[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116757-en":3,"doc-seo-116757-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},116757,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Spherical Geometry in the Eighteenth Century I - Euler, Lagrange and Lambert - Preface and Chronologies","Spherical geometry develops from Ancient Greek foundational treatises through advances by Arab mathematicians in spherical trigonometry, then reaches renewed momentum in the eighteenth century through Euler, Lagrange, and Lambert. The volume presents their complete works translated into English from Latin, French, and German, enriched with explanatory footnotes and synthetic essays that place memoirs within mathematical, historical, and philosophical context. Editorial materials also include biographies and chronologies linking Euler’s major life events with publication dates of works referenced in the collection.","Renzo Caddeo Athanase Papadopoulos Editors  \nSpherical Geometry in the Eighteenth Century I  \nEuler, Lagrange and Lambert  \nSpherical Geometry in the Eighteenth Century I  \nRenzo Caddeo • Athanase Papadopoulos Editors  \nSpherical Geometry  \nin the Eighteenth Century I Euler, Lagrange and Lambert  \nEditors  \nRenzo Caddeo  \nDipartimento di Matematica e Informatica, Palazzo Delle Scienze  \nUniversità degli Studi di Cagliari Cagliari, Italy  \nAthanase Papadopoulos  \nInstitut de Recherche Mathématique Avancée  \nUniversité de Strasbourg et CNRS Strasbourg, France  \nISBN 978-3-032-12465-4 ISBN 978-3-032-12466-1 (eBook)  \n[https://doi.org/10.1007/978-3-032-12466-1](https://doi.org/10.1007/978-3-032-12466-1)  \nMathematics Subject Classification: 01A30, 01A50, 00A30, 01A20, 03A05  \n© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2026  \nThis work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.  \nThe use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.  \nThe publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \nCover illustration: The Astronomer, by Johannes Vermeer (1668) . Musée du Louvre, Paris.  \nThis Springer imprint is published by the registered company Springer Nature Switzerland AG  \nThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland If disposing of this product, please recycle the paper.  \n[. . . ] With silent worship of the Great of old,– The dead, but sceptred, Sovereigns, who  \nstill rule  \nOur spirits from their urns. Lord Byron (Manfred, act 3, scene 4 .)  \nPreface  \nSpherical geometry, as a mathematical field, started in Ancient Greece where several fundamental treatises were written on this subject. Among these works, we mention, in historical order, the Moving sphere by Autolycus of Pitane (fourth century BC), the Spherics by Theodosius of Tripoli (first century BC), and the Spherics by Menelaus of Alexandria (first–second centuries AD) . The next major developmentsin this domain are due to the Arab mathematicians between the tenth and the thirteenth century, and one has to mention here their important work on spherical trigonometry. The Arabs made the connection between the Greek world and the Renaissance in Europe, as far as mathematics is concerned. The next significant progress took place in the eighteenth century and it is due to a large extent to Leonhard Euler, Joseph-Louis Lagrange, and Johann Heinrich Lambert, in the works that are presented in the present volume. Spherical geometry received renewed attention in the nineteenth century by Nikolai Ivanovich Lobachevsky, who lectured intensively on this topic and who realized important analogies between the formulae of spherical trigonometry and those of the new geometry he discovered, which he called “imaginary geometry.” As a matter of fact, Lobachevsky used spherical geometry in his foundational work on hyperbolic geometry. In ","cbCaiimDRaoeZX4r","https://ap.wps.com/l/cbCaiimDRaoeZX4r","pdf",30605414,1,449,"English","en",105,"# Preface\n## Chronologies","[{\"question\":\"What historical development does the preface describe for spherical geometry?\",\"answer\":\"It traces spherical geometry from Ancient Greece, through Arab mathematicians’ work on spherical trigonometry, to major progress in the eighteenth century by Euler, Lagrange, and Lambert, and later renewed study in the nineteenth and twentieth centuries.\"},{\"question\":\"Which mathematicians’ works are included in the volume?\",\"answer\":\"The volume contains the complete works on spherical geometry by Euler, Lagrange, and Lambert.\"},{\"question\":\"How are the translations and supporting materials organized?\",\"answer\":\"Memoirs are translated into English and accompanied by explanatory footnotes and synthetic essays that provide mathematical, historical, and philosophical background, highlighting relationships among the original works.\"}]","Spherical Geometry in the Eighteenth Century I - Euler, Lagrange and Lambert - Preface and Chronologies | PDF",1785666877,1131,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"spherical-geometry-in-the-eighteenth-century-i-euler-lagrange-and-lambert-preface-and-chronologies","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/spherical-geometry-in-the-eighteenth-century-i-euler-lagrange-and-lambert-preface-and-chronologies/116757/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What historical development does the preface describe for spherical geometry?","Question",{"text":75,"@type":76},"It traces spherical geometry from Ancient Greece, through Arab mathematicians’ work on spherical trigonometry, to major progress in the eighteenth century by Euler, Lagrange, and Lambert, and later renewed study in the nineteenth and twentieth centuries.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which mathematicians’ works are included in the volume?",{"text":80,"@type":76},"The volume contains the complete works on spherical geometry by Euler, Lagrange, and Lambert.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the translations and supporting materials organized?",{"text":84,"@type":76},"Memoirs are translated into English and accompanied by explanatory footnotes and synthetic essays that provide mathematical, historical, and philosophical background, highlighting relationships among the original works.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]