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The '''table of chords''', created by the astronomer and geometer '''[[Ptolemy]]''' in [[Egypt]] during the 2nd century AD, is a [[trigonometric table]] in Book&nbsp;I, chapter&nbsp;11 of Ptolemy's ''[[Almagest]]'',<ref name=toomer>{{Citation|title=Ptolemy's Almagest|last1=Toomer|first1=G. J.|authorlink=Gerald J. Toomer|publisher=Princeton University Press|year= 1998|ISBN =0-691-00260-6}}</ref> a treatise on [[mathematical astronomy]].  It is essentially equivalent to a table of values of the [[sine]] function.  It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy (an earlier table of chords by [[Hipparchus]] gave chords only for arcs that were multiples {{nowrap|of 7½° {{=}} &pi;/24 radians}}).<ref>Thurston, [http://books.google.com/books?id=rNpHjqxQQ9oC&pg=PA235#v=onepage&q&f=false pp. 235&ndash;236].</ref>  Several centuries passed before more extensive trigonometric tables were created.
The Christy pocket knife was carried by males and lady and through WWII, and army publish exchanges around the [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%5B%5Biframe+%2F%2Fwww.youtube.com%2Fembed%2FTGXEBKwyrE4+height%3D%22360%22+width%3D%22640%22%5D%5DAs+is+quite+evident+from+the+name%2C+a+Combat+knife+is+generally+utilised+in+a+war+form+predicament+or+for+general+self-defense+purposes.+So+if+you+ever+do+come+across+oneself+in+want+of+a+combat+knife%2C+you+much+better+have+the+ideal+combat+knife+achievable.+Under+no+circumstances+bring+a+knife+to+a+gun+fight%0D%0A%0D%0AThis+is+not+a+recommended+Survival+Knife+per+se+because+it+is+a+folding+survival+knives%2C+but+it+is+a+good+tool+that+I+personally+use+and+carry+in+my+bug-out+pack.+Kershaw+Ken+Onion+Blur+Folding+Survival+Knife+with+Speed+Secure+For+superior+cutting+energy%2C+most+tactical+knife+blades+are+quite+sharp+and+serrated.+The+serration+ought+to+only+be+partial+because+there+are+occasions+where+you+need+to+use+the+straight+portion.+In+addition%2C+usually+check+the+variety+of+tang+applied.+A+tang+is+metal+attached+between+the+blade+and+the+manage.+Knives+with+a+complete+tang+are+often+better+than+those+with+partial+tang+only%2C+simply+because+the+latter+could+later+loosen+right+after+becoming+used+for+a+lengthy+time.+5.six+in+4.four+in+8.eight+Cr17+higher+carbon+stainless+steel%2C+aluminum+%24+four.2+Spyderco+Yojimbo2%0D%0A%0D%0AThe+Cold+Steel+Pistol+Grip+Sword+Cane+is+a+established+old+design+rebuilt+with+new+strategies+and+components.+The+practical+good+quality+of+this+carbon+fiber+and+carbon+steel+piece+surpasses+something+in+my+grandfather%27s+collection+of+antique+blades+and+I%27m+sure+he+would+have+been+fascinated.+Kershaw%27s+Black+Storm%27s+recurved+blade+gives+speedy+cutting+action+for+severing+rope+and+harness+straps+with+a+plain+edge+that%27s+quickly+sharpened+with+ordinary+tools.+With+comfortable+manage+inserts+and+1-handed+manual+opening%2C+it+really+is+a+knife+that%27s+uncomplicated+to+recognize+and+use.+Fixed+Dagger+Tungsten+Black+Blade%2C+G-ten+Green+Deal+with+0121+Ka-Bar+Survival+Complete+Tang+Higher+Carbon+Steel+Blade+Higher+Carbon+Stainless+Steel+Locking+Blade+w%2F+Pistol+Grip+Handle+Kung+Fu+Battle+Prepared+Sharpened+Weapon%0D%0A%0D%0AWhy+do+you+guys+look+to+discountinue+the+finest+knives+you+make%3F+Please+bring+this+knife+back%2C+or+remodel+it+and+bring+it+back.+It+is+perfected+by+craftsmanship%2C+and+the+top+quality+is+outstanding%2C+please+bring+it+back.+From+a+Gerber+fan.+I+carried+an+LHR+via+my+last+deployment+to+Iraq.+Absolutely+the+most+effective+tactical+knife+on+the+market.+So+several+individuals+really+like+this+knife+and+so+do+I.+Bring+it+back%21%21%21+Please%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21%21+I+agree+with+AMIR+the+sheath+requirements+some+operate.+The+exact+same+is+happening+to+my+knife.+DogWood+two+D.++If+you+liked+this+posting+and+you+would+like+to+receive+more+info+with+regards+to+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fbest-combat-knife-fightingtactical-pocket-knives-reviews%2F+http%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fbest-combat-knife-fightingtactical-pocket-knives-reviews%5D+kindly+go+to+our+own+web-site.+Ford+-+OCT+1%2C+2012+for+154+dollars+the+knife+i+good+but+i+consider+that+the+blade+must+be+3+to+4+inches+longer.+True+edge%2C+accurate+knife+Luke+T.+-+FEB+23%2C+2012+Very+best+knife+I+have%21+David+Haynes+-+DEC+21%2C+2011+Chris+Reeve+Knives+Umnumzaan+Assessment%0D%0A%0D%0AC+ombat+knives+have+a+tendency+to+be+a+bit+heavier+and+bulkier+compared+with+conventional+knives+obtainable+in+the+marketplace+today.+On+the+other+hand%2C+these+are+usually+stronger%2C+more+convenient+and+larger.+They+are+effortless+to+carry+around+when+it+is+attached+to+a+pocket+clip+giving+you+a+one-handed+access+in+a+combat+or+fighting+predicament.+It+can+be+made+use+of+as+a+self-defense+tool+in+a+emergency+predicament.+The+first+ever+best+combat+knife+on+the+market+was+Buck+110+Folding+Hunter+in+the+1960s.+It+was+carried+by+soldiers+and+workers+which+grabbed+the+public+focus+as-effectively.+From+that+moment+onwards%2C+now+a+lot+of+well+known+knife+corporation+has+been+producing+tactical+pocket+knife+models.+Overall+Length+%E2%80%93+Full+length+of+knife+handle+and+blade.+Self+Defense+Walking+Cane+w%2F+Weighted+Pommel%0D%0A%0D%0AThis+knife+has+a+four+inch+stainless+steel+blade+that+is+created+particularly+to+be+easy+to+open.+It+gives+quite+a+few+of+the+basics+that+are+necessary+of+a+good+tactical+knife+without+the+need+of+going+overboard.+Additionally%2C+the+knife+is+fairly+light+weight+and+the+truth+that+it+has+an+uncomplicated+open+help+tends+to+make+it+one+of+the+far+more+desirable+merchandise+accessible+on+the+marketplace.+As+you+can+see+choosing+the+ideal+tactical+knife+is+surely+some+thing+that+demands+a+wonderful+deal+of+investigation.+The+knife+that+you+really+should+choose+will+be+totally+dependent+upon+your+precise+desires+and+how+a+lot+you+intend+to+use+the+knife.+When+I+figured+out+how+the+thumb+locked+worked%2C+extracting+the+knife+from+the+sheath+was+a+breeze.+However%2C+I+find+acquiring+back+in+the+sheath+a+challenge+at+times.%0D%0A%0D%0AMeanwhile%2C+a+blade+length+of+not+significantly+less+than+two+and+a+half+inches+and+not+extra+than+four+inches+lengthy+is+ideal+if+you+prefer+putting+the+knife+inside+your+pocket.+In+addition%2C+this+length+is+sufficient+penetrate+the+essential+organs+of+a+individual.+Nonetheless%2C+longer+blades+appear+scarier+than+the+shorter+ones+specially+if+your+opponent+is+bigger+than+you.+Hand+Protection+The+finger+guard+and+the+pommel+are+each+covered+in+matte+gray+coating.+If+you+strategy+on+working+with+your+knife+for%C2%A0hammering%2C+then+you+can+expect+the+coating+of+the+pommel+to+come+off+rather+promptly%2C+therefore+exposing+the+raw+non-stainless+steel.+7.9+3.four+three.9+Sandvik+14C28N+%24%24+four.7+Spyderco+Military+G-10+Created+from+1095+Cro-Van+Steel+%28easy+to+sharpen%2C+holds+an+edge+well%29+THIS+IS+THE+ONLY+KNIFE+You+are+going+to+EVER+Want%0D%0A%0D%0AThe+slimmest+and+lightest+knife+in+our+gallery%2C+the+Snody+Instigator+could+be+the+ultimate+neck+knife.+The+complete-tang+skeletonized+knife+is+crafted+from+AUS-eight+stainless+steel+with+HRC+58-60+hardness.+The+modified+Wharncliff+blade+title+has+a+straight+edge+with+a+single+bevel%2C+and+a+%5Bhttp%3A%2F%2Fwww.Ehow.com%2Fsearch.html%3Fs%3Drounded%2Bspine+rounded+spine%5D+to+give+the+knife+a+talon-sharp+point.+This+knife+is+light+and+sleek+however+powerful.+The+knife+comes+with+a+snap-match+molded+Kydex+thermoplastic+sheath+with+tether+cord.+Knife+weight+is+lowered+and+grip+is+created+by+the+7+strategically+placed+holes+in+the+handle.+The+blade+length+is+3.28+inches%2C+the+overall+length+is+7.38+inches%2C+and+the+knife+weighs+just+2.3+ounces.%0D%0A%0D%0AAs+far+as+your+attacker+is+concerned+this+is+not+a+fight%2C+it+is+an+assassination.+He+is+not+going+to+want+to+stand+there+with+you+and+hack+it+out.+Regrettably%2C+this+is+precisely+the+fantasy+that+quite+a+few+so-named+knife+fighting+instructors+promote.+The+absolute+final+issue+you+want+to+do+is+to+try+to+%22fight.%22+Widespread+sense+tells+us+that+knife+fighting+is+risky.+And+yet%2C+like+a+dog+circling+a+bear%27s+den+-+exactly+where+a+smarter+element+of+it+knows+not+to+wake+that+sleeping+bear%2C+but+an+additional%2C+a+lot+more+instinctive+part+is+urging+it+on+-+quite+a+few+men+and+women+who+train+in+knife+fight+have+the+similar+torn+desires.+1+of+the+biggest+difficulties+goading+these+people+today+is+Do+they+have+what+it+takes%3F%22+In+a+knife+fight%2C+timing+is+right+up+there+close+to+the+major+of+importance.&create=Create globe bought] the Christy knives to our men in service. The Navy issued the Christy knife to pilots as a [http://Www.Federacaodosestivadores.Org.br/userinfo.php?uid=434819 mini-survival] knife, and for a time the US Customs colleges issued the knives to graduates. Followers of pocket knives can now make their own with this handy DIY Pocket Knife Model [http://www.knifecenter.com/knifecenter/index/foldersm.html utility knives] Equipment Put together your individual lockback knife, and while you're making it, you'll find out how this easy mechanism works. You may select from a number of different colours and buy multiple sets to combine and match the colours.<br><br>Like another instruments many types of knife are developed for a sure tasks. Such because the serrated knife that has tooth along the edge of its blade. Its recesses augmented its slicing surface. Not like a straight edge blade knife a serrated knife wants not be sharpened often for the tooth on the knife edge would be the one to penetrate whatever you may be reducing and on the similar time protects the recessed cutting edge from getting [http://wiki.Shol.ru/index.php/Knives_Made_In_The_Usa uninteresting] simply. Case knives also referred to as XX knives characterize a traditionalstyle in pocket knives which makes them differ from different pocket knife companiesmentioned here.<br><br>One in every of my favorite opinions to write down are people who function products that I've a long personal history, with but aren't merchandise that everyone has heard of. I based and bought one of the country’s largest cutlery retailers many years ago, and I have a number of bushels of knives to show my knife bona fides. However despite the dimensions of my knife collection, one knife usually appears to finish up in hand for all manner of tasks Government model Christy Knife. It’s a small knife that is undoubtedly the handiest knife I’ve ever owned.<br><br>For individuals who take [http://sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%0D%0A%0D%0AThis+is+the+kind+of+search+that+you+could+spend+your+entire+life+on+and+nonetheless+not+discover+the+pocket+knife.+So+it%27s+a+search+that+I+launched+into+whereas+knowing+this%2C+being+prepared+to+settle%2C+and+with+a+set+of+normal+criteria+to+information+me.%0D%0A%0D%0AThe+true+great+thing+about+the+knife+doesn%27t+start+to+reveal+itself+until+you+actually+use+it+on+a+daily+basis.+The+short+SpeedSafe+assisted+opening+system+makes+it+a+cinch+to+open+while+you+want+it+and+the+3-inch+blade+proves+just+how+sharp+it+could+stay+after+repeated+use.+And%2C+in+fact%2C+what+could+very+nicely+be+the+best+selling+level+on+this+finely+made+knife+is+that+it+is+manufactured+in+the+USA.+Nothing+beats+that.+Buck+one+hundred+ten+Folding+Hunter+Mild+weight%2C+knife+for+day+by+day-use%2C+good+design+and+such+other+options+makes+Benchmade+Mini+Griptilian+Combo+a+knife+which+is+given+very+excessive+ranking.+Learn+the+opinions+here+Kershaw+Leek+Velocity+safe+and+flipper+opening+techniques%3B+Liner+lock+locking+system+Kershaw+Ken+Onion+Leek+Folding+Knife+with+Velocity+Secure+Two%3B+a+chip+and+a+stab+knife%0D%0A%0D%0AAlso+called+switchblades%2C+automated+knives+require+the+push+of+a+button+or+lever+to+open+the+blade.+Computerized+knives+are+seen+as+harmful+weapons+by+lawmakers+within+the+United+States+and+many+different+international+locations.+Though+outlawed+in+the+late+1950%E2%80%B2s%2C+switchblade+laws+are+under+constant+scrutiny+and+are+often+modified.+Some+states+now+allow+the+use+of+switchblades+beneath+certain+circumstances.+Those+states+wherein+computerized+knives+are+unlawful+in+all+circumstances+are+Colorado%2C+Hawaii%2C+New+Mexico%2C+Virginia%2C+Washington%2C+and+Wisconsin.%0D%0A%0D%0AAnd+with+so+many+a+whole+lot+of+models+of+pocket+knives+on+the+market+ranging+from+%245+to+%24500%2C+it+becomes+very+confusing+for+a+lot+of+people+to+know+which+the+very+best+pocket+knife+suitable+for+them+is.+I+would+love+to+listen+to+your+ideas+on+our+high+pocket+knife+picks.+Be+at+liberty+to+tell+us+when+you+have+any+ideas+relating+to+any+extra+pocket+knife+that+I%27ll+have+missed+out+on.+Blade+size+-+Since+we+are+speaking+Pocket+Knives%2C+you+don%27t+want+a+LARGE+blade+in+your+pocket.+The+best+size+of+the+blade+is+about+3%E2%80%9D+Above+I+listed+corporations+by+way+of+greatest+pocket+knife+And+in+the+event+you%27re+out+tenting%2C+and+you%27ve+forgotten+to+sharpen+up+your+knife+earlier+than+you+left%2C+then+you+can+touch+up+your+knife+blade+very+efficiently+with+either+of+those+pocket+knife+sharpeners.%0D%0A%0D%0ABear+in+mind%2C+knives+are+prohibited+in+schools+or+on+airplanes.+Plus%2C+some+states+prohibit+knife+size.+You+can+buy+an+excellent+knife+for+underneath+%2425.+Greatest+buys+embrace+American-made+pocketknives+%28with+one+or+two+blades%29%2C+genuine+Swiss+Military+knives+%28Victorinox+and+Wenger%29%2C+and+the+Official+Boy+Scout+pocketknife.+If+you+would+like+a+sheath+knife%2C+Canadian+survival+knowledgeable+Mors+Kochanski+recommends+the+carbon-steel+Swedish+Mora+knife.+It+comes+with+a+rugged+Scandinavian-model+sheath+and+prices+underneath+%2415.+Choose+a+knife+as+you%27d+a+buddy.+Be+taught+to+sharpen+it%2C+treat+it+nicely%2C+and+it+will+final+a+lifetime.++If+you+cherished+this+article+and+you+simply+would+like+to+obtain+more+info+with+regards+to+%5Bhttp%3A%2F%2Fwww.Thebestpocketknifereviews.com%2F+best+Folding+Knives+2014%5D+generously+visit+our+web+site.+Owning+a+knife+was+once+a+ceremony+of+passage+for+kids%2C+however+age+isn%27t+the+one+factor+that+is+essential+when+deciding+at+what+age+your+baby+is+prepared+for+a+knife.%0D%0A%0D%0ADeliver+it+out+whenever+you+want+it.+As+I+said+earlier%2C+a+knife+is+a+device%2C+not+a+toy.+I%27ve+minimize+myself+a+couple+of+instances+doing+silly+things+with+a+knife+when+I+used+to+be+youthful.+Despite+the+fact+that+the+foundations+have+been+written+for+boys+and+youngsters%2C+the+Boy+Scouts%27+knife+security+guidelines+are+good+to+study.I+might+highly+recommend++knives.+They+arrive+in+many+styles+with+a+bunch+of+handle+materials+Their+artificial+handles+begin+around+%2430+however+additionally+they+have+knives+with+uncommon+and+beautiful+handles.+My+dad+carries+a+Case+with+a+deal+with+product+of+mammoth+tooth+Initially%2C+that+is+awesome.+Second%2C+it%27s+one+of+the+crucial+beautiful+knives+I%27ve+each+seen.+%5Bhttp%3A%2F%2Fwww.bing.com%2Fsearch%3Fq%3DLanuiole%26form%3DMSNNWS%26mkt%3Den-us%26pq%3DLanuiole+Lanuiole%5D+is+great+as+nicely%0D%0A%0D%0ASpyderco+is+predicated+out+of+Colorado+and+prides+itself+in+making+knives+that+deliver+a+high+quality+performance.+Their+distinctive+ergonomic+designs+make+this+brand+recognizable.+The+FG110+is+a+Basic+American+knife+that+has+stood+the+take+a+look+at+of+time.+This+knife+doesn%27t+have+any+of+the+flash+others+do+but+is+a+solid+strong+knife+that+has+been+round+for+many+years.+It+is+obvious+that+plenty+of+pride+goes+into+building+this+awesome+instrument.+However+serrated+edges+have+their+drawbacks+as+a+result+of+relying+on+the+serration+fashion+and+the+habits+of+the+user%2C+the+serrated+tips+can+chip+off+fairly+easily+and+serrated+edges+are+a+pain+within+the+butt+to+sharpen+unless+you%27ve+gotten+the+right+software.+The+Blade+Girl+carries+a+variety+of+knives+on+your+knife+assortment+or+every+single+day+carry.&create=Create pleasure] in tenting, hunting, prepping or survival getting the most effective survival knife must be a high priority. It is likely one of the greatest instruments you can have at your disposal. Considering the number of issues that the knife can be utilized for, there are several qualities that it is advisable to think about when looking to get the perfect survival knife A good survival knife ought to be straightforward to deal with and must be comfortable in your hand. It ought to have a deal with that matches you appropriately (most top quality knives will) is just not slippery and does not cause scorching spots if you end up chopping or carving.<br><br>Each DIY Pocket Knife Mannequin Kit is a stable coloration (which you can choose), multi-half lockback knife equipment You’ll easily be capable to put this kit collectively and learn the way the lockback knife mechanism works. The DIY Pocket Knife Model Equipment is nice for instructing knife safety (with out the danger of getting stabbed) and for use as a novel letter opener. However we love the custom aspect. Buy a number of colors and you'll combine and match the components. Or, just paint your assembled DIY Pocket Knife Model Kit any way you need. We’d write more, but we've to get painting.<br><br>Knife is likely one of the needed utility in our day by day life. There are various forms of knives accessible available in the market. Many numbers of knife makers are there to construct in unique design that you just want to purchase. Right here is the article for you providing the perfect data on knives and completely different knife makers with a short description. Even you should not forget to seek out out each considered one of these unique knives at their very own web site comfortably. When you determine to use more tools than just a knife , you may be entering the world of detailed wooden carving. Simply clarifying that for you.<br><br>The tang design can be something one should think about when looking for a survival knife This influences the energy of the knife A tang that extends all the way to the tip of the blade is stronger in comparison with one that does not. I would not even take into account a knife with [http://www.thebestpocketknifereviews.com/ Www.Thebestpocketknifereviews.com] no full tang for severe work. W. R. Case & Sons Cutlery Company is a primary instance of the true American approach. It is not just about cash, but extra about folks, ideas, and quality. I consider it is Case's true American approach that has led them to such success.
 
== The chord function and the table ==
 
[[File:Chords.svg|thumb|right|400px|Example: The length of the chord subtending a {{nowrap|109.5° arc}} is approximately&nbsp;98.]]
 
A [[chord (geometry)|chord]] of a [[circle]] is a line segment whose endpoints are on the circle. Ptolemy used a circle whose diameter is&nbsp;120.  He tabulated the length of a chord whose endpoints are separated by an arc of ''n''&nbsp;degrees, for ''n'' ranging from 1/2 to 180 by increments of&nbsp;1/2.  In modern notation, the length of the chord corresponding to an arc of ''&theta;''&nbsp;degrees is
 
: <math> \mathrm{chord}\ \theta = 120 \, \sin\left(\frac{\theta^\circ}{2}\right) = 60 \cdot \left( 2 \, \sin\left(\frac{\pi\theta}{360}\right) \right).</math>
 
As ''θ'' goes from 0 to 180, the chord of a ''θ''° arc goes from 0 to&nbsp;120.  For tiny arcs, the chord is to the arc as ''&pi;'' is to&nbsp;3, or more precisely, the ratio can be made as close as desired to ''&pi;''/3&nbsp;≈&nbsp;1.04719755 by making ''θ'' small enough. Thus, for the arc of (1/2)°, the chord is slightly more than the arc.  As the arc increases, the ratio of the chord to the arc decreases.  When the arc reaches 60°, the chord is exactly equal to the arc, i.e. chord&nbsp;60°&nbsp;=&nbsp;60.  For arcs of more than 60°, the chord is less than the arc, until an arc of 180° is reached, when the chord is only&nbsp;120.
 
The fractional parts of chord lengths were expressed in [[sexagesimal]], i.e. base-60, numerals.  For example, where the length of a chord subtended by a 112° arc is reported to be&nbsp;99<sup>p</sup>&nbsp;&nbsp;29'&nbsp;&nbsp;5", it has a length of
 
: <math> 99 + \frac{29}{60} + \frac{5}{60^2}, </math>
 
rounded to the nearest&nbsp;1/60<sup>2</sup>.<ref name=toomer>{{Citation|title=Ptolemy's Almagest|last1=Toomer|first1=G. J.|publisher=Princeton University Press|year= 1998|ISBN =0-691-00260-6}}</ref>
 
After the columns for the arc and the chord, a third column is labeled "sixtieths".  For an arc of&nbsp;''θ''°, the entry in the "sixtieths" column is
 
: <math> \frac{\mathrm{chord} \left(\theta + \tfrac12 \right)^\circ - \mathrm{chord} \left( \theta^\circ\right)}{1/2}. </math>
 
This is the average number of sixtieths that must be added to chord(''θ''°) for each 1° increase in&nbsp;''θ''°, between the entry for&nbsp;''θ''° and that for&nbsp;(''θ''&nbsp;+&nbsp;½)°.  Thus, it is used for [[linear interpolation]].  Glowatzki and Göttsche showed that Ptolemy must have calculated chords to five sexigesimal places in order to achieve the degree of accuracy found in the "sixtieths" column.<ref>Ernst Glowatzki and Helmut Göttsche, ''Die Sehnentafel des Klaudios Ptolemaios. Nach den historischen Formelplänen neuberechnet.'', München, 1976.</ref>
 
: <math>
\begin{array}{|l|rrr|rrr|}
\hline
\text{arc} & \text{chord} & & & \text{sixtieths} & & \\
\hline
{}\,\,\,\,\,\,\,\,\,\, \tfrac12 &  0 & 31 & 25 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, 1 & 1 & 2 & 50 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, 1\tfrac12 & 1 & 34 & 15 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
109 & 97 & 41 & 38 & 0 & 36 & 23 \\
109\tfrac12 & 97 & 59 & 49 & 0 & 36 & 9 \\
110 & 98 & 17 & 54 & 0 & 35 & 56 \\
110\tfrac12 & 98 & 35 & 52 & 0 & 35 & 42\\
111 & 98 & 53 & 43 & 0 & 35 & 29 \\
111\tfrac12 & 99 & 11 & 27 & 0 & 35 & 15 \\
112 & 99 & 29 & 5 & 0 & 35 & 1\\
112\tfrac12 & 99 & 46 & 35 & 0 & 34 & 48 \\
113 & 100 & 3 & 59 & 0 & 34 & 34 \\
{}\,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
179 & 119 & 59 & 44 & 0 & 0 & 25 \\
179\frac12 & 119 & 59 & 56 & 0 & 0 & 9 \\
180 & 120 & 0 & 0 & 0 & 0 & 0 \\
\hline
\end{array}
</math>
 
== How Ptolemy computed chords ==
 
Chapter 10 of Book&nbsp;I of the ''Almagest'' presents [[Euclidean geometry|geometric]] theorems used for computing chords. Ptolemy used geometric reasoning based on Proposition&nbsp;10 of Book&nbsp;XIII of [[Euclid|Euclid's]] ''[[Euclid's Elements|Elements]]'' to find the chords of 72° and&nbsp;36°.  That Proposition states that if an equilateral [[pentagon]] is inscribed in a circle, then the area of the square on the side of the pentagon equals the sum of the areas of the squares on the sides of the [[hexagon]] and the [[decagon]] inscribed in the same circle.
 
He used [[Ptolemy's theorem]] on quadrilaterals inscribed in a circle to derive formulas for the chord of a half-arc, the chord of the sum of two arcs, and the chord of a difference of two arcs. The theorem states that for a [[quadrilateral]] inscribed in a [[circle]], the product of the lengths of the diagonals equals the sum of the products of the two pairs of opposite sides. The derivations of trigonometric identities rely on a [[cyclic quadrilateral]] in which one side is a diameter of the circle.
 
To find the chords of arcs of 1°&nbsp; and ½° he used approximations based on [[Aristarchus' inequality]]. The inequality states that for arcs &alpha; and &beta;, if 0&nbsp;<&nbsp;&beta;&nbsp;<&nbsp;&alpha;&nbsp;<&nbsp;90°, then
 
: <math> \frac{\sin \alpha}{\sin \beta} < \frac\alpha\beta < \frac{\tan\alpha}{\tan\beta}.</math>
 
Ptolemy showed that for arcs of 1°&nbsp;and ½°, the approximations correctly give the first two sexigesimal places after the integer part.
 
== The numeral system and the appearance of the untranslated table ==
 
{{main|Greek numerals}}
 
Lengths of arcs of the circle, in degrees, and the integer parts of chord lengths, were expressed in a base-10 [[numeral system]] that used 21 of the letters of the [[Greek alphabet]] with the meanings given in the following table, and a symbol, "∠'&thinsp;", that means&nbsp;1/2. Two of the letters, labeled "archaic" below, had not been in use in the Greek language for some centuries before the ''Almagest'' was written.
: <math>
\begin{array}{|rlr|rlr|rlr|}
\hline
\alpha & \mathrm{alpha} &  1 & \iota & \mathrm{iota} & 10 & \varrho & \mathrm{rho} & 100 \\  \beta & \mathrm{beta} & 2 & \kappa & \mathrm{kappa} & 20 & & & \\  \gamma & \mathrm{gamma} & 3 & \lambda & \mathrm{lambda} & 30 & & & \\  \delta & \mathrm{delta} & 4 & \mu & \mathrm{mu} & 40 & & & \\  \varepsilon & \mathrm{epsilon} & 5 & \nu & \mathrm{nu} & 50 & & & \\  \stigma & \mathrm{stigma\ (archaic)} & 6 & \xi & \mathrm{xi} & 60 & & & \\  \zeta & \mathrm{zeta} & 7 & \omicron & \mathrm{omicron} & 70 & & & \\  \eta & \mathrm{eta} & 8 & \pi & \mathrm{pi} & 80 & & & \\  \vartheta & \mathrm{theta} & 9 & \koppa & \mathrm{koppa\ (archaic)} & 90 & & & \\  \hline
\end{array}
</math>
Thus, for example, an arc of {{frac|143|1|2}}&deg; is expressed as <math>\varrho\mu\gamma\angle'</math>.
 
The fractional parts of chord lengths required great accuracy, and were given in two columns in the table: the first giving an integer multiple of&nbsp;1/60, in the range 0–59, the second an integer multiple of&nbsp;1/60<sup>2</sup>&nbsp;=&nbsp;1/3600, also in the range 0–59.
 
Thus in  Heiberg's [http://www.wilbourhall.org/pdfs/HeibergAlmagestComplete.pdf edition of the ''Almagest'' with the table of chords on pages 48–63], the beginning of the table, corresponding to arcs from 1/2&deg; through {{frac|7|1|2}}&deg;, looks like this:
: <math>
\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\
\begin{array}{|l|} \hline \angle' \\ \alpha \\  \alpha\;\angle' \\  \hline\beta \\  \beta\;\angle' \\  \gamma \\  \hline\gamma\;\angle' \\  \delta \\  \delta\;\angle' \\  \hline\varepsilon \\  \varepsilon\;\angle' \\  \stigma \\  \hline\stigma\;\angle' \\  \zeta \\  \zeta\;\angle' \\  \hline \end{array} & \begin{array}{|r|r|r|} \hline\circ & \lambda\alpha & \kappa\varepsilon \\  \alpha & \beta & \nu \\  \alpha & \lambda\delta & \iota\varepsilon \\  \hline \beta & \varepsilon & \mu \\  \beta & \lambda\zeta & \delta \\  \gamma & \eta & \kappa\eta \\  \hline \gamma & \lambda\vartheta & \nu\beta \\  \delta & \iota\alpha & \iota\stigma \\  \delta & \mu\beta & \mu \\  \hline \varepsilon & \iota\delta & \delta \\  \varepsilon & \mu\varepsilon & \kappa\zeta \\  \stigma & \iota\stigma & \mu\vartheta \\  \hline \stigma & \mu\eta & \iota\alpha \\  \zeta & \iota\vartheta & \lambda\gamma \\  \zeta & \nu & \nu\delta \\  \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \nu \\  \hline \circ & \alpha & \beta & \nu \\  \circ & \alpha & \beta & \mu\eta \\  \circ & \alpha & \beta & \mu\eta \\  \hline\circ & \alpha & \beta & \mu\eta \\  \circ & \alpha & \beta & \mu\zeta \\  \circ & \alpha & \beta & \mu\zeta \\  \hline \circ & \alpha & \beta & \mu\stigma \\  \circ & \alpha & \beta & \mu\varepsilon \\  \circ & \alpha & \beta & \mu\delta \\  \hline \circ & \alpha & \beta & \mu\gamma \\  \circ & \alpha & \beta & \mu\beta \\  \circ & \alpha & \beta & \mu\alpha \\  \hline \end{array}
\end{array}
</math>
 
Later in the table, one can see the base-10 nature of the numerals expressing the integer parts of the arc and the chord length. Thus an arc of 85&deg; is written as <math>\pi\varepsilon</math> (<math>\pi</math> for 80 and <math>\varepsilon</math> for 5) and not broken down into 60&nbsp;+&nbsp;25, and the corresponding chord length of 81 plus a fractional part begins with <math>\pi\alpha</math>, likewise not broken into 60&nbsp;+&nbsp;21. But the fractional part, 4/60&nbsp;+&nbsp;15/60<sup>2</sup>, is written as <math>\delta</math>, for 4, in the 1/60 column, followed by <math>\iota\varepsilon</math>, for 15, in the 1/60<sup>2</sup> column.
: <math>
\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\
\begin{array}{|l|} \hline \pi\delta\angle' \\  \pi\varepsilon \\  \pi\varepsilon\angle' \\  \hline  \pi\stigma \\  \pi\stigma\angle' \\  \pi\zeta \\  \hline \end{array} & \begin{array}{|r|r|r|} \hline \pi & \mu\alpha & \gamma \\  \pi\alpha & \delta & \iota\varepsilon \\  \pi\alpha & \kappa\zeta & \kappa\beta \\  \hline \pi\alpha & \nu & \kappa\delta \\  \pi\beta & \iota\gamma & \iota\vartheta \\  \pi\beta & \lambda\stigma & \vartheta \\  \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\  \circ & \circ & \mu\stigma & \iota\delta \\  \circ & \circ & \mu\stigma & \gamma \\  \hline \circ & \circ & \mu\varepsilon & \nu\beta \\  \circ & \circ & \mu\varepsilon & \mu \\  \circ & \circ & \mu\varepsilon & \kappa\vartheta \\  \hline \end{array}
\end{array}
</math>
The table has 45 lines on each of eight pages—thus 360 lines.
 
== Notes and references ==
{{Reflist}}
 
* {{Citation|title=Episodes from the Early History of Mathematics|last1=Aaboe|first1=Asger|authorlink=Asger Aaboe|publisher=Mathematical Association of America|year= 1997|ISBN =978-0-88385-613-0}}
 
* {{Citation|title=Greek Science in Antiquity|last1=Clagett|first1=Marshall|authorlink=Marshall Clagett |publisher=Courier Dover Publications|year=2002|ISBN =978-0-8369-2150-2}}
 
* {{Citation|title=A History of Ancient Mathematical Astronomy|last1=Neugebauer|first1=Otto|authorlink=Otto Neugebauer|publisher=Springer-Verlag|year= 1975|ISBN =978-0-387-06995-1}}
 
* {{Citation|title=Early Astronomy|last1=Thurston|first1=Hugh|authorlink=Hugh Thurston |publisher=Springer|year=1996|ISBN =978-0-387-94822-5}}
 
== External links ==
* [[Johan Ludvig Heiberg (historian)|J. L. Heiberg's]] [http://www.wilbourhall.org/pdfs/HeibergAlmagestComplete.pdf edition of the ''Almagest'' with the table of chords on pages 48–63]
* [http://hypertextbook.com/eworld/chords.shtml Ptolemy's Table of Chords: Trigonometry in the Second Century, by Glenn Elert]
 
[[Category:Trigonometry]]
[[Category:History of mathematics|Trigonometry]]
[[Category:History of astronomy]]
[[Category:Elementary special functions]]

Latest revision as of 14:27, 19 December 2014

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