{"id":12397,"date":"2015-07-03T01:32:07","date_gmt":"2015-07-03T00:32:07","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=12397"},"modified":"2026-04-24T18:04:49","modified_gmt":"2026-04-24T17:04:49","slug":"aristarco-de-samos-sobre-os-tamanhos-e-distancias-entre-o-sol-e-a-lua","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=12397","title":{"rendered":"Aristarco de Samos: Sobre os tamanhos e dist\u00e2ncias entre o Sol e a Lua"},"content":{"rendered":"<div class=\"seriesmeta\">This entry is part 5 of 6 in the series <a href=\"https:\/\/www.acasinhadamatematica.pt\/?series=af-cfaoa\" class=\"series-640\" title=\"AF \u2013 CFAOA\">AF \u2013 CFAOA<\/a><\/div><h5>Aristarco de Samos<\/h5>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12403\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12403\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos.jpg\" data-orig-size=\"395,1024\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Aristarco de Samos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos-395x1024.jpg\" class=\"alignright wp-image-12403\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos-116x300.jpg\" alt=\"Aristarco de Samos\" width=\"160\" height=\"415\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos-116x300.jpg 116w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/AristacoSamos.jpg 395w\" sizes=\"auto, (max-width: 160px) 100vw, 160px\" \/><\/a><a href=\"https:\/\/en.wikipedia.org\/wiki\/Aristarchus_of_Samos\" target=\"_blank\" rel=\"noopener\">Aristarco de Samos<\/a>, astr\u00f3nomo e matem\u00e1tico grego que defendia que a Terra roda sobre o seu eixo e gira em torno do Sol, viveu entre 310 e 230 a.C. Foi o primeiro a formular a hip\u00f3tese de <a href=\"https:\/\/en.wikipedia.org\/wiki\/Nicolaus_Copernicus\" target=\"_blank\" rel=\"noopener\">Cop\u00e9rnico<\/a>, que foi abandonada at\u00e9 o pr\u00f3prio Cop\u00e9rnico a ter estabelecido no s\u00e9c. XVI, propondo que corpos menores circundariam corpos maiores e que seria a Terra que se moveria em torno do Sol.\u00a0Temos o melhor testemunho poss\u00edvel na declara\u00e7\u00e3o precisa de um grande contempor\u00e2neo, <a href=\"https:\/\/pt.wikipedia.org\/wiki\/Arquimedes\" target=\"_blank\" rel=\"noopener\">Arquimedes<\/a>. No tratado <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/The_Sand_Reckoner\" target=\"_blank\" rel=\"noopener\">O Contador de Areia<\/a><\/em>, Arquimedes tem esta passagem:<\/p>\n<blockquote><p>Voc\u00ea [<a href=\"https:\/\/en.wikipedia.org\/wiki\/Gelo,_son_of_Hiero_II\" target=\"_blank\" rel=\"noopener\">Rei Gelo<\/a>] est\u00e1 ciente de que &#8220;universo&#8221; \u00e9 o nome dado pela maioria dos astr\u00f3nomos \u00e0 esfera cujo centro \u00e9 o centro da terra, enquanto o seu raio \u00e9 igual \u00e0 linha reta entre o centro do sol e o centro da terra. Esta \u00e9 a descri\u00e7\u00e3o comum, como j\u00e1 ouviu dos astr\u00f3nomos. Mas Aristarco escreveu um livro, onde aparece, como uma consequ\u00eancia das suposi\u00e7\u00f5es feitas, que o universo \u00e9 muitas vezes maior do que o &#8220;universo&#8221; que acabei de mencionar. As suas hip\u00f3teses s\u00e3o que as estrelas fixas e o sol permanecem im\u00f3veis, que a terra gira em torno do sol na circunfer\u00eancia de um c\u00edrculo, com o sol no meio da \u00f3rbita, e que a esfera das estrelas fixas, situada sobre o mesmo centro, como o sol, \u00e9 t\u00e3o grande que o c\u00edrculo em que ele sup\u00f5e que a terra gira tem uma propor\u00e7\u00e3o para a dist\u00e2ncia das estrelas fixas como o centro da esfera tem para a sua superf\u00edcie.<\/p><\/blockquote>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" width=\"640\" height=\"480\" src=\"https:\/\/www.youtube-nocookie.com\/embed\/fqgpgsOHNDg?rel=0\" frameborder=\"0\" allowfullscreen><\/iframe><br \/>\n<a href=\"https:\/\/pt.wikipedia.org\/wiki\/Carl_Sagan\" target=\"_blank\" rel=\"noopener\">Carl Sagan<\/a>: Aristarco e heliocentrismo<\/p>\n<h5>Sobre os tamanhos e as dist\u00e2ncias do Sol e da Lua<\/h5>\n<div id=\"attachment_12404\" style=\"width: 250px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-12404\" data-attachment-id=\"12404\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12404\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes.jpg\" data-orig-size=\"748,1295\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"OnSizes\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;&amp;#8220;Moon, Earth, and Sun diagrammed in Aristarchus\u2019s On the Sizes and Distances of the Sun and Moon&amp;#8221;. Photograph. Encyclop\u00e6dia Britannica Online. Web. 07 Jul. 2015.&lt;br \/&gt;\n&lt;http:\/\/www.britannica.com\/biography\/Aristarchus-of-Samos\/images-videos\/Diagram-of-the-Moon-Earth-and-Sun-in-a-1572\/192059&gt;&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes-591x1024.jpg\" class=\"wp-image-12404\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes-173x300.jpg\" alt=\"&quot;Moon, Earth, and Sun diagrammed in Aristarchus\u2019s On the Sizes and Distances of the Sun and Moon&quot;.\" width=\"240\" height=\"416\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes-173x300.jpg 173w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes-591x1024.jpg 591w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/OnSizes.jpg 748w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><p id=\"caption-attachment-12404\" class=\"wp-caption-text\">&#8220;Moon, Earth, and Sun diagrammed in Aristarchus\u2019s On the Sizes and Distances of the Sun and Moon&#8221; &#8211; Encyclop\u00e6dia Britannica Online<\/p><\/div>\n<p>O Tratado <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/On_the_Sizes_and_Distances_(Aristarchus)\" target=\"_blank\" rel=\"noopener\">Sobre os tamanhos e as dist\u00e2ncias do Sol e da Lua<\/a><\/em> \u00e9 o \u00fanico trabalho de Aristarco que chegou at\u00e9 n\u00f3s. A par de outros autores da antiguidade, a maioria das coisas que conhecemos de Aristarco prov\u00eam de cita\u00e7\u00f5es em obras de outros autores.<\/p>\n<p>Neste tratado n\u00e3o encontramos qualquer vest\u00edgio da hip\u00f3tese helioc\u00eantrica, provavelmente por esse trabalho ter sido escrito antes da hip\u00f3tese formulada no livro referido por Arquimedes. No entanto, a geometria do tratado n\u00e3o \u00e9 afetada pela diferen\u00e7a entre as hip\u00f3teses geoc\u00eantrica e helioc\u00eantrica.<\/p>\n<p>Aristarco desenvolveu o seu tratado <em>Sobre os tamanhos e as dist\u00e2ncias do Sol e da Lua<\/em> no estilo axiom\u00e1tico de <em><a href=\"http:\/\/aleph0.clarku.edu\/~djoyce\/java\/elements\/\" target=\"_blank\" rel=\"noopener\">Os Elementos<\/a><\/em> de <a href=\"https:\/\/en.wikipedia.org\/wiki\/Euclid\" target=\"_blank\" rel=\"noopener\">Euclides<\/a>, iniciando com um conjunto de hip\u00f3teses e terminando com um conjunto de proposi\u00e7\u00f5es, que s\u00e3o as consequ\u00eancias l\u00f3gicas das hip\u00f3teses formuladas.<\/p>\n<p>A imprecis\u00e3o dos pressupostos, contudo, n\u00e3o prejudica o interesse matem\u00e1tico da investiga\u00e7\u00e3o subsequente. Encontramos a sequ\u00eancia l\u00f3gica de proposi\u00e7\u00f5es e o rigor absoluto de demonstra\u00e7\u00e3o caracter\u00edstica da geometria Grega. A forma e o estilo do livro s\u00e3o absolutamente cl\u00e1ssicas, como conv\u00e9m ao per\u00edodo entre Euclides e Arquimedes.<\/p>\n<p>O conte\u00fado do ponto de vista matem\u00e1tico n\u00e3o \u00e9 menos interessante, porque temos aqui o primeiro esp\u00e9cime sobrevivente da geometria pura usada com um objeto trigonom\u00e9trico, que \u00e9 uma esp\u00e9cie de precursor da <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Measurement_of_a_Circle\" target=\"_blank\" rel=\"noopener\">Medida do C\u00edrculo<\/a><\/em> de Arquimedes. Aristarco realmente n\u00e3o avalia as raz\u00f5es trigonom\u00e9tricas (senos, cossenos, etc., que na \u00e9poca de Aristarco ainda n\u00e3o tinham sido inventadas, nem t\u00e3o pouco existia uma aproxima\u00e7\u00e3o razoavelmente de \u03c0 [Arquimedes foi quem primeiro obteve o valor 22\/7]) de que dependem as propor\u00e7\u00f5es dos tamanhos e dist\u00e2ncias a serem obtidas; ele determina limites entre os quais se encontram e isso por meio de certas proposi\u00e7\u00f5es que ele assume sem provas e que, portanto, devem ter sido conhecidas dos matem\u00e1ticos da sua \u00e9poca.<\/p>\n<h5>Hip\u00f3teses<\/h5>\n<p>No in\u00edcio do tratado, Aristarco formula\u00a0as seguintes hip\u00f3teses:<\/p>\n<ol>\n<li><em>Que a lua recebe a sua frente de luz do sol.<\/em><\/li>\n<li><em>Que a terra est\u00e1 na rela\u00e7\u00e3o de um ponto e centro para a esfera em que a lua se move.<\/em><\/li>\n<li><em>Que, quando a lua aparece para n\u00f3s pela metade <\/em>(quarto crescente ou quarto minguante)<em>, o grande c\u00edrculo que divide o escuro e as partes claras da lua est\u00e1 na dire\u00e7\u00e3o dos nossos olhos.<\/em><\/li>\n<li><em>Que, quando a lua aparece para n\u00f3s pela metade, a sua dist\u00e2ncia <\/em>(angular)<em> do Sol \u00e9, ent\u00e3o, um quadrante menos um trig\u00e9simo de um quadrante <\/em>(isto \u00e9, 90\u00ba &#8211; 90\u00ba\/30 = 87\u00ba)<em>.<\/em><\/li>\n<li><em>Que a largura\u00a0da sombra (da Terra) \u00e9 (a) de duas luas.<\/em><\/li>\n<li><em>Que a lua subtende uma parte de 1\/15 de um signo do zod\u00edaco <\/em>(isto \u00e9, 30\u00ba\/15 = 2\u00ba)<em>.<\/em><\/li>\n<\/ol>\n<h5>Proposi\u00e7\u00f5es<\/h5>\n<p>Enunciadas as hip\u00f3teses, Aristarco prossegue dizendo que estamos agora em condi\u00e7\u00f5es de provar as seguintes proposi\u00e7\u00f5es:<\/p>\n<ol>\n<li><em>A dist\u00e2ncia do sol \u00e0 terra \u00e9 maior do que dezoito vezes, mas menos do que vinte vezes, a dist\u00e2ncia da lua \u00e0 terra <\/em>(isto est\u00e1 no seguimento da hip\u00f3tese sobre a lua pela metade)<em>. <\/em>[Proposi\u00e7\u00e3o 7]<\/li>\n<li><em>O di\u00e2metro do sol tem a mesma raz\u00e3o (como acima descrito) para o di\u00e2metro da lua. <\/em>[Proposi\u00e7\u00e3o 9]<\/li>\n<li><em>O di\u00e2metro do sol tem para o di\u00e2metro da terra numa raz\u00e3o maior do que aquela que tem 19 para 3, mas menor do que a que tem 43 para 6 <\/em>(isto est\u00e1 no seguimento da raz\u00e3o descoberta entre as dist\u00e2ncias, a hip\u00f3tese sobre a sombra e a hip\u00f3tese de que a lua subtende uma parte de 1\/15 de um signo do zod\u00edaco)<em>. <\/em>[Proposi\u00e7\u00e3o 15]<\/li>\n<\/ol>\n<p>As proposi\u00e7\u00f5es que cont\u00eam estes resultados s\u00e3o as Proposi\u00e7\u00f5es 7, 9 e 15.<\/p>\n<p>O tratado prossegue com a demonstra\u00e7\u00e3o de 18 proposi\u00e7\u00f5es (<a href=\"https:\/\/archive.org\/stream\/aristarchusofsam00heat#page\/354\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Heath 1913, p\u00e1g. 355-411<\/a>).<\/p>\n<p>Algumas ilustra\u00e7\u00f5es interativas das proposi\u00e7\u00f5es <em>Sobre os tamanhos e as dist\u00e2ncias do Sol e da Lua<\/em> podem ser encontradas <a href=\"https:\/\/en.wikipedia.org\/wiki\/On_the_Sizes_and_Distances_(Aristarchus)#Illustrations\" target=\"_blank\" rel=\"noopener\">aqui<\/a>. <span style=\"font-size: 8pt;\">(https:\/\/en.wikipedia.org\/wiki\/On_the_Sizes_and_Distances_(Aristarchus)#Illustrations)<\/span><\/p>\n<h5>A imprecis\u00e3o dos pressupostos e o\u00a0m\u00e9todo do tratado<\/h5>\n<p>Arquimedes tamb\u00e9m refere que foi Aristarco que descobriu que o di\u00e2metro angular aparente do sol \u00e9 de cerca de 1\/720 do c\u00edrculo do <a href=\"https:\/\/en.wikipedia.org\/wiki\/Zodiac\" target=\"_blank\" rel=\"noopener\">zod\u00edaco<\/a>, ou seja, meio grau. N\u00e3o sabemos como chegou a esse valor\u00a0bastante preciso, mas, como ele \u00e9 creditado com a inven\u00e7\u00e3o do <em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Scaphe\" target=\"_blank\" rel=\"noopener\">\u03c3\u03ba\u03ac\u03c6\u03b7<\/a><\/em>, pode ter usado este instrumento para o efeito. Mas aqui novamente a descoberta deve, aparentemente, ter sido mais tarde do que o tratado <em>Sobre tamanhos e dist\u00e2ncias,<\/em> pois o valor do \u00e2ngulo que est\u00e1 l\u00e1 assumido \u00e9 2\u00b0 (Hip\u00f3tese 6). Como Aristarco veio a assumir um valor t\u00e3o excessivo \u00e9 incerto. Como a matem\u00e1tica do seu tratado n\u00e3o \u00e9 dependente do valor tomado, 2\u00b0 pode ter sido assumido meramente a t\u00edtulo de ilustra\u00e7\u00e3o; ou pode ter sido um palpite sobre o di\u00e2metro aparente feito antes de ter pensado em tentar mensur\u00e1-lo. Aristarco assumiu que os di\u00e2metros angulares do sol e da lua no centro da terra s\u00e3o iguais.<\/p>\n<p>Na Hip\u00f3tese 5, Aristarco toma o di\u00e2metro da sombra da terra (no lugar onde a lua passa por ela no momento de um eclipse) duas vezes maior que a lua. O valor 2 para essa raz\u00e3o foi baseado presumivelmente sobre o comprimento observado nos eclipses mais longos em registro.<sup>(1)<\/sup> <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hipparchus\" target=\"_blank\" rel=\"noopener\">Hiparco<\/a>, como aprendemos de <a href=\"https:\/\/en.wikipedia.org\/wiki\/Ptolemy\" target=\"_blank\" rel=\"noopener\">Ptolomeu<\/a><sup>(2)<\/sup>, considerou a raz\u00e3o 2 \u00bd para o momento em que a lua est\u00e1 em dist\u00e2ncia m\u00e9dia nas conjun\u00e7\u00f5es; Ptolomeu escolheu o momento em que a lua est\u00e1 em sua maior dist\u00e2ncia, e fez a raz\u00e3o\u00a0insensivelmente menos de 2 3\/5 (um pouco grande demais).<sup>(3)<\/sup><\/p>\n<p>O m\u00e9todo do tratado depende apenas da observa\u00e7\u00e3o, que \u00e9 a terceira &#8216;hip\u00f3tese&#8217; de Aristarco, que &#8216;quando a lua aparece para n\u00f3s pela metade, o grande c\u00edrculo que divide o escuro e as partes claras da lua est\u00e1 na dire\u00e7\u00e3o de nossos olhos&#8217;; o efeito disto (desde que a lua receba a luz do sol) \u00e9 que, no momento da dicotomia, os centros do sol, lua e terra formam um tri\u00e2ngulo ret\u00e2ngulo no centro da lua. Foram necess\u00e1rias duas outras hip\u00f3teses: primeiro, uma estimativa do tamanho do \u00e2ngulo deste tri\u00e2ngulo no centro da terra no momento da dicotomia: este, Aristarco assumiu (hip\u00f3tese 4) ser \u2018um quadrante menos um trig\u00e9simo de um quadrante\u2019, isto \u00e9 87\u00b0, novamente uma estimativa imprecisa, o valor verdadeiro ser\u00e1 \u00a089\u00b0 50&#8242;; em segundo lugar, uma estimativa da largura da sombra da terra onde a lua a atravessa: esta \u00e9 assumida ser &#8216;a largura de duas luas&#8217; (Hip\u00f3tese 5).<\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Paul_Tannery\" target=\"_blank\" rel=\"noopener\">Tannery<\/a><sup>(4)<\/sup> mostra, de uma maneira interessante, a conex\u00e3o entre (1) a estimativa (Hip\u00f3tese 4) que a dist\u00e2ncia angular entre o sol e a lua visto da terra no momento em que a lua aparece a metade \u00e9 87\u00ba, o complemento de 3\u00ba, (2) a estimativa (Hip\u00f3tese 5) de 2 para a raz\u00e3o entre o di\u00e2metro da sombra da terra e o di\u00e2metro da lua, e (3) a raz\u00e3o (superior a 18 para 1 e menos de 20 para 1) do di\u00e2metro de o sol para o di\u00e2metro da lua conforme obtida nas Proposi\u00e7\u00f5es 7 e 9 do tratado.<\/p>\n<p>A anima\u00e7\u00e3o seguinte servir\u00e1 para indicar muito aproximadamente as posi\u00e7\u00f5es relativas do Sol, da Terra e da Lua durante um eclipse lunar, no momento em que a Lua est\u00e1 no meio da sombra da Terra.<\/p>\n<ul>\n<li>\\(S\\): raio da \u00f3rbita do Sol<\/li>\n<li>\\(L\\): raio da \u00f3rbita da Lua<\/li>\n<li>\\(s\\): raio do Sol<\/li>\n<li>\\(t\\): raio da Terra<\/li>\n<li>\u00a0\\(l\\): raio da Lua<\/li>\n<li>\\(D\\): dist\u00e2ncia do centro da Terra ao v\u00e9rtice do cone da sombra da Terra<\/li>\n<li>\u00a0\\(d\\): raio da sombra da Terra \u00e0 dist\u00e2ncia da Lua<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p style=\"text-align: center;\">Os pontos podem ser deslocados.<\/p>\n<p>Pela semelhan\u00e7a de tri\u00e2ngulos, temos:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{D}{S} = \\frac{t}{{s &#8211; t}}}&amp;{\\left( 1 \\right)}&amp;{}&amp;{\\rm{e}}&amp;{}&amp;{\\frac{d}{y} = \\frac{{D &#8211; L}}{D}}&amp;{\\left( 2 \\right)}\\end{array}\\]<\/p>\n<p>Da equa\u00e7\u00e3o (1), resulta:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{D = \\frac{t}{{s &#8211; t}} \\times S}&amp;{\\left( 3 \\right)}\\end{array}\\]<\/p>\n<p>Resolvendo a equa\u00e7\u00e3o (2) em ordem a $D$, vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{d}{y} = \\frac{{D &#8211; L}}{D}}&amp; \\Leftrightarrow &amp;{D \\times d = y \\times \\left( {D &#8211; L} \\right)}\\\\{}&amp; \\Leftrightarrow &amp;{D \\times \\left( {y &#8211; d} \\right) = y \\times L}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{D = \\frac{{y \\times L}}{{y &#8211; d}}}&amp;{\\left( 4 \\right)}\\end{array}}\\end{array}\\]<\/p>\n<p>Eliminando $D$ entre as equa\u00e7\u00f5es (3) e (4), vem:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{{t \\times S}}{{s &#8211; t}} = \\frac{{y \\times L}}{{y &#8211; d}}}&amp; \\Leftrightarrow &amp;{\\frac{S}{{y\\left( {s &#8211; t} \\right)}} = \\frac{L}{{t\\left( {y &#8211; d} \\right)}}}\\\\{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{c}}{\\frac{L}{S} = \\frac{{t\\left( {y &#8211; d} \\right)}}{{y\\left( {s &#8211; t} \\right)}}}&amp;{\\left( 5 \\right)}\\end{array}}\\end{array}\\]<\/p>\n<p>Usando a observa\u00e7\u00e3o de que os tamanhos aparentes do Sol e da Lua s\u00e3o iguais, \\(\\frac{L}{S} = \\frac{l}{s}\\), resulta a partir de (5):<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{l}{s} = \\frac{{t\\left( {y &#8211; d} \\right)}}{{y\\left( {s &#8211; t} \\right)}}}&amp; \\Leftrightarrow &amp;{\\frac{{y\\left( {s &#8211; t} \\right)}}{s} = \\frac{{t\\left( {y &#8211; d} \\right)}}{l}}&amp; \\Leftrightarrow &amp;{y &#8211; y\\frac{t}{s} = y\\frac{t}{l} &#8211; t\\frac{d}{l}}&amp; \\Leftrightarrow \\\\{}&amp; \\Leftrightarrow &amp;{y\\frac{t}{l} + y\\frac{t}{s} = y + t\\frac{d}{l}}&amp; \\Leftrightarrow &amp;{\\frac{t}{l} + \\frac{t}{s} = 1 + \\frac{t}{y} \\times \\frac{d}{l}}&amp;{\\left( 6 \\right)}\\end{array}\\]<\/p>\n<p>Considerando \\(n = \\frac{d}{l}\\) e \\(\\frac{t}{y} = \\Delta \\), resultam:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{t}{l} + \\frac{t}{s} = 1 + \\Delta \\times n}&amp;{\\left( 7 \\right)}\\end{array}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{D}{L} = \\frac{y}{{y &#8211; d}}}&amp; \\Leftrightarrow &amp;{\\frac{D}{L} = \\frac{1}{{1 &#8211; \\frac{d}{y}}}}&amp; \\Leftrightarrow &amp;{\\frac{D}{L} = \\frac{1}{{1 &#8211; \\frac{1}{y} \\times d}}}&amp; \\Leftrightarrow &amp;{\\frac{D}{L} = \\frac{1}{{1 &#8211; \\frac{\\Delta }{t} \\times n \\times l}}}&amp; \\Leftrightarrow &amp;{\\frac{D}{L} = \\frac{1}{{1 &#8211; \\Delta \\times n \\times \\frac{l}{t}}}}&amp;{\\left( 8 \\right)}\\end{array}\\]<\/p>\n<p>Como \\(\\frac{t}{y} = \\Delta = \\sqrt {1 &#8211; {{\\left( {\\frac{{s &#8211; t}}{S}} \\right)}^2}} \\approx 0,9999894227 \\approx 1\\) (valor moderno, calculado abaixo), as equa\u00e7\u00f5es anteriores podem apresentar-se de forma bastante aproximada:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{t}{l} + \\frac{t}{s} = 1 + n}&amp;{\\left( 9 \\right)}\\end{array}\\]<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{D}{L} = \\frac{1}{{1 &#8211; n \\times \\frac{l}{t}}}}&amp;{\\left( {10} \\right)}\\end{array}\\]<\/p>\n<p>Dado que os eclipses do Sol ocorrem pela interposi\u00e7\u00e3o da Lua, ent\u00e3o \\(S &gt; L\\), pelo que \\(s &gt; l\\). Os antigos sabiam, tamb\u00e9m, que o Sol \u00e9 maior do que a Terra, pelo que \\(s &gt; t\\). Da equa\u00e7\u00e3o (9) resulta que \\(\\frac{t}{l} &gt; n\\) (porque \\(\\frac{t}{s} &lt; 1\\)), por isso a Lua \u00e9 mais pequena do que a Terra.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12401\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12401\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a.png\" data-orig-size=\"836,361\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Lua pela metade\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a.png\" class=\"alignright wp-image-12401\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a-300x130.png\" alt=\"Lua pela metade\" width=\"500\" height=\"216\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a-300x130.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_2a.png 836w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a>Agora vamos supor que \\(\\delta \\) \u00e9 a amplitude do \u00e2ngulo subtenso no centro do Sol pela dist\u00e2ncia entre a Lua e a Terra no instante em que a Lua aparece pela metade, isto \u00e9 quando a Terra, Sol e Lua formam um tri\u00e2ngulo ret\u00e2ngulo com o \u00e2ngulo reto no centro da Lua.<\/p>\n<p>Seja\u00a0\\(x = \\frac{S}{L} = \\frac{s}{l} = \\frac{1}{{{\\mathop{\\rm sen}\\nolimits} \\delta }}\\).<\/p>\n<p>Obtemos ent\u00e3o de (9), considerando \\(l = \\frac{s}{x}\\):<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{{t \\times x}}{s} + \\frac{t}{s} = 1 + n}&amp; \\Leftrightarrow &amp;{\\frac{t}{s}\\left( {x + 1} \\right) = 1 + n}&amp; \\Leftrightarrow &amp;{\\frac{s}{t} = \\frac{{x + 1}}{{n + 1}}}&amp;{\\left( {10} \\right)}\\end{array}\\]<\/p>\n<p>Considerando agora \\(s = l \\times x\\), temos:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}{\\frac{t}{l} + \\frac{t}{{l \\times x}} = 1 + n}&amp; \\Leftrightarrow &amp;{\\frac{{t \\times \\left( {x + 1} \\right)}}{{l \\times x}} = 1 + n}&amp; \\Leftrightarrow &amp;{\\frac{t}{l} = \\frac{{n + 1}}{{x + 1}} \\times x}&amp;{\\left( {11} \\right)}\\end{array}\\]<\/p>\n<p>Agora se considerarmos \\(x\\left( { = \\frac{s}{l}} \\right)\\) igual a $19$, o valor m\u00e9dio de Aristarco, e \\(n = 2\\), essas f\u00f3rmulas d\u00e3o:<\/p>\n<p><span style=\"color: #333399; background-color: #ffffff;\">\\[\\begin{array}{*{20}{c}}{\\frac{s}{l} = 19}&amp;{}&amp;{\\frac{s}{t} = \\frac{{20}}{3} = 6,\\left( 7 \\right)}&amp;{}&amp;{\\frac{t}{l} = \\frac{{57}}{{20}} = 2,85}&amp;{}&amp;{\\delta \u00a0= {{{\\mathop{\\rm sen}\\nolimits} }^{ &#8211; 1}}\\frac{1}{{19}} \\approx 3^\\circ \\;1&#8217;\\;1&#8221;}\\end{array}\\]<\/span><\/p>\n<p>O objetivo de Tannery \u00e9 provar que o m\u00e9todo do tratado n\u00e3o foi inventado por Aristarco mas por <a href=\"https:\/\/en.wikipedia.org\/wiki\/Eudoxus_of_Cnidus\" target=\"_blank\" rel=\"noopener\">Eudoxo<\/a>. Sabemos em primeiro lugar, a partir de Arist\u00f3teles, que em meados do s\u00e9culo IV (a.C.) j\u00e1 tinham come\u00e7ado especula\u00e7\u00f5es matem\u00e1ticas sobre os tamanhos e dist\u00e2ncias do sol e da lua. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Aristotle\" target=\"_blank\" rel=\"noopener\">Arist\u00f3teles<\/a><sup>(5)<\/sup> diz:<\/p>\n<blockquote><p>Al\u00e9m disso, se os factos como mostrado nos teoremas de astronomia est\u00e3o corretos, e o tamanho do sol \u00e9 maior do que o da terra, enquanto a dist\u00e2ncia das estrelas \u00e0 terra \u00e9 muitas vezes maior do que a dist\u00e2ncia ao sol, assim como a dist\u00e2ncia do sol \u00e0 terra \u00e9 muitas vezes maior do que a da lua, o cone de marca\u00e7\u00e3o da converg\u00eancia dos raios do sol (depois de passarem a terra) ter\u00e1 o seu v\u00e9rtice n\u00e3o muito longe da terra, e a sombra da terra, a que chamamos noite, portanto, n\u00e3o ir\u00e1 alcan\u00e7ar as estrelas, mas todas as estrelas v\u00e3o estar necessariamente na vis\u00e3o do sol, e nenhuma delas ser\u00e1 bloqueada pela terra.<\/p><\/blockquote>\n<p>Eudoxo foi a primeira pessoa a desenvolver cientificamente a hip\u00f3tese de que o sol e a lua permanecem a uma dist\u00e2ncia constante da terra, respetivamente, e esta \u00e9 a hip\u00f3tese de Aristarco. Al\u00e9m disso, somos informados por Arquimedes que Eudoxo tinha estimado a raz\u00e3o entre o di\u00e2metro do sol e o da lua em 9:1, F\u00eddias, pai de Arquimedes, em 12:1, e Aristarco num valor entre 18:1 e 20:1. Por conseguinte, no pressuposto de que Eudoxo e F\u00eddias tomaram \\(n = 2\\) nas f\u00f3rmulas acima, como fez Aristarco, podemos fazer a seguinte tabela:<\/p>\n<table class=\"aligncenter\" style=\"width: 60%;\" border=\"rgb(0, 0, 0)\">\n<tbody>\n<tr>\n<td style=\"border-color: #000000;\"><\/td>\n<td style=\"border-color: #000000;\">\\(\\frac{s}{l}\\)<\/td>\n<td style=\"border-color: #000000;\">\\(\\frac{s}{t}\\)<\/td>\n<td style=\"border-color: #000000;\">\\(\\frac{t}{l}\\)<\/td>\n<td style=\"border-color: #000000;\">\\(\\delta \\)<br \/>\n(valor calculado)<\/td>\n<\/tr>\n<tr>\n<td style=\"border-color: #000000;\">Eudoxo<\/td>\n<td style=\"border-color: #000000;\">\\(9\\)<\/td>\n<td style=\"border-color: #000000;\">\\(3,\\left( 3 \\right)\\)<\/td>\n<td style=\"border-color: #000000;\">\\(2,7\\)<\/td>\n<td style=\"border-color: #000000;\">\\(6^\\circ \\;22&#8217;\\;46&#8221;\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"border-color: #000000;\">F\u00eddias<\/td>\n<td style=\"border-color: #000000;\">\\(12\\)<\/td>\n<td style=\"border-color: #000000;\">\\(4,\\left( 3 \\right)\\)<\/td>\n<td style=\"border-color: #000000;\">\\(2,76923\\)<\/td>\n<td style=\"border-color: #000000;\">\\(4^\\circ \\;46&#8217;\\;49&#8221;\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"border-color: #000000;\">Aristarco<\/td>\n<td style=\"border-color: #000000;\">\\(19\\)<br \/>\n(m\u00e9dia)<\/td>\n<td style=\"border-color: #000000;\">\\(6,\\left( 6 \\right)\\)<\/td>\n<td style=\"border-color: #000000;\">\\(2,85\\)<\/td>\n<td style=\"border-color: #000000;\">\\(3^\\circ \\;1&#8217;\\;1&#8221;\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Por isso, diz Tannery, enquanto que Aristarco tomou 3\u00b0 como o valor de \\(\\delta \\), Eudoxo provavelmente tomou 6\u00b0 ou 1\/5 de um signo do zod\u00edaco, e F\u00eddias 5\u00b0 ou 1\/6 de um signo. &#8216;Eu n\u00e3o posso acreditar que esses valores foram deduzidos a partir de observa\u00e7\u00f5es diretas da dist\u00e2ncia angular. Os instrumentos necess\u00e1rios eram com toda a probabilidade inexistentes no s\u00e9culo IV. Mas Eudoxo poderia, no dia da dicotomia, marcar as posi\u00e7\u00f5es do sol e da lua no zod\u00edaco, e tentar observar a que horas a dicotomia ocorreu. As avalia\u00e7\u00f5es envolvem um erro de cerca de 12 horas para Eudoxo, dez para F\u00eddias, e seis para Aristarco. Parece que todos eles procuraram limites superiores para \\(\\delta \\). Ser\u00e1 de notar que o valor de \\(\\delta \\) afeta especialmente os valores das raz\u00f5es \\(\\frac{s}{l}\\), \\(\\frac{s}{t}\\); a raz\u00e3o \\(\\frac{t}{l}\\), pelo contr\u00e1rio, depende principalmente do valor de $n$.<sup>(6)<\/sup><\/p>\n<p>Vendo, no entanto, que os \u00fanicos dados na tabela acima que s\u00e3o comprovados, na verdade, s\u00e3o os tr\u00eas na primeira coluna, os 3\u00b0 de Aristarco, e os resultados obtidos por Aristarco na base dos seus pressupostos, parece uma hip\u00f3tese altamente especulativa supor que Eudoxo come\u00e7asse com 6\u00b0, e F\u00eddias com 5\u00b0, como Aristarco fez\u00a0com 3\u00b0, e, em seguida, deduzisse a raz\u00e3o entre o di\u00e2metro do sol e o da lua precisamente pelo m\u00e9todo de Aristarco.<\/p>\n<p><sup>(1)<\/sup> Tannery, <em><a href=\"https:\/\/archive.org\/stream\/recherchessurlhi00tannuoft#page\/224\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Recherches sur l\u2019histoire de l\u2019astronomie ancienne<\/a><\/em>, p. 225.<\/p>\n<p><sup>(2)<\/sup> Ptolemy, <em><a href=\"https:\/\/archive.org\/stream\/syntaxismathema00ptolgoog#page\/n335\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Syntaxis<\/a><\/em>, iv. 9, p. 327. 3-4, Heib.<\/p>\n<p><sup>(3)<\/sup> Ibid., v. 14, p. 421. 12-13.<\/p>\n<p><sup>(4)<\/sup> Tannery in <em><a href=\"https:\/\/archive.org\/stream\/mmoiresdelasoci00bordgoog#page\/n365\/mode\/2up\" target=\"_blank\" rel=\"noopener\">M\u00e9moires de la Soci\u00e9t\u00e9 des sciences physiques et naturelles de Bordeaux<\/a><\/em>, 2<sup>e<\/sup> s\u00e9rie, v, 1883, pp. 241-3 ; <em>M\u00e9moires scientifiques<\/em>, ed. Heiberg and Zeuthen, i, 1912, pp. 376-9.<\/p>\n<p>(5) Arist. <em>Meteorologica<\/em>, i. 8, 345 b 1-9.<\/p>\n<p><sup>(6)<\/sup> Tannery, <em><a href=\"https:\/\/archive.org\/stream\/mmoiresdelasoci00bordgoog#page\/n367\/mode\/2up\" target=\"_blank\" rel=\"noopener\">M\u00e9moires de la Soci\u00e9t\u00e9 des sciences fihys. et nat. de Bordeaux<\/a><\/em>, 2<sup>e<\/sup> s\u00e9rie, v, 1883, pp. 243-4 ; <em>M\u00e9moires scientifiques<\/em>, ed. Heiberg and Zeuthen, i, p. 379.<\/p>\n<p>Adaptado de:<\/p>\n<ul>\n<li><a href=\"https:\/\/archive.org\/stream\/aristarchusofsam00heatuoft#page\/328\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus\u2019s Treatise on the sizes and distances of the sun and moon, by Heath, Thomas Little, Sir, 1861-1940; Aristarchus, of Samos. On the sizes and distances of the sun and moon. English &amp; Greek. 1913. P\u00e1g. 328-332.<\/a><\/li>\n<li><a href=\"https:\/\/archive.org\/stream\/historyofgreekm02heat#page\/n15\/mode\/2up\" target=\"_blank\" rel=\"noopener\">A HISTORY OF GREEK MATHEMATICS, BY SIR THOMAS HEATH, VOLUME II \u2013 FROM ARISTARCHUS TO DIOPHANTUS, OXFORD, AT THE CLARENDON PRESS 1921.\u00a0P\u00e1g. 1-6.<\/a><\/li>\n<\/ul>\n<p><ul id='GTTabs_ul_12397' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_12397' class='GTTabs_curr'><a  id=\"12397_0\" onMouseOver=\"GTTabsShowLinks('C\u00e1lculo de\u00a0\u0394'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>C\u00e1lculo de\u00a0\u0394<\/a><\/li>\n<li id='GTTabs_li_1_12397' ><a  id=\"12397_1\" onMouseOver=\"GTTabsShowLinks('ARISTARCHUS OF SAMOS'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>ARISTARCHUS OF SAMOS<\/a><\/li>\n<li id='GTTabs_li_2_12397' ><a  id=\"12397_2\" onMouseOver=\"GTTabsShowLinks('HYPOTHESES AND PROPOSITIONS'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>HYPOTHESES AND PROPOSITIONS<\/a><\/li>\n<li id='GTTabs_li_3_12397' ><a  id=\"12397_3\" onMouseOver=\"GTTabsShowLinks('THE PROPOSITIONS'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>THE PROPOSITIONS<\/a><\/li>\n<li id='GTTabs_li_4_12397' ><a  id=\"12397_4\" onMouseOver=\"GTTabsShowLinks('As conclus\u00f5es de Paul Tannery'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>As conclus\u00f5es de Paul Tannery<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_12397'>\n<span class='GTTabs_titles'><b>C\u00e1lculo de\u00a0\u0394<\/b><\/span><\/p>\n<h5>C\u00e1lculo de \u0394<\/h5>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12400\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12400\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a.png\" data-orig-size=\"1257,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Eclipse lunar\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a-1024x407.png\" class=\"aligncenter wp-image-12400 size-large\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a-1024x407.png\" alt=\"Eclipse lunar\" width=\"720\" height=\"286\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a-1024x407.png 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a-300x119.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarco_1a.png 1257w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/a>Tendo em considera\u00e7\u00e3o que os dois tri\u00e2ngulos ret\u00e2ngulos mais \u00e0 direita s\u00e3o semelhantes e aplicando o Teorema de Pit\u00e1goras no tri\u00e2ngulo superior, temos:<\/p>\n<p>\\[\\Delta \u00a0= \\frac{t}{y} = \\frac{{\\sqrt {{D^2} &#8211; {t^2}} }}{D} = \\sqrt {1 &#8211; {{\\left( {\\frac{t}{D}} \\right)}^2}} \\]<\/p>\n<p>Eliminando a vari\u00e1vel $D$, por considera\u00e7\u00e3o da equa\u00e7\u00e3o\u00a0\\(\\begin{array}{*{20}{c}}{D = \\frac{t}{{s &#8211; t}} \\times S}&amp;{\\left( 3 \\right)}\\end{array}\\), vem:<\/p>\n<p>\\[\\Delta \u00a0= \\sqrt {1 &#8211; {{\\left( {t \\times \\frac{{s &#8211; t}}{{t \\times S}}} \\right)}^2}} \u00a0= \\sqrt {1 &#8211; {{\\left( {\\frac{{s &#8211; t}}{S}} \\right)}^2}} \\]<\/p>\n<p>Os valores modernos s\u00e3o, aproximadamente:<\/p>\n<ul>\n<li>\\(t = 6371\\) km (m\u00e9dio)<\/li>\n<li>\\(s = 109 \\times t\\)<\/li>\n<li>\\(S = 1,496 \\times {10^8}\\) km (m\u00e9dia)<\/li>\n<\/ul>\n<p>\\[\\Delta \u00a0\\approx \\sqrt {1 &#8211; {{\\left( {\\frac{{108 \\times 6371}}{{1,496 \\times {{10}^8}}}} \\right)}^2}} \u00a0\\approx 0,9999894227 \\approx 1\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(1,12397)'>ARISTARCHUS OF SAMOS &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_12397'>\n<span class='GTTabs_titles'><b>ARISTARCHUS OF SAMOS<\/b><\/span><\/p>\n<h5>ARISTARCHUS OF SAMOS<\/h5>\n<p>HISTORIANS\u00a0of mathematics have, as a rule, given too little attention to Aristarchus of Samos. The reason is no doubt that he was an astronomer, and therefore it might be supposed that his work would have no sufficient interest for the mathematician. The Greeks knew better; they called him Aristarchus &#8216;the mathematician&#8217;, to distinguish him from the host of other Aristarchuses; he is also included by Vitruvius among the few great men who possessed an equally profound knowledge of all branches of science, geometry, astronomy, music, &amp;c.<\/p>\n<blockquote><p>Men of this type are rare, men such as were, in times past, Aristarchus of Samos, Philolaus and Archytas of Tarentum, Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and Scopinas of Syracuse, who left to posterity many mechanical and gnomonic appliances which they invented and explained on mathematical (lit. &#8216;numerical&#8217;) principles.\u00a0<sup>(1)<\/sup><\/p><\/blockquote>\n<p>That Aristarchus was a very capable geometer is proved by his extant work <em>On the sizes and distances of the Sun and Moon<\/em> which will be noticed later in this chapter: in the mechanical line he is credited with the discovery of an improved sun-dial, the so-called <em>\u03c3\u03ba\u03ac\u03c6\u03b7<\/em>, which had, not a plane, but a concave hemispherical surface, with a pointer erected vertically in the middle throwing shadows and so enabling the direction and the height of the sun to be read off by means of lines marked on the surface of the hemisphere. He also wrote on vision, light and colours. His views on the latter subjects were no doubt largely influenced by his master, Strato of Lampsacus; thus Strato held that colours were emanations from bodies, material molecules, as it were, which imparted to the intervening air the same colour as that possessed by the body, while Aristarchus said that colours are &#8216;shapes or forms stamping the air with impressions like themselves, as it were&#8217;, that &#8216;colours in darkness have no colouring&#8217;, and that &#8216;light is the colour impinging on a substratum&#8217;.<\/p>\n<p>Two facts enable us to fix Aristarchus&#8217;s date approximately. In 281\/280 B.C. he made an observation of the summer solstice; and a book of his, presently to be mentioned, was published before the date of Archimedes&#8217;s <em>Psammites<\/em> or <em>Sand-reckoner<\/em>, a work written before 216 B.C. Aristarchus, therefore, probably lived <em>circa<\/em> 310-230 B.C., that is, he was older than Archimedes by about 25 years.<\/p>\n<p>To Aristarchus belongs the high honour of having been the first to formulate the Copernican hypothesis, which was then abandoned again until it was revived by Copernicus himself. His claim to the title of &#8216;the ancient Copernicus&#8217; is still, in my opinion, quite unshaken, notwithstanding the ingenious and elaborate arguments brought forward by Schiaparelli to prove that it was Heraclides of Pontus who first conceived the heliocentric idea. Heraclides is (along with one Ecphantus, a Pythagorean) credited with having been the first to hold that the earth revolves about its own axis every 24 hours, and he was the first to discover that Mercury and Venus revolve, like satellites, about the sun. But though this proves that Heraclides came near, if he did not actually reach, the hypothesis of Tycho Brahe, according to which the earth was in the centre and the rest of the system, the sun with the planets revolving round it, revolved round the earth, it does not suggest that he moved the earth away from the centre. The contrary is indeed stated by A\u00ebtius, who says that &#8216;Heraclides and Ecphantus make the earth move, <em>not in the sense of translation<\/em>, but by way of turning on an axle, like a wheel, from west to east, about its own centre&#8217; <sup>(2)<\/sup>. None of the champions of Heraclides have been able to meet this positive statement. But we have conclusive evidence in favour of the claim of Aristarchus; indeed, ancient testimony is unanimous on the point. Not only does Plutarch tell us that Cleanthes held that Aristarchus ought to be indicted for the impiety of &#8216;putting the Hearth of the Universe in motion&#8217; <sup>(3)<\/sup>; we have the best possible testimony in the precise statement of a great contemporary, Archimedes. In the <em>Sand-reckoner<\/em> Archimedes has this passage.<\/p>\n<blockquote><p>You [King Gelon] are aware that &#8220;universe&#8221; is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers. But Aristarchus brought out <strong><em>a book consisting of certain hypotheses<\/em><\/strong>, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the &#8220;universe&#8221; just mentioned. His hypotheses are that <strong><em>the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit<\/em><\/strong>, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.<\/p><\/blockquote>\n<p>(The last statement is a variation of a traditional phrase, for which there are many parallels (cf. Aristarchus&#8217;s Hypothesis 2 &#8216;that the earth is in the relation of a point and centre to the sphere in which the moon moves&#8217;), and is a method of saying that the &#8216;universe&#8217; is infinitely great in relation not merely to the size of the sun but even to the orbit of the earth in its revolution about it ; the assumption was necessary to Aristarchus in order that he might not have to take account of parallax.)<\/p>\n<p>Plutarch, in the passage referred to above, also makes it clear that Aristarchus followed Heraclides in attributing to the earth the daily rotation about its axis. The bold hypothesis of Aristarchus found few adherents. Seleucus, of Seleucia on the Tigris, is the only convinced supporter of it of whom we hear, and it was speedily abandoned altogether, mainly owing to the great authority of Hipparchus. Nor do we find any trace of the heliocentric hypothesis in Aristarchus&#8217;s extant work <em>On the sizes and distances of the Sun and Moon<\/em>. This is presumably because that work was written before the hypothesis was formulated in the book referred to by Archimedes. The geometry of the treatise is, however, unaffected by the difference between the hypotheses.<\/p>\n<p>Archimedes also says that it was Aristarchus who discovered that the apparent angular diameter of the sun is about 1\/720th part of the zodiac circle, that is to say, half a degree. We do not know how he arrived at this pretty accurate figure: but, as he is credited with the invention of the <em>\u03c3\u03ba\u03ac\u03a6\u03b7<\/em>, he may have used this instrument for the purpose. But here again the discovery must apparently have been later than the treatise <em>On sizes and distances<\/em>, for the value of the subtended angle is there assumed to be 2\u00b0 (Hypothesis 6). How Aristarchus came to assume a value so excessive is uncertain. As the mathematics of his treatise is not dependent on the actual value taken, 2\u00b0 may have been assumed merely by way of illustration; or it may have been a guess at the apparent diameter made before he had thought of attempting to measure it. Aristarchus assumed that the angular diameters of the sun and moon at the centre of the earth are equal.<\/p>\n<p>The method of the treatise depends on the just observation, which is Aristarchus&#8217;s third &#8216;hypothesis&#8217;, that &#8216;when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye&#8217;; the effect of this (since the moon receives its light from the sun), is that at the time of the dichotomy the centres of the sun, moon and earth form a triangle right-angled at the centre of the moon. Two other assumptions were necessary: first, an estimate of the size of the angle of the latter triangle at the centre of the earth at the moment of dichotomy: this Aristarchus assumed (Hypothesis 4) to be &#8216;less than a quadrant by one-thirtieth of a quadrant&#8217;, i. e. 87\u00b0, again an inaccurate estimate, the true value being 89\u00b0 50&#8242;; secondly, an estimate of the breadth of the earth&#8217;s shadow where the moon traverses it: this he assumed to be &#8216;the breadth of two moons&#8217; (Hypothesis 5).<\/p>\n<p>The inaccuracy of the assumptions does not, however, detract from the mathematical interest of the succeeding investigation. Here we find the logical sequence of propositions and the absolute rigour of demonstration characteristic of Greek geometry; the only remaining drawback would be the practical difficulty of determining the exact moment when the moon &#8216;appears to us halved&#8217;. The form and style of the book are thoroughly classical, as befits the period between Euclid and Archimedes; the Greek is even remarkably attractive. The content from the mathematical point of view is no less interesting, for we have here the first specimen extant of pure geometry used with a <em>trigonometrical<\/em> object, in which respect it is a sort of forerunner of Archimedes&#8217;s <em>Measurement of a Circle<\/em>. Aristarchus does not actually evaluate the trigonometrical ratios on which the ratios of the sizes and distances to be obtained depend; he finds limits between which they lie, and that by means of certain propositions which he assumes without proof, and which therefore must have been generally known to mathematicians of his day. These propositions are the equivalents of the statements that,<\/p>\n<ol>\n<li>if \u03b1 is what we call the circular measure of an angle \u00a0and \u03b1 is less than \\(\\frac{1}{2}\\pi \\), then the ratio \\(\\frac{{\\sin \\alpha }}{\\alpha }\\) <em>decreases<\/em>, and the ratio \\(\\frac{{\\tan \\alpha }}{\\alpha }\\) <em>increases<\/em>, as \u03b1 increases from 0 to \\(\\frac{1}{2}\\pi \\);<\/li>\n<li>if \u03b2 be the circular measure of another angle less than \\(\\frac{1}{2}\\pi \\), then<\/li>\n<\/ol>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{{\\sin \\alpha }}{{\\sin \\beta }}}&amp; &lt; &amp;{\\frac{\\alpha }{\\beta }}&amp; &lt; &amp;{\\frac{{\\tan \\alpha }}{{\\tan \\beta }}}\\end{array}\\]<\/p>\n<p>Aristarchus of course deals, not with actual circular measures, sines and tangents, but with angles (expressed not in degrees but as fractions of right angles), arcs of circles and their chords. Particular results obtained by Aristarchus are the equivalent of the following:<\/p>\n<p>\\[\\begin{array}{*{20}{l}}{\\frac{1}{{18}}}&amp; &gt; &amp;{\\sin 3^\\circ }&amp; &gt; &amp;{\\frac{1}{{20}}}&amp;{}&amp;{}&amp;{\\left[ {{\\rm{Prop}}{\\rm{. 7}}} \\right]}\\\\{\\frac{1}{{45}}}&amp; &gt; &amp;{\\sin 1^\\circ }&amp; &gt; &amp;{\\frac{1}{{60}}}&amp;{}&amp;{}&amp;{\\left[ {{\\rm{Prop}}{\\rm{. 11}}} \\right]}\\\\1&amp; &gt; &amp;{\\cos 1^\\circ }&amp; &gt; &amp;{\\frac{{89}}{{90}}}&amp;{}&amp;{}&amp;{\\left[ {{\\rm{Prop}}{\\rm{. 12}}} \\right]}\\\\1&amp; &gt; &amp;{{{\\cos }^2}1^\\circ }&amp; &gt; &amp;{\\frac{{44}}{{45}}}&amp;{}&amp;{}&amp;{\\left[ {{\\rm{Prop}}{\\rm{. 13}}} \\right]}\\end{array}\\]<\/p>\n<p>The book consists of eighteen propositions. Beginning with six hypotheses to the effect already indicated, Aristarchus declares that he is now in a position to prove<\/p>\n<ol>\n<li>that the distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth;<\/li>\n<li>that the diameter of the sun has the same ratio as aforesaid to the diameter of the moon;<\/li>\n<li>that the diameter of the sun has to the diameter of the earth a ratio greater than 19:3, but less than 43:6.<\/li>\n<\/ol>\n<p>The propositions containing these results are Props. 7, 9 and 15.<\/p>\n<p><sup>(1)<\/sup> Vitruvius, <em>De architecture<\/em>, i. 1. 16.<\/p>\n<p><sup>(2)<\/sup> A\u00ebt. iii. 13. 3, <em>Vors<\/em>. i <sup>3<\/sup> , p. 341. 8.<\/p>\n<p><sup>(3)<\/sup> Plutarch, <em>De facie in orbe lunae<\/em>, c. 6, pp. 922 F-923 A.<\/p>\n<ul>\n<li>Extra\u00eddo de <a href=\"https:\/\/archive.org\/details\/historyofgreekm02heat\" target=\"_blank\" rel=\"noopener\">A HISTORY OF GREEK MATHEMATICS, BY SIR THOMAS HEATH, VOLUME II &#8211; FROM ARISTARCHUS TO DIOPHANTUS, OXFORD, AT THE CLARENDON PRESS 1921<\/a><\/li>\n<\/ul>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(0,12397)'>&lt;&lt; C\u00e1lculo de\u00a0\u0394<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(2,12397)'>HYPOTHESES AND PROPOSITIONS &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_12397'>\n<span class='GTTabs_titles'><b>HYPOTHESES AND PROPOSITIONS<\/b><\/span><\/p>\n<h5>HYPOTHESES AND PROPOSITIONS<\/h5>\n<ol>\n<li><em>That the moon receives its light front the sun<\/em>.<\/li>\n<li><em>That the earth is in the relation of a point and centre to the sphere in which the moon moves<\/em>.<sup>(1)<\/sup><\/li>\n<li><em>That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye<\/em>.<sup>(2)<\/sup><\/li>\n<li><em>That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant<\/em>.<sup>(3)<\/sup><\/li>\n<li><em>That the breadth of the (earth&#8217;s) shadow is (that) of two moons<\/em>.<\/li>\n<li><em>That the moon subtends one fifteenth part of a sign of the zodiac<\/em>.<sup>(4)<\/sup><\/li>\n<\/ol>\n<p>We are now in a position to prove the following propositions:<\/p>\n<ol>\n<li><em>The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon (from the earth); <\/em>this follows from the hypothesis about the halved moon.<\/li>\n<li><em>The diameter of the sun has the same ratio (as aforesaid) to the diameter of the moon<\/em>.<sup>(5)<\/sup><\/li>\n<li><em>The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6; <\/em>this follows from the ratio thus discovered between the distances, the hypothesis about the shadow, and the hypothesis that the moon subtends one fifteenth part of a sign of the zodiac.<\/li>\n<\/ol>\n<p><sup>(1)<\/sup> Literally \u2018the sphere of the moon\u2019.<\/p>\n<p><sup>(2)<\/sup> Literally &#8216;<em>verges<\/em> towards our eye&#8217;, the word &#8216;<em>\u03bd\u03b5\u1f7b\u03b5\u03b9\u03bd<\/em>&#8216;\u00a0meaning to &#8216;verge&#8217; or \u2018incline\u2019. What is meant is that the plane of the great circle in question passes through the observer&#8217;s eye or, in other words, that his eye and the great circle are in one plane (cf. Aristarchus&#8217;s own explanation in the enunciation of Prop. 5).<\/p>\n<p><sup>(3)<\/sup> I.e. is less than 90\u00ba by 1\/30th of 90\u00ba or 3\u00ba, and is therefore equal to 87\u00ba.<\/p>\n<p><sup>(4)<\/sup> I. e. 1\/15th of 30\u00ba, or 2\u00b0. Archimedes in his <em>Sand-reckoner<\/em> (Archimedes, ed. Heiberg, ii, p. 248, 19) says that Aristarchus &#8216;discovered that the sun appeared to be about 1\/720th part of the circle of the zodiac&#8217;; that is, Aristarchus discovered (evidently at a date later than that of our treatise) the much more correct value of 1\/2\u00b0 for the angular diameter of the sun or moon (for he maintained that both had the same angular diameter: cf. Prop. 8). Archimedes himself in the same place describes a rough method of observation by which he inferred that the diameter of the sun was less than 1\/164th part, and greater than 1\/200th part, of a right angle. Cf. pp. 311-12 <em>ante<\/em>.<\/p>\n<p><sup>(5)<\/sup> Pappus gives this second result immediately after the first result, i. e. before the parenthesis &#8216;this follows from the hypothesis . . .\u2019. This arrangement seems at first sight more appropriate, and Nizze alters his text accordingly. But I think it better to follow the above order which is that of the MSS. and Wallis. One consideration which weighs with me is that the second result does not follow from the hypothesis of the halved moon alone; it depends on another assumption also, namely, that the sun and the moon have the same apparent angular diameter (see Prop. 8).<\/p>\n<ul>\n<li>Extra\u00eddo de <a href=\"https:\/\/archive.org\/details\/aristarchusofsam00heat\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus&#8217;s Treatise on the sizes and distances of the sun and moon,\u00a0by Heath, Thomas Little, Sir, 1861-1940; Aristarchus, of Samos. On the sizes and distances of the sun and moon. English &amp; Greek. 1913<\/a><\/li>\n<\/ul>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(1,12397)'>&lt;&lt; ARISTARCHUS OF SAMOS<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(3,12397)'>THE PROPOSITIONS &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_12397'>\n<span class='GTTabs_titles'><b>THE PROPOSITIONS<\/b><\/span><\/p>\n<h5>THE PROPOSITIONS<\/h5>\n<p><strong>Proposition 1.<\/strong><br \/>\n<em>Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.<\/em><\/p>\n<p><strong>Proposition 2.<\/strong><br \/>\n<em>If a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere.<\/em><\/p>\n<p><strong>Proposition 3.<\/strong><br \/>\n<em>The circle in the moon which divides the dark and the bright portions is least when the cone comprehending both the sun and the moon has its vertex at our eye.<\/em><\/p>\n<p><strong>Proposition 4.<\/strong><br \/>\n<em>The circle which divides the dark and the bright portions in the moon is not perceptibly different from a great circle in the moon.<\/em><\/p>\n<p><strong>Proposition 5.<\/strong><br \/>\n<em>When the moon appears to us halved, the great circle parallel to the circle which divides the dark and the bright portions in the moon is then in the direction of our eye; that is to say, the great circle parallel to the dividing circle and our eye are in one plane.<\/em><\/p>\n<p><strong>Proposition 6.<\/strong><br \/>\n<em>The moon moves (in an orbit) lower than (that of) the sun, and, when it is halved, is distant less than a quadrant from the sun.<\/em><\/p>\n<p><strong>Proposition 7.<\/strong><br \/>\n<em>The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth.<\/em><\/p>\n<p><strong>Proposition 8.<\/strong><br \/>\n<em>When the sun is totally eclipsed, the sun and the moon are then comprehended by one and the same cone which has its vertex at our eye.<\/em><\/p>\n<p><strong>Proposition 9.<\/strong><br \/>\n<em>The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon.<\/em><\/p>\n<p><strong>Proposition 10.<\/strong><br \/>\n<em>The sun has to the moon a ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1.<\/em><\/p>\n<p><strong>Proposition 11.<\/strong><br \/>\n<em>The diameter of the moon is less than 2\/45ths, but greater than 1\/30th, of the distance of the centre of the moon from our eye.<\/em><\/p>\n<p><strong>Proposition 12.<\/strong><br \/>\n<em>The diameter of the circle which divides the dark and the bright portions in the moon is less than the diameter of the moon, but has to it a ratio greater than that which 89 has to 90.<\/em><\/p>\n<p><strong>Proposition 13.<\/strong><br \/>\n<em>The straight line subtending the portion intercepted within the earth&#8217;s shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1\/9th part of the diameter of the sun, but has to it a ratio greater than that which 21 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10125.<\/em><\/p>\n<p><strong>Proposition 14.<\/strong><br \/>\n<em>The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth&#8217;s shadow a ratio greater than that which 675 has to 1.<\/em><\/p>\n<p><strong>Proposition 15.<\/strong><br \/>\n<em>The diameter of the sun has, to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6.<\/em><\/p>\n<p><strong>Proposition 16.<\/strong><br \/>\n<em>The sun has to the earth a ratio greater than that which 6859 has to 27, but less than that which 79507 has to 216.<\/em><\/p>\n<p><strong>Proposition 17.<\/strong><br \/>\n<em>The diameter of the earth is to the diameter of the moon in a ratio greater than that which 108 has to 43, but less than that which 60 has to 19.<\/em><\/p>\n<p><strong>Proposition 18.<\/strong><br \/>\n<em>The earth is to the moon in a ratio greater than that which 1259712 has to 79507, but less than that which 216000 has to 6859.<\/em><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(2,12397)'>&lt;&lt; HYPOTHESES AND PROPOSITIONS<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(4,12397)'>As conclus\u00f5es de Paul Tannery &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_4_12397'>\n<span class='GTTabs_titles'><b>As conclus\u00f5es de Paul Tannery<\/b><\/span><\/p>\n<h5>As conclus\u00f5es de Paul Tannery<\/h5>\n<p>No final do estudo publicado na Mem\u00f3ria\u00a0<em><a href=\"https:\/\/archive.org\/stream\/mmoiresdelasoci00bordgoog#page\/n361\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Aristarque de Samos<\/a><\/em>, Paul Tannery conclui:<\/p>\n<blockquote><p>Je r\u00e9sumerai comme suit les principales conclusions de cette \u00e9tude:<\/p>\n<p>1.\u00b0 Les anciens n&#8217;ont connu pour la d\u00e9termination des distances du Soleil et de la Lune qu&#8217;une seule m\u00e9thode, dont l\u2019invention doit \u00eatre attribu\u00e9e \u00e0 Eudoxe, qui fut le v\u00e9ritable fondateur de l\u2019astronomie th\u00e9orique.<\/p>\n<p>2.\u00ba Cette m\u00e9thode supposait la d\u00e9termination de deux \u00e9l\u00e9ments dont l&#8217;un, le diam\u00e8tre du cercle d&#8217;ombre de la Terre, peut \u00eatre fix\u00e9 avec assez de pr\u00e9cision par l&#8217;observation des \u00e9clipses lunaires, mais dont l&#8217;autre, la distance angulaire du Soleil et de la Lune au moment de la dichotomie, ne put jamais \u00eatre mesur\u00e9 en r\u00e9alit\u00e9 avec quelque semblant d&#8217;approximation.<\/p>\n<p>3.\u00ba Comme calcul, cette m\u00e9thode \u00e9tait de fait tr\u00e8s simple, si l&#8217;on se contentait d&#8217;une approximation en rapport avec l&#8217;incertitude des donn\u00e9es. Le r\u00f4le d&#8217;\u00c2ristarque fut de lui donner une rigueur g\u00e9om\u00e9trique, mais le d\u00e9faut de la trigonom\u00e9trie l&#8217;obligea \u00e0 exag\u00e9rer les limites des erreurs provenant du calcul.<\/p>\n<p>4.\u00ba Pour d\u00e9terminer les dimensions des deux astres par rapport \u00e0 la Terre, il fallait de plus conna\u00eetre leur diam\u00e8tre apparent et le rapport de la circonf\u00e9rence au diam\u00e8tre. Avant \u00c2rchim\u00e8de, les Grecs savaient seulement que ce rapport \u00e9tait compris entre 3 e 3 1\/3.<\/p>\n<p>5.\u00ba Du vice irr\u00e9m\u00e9diable de la m\u00e9thode, il s&#8217;ensuivit que la distance et les dimensions du Soleil furent toujours calcul\u00e9es d&#8217;une fa\u00e7on absolument erron\u00e9e, tandis que pour la Lune ces \u00e9l\u00e9ments purent \u00eatre d\u00e9termin\u00e9s avec une approximation de plus en plus satisfaisante.<\/p><\/blockquote>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_12397' onClick='GTTabs_show(3,12397)'>&lt;&lt; THE PROPOSITIONS<\/a><\/span><\/div><\/div>\n\n<\/p>\n<p>Fontes:<\/p>\n<ul>\n<li><a href=\"https:\/\/archive.org\/stream\/aristarchusofsam00heatuoft#page\/328\/mode\/2up\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus\u2019s Treatise on the sizes and distances of the sun and moon, by Heath, Thomas Little, Sir, 1861-1940; Aristarchus, of Samos. On the sizes and distances of the sun and moon. English &amp; Greek. 1913. P\u00e1g. 328-411.<\/a><\/li>\n<li><a href=\"https:\/\/archive.org\/stream\/historyofgreekm02heat#page\/n15\/mode\/2up\" target=\"_blank\" rel=\"noopener\">A HISTORY OF GREEK MATHEMATICS, BY SIR THOMAS HEATH, VOLUME II \u2013 FROM ARISTARCHUS TO DIOPHANTUS, OXFORD, AT THE CLARENDON PRESS 1921.\u00a0P\u00e1g. 1-6.<\/a><\/li>\n<li>Tannery:\u00a0<em><a href=\"https:\/\/archive.org\/stream\/mmoiresdelasoci00bordgoog#page\/n361\/mode\/2up\" target=\"_blank\" rel=\"noopener\">M\u00e9moires de la Soci\u00e9t\u00e9 des sciences physiques et naturelles de Bordeaux<\/a><\/em>, 2<sup>e<\/sup> s\u00e9rie, v, 1883, pp. 237-258<\/li>\n<li>J. L. Berggren and Nathan Sidoli \u2013 <a href=\"https:\/\/web.archive.org\/web\/20110428064826\/http:\/\/individual.utoronto.ca\/acephalous\/Berggren_Sidoli_2007a.pdf\" target=\"_blank\" rel=\"noopener\">Aristarchus\u2019s On the Sizes and Distances of the Sun and the Moon: Greek and Arabic Texts<\/a><\/li>\n<li>Wikip\u00e9dia \u2013 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Aristarchus_of_Samos\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos<\/a><\/li>\n<li>School of Mathematics and Statistics, University of St Andrews, Scotland: <a href=\"https:\/\/mathshistory.st-andrews.ac.uk\/Biographies\/Aristarchus\/\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos<\/a><\/li>\n<li>Encyclop\u00e6dia Britannica \u2013 <a href=\"http:\/\/www.britannica.com\/biography\/Aristarchus-of-Samos\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos<\/a><\/li>\n<li>Hellenic World encyclopaedia\u00a0\u2013 <a href=\"http:\/\/www.hellenicaworld.com\/Greece\/Person\/en\/AristarchusSamos.html\" target=\"_blank\" rel=\"noopener\">Aristarchus of Samos<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<div class=\"seriesmeta\">This entry is part 5 of 6 in the series <a href=\"https:\/\/www.acasinhadamatematica.pt\/?series=af-cfaoa\" class=\"series-640\" title=\"AF \u2013 CFAOA\">AF \u2013 CFAOA<\/a><\/div><p>Aristarco de Samos Aristarco de Samos, astr\u00f3nomo e matem\u00e1tico grego que defendia que a Terra roda sobre o seu eixo e gira em torno do Sol, viveu entre 310 e 230 a.C. Foi o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21312,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[413,411,4,3],"tags":[417,412,9,80],"series":[640],"class_list":["post-12397","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-af-cfaoa","category-astronomia","category-ciencia-e-tecnologia","category-matematica","tag-aristarco-de-samos","tag-astronomia","tag-historia-da-matematica","tag-matematica-2","series-af-cfaoa"],"views":6885,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2015\/07\/Aristarchus_working_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/12397","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=12397"}],"version-history":[{"count":1,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/12397\/revisions"}],"predecessor-version":[{"id":27685,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/12397\/revisions\/27685"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21312"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12397"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=12397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}