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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="jbAYiHiocRN1OqaOSYxV87sy9IlIjK0zr2hb2TIozLP9D5Swcilv0NKtlXAjI6e92fFTRkgNhbDGuYbZBNqFMw" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/35766872/Heuristic_analogy_in_Ars_Conjectandi_From_Archimedes_De_Circuli_Dimensione_to_Bernoullis_theorem" /> <meta name="twitter:title" content="Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem" /> <meta name="twitter:description" content="This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective" /> <meta name="twitter:image" content="https://0.academia-photos.com/239526/1161627/10798628/s200_daniel.campos.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/35766872/Heuristic_analogy_in_Ars_Conjectandi_From_Archimedes_De_Circuli_Dimensione_to_Bernoullis_theorem" /> <meta property="og:title" content="Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective" /> <meta property="article:author" content="https://brooklyn-cuny.academia.edu/DanielCampos" /> <meta name="description" content="This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective" /> <meta name="robots" content="noindex" /> <title>Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem</title> <link rel="canonical" href="https://www.academia.edu/35766872/Heuristic_analogy_in_Ars_Conjectandi_From_Archimedes_De_Circuli_Dimensione_to_Bernoullis_theorem" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '7794cd1de67d4501e25b3d8f9c72911f83cf4fa2'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1741108938000); window.Aedu.timeDifference = new Date().getTime() - 1741108938000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective into the question of problem-solving methods in mathematics while also contributing to the comprehension of the historical development of mathematical probability. It argues that Bernoulli proved his theorem by a process of mathematical experimentation in which the central heuristic strategy was analogy. In this context, the analogy functioned as an experimental hypothesis. The article expounds, first, Bernoulli\u0026amp;#39;s reasoning for proving his theorem, describing it as a process of experimentation in which hypothesis-making is crucial. Next, it investigates the analogy between his reasoning and Archimedes\u0026amp;#39; approximation of the value of 蟺, by clarifying both Archimedes\u0026amp;#39; own experimental approach to the said approximation and its heuristic influence on Bernoulli\u0026amp;#39;s problem-solving strategy. The discussion includes some general considerations about analogy as a heuristic technique to make experimental hypotheses in mathematics.","author":[{"@context":"https://schema.org","@type":"Person","name":"Daniel Campos","url":"https://brooklyn-cuny.academia.edu/DanielCampos","image":"https://0.academia-photos.com/239526/1161627/10798628/s200_daniel.campos.jpg","sameAs":[]}],"contributor":[],"dateCreated":"2018-01-26","dateModified":"2018-01-26","headline":"Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem","identifier":{"@type":"PropertyValue","propertyID":"DOI","value":"10.1016/j.shpsa.2017.12.002"},"image":"https://a.academia-assets.com/images/blank-paper.jpg","inLanguage":"en","keywords":["History of Mathematics","Philosophy Of Mathematics","Charles S. 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{"work":{"id":35766872,"created_at":"2018-01-26T12:58:18.948-08:00","from_world_paper_id":null,"updated_at":"2021-01-17T20:00:21.513-08:00","_data":{"doi":"10.1016/j.shpsa.2017.12.002","abstract":"This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective into the question of problem-solving methods in mathematics while also contributing to the comprehension of the historical development of mathematical probability. It argues that Bernoulli proved his theorem by a process of mathematical experimentation in which the central heuristic strategy was analogy. In this context, the analogy functioned as an experimental hypothesis. The article expounds, first, Bernoulli's reasoning for proving his theorem, describing it as a process of experimentation in which hypothesis-making is crucial. Next, it investigates the analogy between his reasoning and Archimedes' approximation of the value of 蟺, by clarifying both Archimedes' own experimental approach to the said approximation and its heuristic influence on Bernoulli's problem-solving strategy. The discussion includes some general considerations about analogy as a heuristic technique to make experimental hypotheses in mathematics.","publication_name":"Studies in History and Philosophy of Science"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"low","language":"en","title":"Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem","broadcastable":false,"draft":null,"has_indexable_attachment":false,"indexable":true,"seo_quality":null}}["work"]; window.loswp.workCoauthors = [239526]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "grid"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="loswp-grid--container"><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-landing_url="https://www.academia.edu/35766872/Heuristic_analogy_in_Ars_Conjectandi_From_Archimedes_De_Circuli_Dimensione_to_Bernoullis_theorem" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="above-fold js-swp-splash-above-fold"><div class="work-card--container js-swp-control-work-card" data-entity-id="35766872"><div class="work-cover--wrapper"><div class="work-cover--container"><div class="work-cover--no-attachment-container js-swp-splash-paper-cover"><div class="work-cover--file-icon-wrapper"><img alt="paper cover icon" src="//a.academia-assets.com/images/single_work_splash/adobe.icon.svg" /></div><div class="work-cover--title js-swp-splash-paper-cover-page-title">Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem</div><br /><div style="margin-top: 170px"><button class="work-cover--request-pdf-button js-request-pdf-button"><svg style="width: 14px; margin-right: 8px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="envelope" class="svg-inline--fa fa-envelope fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"></path></svg>Request PDF</button></div></div></div></div><div class="work-card--content"><h1 class="work-card--title">Heuristic analogy in Ars Conjectandi: From Archimedes' De Circuli Dimensione to Bernoulli's theorem</h1><div class="work-card--pub-info">Studies in History and Philosophy of Science</div><a class="js-profile-link work-card--single-author" data-has-card-for-user="239526" href="https://brooklyn-cuny.academia.edu/DanielCampos?swp=tc-au-35766872"><div class="work-card--single-author-photo"><img class="Avatar--circle s65 Avatar--circle s65 Avatar--xxxs" alt="Daniel Campos" shape="circle" border="0" onerror="if (this.src != '//a.academia-assets.com/images/s65_no_pic.png') this.src = '//a.academia-assets.com/images/s65_no_pic.png';" width="65" height="65" src="https://0.academia-photos.com/239526/1161627/10798628/s65_daniel.campos.jpg" /></div><div class="work-card--author-name-wrapper" style="max-width: 125px; font-weight: 500; color: #4b4b4b;"><div class="u-textTruncate"><span itemprop="author">Daniel Campos</span></div></div></a><script data-card-contents-for-user="239526" type="text/json">{"id":239526,"first_name":"Daniel","middle_initials":null,"last_name":"Campos","page_name":"DanielCampos","domain_name":"brooklyn-cuny","created_at":"2010-08-28T03:46:15.604-07:00","display_name":"Daniel Campos","url":"https://brooklyn-cuny.academia.edu/DanielCampos","photo":"https://0.academia-photos.com/239526/1161627/10798628/s65_daniel.campos.jpg","has_photo":true,"department":{"id":32074,"name":"Philosophy","url":"https://brooklyn-cuny.academia.edu/Departments/Philosophy/Documents","university":{"id":5171,"name":"Brooklyn College of CUNY","url":"https://brooklyn-cuny.academia.edu/"}},"position":"Faculty Member","position_id":1,"interests":[{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics"},{"id":3495,"name":"American Philosophy","url":"https://www.academia.edu/Documents/in/American_Philosophy"},{"id":13110,"name":"Charles S. Peirce","url":"https://www.academia.edu/Documents/in/Charles_S._Peirce"},{"id":803,"name":"Philosophy","url":"https://www.academia.edu/Documents/in/Philosophy"},{"id":981294,"name":"Sport Philosophy","url":"https://www.academia.edu/Documents/in/Sport_Philosophy"},{"id":820,"name":"Latin American Philosophy","url":"https://www.academia.edu/Documents/in/Latin_American_Philosophy"},{"id":14189,"name":"Philosophy of Sport","url":"https://www.academia.edu/Documents/in/Philosophy_of_Sport"},{"id":14167,"name":"Philosophy of Culture","url":"https://www.academia.edu/Documents/in/Philosophy_of_Culture"}]}</script><div class="work-card--no-attachment-details"><div style="font-weight: 700; font-size: 14px;">Abstract</div><div class="work-card--abstract js-swp-splash-abstract">This article investigates the way in which Jacob Bernoulli proved the main mathematical theorem that undergirds his art of conjecturing鈥攖he theorem that founded, historically, the field of mathematical probability. It aims to contribute a perspective into the question of problem-solving methods in mathematics while also contributing to the comprehension of the historical development of mathematical probability. It argues that Bernoulli proved his theorem by a process of mathematical experimentation in which the central heuristic strategy was analogy. In this context, the analogy functioned as an experimental hypothesis. The article expounds, first, Bernoulli's reasoning for proving his theorem, describing it as a process of experimentation in which hypothesis-making is crucial. Next, it investigates the analogy between his reasoning and Archimedes' approximation of the value of 蟺, by clarifying both Archimedes' own experimental approach to the said approximation and its heuristic influence on Bernoulli's problem-solving strategy. The discussion includes some general considerations about analogy as a heuristic technique to make experimental hypotheses in mathematics.</div></div><div class="request-upload--container"><p class="request-upload--title">Daniel Campos hasn't uploaded this paper.</p><div class="request-upload--info-text"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="info-circle" class="request-upload--info-icon svg-inline--fa fa-info-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z"></path></svg><p class="no-margin hide-on-small-mobile">Let Daniel know you want this paper to be uploaded.</p><p class="no-margin hide-above-small-mobile">Ask for this paper to be uploaded.</p></div><button class="work-cover--request-pdf-button small js-request-pdf-button hide-on-desktop"><svg style="width: 14px; margin-right: 8px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="envelope" class="svg-inline--fa fa-envelope fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"></path></svg>Request PDF</button></div></div></div></div><div class="related-content-container--mobile"><div class="mobile-ri-container--header-small"><p class="ds2-5-body-md-bold">Related topics</p></div><div class="mobile-ri-container--header-medium"><h4 class="ds2-5-heading-sans-serif-sm">Related topics</h4></div><div class="sidebar-redesign--ri-container" data-impression-entity-id="35766872" data-impression-entity-type="2" data-impression-source="loswp--mobile-research-interests"><style type="text/css">.related-research-interest--pill { box-sizing: border-box; /* Auto layout */ display: flex; flex-direction: row; justify-content: center; padding: 8px 16px; background: #FFFFFF; border: 1px solid #DDDDE2; border-radius: 100px; /* Inside auto layout */ color: #222233; } .related-research-interest--pill:hover { cursor: pointer; border: 1px solid #0057EB; color: #0057EB; text-decoration: none; }</style><a class="js-related-research-interest related-research-interest--pill" data-entity-id="2084" rel="nofollow" href="https://www.academia.edu/Documents/in/History_of_Mathematics">History of Mathematics</a><style type="text/css">.related-research-interest--pill { box-sizing: border-box; /* Auto layout */ display: flex; flex-direction: row; justify-content: center; padding: 8px 16px; background: #FFFFFF; border: 1px solid #DDDDE2; border-radius: 100px; /* Inside auto layout */ color: #222233; } .related-research-interest--pill:hover { cursor: pointer; border: 1px solid #0057EB; color: #0057EB; text-decoration: none; }</style><a class="js-related-research-interest related-research-interest--pill" data-entity-id="2509" rel="nofollow" href="https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics">Philosophy Of Mathematics</a><style type="text/css">.related-research-interest--pill { box-sizing: border-box; /* Auto layout */ display: flex; flex-direction: row; justify-content: center; padding: 8px 16px; background: #FFFFFF; border: 1px solid #DDDDE2; border-radius: 100px; /* Inside auto layout */ color: #222233; } .related-research-interest--pill:hover { cursor: pointer; border: 1px solid #0057EB; color: #0057EB; text-decoration: none; }</style><a class="js-related-research-interest related-research-interest--pill" data-entity-id="13110" rel="nofollow" href="https://www.academia.edu/Documents/in/Charles_S._Peirce">Charles S. 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