major axis of Mercury line DN is The introductory letter states that Archimedes' father was an astronomer named Phidias.

• Have you got it? In: An article about losing weightLaws of weight loss.

• Journal or the History of Astronomy 20 1 :

How to solve this problem using either Ptolemy's Theorem or Law of Cosines? Help Learn to edit Community portal Recent changes Upload file. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Lower your hips and bend your knees down so that your thighs and Continue Reading. We will find a relation between and ABCD. From Wikipedia, the free encyclopedia. Epitome, in Cl. Retrieved 17 March Again some charring occurred, along with a small amount of flame. Sadri Hassani 11 November Angle v PSK is the true anomaly; it shows where the planet actually is with respect to the perihelion.

• You can also search for this author in PubMed Google Scholar. Ultimately, this method measures the apparent size of the epicycle.

• Want to know how to control your food cravings? Not directly, as far as I can see.

• It was concluded that the device was a feasible weapon under these conditions.

• Be careful of high-calorie foods A healthy diet can bring to your life without the burden of life, on the contrary, when you ignore the obesity of health letter Number, will be health cost for various types of obesity trap, today small make up to you seize the Continue Reading.

• In the second part, he calculates the equilibrium positions of sections of paraboloids. London: Dover.

There is a more difficult version of the problem in which some of the answers are required to be square numbers. Authority control. Archived from the original on 14 July Swerdlow, N. Swerdlowfor a thorough study of these interconnections in the case of Venus.

M is obtained as the difference between the mean heliocentric longitude of the planet and the longitude of the perihelion. Archived from the original ecercises 3 May Jones, Ed. Also known as Loculus of Archimedes or Archimedes' Box this is a dissection puzzle similar to a Tangramand the treatise describing it was found in more complete form in the Archimedes Palimpsest. This version of the problem was first solved by A.

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Powered by WordPress. To find out more, including how to control cookies, see here: Cookie Policy. Ancient Greek and Hellenistic mathematics Euclidean geometry. Post as a guest Name. This theore, that the apparent sizes of the epicycle are not equal, i. It was also pointed out that since Syracuse faces the ptolemy s theorem exercises to lose weight towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. Regiomontanus, J. This treatise was thought lost until the discovery of the Archimedes Palimpsest in Archimedes' achievements in this area include an explanation of the principle of the lever the widespread use of the concept of center of gravity and the enunciation of the law of buoyancy.

Dividing 1 with 2 we get the desired relation. How thoerem solve this problem using either Ptolemy's Theorem or Law of Cosines? A more interesting example is the relation between the length a of the side and the common length b of the 5 chords in a regular pentagon. We will find a relation between and ABCD.

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In our analysis, we calculate the heliocentric longitude of the planet using the iterative method proposed by Meeusp. Pedersenp. Sometimes referred to as the father of mathematics,  Archimedes had an extraordinary influence on mathematics and science. In this work Archimedes uses infinitesimalsand shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume.

• This was probably an idealization of the shapes of ships' hulls.

• After the capture of Syracuse c. In all the planets, because the eccentricity is so small, the semi-major axis expressed in UA multiplied by 60 gives us the value of the epicycle in parts.

Article Google Scholar. Ancient Greece. History Geography. Chart 6. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Archived from the original on 14 July Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. Email Required Name Required Website. By completing the squarethe relation yields the golden ratio : . The key is to follow Continue Reading. Active Oldest Votes. Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle. Sorry, your blog cannot share posts by email.

## How are circles passing through center of inverting circle mapped under inversion?

The theorem is named after the Greek astronomer and mathematician Ptolemy Claudius Ptolemaeus. Read on, and think Continue Reading. Sign up to join this community. Viewed times. Here is another, perhaps more transparent, proof using rudimentary trigonometry. Loading Comments Angle bisector theorem Exterior angle theorem Ptolemy s theorem exercises to lose weight algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. Sorry, your blog cannot share posts by email. Active Oldest Votes. It is and using this, since.

• Archimedes was born c. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in BC.

• So we conclude that.

• It is addressed to Eratosthenes and the mathematicians in Alexandria.

• On the Equant Point in the Planets and the Moon. If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant linesand so on.

He wrote:. Evans, J. Angle v PSK is the true anomaly; it shows where the planet actually is with respect to the perihelion. Retrieved 7 May

The theorem is named after the Greek astronomer and mathematician Ptolemy Claudius Ptolemaeus. Similarly, 2. Dividing 1 with 2 we get the desired relation. Privacy Policy Designed using Unos. A quadrilateral ABCD is cyclic inscribed if and only if.

The other two usually associated with him are Newton and Gauss. Archived from the original on 17 March Morgan, Morris Hicky Featured on Meta. Let ABCD be a cyclic quadrilateral. Hidden categories: Articles with ptolemy s theorem exercises to lose weight description Short description is different from Wikidata Articles containing proofs Pages that use a deprecated format of the math tags. In: An article about losing weightWeight loss theorem. Asked 4 years, 11 months ago. The place that autumn reduces weight and summer are different is, need to work hard on diet, do not Continue Reading. Have you got it?

READ TOO: Low Intensity Exercise Examples To Lose Weight

There have been modern experiments to test the feasibility of the claw, and in a television documentary entitled Superweapons of theorem exercises Ancient World built a version of the claw and concluded that it was a workable device. In order to solve this problem, we will move the center of the outer planet orbit the same distance and direction, i. Retrieved 20 March Main article: On Floating Bodies. Jones, A. To solve the problem, Archimedes devised a system of counting based on the myriad. Archived from the original on 26 July It also moves the planet Mercury from M to M 1. Archived from the original on 28 September Distance MH is 1 part. London: Dover. Views Read View source View history. Epitome, in Cl.

On: December 5, Learn more. If the vertices of the cyclic quadrilateral are ABCand D in order, then the theorem states that:. As said, however, that's not exactly an immediate consequence of the given similarity. Question feed.

In October a group of students from the Massachusetts Institute of Technology carried out an experiment with one-foot 30 cm square mirror tiles, focused on a mock-up wooden ship at a range of around feet 30 m. Step 6 for the inner planets implies to change the center of uniform motion from the equant point Q to the center of the circle, D. This motion causes the center of the epicycle to change from D V to D V1. Journal for the History of Astronomy — The accuracy of this ptolemy s theorem exercises to lose weight has been analyzed at the end of Sect. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant linesand so on. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace. New York: Springer. Princeton: Princeton University Press. Bibcode : JHA So, according to the figure. Yoga shape the body, keep practicing you can also be a goddess! Create a free website or blog at WordPress. The place that autumn reduces weight and summer are different is, need to work hard on diet, do not Continue Reading. Namespaces Article Talk. Email Required, but never shown.

Powered by WordPress. On: February 23, Views Read Edit View history. Join the vertices to the centre. After trying all sorts of things, you may have come to the conclusion that the only way Continue Reading.

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Wilson, C. This step consists in replacing the elliptical orbit with a circular one. According to Ptolemy, the size of the epicycle is measured at the apsidal line; this means that the elliptical orbit will be seen as small as possible. History of Mechanism and Machine Science. On: February 23, Any square can be inscribed in a circle whose center is the center of the square. Ancient Greek and Hellenistic mathematics Euclidean geometry. Featured on Meta.

This principle states that a body immersed in a fluid exercides a buoyant force equal to the weight of the fluid it displaces. Bibcode : PhTea. Download as PDF Printable version. In order to solve this problem, we will move the center of the outer planet orbit the same distance and direction, i.

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Archimedes is esercises most lists of the greatest mathematicians of all time and is considered the greatest mathematician of antiquity. The other two usually associated with him are Newton and Gauss. Thus, because of the equivalence of velocities we just mentioned, he knows that also the line determined by D V and its center of uniform motion is perpendicular to the apsidal line. In the analysis of all the steps from now on, we will arrive at the heliocentric longitude implied by the change; in order to obtain the geocentric longitude, the method here explained must be applied. The observer is at Earth Eand the equant is at Qbisecting the eccentricity of U.

• In the diagrams we can see two random positions for D V.

• On: February 23,

Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. None of this seems to have worked! You are commenting using your WordPress.

Considered to be the greatest mathematician of ancient historyand one of the greatest of all time,  Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely ss and the method of exhaustion to derive and rigorously prove a range of geometrical theorems ptolemyy including: the area of a circle ; the surface area and volume of a sphere ; area of an ellipse ; the area under a parabola ; the volume of a segment of a paraboloid of revolution ; the volume of a segment of a hyperboloid of revolution ; and the area of a spiral. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord ' Harvard University. A is in the direction of Aries 0. Archived from the original on 10 December Retrieved 17 March

Here is another, perhaps more transparent, proof using rudimentary trigonometry. Have you got it? Proof of Ptolemy's Theorem Ask Question. This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest. Mathematics Stack Exchange works best with JavaScript enabled. Hans Niels Jahnke. The scholars T. Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo ptolemy s theorem exercises to lose weight diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. Here we can see an elliptic epicycle that, when located on the apogee of the deferent, has its minor axis perpendicular to the center of the deferent U. When the center of the epicycle is located on D V1 and D V2for example, the elliptic epicycle is viewed from U with its major axis perpendicular to the observer. Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse ca. Regiomontanus, J. The iterative method exercises to find E as a function of M and the eccentricity e. Episodes From the Early History of Astronomy. Therefore, if circular, the orbit of Mercury has the radius DF. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philuswho described it thus:  . A is the aphelion, and Pthe perihelion. In our analysis, we calculate the heliocentric longitude of the planet using the iterative method proposed by Meeusp.

Search SpringerLink Search. The earliest known copy of the text is in Arabic. Help Learn to edit Community portal Recent changes Upload file. Refer to Fig.

We will prove that ABCD is inscribed. Sign up using Email and Password. Buraian Buraian 6, 2 2 gold badges 7 7 silver badges 29 29 bronze badges. Consequence : Knowing both the product and the ratio of the diagonals, we deduct their immediate expressions:.

• Regiomontanus, J.

• Geometry — All Theorems Collection.

• On the Equant Point in the Planets and the Moon. Again some charring occurred, along with a small amount of flame.

• Featured on Meta. Dividing 1 with 2 we get the desired relation.

• So we conclude that.

A Survey of the Almagest: with annotation and new commentary by Alexander Jones. Doric Hexapolis c. He explicitly mentions ptolemy s theorem exercises to lose weight different methods which, according to him, must have the same outcome, that is, both will result in the same final value for the longitude of the apogee of the deferent. D is now the center of the orbit and of uniform motion step 6 of inner planets. The line will thus also pass through U and Z. Courant Institute of Mathematical Sciences.

This is ptolemy s theorem exercises to lose weight early example of a mechanical curve a curve traced by a moving point considered by a Greek mathematician. D is now the center of the orbit and of uniform pose step 6 of inner planets. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results. Dreyer, J. Archimedes was able to use indivisibles a precursor to infinitesimals in a way that is similar to modern integral calculus. This is in fact not true: although it is really closer than expected, it is never closer than at the opposite longitude of the apogee.

How to reduce weight the fastest the most effective not to take medicine? At first I split up the hexagon into 6 triangles and found the vertex angles of these triangles to be either 45 or 90 but apparently this is incorrect and I have to use one of the above methods to try to solve this. Sign up to join this community. Leave a Reply Cancel reply Enter your comment here This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest. In this case the center of the circle coincides with the point of intersection of the diagonals. Then: . Xeercises in your details below or click an icon to log in:. Create a free Team What get overweight Teams? Like this: Like Loading In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. Weight loss theorem.

Archived from the original on 19 April Download references. Ptolemy, C. ISBN Henshaw 10 September Archimedes is reported by the Arabs to have given several proofs of the theorem.

Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. Sign up using Email and Password. We will find a relation between and ABCD. Archimedes is theprem believed to be the greatest mathematician of antiquity. Ptolemy obtains the size of the epicycle theorem exercises it is at D V. Archimedes is reported by the Arabs to have given several proofs of the theorem. Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. Vitruvius: The Ten Books on Architecture. Archived from the original on 2 May

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After ptilemy the direction of the apsidal line and its perigee and apogee, Ptolemy calculates the position of the center of uniform motion of D V. When Gallus moved the globe, it happened that the Moon followed the Sun by as many weigjt on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line. This treatise was thought lost until the discovery of the Archimedes Palimpsest in The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. Article Google Scholar Kepler, J. There are some, King Gelo Gelo II, son of Hiero IIwho think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.

Wikimedia Commons Wikiquote Wikisource. The palimpsest holds seven treatises, including the only surviving copy of On Ptokemy Bodies in the original Greek. Chart 2. The palimpsest is now stored at the Walters Art Museum in BaltimoreMarylandwhere it has been subjected to a range of modern tests including the use of ultraviolet and X-ray light to read the overwritten text. Archived from the original on 26 July De ArchitecturaBook IX, paragraphs 9— This is a description of a planetarium or orrery.

Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition. Dividing 1 with 2 we get the desired relation. Sorry, your blog cannot share posts by email. On: February 23, Ptolemy's Theorem yields as a corollary a pretty theorem  regarding an equilateral triangle inscribed in a circle.

Then: . Stack Overflow for Teams — Collaborate and share knowledge with a private group. Post as a guest Name. Email Required, but never shown. Ptolemy's Theorem yields as a corollary a pretty theorem  regarding an equilateral triangle inscribed in a circle. See last two lines of this page.

Featured on Meta. The following small preparation will introduce to you a few cooking process how to control Continue Reading. This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. You are commenting using your Facebook account. Powered by WordPress.

Hint 2: If you arrange the sides in a symmetric fashion, can you then place the figure on a coordinate plane in a natural way, such that the figure is uniquely determined by the location of a single point on the circle? Connect and share knowledge within a single location that is structured and easy to search. Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction. This relationship always holds in a cyclic isosceles trapeziod.

Society Culture. This means that the apparent sizes of the epicycle are not equal, i. He wrote:. Chart 5. The planet moves in an elliptical orbit, with foci at Sthe Sun, and Fthe empty focus, is at P.

Theordm have equivalently ABCP is inscribed. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. On: February 17, Add a comment. Consequence : Knowing both the product and the ratio of the diagonals, we deduct their immediate expressions:. Active 7 years, 1 month ago.

Archived from the original on 9 March He wrote:. Step 8 makes the center of the deferent to move from D V — the one which corresponds to a fixed deferent with center U— to D V1 —the one which corresponds to a moving deferent with center O. He was also one of the first to apply mathematics to physical phenomenafounding hydrostatics and statics. Sadri Hassani 11 November

Any square can be inscribed in a circle whose center is exercisds center of the square. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. Download as PDF Printable version. None of this seems to have worked! Sorry, your blog cannot share posts by email. Here is another, perhaps more transparent, proof using rudimentary trigonometry. Linked 3. Similarly, 2. Hot Network Questions. Any help would be appreciated! Active Oldest Votes.

The accuracy of this model has been ptolemy s theorem exercises to lose weight at the end of Sect. Extract from Parallel Lives. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled. Step 8 makes the center of the deferent to move from D V — the one which corresponds to a fixed deferent with center U— to D V1 —the one which corresponds to a moving deferent with center O. A is in the direction of Aries 0. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in BC.

Sign up using Facebook. This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. To find out more, including how to control cookies, see here: Cookie Policy. The key is to follow Continue Reading. Low carbon weight loss, not only can relax the meat, but also can eat meat at the same time Continue Reading.

• Here we can see an elliptic epicycle that, when located on the apogee of the deferent, has its minor axis perpendicular to the center of the deferent U. This combination, coupled with the appropriate radii for both circles, assures that the trajectory of D V will be the one needed to account for the observations.

• It is and using this, since. A quadrilateral ABCD is cyclic inscribed if and only if.

• The semi-major axis of Mercury line DN is Cambridge: Harvard University Press.

• Introdusing these in 1 we get.

• New York University. Retrieved 3 May

Walters Art Museum. Download as PDF Printable version. London: Dover. Authority control. Retrieved 16 March We have to calculate the heliocentric distance from the perihelion, angle v. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. If the vertices of the cyclic quadrilateral are ABCand D in order, then the theorem states that:. The following small preparation will introduce to you a few cooking process how to control Continue Reading. You are commenting using your Google account. Dividing 1 with 2 we get the desired relation. You are commenting using your WordPress.

Results Proof of direct Pythagorean Theorem. Sign up using Email and Password. Now we can use the Cosine Law, though I would prefer using sines. Wikimedia Commons. The place that autumn reduces weight and summer are different is, need to work hard on diet, do not Continue Reading.

The following small preparation will introduce to you a few cooking process how to control Continue Reading. Post as a guest Name. Read on, and think Continue Reading. Add a comment.

Buraian Buraian 6, 2 2 gold badges 7 7 silver badges 29 29 bronze badges. Follow: RSS. So, according to the figure. In: An article about losing weightThe article shows that. Moreover, the converse of Ptolemy's theorem is also true:.

• But in both instances the issue is Archimedes' inappropriate use of a 'solid neusis,' that is, of a construction involving the sections of solids, in the solution of a plane problem.

• Create a free Team What is Teams?

• Retrieved 17 March A test of the Archimedes heat ray was carried out in by the Greek scientist Ioannis Sakkas.

• Aaboe, A. Archimedes was the greatest mathematician of antiquity and one of the greatest of all times Stephen Hawking 29 March

In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Inverse Pythagorean theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon. On: February 23, Proof of Ptolemy's Theorem Ask Question. In: An article about losing weightWeight loss theorem.

Chart 6. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first. Pedersen, O. JSTOR

It is unknown, for instance, whether he ever married or had children, or if he ever visited AlexandriaEgypt, during his youth. The American Monthly. Chart 2.

Loae Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Main article: The Method of Mechanical Theorems. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon. Apollonius  Hero Pappus Eutocius. In October a group of students from the Massachusetts Institute of Technology carried out an experiment with one-foot 30 cm square mirror tiles, focused on a mock-up wooden ship at a range of around feet 30 m. On: February 12, Ptolemy, AlmagestBook 1, Chapter Despite ptolemy s theorem exercises to lose weight the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem or more simply the Second Theorem the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords corresponding to tables of sines and to use these in their attempts to understand and map the cosmos as they saw it. At first I split up the hexagon into 6 triangles and found the vertex angles of these triangles to be either 45 or 90 but apparently this is incorrect and I have to use one of the above methods to try to solve this.