Elsődleges fülek

Devel Load (with references)

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          ]
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          ]
        ]
        "metatag" => array:3 [
          0 => array:2 [
            "tag" => "meta"
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              "content" => "Szabadság - office1 | Szabadság"
            ]
          ]
          1 => array:2 [
            "tag" => "meta"
            "attributes" => array:2 [
              "name" => "description"
              "content" => "Szabadság"
            ]
          ]
          2 => array:2 [
            "tag" => "link"
            "attributes" => array:2 [
              "rel" => "canonical"
              "href" => "https://devel.szabadsag.ro/user/2721925"
            ]
          ]
        ]
        "path" => array:1 [
          0 => array:1 [
            "langcode" => "hu"
          ]
        ]
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            "url" => null
            "guid" => null
            "hash" => "d194ca1ee122ae6017ecd94629a9c59d"
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              "type" => array:1 [
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                    "langcode" => "hu"
                    "status" => true
                    "dependencies" => array:2 [
                      "config" => array:3 [
                        0 => "field.field.user.user.field_keresztnev"
                        1 => "field.field.user.user.field_vezeteknev"
                        2 => "field.field.user.user.user_picture"
                      ]
                      "module" => array:3 [
                        0 => "feeds_tamper"
                        1 => "phpass"
                        2 => "user"
                      ]
                    ]
                    "third_party_settings" => array:1 [
                      "feeds_tamper" => array:1 [
                        "tampers" => array:4 [
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                            "default_value" => "1"
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                            "plugin" => "default_value"
                            "source" => "active"
                            "weight" => 0
                            "label" => "Set value or default value"
                          ]
                          "9004325e-27ff-419d-95e4-c2ea86b8f8e9" => array:6 [
                            "text" => "[lastname] [firstname] - [username]"
                            "uuid" => "9004325e-27ff-419d-95e4-c2ea86b8f8e9"
                            "plugin" => "rewrite"
                            "source" => "username"
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                            "label" => "Rewrite"
                          ]
                          "779f897e-de8c-490e-9c66-7b3d6e9aedeb" => array:6 [
                            "text" => "https://uj.szabadsag.ro/sites/default/files/img/[image_id]"
                            "uuid" => "779f897e-de8c-490e-9c66-7b3d6e9aedeb"
                            "plugin" => "rewrite"
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                            "label" => "Rewrite"
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                            "character" => ""
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                            "plugin" => "trim"
                            "source" => "username"
                            "weight" => 1
                            "label" => "Characters to trim"
                          ]
                        ]
                      ]
                    ]
                    "id" => "user_importer"
                    "label" => "User importer"
                    "description" => ""
                    "help" => ""
                    "import_period" => -1
                    "fetcher" => "upload"
                    "fetcher_configuration" => array:2 [
                      "allowed_extensions" => "txt csv tsv xml opml"
                      "directory" => "public://feeds"
                    ]
                    "parser" => "csv"
                    "parser_configuration" => array:3 [
                      "delimiter" => ","
                      "no_headers" => false
                      "line_limit" => 100
                    ]
                    "processor" => "entity:user"
                    "processor_configuration" => array:11 [
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                      "langcode" => "hu"
                      "insert_new" => 1
                      "update_existing" => 2
                      "update_non_existent" => "_keep"
                      "skip_hash_check" => true
                      "authorize" => false
                      "revision" => false
                      "expire" => -1
                      "owner_feed_author" => false
                      "owner_id" => 0
                    ]
                    "custom_sources" => array:10 [
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                        "value" => "userid"
                        "label" => "userid"
                        "machine_name" => "userid"
                        "type" => "csv"
                      ]
                      "username" => array:4 [
                        "value" => "username"
                        "label" => "username"
                        "machine_name" => "username"
                        "type" => "csv"
                      ]
                      "created" => array:4 [
                        "value" => "created"
                        "label" => "created"
                        "machine_name" => "created"
                        "type" => "csv"
                      ]
                      "email" => array:4 [
                        "value" => "email"
                        "label" => "email"
                        "machine_name" => "email"
                        "type" => "csv"
                      ]
                      "lastname" => array:4 [
                        "value" => "lastname"
                        "label" => "lastname"
                        "machine_name" => "lastname"
                        "type" => "csv"
                      ]
                      "firstname" => array:4 [
                        "value" => "firstname"
                        "label" => "firstname"
                        "machine_name" => "firstname"
                        "type" => "csv"
                      ]
                      "password" => array:4 [
                        "value" => "password"
                        "label" => "password"
                        "machine_name" => "password"
                        "type" => "csv"
                      ]
                      1 => array:4 [
                        "value" => "1"
                        "label" => "1"
                        "machine_name" => "1"
                        "type" => "blank"
                      ]
                      "active" => array:4 [
                        "value" => "active"
                        "label" => "active"
                        "machine_name" => "active"
                        "type" => "blank"
                      ]
                      "image_id" => array:4 [
                        "value" => "image_id"
                        "label" => "image_id"
                        "machine_name" => "image_id"
                        "type" => "csv"
                      ]
                    ]
                    "mappings" => array:9 [
                      0 => array:4 [
                        "target" => "uid"
                        "map" => array:1 [
                          "value" => "userid"
                        ]
                        "settings" => []
                        "unique" => array:1 [
                          "value" => 1
                        ]
                      ]
                      1 => array:4 [
                        "target" => "name"
                        "map" => array:1 [
                          "value" => "username"
                        ]
                        "settings" => []
                        "unique" => []
                      ]
                      2 => array:3 [
                        "target" => "created"
                        "map" => array:1 [
                          "value" => "created"
                        ]
                        "settings" => array:1 [
                          "timezone" => "UTC"
                        ]
                      ]
                      3 => array:4 [
                        "target" => "field_vezeteknev"
                        "map" => array:1 [
                          "value" => "lastname"
                        ]
                        "settings" => array:1 [
                          "language" => null
                        ]
                        "unique" => []
                      ]
                      4 => array:4 [
                        "target" => "field_keresztnev"
                        "map" => array:1 [
                          "value" => "firstname"
                        ]
                        "settings" => array:1 [
                          "language" => null
                        ]
                        "unique" => []
                      ]
                      5 => array:3 [
                        "target" => "pass"
                        "map" => array:1 [
                          "value" => "password"
                        ]
                        "settings" => array:1 [
                          "pass_encryption" => "md5"
                        ]
                      ]
                      6 => array:3 [
                        "target" => "status"
                        "map" => array:1 [
                          "value" => "active"
                        ]
                        "settings" => []
                      ]
                      7 => array:4 [
                        "target" => "mail"
                        "map" => array:1 [
                          "value" => "email"
                        ]
                        "settings" => array:1 [
                          "defuse" => false
                        ]
                        "unique" => []
                      ]
                      8 => array:3 [
                        "target" => "user_picture"
                        "map" => array:3 [
                          "target_id" => "image_id"
                          "alt" => ""
                          "title" => ""
                        ]
                        "settings" => array:5 [
                          "language" => ""
                          "reference_by" => "fid"
                          "autocreate" => 0
                          "existing" => "2"
                          "autocreate_bundle" => false
                        ]
                      ]
                    ]
                  ]
                  "bundle" => "feeds_feed_type"
                ]
              ]
              "title" => array:1 [
                0 => array:1 [
                  "value" => "Felhasználó import"
                ]
              ]
              "uid" => array:1 [
                0 => array:3 [
                  "target_id" => "1"
                  "entity" => array:34 [
                    "uid" => array:1 [
                      0 => array:1 [
                        "value" => "1"
                      ]
                    ]
                    "uuid" => array:1 [
                      0 => array:1 [
                        "value" => "cdb88c70-80ba-4b0c-8ce8-25cc7b33f212"
                      ]
                    ]
                    "langcode" => array:1 [
                      0 => array:1 [
                        "value" => "hu"
                      ]
                    ]
                    "preferred_langcode" => array:1 [
                      0 => array:1 [
                        "value" => "hu"
                      ]
                    ]
                    "preferred_admin_langcode" => []
                    "name" => array:1 [
                      0 => array:1 [
                        "value" => "Kopacz Levente"
                      ]
                    ]
                    "pass" => array:1 [
                      0 => array:1 [
                        "value" => "$2y$10$ocHDQals4xLA7LRI6CLYHOQYQInSh4GtN0ofs/6Snaj5kgYKxjSIy"
                      ]
                    ]
                    "mail" => array:1 [
                      0 => array:1 [
                        "value" => "office@imageservices.ro"
                      ]
                    ]
                    "timezone" => array:1 [
                      0 => array:1 [
                        "value" => "Europe/Bucharest"
                      ]
                    ]
                    "status" => array:1 [
                      0 => array:1 [
                        "value" => "1"
                      ]
                    ]
                    "created" => array:1 [
                      0 => array:1 [
                        "value" => "1685971478"
                      ]
                    ]
                    "changed" => array:1 [
                      0 => array:1 [
                        "value" => "1747237364"
                      ]
                    ]
                    "access" => array:1 [
                      0 => array:1 [
                        "value" => "1754321991"
                      ]
                    ]
                    "login" => array:1 [
                      0 => array:1 [
                        "value" => "1754314741"
                      ]
                    ]
                    "init" => array:1 [
                      0 => array:1 [
                        "value" => "office@imageservices.ro"
                      ]
                    ]
                    "roles" => array:2 [
                      0 => array:3 [
                        "target_id" => "content_editor"
                        "entity" => array:10 [
                          "uuid" => "e0497171-53d3-46c3-960a-e1c878e4b7c5"
                          "langcode" => "hu"
                          "status" => true
                          "dependencies" => array:2 [
                            "config" => array:14 [
                              0 => "filter.format.basic_html"
                              1 => "filter.format.full_html"
                              2 => "node.type.aprohirdetes"
                              3 => "node.type.article"
                              4 => "node.type.extra"
                              5 => "node.type.nevnapok"
                              6 => "node.type.newspaper"
                              7 => "node.type.packages"
                              8 => "node.type.page"
                              9 => "node.type.partner"
                              10 => "node.type.reklam"
                              11 => "taxonomy.vocabulary.aprohirdetes_kategoriak"
                              12 => "taxonomy.vocabulary.rovat"
                              13 => "taxonomy.vocabulary.tags"
                            ]
                            "module" => array:14 [
                              0 => "comment"
                              1 => "custom_tags_block"
                              2 => "field_permissions"
                              3 => "file"
                              4 => "filter"
                              5 => "membercheck"
                              6 => "node"
                              7 => "path"
                              8 => "scheduler"
                              9 => "shortcut"
                              10 => "statistics"
                              11 => "system"
                              12 => "taxonomy"
                              13 => "toolbar"
                            ]
                          ]
                          "_core" => array:1 [
                            "default_config_hash" => "Wur9kcEOwY1Jal81NssKnz3RhVJxAvBwyWQBGcA_1Go"
                          ]
                          "id" => "content_editor"
                          "label" => "Content editor"
                          "weight" => -7
                          "is_admin" => false
                          "permissions" => array:117 [
                            0 => "access administration pages"
                            1 => "access content"
                            2 => "access content overview"
                            3 => "access files overview"
                            4 => "access shortcuts"
                            5 => "access site in maintenance mode"
                            6 => "access taxonomy overview"
                            7 => "access toolbar"
                            8 => "access user profiles"
                            9 => "administer comments"
                            10 => "administer member order entities"
                            11 => "administer nodes"
                            12 => "bypass node access"
                            13 => "create article content"
                            14 => "create extra content"
                            15 => "create field_premium_tartalom"
                            16 => "create field_public_author"
                            17 => "create nevnapok content"
                            18 => "create packages content"
                            19 => "create page content"
                            20 => "create partner content"
                            21 => "create reklam content"
                            22 => "create terms in aprohirdetes_kategoriak"
                            23 => "create terms in rovat"
                            24 => "create terms in tags"
                            25 => "create url aliases"
                            26 => "delete all revisions"
                            27 => "delete any aprohirdetes content"
                            28 => "delete any article content"
                            29 => "delete any extra content"
                            30 => "delete any file"
                            31 => "delete any nevnapok content"
                            32 => "delete any newspaper content"
                            33 => "delete any packages content"
                            34 => "delete any page content"
                            35 => "delete any partner content"
                            36 => "delete any reklam content"
                            37 => "delete own aprohirdetes content"
                            38 => "delete own article content"
                            39 => "delete own extra content"
                            40 => "delete own files"
                            41 => "delete own nevnapok content"
                            42 => "delete own newspaper content"
                            43 => "delete own packages content"
                            44 => "delete own page content"
                            45 => "delete own partner content"
                            46 => "delete own reklam content"
                            47 => "delete terms in aprohirdetes_kategoriak"
                            48 => "delete terms in rovat"
                            49 => "delete terms in tags"
                            50 => "edit any aprohirdetes content"
                            51 => "edit any article content"
                            52 => "edit any extra content"
                            53 => "edit any nevnapok content"
                            54 => "edit any newspaper content"
                            55 => "edit any packages content"
                            56 => "edit any page content"
                            57 => "edit any partner content"
                            58 => "edit any reklam content"
                            59 => "edit field_fizetve"
                            60 => "edit field_premium_tartalom"
                            61 => "edit field_public_author"
                            62 => "edit field_seeded"
                            63 => "edit own aprohirdetes content"
                            64 => "edit own article content"
                            65 => "edit own comments"
                            66 => "edit own extra content"
                            67 => "edit own field_premium_tartalom"
                            68 => "edit own field_public_author"
                            69 => "edit own nevnapok content"
                            70 => "edit own newspaper content"
                            71 => "edit own packages content"
                            72 => "edit own page content"
                            73 => "edit own partner content"
                            74 => "edit own reklam content"
                            75 => "edit terms in aprohirdetes_kategoriak"
                            76 => "edit terms in rovat"
                            77 => "edit terms in tags"
                            78 => "link to any page"
                            79 => "manage custom tags block settings"
                            80 => "revert all revisions"
                            81 => "revert aprohirdetes revisions"
                            82 => "revert article revisions"
                            83 => "revert extra revisions"
                            84 => "revert nevnapok revisions"
                            85 => "revert packages revisions"
                            86 => "revert page revisions"
                            87 => "revert partner revisions"
                            88 => "revert reklam revisions"
                            89 => "schedule publishing of nodes"
                            90 => "use text format basic_html"
                            91 => "use text format full_html"
                            92 => "view all revisions"
                            93 => "view aprohirdetes revisions"
                            94 => "view article revisions"
                            95 => "view extra revisions"
                            96 => "view field_fizetve"
                            97 => "view field_kiemelt_ido"
                            98 => "view field_premium_ervenyesseg"
                            99 => "view field_premium_tartalom"
                            100 => "view field_print_elofizetes"
                            101 => "view field_seeded"
                            102 => "view member order entities"
                            103 => "view nevnapok revisions"
                            104 => "view own field_fizetve"
                            105 => "view own field_kiemelt_ido"
                            106 => "view own field_premium_tartalom"
                            107 => "view own field_public_author"
                            108 => "view own field_seeded"
                            109 => "view own unpublished content"
                            110 => "view packages revisions"
                            111 => "view page revisions"
                            112 => "view partner revisions"
                            113 => "view post access counter"
                            114 => "view reklam revisions"
                            115 => "view scheduled content"
                            116 => "view the administration theme"
                          ]
                        ]
                        "bundle" => "user_role"
                      ]
                      1 => array:3 [
                        "target_id" => "administrator"
                        "entity" => array:10 [
                          "uuid" => "6b72c4a4-06c6-4b76-9694-e7f898c8188c"
                          "langcode" => "hu"
                          "status" => true
                          "dependencies" => array:1 [
                            "module" => array:1 [
                              0 => "field_permissions"
                            ]
                          ]
                          "_core" => array:1 [
                            "default_config_hash" => "OeKGIkmZA_c-t6QLH81WNQx8gDCc1MRmxaTuQgxBByU"
                          ]
                          "id" => "administrator"
                          "label" => "Adminisztrátor"
                          "weight" => -5
                          "is_admin" => true
                          "permissions" => array:5 [
                            0 => "create field_oldalak"
                            1 => "edit field_oldalak"
                            2 => "edit own field_oldalak"
                            3 => "view field_oldalak"
                            4 => "view own field_oldalak"
                          ]
                        ]
                        "bundle" => "user_role"
                      ]
                    ]
                    "default_langcode" => array:1 [
                      0 => array:1 [
                        "value" => "1"
                      ]
                    ]
                    "metatag" => array:3 [
                      0 => array:2 [
                        "tag" => "meta"
                        "attributes" => array:2 [
                          "name" => "title"
                          "content" => "Kopacz Levente | Szabadság"
                        ]
                      ]
                      1 => array:2 [
                        "tag" => "meta"
                        "attributes" => array:2 [
                          "name" => "description"
                          "content" => "Szabadság"
                        ]
                      ]
                      2 => array:2 [
                        "tag" => "link"
                        "attributes" => array:2 [
                          "rel" => "canonical"
                          "href" => "https://devel.szabadsag.ro/user/1"
                        ]
                      ]
                    ]
                    "path" => array:1 [
                      0 => array:1 [
                        "langcode" => "hu"
                      ]
                    ]
                    "feeds_item" => []
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      "value" => "<p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><span style="font-family: nerisblack; font-size: 26px; line-height: 32px; margin-bottom: 26px; margin-top: 8px;"><strong>Tenger a poh&aacute;rban</strong></span></span><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Amikor a h&eacute;tk&ouml;znapi ember meghallja a <em>matematika</em>, <em>matematikus</em> szavakat, amolyan &bdquo;szent borzad&aacute;ly&rdquo; fogja el t&#337;l&uuml;k. <em>Hatv&aacute;nyozottan</em> igaz ez Bolyai J&aacute;nossal &eacute;s munk&aacute;ss&aacute;g&aacute;val kapcsolatban. Mind Bolyai &eacute;let&eacute;t &eacute;s munk&aacute;j&aacute;t, mind az ezekkel foglalkoz&oacute; kutat&aacute;sokat &eacute;s az eddig megjelent k&ouml;tetek mennyis&eacute;g&eacute;t illet&#337;en, nemk&uuml;l&ouml;nben az &eacute;vfordul&oacute; kapcs&aacute;n tartott sz&aacute;mos rendezv&eacute;ny hat&aacute;s&aacute;ra a borzong&aacute;son k&iacute;v&uuml;l olyan &eacute;rz&eacute;s&uuml;nk is t&aacute;mad, mintha a tengert akarn&aacute;nk belet&ouml;lteni egyetlen poh&aacute;rba. B&aacute;rmekkor&aacute;ra n&ouml;veln&eacute;nk a poh&aacute;r m&eacute;ret&eacute;t... B&aacute;torkodjunk az&eacute;rt m&eacute;gis kicsit k&ouml;zelebb l&eacute;pni, megk&oacute;stolni egy cseppet a tenger viz&eacute;b&#337;l, belehallgatni, mint morajlik a hatalmas v&iacute;zt&ouml;meg a f&uuml;l&uuml;nkh&ouml;z emelt kagyl&oacute; &uuml;reg&eacute;ben.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Mindenekel&#337;tt foglaljuk &ouml;ssze r&ouml;viden, mi is az az <em>Appendix. A t&eacute;r tudom&aacute;nya,</em> &eacute;s miben &aacute;ll a tudom&aacute;nyt&ouml;rt&eacute;neti jelent&#337;s&eacute;ge. A gondolkod&oacute; embert mindig is foglalkoztatta az a k&eacute;rd&eacute;s, hogy milyen az &#337;t k&ouml;r&uuml;lvev&#337; vil&aacute;g. Az anyagi vil&aacute;g t&eacute;rbeli form&aacute;it, a k&ouml;zt&uuml;k lev&#337; &ouml;sszef&uuml;gg&eacute;seket a geometria tanulm&aacute;nyozza. Ezen a ter&uuml;leten az &oacute;g&ouml;r&ouml;g matematika m&aacute;r Kr. e. a 3. sz&aacute;zadban rendk&iacute;v&uuml;l magas sz&iacute;nvonalat &eacute;rt el Euklid&eacute;sz <em>Elemek </em>c&iacute;m&#369; k&ouml;nyv&eacute;ben. Amint azt Szenkovits Ferenc matematikus, csillag&aacute;sz, a Babe&#351;&ndash;Bolyai Tudom&aacute;nyegyetem tan&aacute;r&aacute;nak november 16-i kolozsv&aacute;ri el&#337;ad&aacute;s&aacute;b&oacute;l megtudhattuk, t&ouml;bb ezer &eacute;ve foglalkoztatta a matematikusokat a p&aacute;rhuzamosok probl&eacute;m&aacute;ja: mi t&ouml;rt&eacute;nik, ha adott egyeneshez egy k&uuml;ls&#337; ponton kereszt&uuml;l egyeneseket h&uacute;zunk, ezek mind metszik-e az eredeti egyenest. M&aacute;sk&eacute;ppen fogalmazva: h&aacute;ny p&aacute;rhuzamos egyenest rajzolhatunk egy adott egyeneshez egy k&uuml;ls&#337; pontj&aacute;n &aacute;t? A tapasztalat azt mutatja: egyetlenegyet. De val&oacute;ban &iacute;gy van-e? Euklid&eacute;sz &uacute;gy oldotta fel a probl&eacute;m&aacute;t, hogy a p&aacute;rhuzamoss&aacute;got hat&aacute;rhelyzetnek tekintette, ezt posztul&aacute;tumk&eacute;nt deklar&aacute;lta, &eacute;s erre &eacute;p&iacute;tette fel geometriai rendszer&eacute;t.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><img alt="" src="/sites/default/files/oldfiles/f1%2BAppend.jpg"></span><strong>Az <em>Appendix</em> eredti, latin nyelv&#369; kiad&aacute;s&aacute;nak egyik oldala &eacute;s &aacute;br&aacute;i kihajthat&oacute; lapokon</strong></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Az &eacute;vsz&aacute;zadok sor&aacute;n sok matematikus azt gondolta, hogy a p&aacute;rhuzamoss&aacute;gi axi&oacute;ma bizony&iacute;that&oacute;, m&aacute;s, igaz &aacute;ll&iacute;t&aacute;sokb&oacute;l levezethet&#337;. Bolyai Farkas is vizsg&aacute;lta ezt a k&eacute;rd&eacute;st sok &eacute;vtizeden &aacute;t, &aacute;m nem jutott eredm&eacute;nyre &ndash; bar&aacute;tja, Gauss, a &bdquo;matematika fejedelme&rdquo; hib&aacute;kat tal&aacute;lt a levezet&eacute;seiben. Ez&eacute;rt Farkas &oacute;vta a fi&aacute;t att&oacute;l, hogy a paralell&aacute;kkal foglalkozzon, legal&aacute;bbis arra intette, az &#337; &uacute;tj&aacute;n ne menjen. &Aacute;m J&aacute;nos m&aacute;r B&eacute;csben, katonai akad&eacute;miai &eacute;vei alatt elkezdett gondolkodni a probl&eacute;m&aacute;n. A matematika t&ouml;rt&eacute;net&eacute;ben m&aacute;r a 18. sz&aacute;zadban &aacute;ll&iacute;tott&aacute;k, hogy a p&aacute;rhuzamosok feltev&eacute;s&eacute;t el lehetne vetni, de Bolyai J&aacute;nos volt az els&#337;, aki ezt meg is tette, &eacute;s azt tapasztalta, hogy a p&aacute;rhuzamoss&aacute;gi axi&oacute;m&aacute;t&oacute;l f&uuml;ggetlen&uuml;l is nagyon sok t&eacute;tel bebizony&iacute;that&oacute;. Ezek &ouml;sszess&eacute;g&eacute;t nevezz&uuml;k ma abszol&uacute;t geometri&aacute;nak, ezt foglalta &ouml;ssze Bolyai az <em>Appendix</em>ben &ndash; magyar&aacute;zta Szenkovits Ferenc. Bolyai m&#369;ve egy&eacute;bk&eacute;nt csup&aacute;n az&eacute;rt viseli ezt a nevet &ndash; az <em>appendix</em> sz&oacute; f&uuml;ggel&eacute;ket jelent &ndash;, mert J&aacute;nos munk&aacute;ja &eacute;desapja tank&ouml;nyv&eacute;nek, a <em>Tentamen</em>nek a f&uuml;ggel&eacute;kek&eacute;nt&nbsp;jelent meg el&#337;sz&ouml;r 1832-ben.&nbsp;</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Bolyai megt&ouml;rte az euklideszi m&eacute;rtan egyeduralm&aacute;t, &aacute;m a tapasztalat nem igazolta azonnal a nemeuklideszi geometri&aacute;t. Mi haszna ezen t&eacute;relm&eacute;letnek, ha nincs gyakorlati alkalmaz&aacute;sa &ndash; k&eacute;rdezhett&eacute;k sokan Bolyai kor&aacute;ban. Ma m&aacute;r tudjuk: Bolyai a nemeuklideszi geometria felfedez&eacute;s&eacute;vel hozz&aacute;j&aacute;rult a t&eacute;r felfog&aacute;s&aacute;nak &eacute;s vil&aacute;gk&eacute;p&uuml;nknek a megv&aacute;ltoz&aacute;s&aacute;hoz. A modern matematika 19. &eacute;s 20. sz&aacute;zadi fejl&#337;d&eacute;se nagym&eacute;rt&eacute;kben k&ouml;sz&ouml;nhet&#337; az &#337; munk&aacute;ss&aacute;g&aacute;nak, amely utat nyitott a 20. sz&aacute;zadi modern fizika elm&eacute;letei sz&aacute;m&aacute;ra is. Ekkor &eacute;s ezut&aacute;n sz&uuml;lettek Bolyai elm&eacute;let&eacute;nek alkalmaz&aacute;sai: a relativit&aacute;selm&eacute;let, az &#369;rkutat&aacute;s, a legkorszer&#369;bb anyagtudom&aacute;nyok, az orvostudom&aacute;ny sz&aacute;mos felfedez&eacute;se mind-mind Bolyai korszakalkot&oacute; elm&eacute;let&eacute;re is &eacute;p&uuml;l.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">&bdquo;Bolyai J&aacute;nos egyetlen nyomtat&aacute;sban megjelent, t&ouml;m&ouml;r latin nyelven &iacute;rt m&#369;ve a matematikairodalom csod&aacute;latos gy&ouml;ngyszeme. L&aacute;ngelme tudom&aacute;nyt forradalmas&iacute;t&oacute; alkot&aacute;sa.&rdquo; &ndash; mondja K&aacute;rteszi Ferenc matematikus az <em>Appendix </em>&uacute;j kiad&aacute;s&aacute;nak bevezet&#337;j&eacute;ben. Szenkovits Ferenc szerint ezt a napot a magyar matematika sz&uuml;let&eacute;snapj&aacute;nak is tekinthetn&eacute;nk, hiszen ez a legkimagasl&oacute;bb eredm&eacute;ny, amit valaha magyar matematikus el&eacute;rt.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><span style="font-family: nerisblack; font-size: 26px; line-height: 32px; margin-bottom: 26px; margin-top: 8px;"><strong>Temesv&aacute;r &ndash;&nbsp;</strong><strong>tudom&aacute;nyos el&#337;ad&aacute;sok</strong></span></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Temesv&aacute;r id&eacute;n b&uuml;szk&eacute;n viselhette az <em>Eur&oacute;pa kultur&aacute;lis f&#337;v&aacute;rosa</em> c&iacute;met, amelyhez mi sem illik jobban, mint a zseni&aacute;lis matematikusr&oacute;l, Bolyai J&aacute;nosr&oacute;l val&oacute; megeml&eacute;kez&eacute;s &ndash; ann&aacute;l ink&aacute;bb, mivel Bolyai a b&eacute;csi hadm&eacute;rn&ouml;ki akad&eacute;mia elv&eacute;gz&eacute;se ut&aacute;n h&aacute;rom &eacute;ven kereszt&uuml;l a temesv&aacute;ri er&#337;d&iacute;t&eacute;si igazgat&oacute;s&aacute;gon dolgozott, &eacute;s 1823. november 3-&aacute;n &ndash; mind&ouml;ssze huszonegy &eacute;vesen &ndash; e v&aacute;rosb&oacute;l &iacute;rta &eacute;desapj&aacute;nak, Bolyai Farkasnak azt a levelet, amelyben el&#337;sz&ouml;r adott h&iacute;rt arr&oacute;l, hogy &uacute;j geometriai rendszer megalkot&aacute;s&aacute;n dolgozik. Eml&eacute;k&eacute;t &ouml;t (rom&aacute;n, magyar, n&eacute;met, angol, szerb) nyelv&#369; t&aacute;bla &#337;rzi Temesv&aacute;r belv&aacute;ros&aacute;ban.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><img alt="" src="/sites/default/files/oldfiles/f3%2BTEM%2BUj%2BEvezred.jpg"></span><strong>A temesv&aacute;ri <em>Bolyai 200</em> m&#369;v&eacute;szeti &eacute;s tudom&aacute;nyos fesztiv&aacute;l rendezv&eacute;nyeinek f&#337; helysz&iacute;ne a Makovecz Imre tervezte &Uacute;j Ezred&eacute;v Reform&aacute;tus K&ouml;zpont volt. (Fot&oacute;: Orosz S&aacute;ndor)</strong></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A november elej&eacute;n Temesv&aacute;ron lezajlott <em>Bolyai 200 </em>m&#369;v&eacute;szeti &eacute;s tudom&aacute;nyos fesztiv&aacute;l f&#337;szervez&#337;je a temesv&aacute;ri Bart&oacute;k B&eacute;la Alap&iacute;tv&aacute;ny volt<em>, </em>t&aacute;rsszervez&#337;k az MTA Kolozsv&aacute;ri Akad&eacute;miai Bizotts&aacute;g&aacute;nak b&aacute;ns&aacute;gi munkacsoportja, az Erd&eacute;lyi M&uacute;zeum-Egyes&uuml;let, a Szimmetria Alap&iacute;tv&aacute;ny &eacute;s az &Oacute;budai Egyetem. Lelkes, idej&uuml;ket, erej&uuml;ket a k&ouml;z&ouml;ss&eacute;gi c&eacute;lnak szentel&#337; munkat&aacute;rsaik k&ouml;z&uuml;l k&uuml;l&ouml;n elismer&eacute;s &eacute;s k&ouml;sz&ouml;net illeti a k&eacute;t f&#337;szervez&#337;t, Nagy-Gy&ouml;rgy Tam&aacute;st, a Temesv&aacute;ri M&#369;szaki Egyetem tan&aacute;r&aacute;t, &eacute;p&iacute;t&eacute;szm&eacute;rn&ouml;k&ouml;t &eacute;s Erdei Ildik&oacute; tan&aacute;rt, a Bart&oacute;k B&eacute;la-l&iacute;ceum igazgat&oacute;j&aacute;t, a Bart&oacute;k Alap&iacute;tv&aacute;ny eln&ouml;k&eacute;t. A rendezv&eacute;nyek z&ouml;m&eacute;nek a Makovecz Imre tervezte &Uacute;j Ezred&eacute;v Reform&aacute;tus K&ouml;zpont adott otthont. A h&aacute;romnapos el&#337;ad&aacute;s-sorozat nem kiz&aacute;r&oacute;lag matematika&ndash;fizika konferencia volt, hiszen sz&aacute;mos olyan rendezv&eacute;nyt is mag&aacute;ba foglalt, amely bemutatta a Bolyaiak sz&eacute;les k&ouml;r&#369; munk&aacute;ss&aacute;g&aacute;t, amelyet a kert&eacute;szet, a bor&aacute;szat, az erd&eacute;szet, a kemence&eacute;p&iacute;t&eacute;s, az irodalom, a zeneelm&eacute;let, a nyelv&eacute;szet &eacute;s az &bdquo;&uuml;dvtan&rdquo; ter&eacute;n kifejtett e k&eacute;t kiv&eacute;telesen tehets&eacute;ges, sokoldal&uacute; gondolkod&oacute;, apa &eacute;s fia, Farkas &eacute;s J&aacute;nos. Tov&aacute;bb&aacute; nagyon j&oacute; alkalmat ny&uacute;jtott a konferencia t&ouml;bb kiv&aacute;l&oacute; Bolyai-kutat&oacute;r&oacute;l val&oacute; megeml&eacute;kez&eacute;sre, &eacute;s a megkezdett kutat&oacute;munka folytat&aacute;s&aacute;hoz sz&uuml;ks&eacute;ges &uacute;t kijel&ouml;l&eacute;s&eacute;re.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A konferenci&aacute;ra sz&aacute;mos hazai megh&iacute;vott el&#337;ad&oacute; mellett Magyarorsz&aacute;gr&oacute;l, B&eacute;csb&#337;l &eacute;s Kijevb&#337;l is &eacute;rkeztek kutat&oacute;k, s&#337;t az Egyes&uuml;lt &Aacute;llamokb&oacute;l is bekapcsol&oacute;dtak Zoomon kereszt&uuml;l, akik k&uuml;l&ouml;nb&ouml;z&#337; tudom&aacute;nyter&uuml;letek oldal&aacute;r&oacute;l k&ouml;zel&iacute;tettek Bolyai munk&aacute;ss&aacute;g&aacute;hoz. Az al&aacute;bbiakban megeml&iacute;t&uuml;nk n&eacute;h&aacute;nyat az elhangzott el&#337;ad&aacute;sok k&ouml;z&uuml;l, a teljess&eacute;g ig&eacute;nye n&eacute;lk&uuml;l. Matematikai, illetve matematikat&ouml;rt&eacute;neti el&#337;ad&aacute;sok:<em> Euklid&eacute;sz, Bolyai &eacute;s a geometriai szeml&eacute;let&uuml;nk </em>(Szilassi Lajos docens, Szegedi Tudom&aacute;nyegyetem)<em>, Bolyai, Gauss &eacute;s Lobacsevszkij eml&eacute;kezete </em>(Jenkovszky L&aacute;szl&oacute;, Kijevi Elm&eacute;leti Fizika Int&eacute;zet),<em> Halsted amerikai matematikus &eacute;s a Bolyaiak </em>(K&aacute;sa Zolt&aacute;n professor emeritus, Sapientia Erd&eacute;lyi Magyar Tudom&aacute;nyegyetem). Interdiszciplin&aacute;ris el&#337;ad&aacute;sok:<em> A Bolyaiak a tudom&aacute;nyban &eacute;s a m&#369;v&eacute;szetben &ndash; &eacute;s szerte a vil&aacute;gban</em> (Nagy D&eacute;nes, a Nemzetk&ouml;zi Szimmetria T&aacute;rsas&aacute;g eln&ouml;ke), <em>A k&eacute;t Bolyai csillag&aacute;szata</em> (Szenkovits Ferenc docens, Babe&#351;&ndash;Bolyai Tudom&aacute;nyegyetem), <em>Bolyai J&aacute;nos &eacute;s a gravit&aacute;ci&oacute;, ahogy G&aacute;bos Zolt&aacute;n l&aacute;tta</em> (Nagy L&aacute;szl&oacute; akad&eacute;mikus, a Kolozsv&aacute;ri Akad&eacute;miai Bizotts&aacute;g eln&ouml;ke), <em>A matematika &eacute;s az interdiszciplinarit&aacute;s szerepe a szem&eacute;lyre szabott orvosl&aacute;sban</em> (Kov&aacute;cs Levente professzor, az &Oacute;budai Egyetem rektora).</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><img alt="" src="/sites/default/files/oldfiles/f4%2BTEM%2Besemeny.jpg" style="cursor: default; -webkit-tap-highlight-color: rgba(0, 0, 0, 0); max-width: 100%; height: auto; vertical-align: middle; border: 0px; color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px; background-color: rgb(255, 255, 255);"></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><strong style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">A temesv&aacute;ri&nbsp;<em>Bolyai 200</em>&nbsp;rendezv&eacute;nysorozat plak&aacute;tja, a Z gener&aacute;ci&oacute;s m&eacute;diam&#369;helyen k&eacute;sz&uuml;lt munk&aacute;k bemutat&aacute;sa &eacute;s egyik d&iacute;jazott di&aacute;kcsapat, mellett&uuml;k Erdei Ildik&oacute;, a Bart&oacute;k Alap&iacute;tv&aacute;ny eln&ouml;ke&nbsp;(Fot&oacute;k: </strong><strong>facebook.com/bolyai200/ Orosz S&aacute;ndor</strong><strong style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">)</strong><span style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">&nbsp; &nbsp;</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">&bdquo;Az esem&eacute;ny c&eacute;lja az interdiszciplin&aacute;ris gondolkod&aacute;s sz&uuml;ks&eacute;gess&eacute;g&eacute;nek hangs&uacute;lyoz&aacute;sa, kiemelve a kultur&aacute;lis nevel&eacute;st, a tehets&eacute;ggondoz&aacute;s fontoss&aacute;g&aacute;t, amelyben mind a csal&aacute;d, mind az oktat&aacute;si int&eacute;zm&eacute;nyek jelent&#337;s szerepet kell j&aacute;tszanak&rdquo; &ndash; olvashattuk a fesztiv&aacute;l honlapj&aacute;n. Ennek jegy&eacute;ben k&ouml;z&eacute;piskol&aacute;s di&aacute;koknak k&uuml;l&ouml;n tev&eacute;kenys&eacute;get &ndash; Z gener&aacute;ci&oacute;s m&eacute;diam&#369;helyt &eacute;s versenyt &ndash; hirdettek a szervez&#337;k. A jelentkez&#337; iskol&aacute;k k&ouml;z&uuml;l hat csapatot h&iacute;vtak meg a programra: a marosv&aacute;s&aacute;rhelyi Bolyai Farkas-l&iacute;ceum k&eacute;t, valamint a nagyenyedi Bethlen G&aacute;bor Koll&eacute;gium, az aradi Csiky Gergely F&#337;gimn&aacute;zium, a temesv&aacute;ri Bart&oacute;k B&eacute;la-l&iacute;ceum &eacute;s a zentai Bolyai J&aacute;nos Tehets&eacute;ggondoz&oacute; Gimn&aacute;zium egy-egy csapat&aacute;t. A h&aacute;romtag&uacute; csapatok mentortan&aacute;raikkal egy&uuml;tt r&eacute;szt vettek a tehets&eacute;ggondoz&oacute; m&#369;helyen, illetve a fesztiv&aacute;l programjain, ezzel p&aacute;rhuzamosan pedig olyan kreat&iacute;v alkot&aacute;st kellett kidolgozniuk, amely Bolyai J&aacute;nos &eacute;let&eacute;t, szerte&aacute;gaz&oacute; munk&aacute;ss&aacute;g&aacute;t vagy &uuml;zenet&eacute;t n&eacute;pszer&#369;s&iacute;ti. Az elk&eacute;sz&uuml;lt munk&aacute;kat d&iacute;jazt&aacute;k, a v&eacute;gleges v&aacute;ltozatot pedig k&ouml;zz&eacute;tett&eacute;k a projekt honlapj&aacute;n.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A rendezv&eacute;nysorozat honlapj&aacute;n (https://bolyai200.temesvaros.ro/), illetve Facebook-oldal&aacute;n (https://www.facebook.com/bolyai200) tov&aacute;bbi &eacute;rdekes olvasnival&oacute;kat &eacute;s f&eacute;nyk&eacute;peket tal&aacute;lhatnak az &eacute;rdekl&#337;d&#337;k.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><span style="font-family: nerisblack; font-size: 26px; line-height: 32px; margin-bottom: 26px; margin-top: 8px;"><strong>Kolozsv&aacute;r &ndash;&nbsp;</strong><strong>Bolyai-eml&eacute;kki&aacute;ll&iacute;t&aacute;s</strong></span></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A Sapientia Erd&eacute;lyi Magyar Tudom&aacute;nyegyetem ki&aacute;ll&iacute;t&aacute;ssal eml&eacute;kezett meg a h&iacute;res matematikus &eacute;let&eacute;r&#337;l &eacute;s munk&aacute;ss&aacute;g&aacute;r&oacute;l. K&aacute;sa Zolt&aacute;n, a Sapientia EMTE professzora k&eacute;t el&#337;ad&aacute;st is tartott Temesv&aacute;ron, november 13-&aacute;n pedig &#337; nyitotta meg a saj&aacute;t Bolyai-gy&#369;jtem&eacute;ny&eacute;nek tekint&eacute;lyes r&eacute;sz&eacute;t bemutat&oacute; ki&aacute;ll&iacute;t&aacute;st.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Az anyag egyik r&eacute;sze Bolyai J&aacute;nos &eacute;letrajzi adataib&oacute;l, illetve a k&uuml;l&ouml;nb&ouml;z&#337; m&#369;v&eacute;szek &aacute;ltal k&eacute;sz&iacute;tett portr&eacute;it, szobrait &aacute;br&aacute;zol&oacute; f&eacute;nyk&eacute;pekb&#337;l &eacute;s sz&aacute;mos k&eacute;ziratos dokumentum m&aacute;solat&aacute;b&oacute;l tev&#337;d&ouml;tt &ouml;ssze. Az ut&oacute;bbiak a kolozsv&aacute;ri lev&eacute;lt&aacute;rb&oacute;l, a Magyar Tudom&aacute;nyos Akad&eacute;mia lev&eacute;lt&aacute;r&aacute;b&oacute;l vagy r&eacute;gi &uacute;js&aacute;gokb&oacute;l sz&aacute;rmaznak, mint p&eacute;ld&aacute;ul az anyak&ouml;nyvi bejegyz&eacute;s J&aacute;nos keresztel&eacute;s&eacute;r&#337;l, r&eacute;szlet J&aacute;nos h&iacute;res temesv&aacute;ri level&eacute;b&#337;l, egy marosv&aacute;s&aacute;rhelyi hangverseny hirdet&eacute;se 1843 m&aacute;jus&aacute;b&oacute;l, amelyen Bolyai Paganini-muzsik&aacute;t j&aacute;tszott heged&#369;n, vagy David Hilbert n&eacute;met matematikus k&ouml;sz&ouml;n&#337;levele abb&oacute;l az alkalomb&oacute;l, hogy 1910-ben a Magyar Tudom&aacute;nyos Akad&eacute;mia neki &iacute;t&eacute;lte a Bolyai-d&iacute;jat (nemzetk&ouml;zi, &ouml;t&eacute;vente oda&iacute;t&eacute;lend&#337;, 10 ezer korona jutalommal j&aacute;r&oacute; kit&uuml;ntet&eacute;s, amelyet Bolyai sz&uuml;let&eacute;s&eacute;nek 100. &eacute;vfordul&oacute;j&aacute;ra alap&iacute;tottak kiemelked&#337; matematikai munk&aacute;k d&iacute;jaz&aacute;s&aacute;ra). Az anyag t&ouml;bbi r&eacute;sz&eacute;t a Bolyai &eacute;let&eacute;t &eacute;s munk&aacute;ss&aacute;g&aacute;t, &eacute;desapj&aacute;val val&oacute; levelez&eacute;s&eacute;t bemutat&oacute;, ezek kutat&aacute;s&aacute;t feldolgoz&oacute; k&ouml;nyvek, foly&oacute;iratok k&eacute;pezt&eacute;k &ndash; k&ouml;zel &ouml;tven k&ouml;tet, amelyek t&ouml;bbs&eacute;ge magyar nyelv&#369;, de akadt k&ouml;z&ouml;tt&uuml;k rom&aacute;n, angol, n&eacute;met, orosz &eacute;s jap&aacute;n m&#369; is.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><img alt="" src="/sites/default/files/oldfiles/f5%2BKVAR%2Bkiallit.jpg" style="cursor: default; -webkit-tap-highlight-color: rgba(0, 0, 0, 0); max-width: 100%; height: auto; vertical-align: middle; border: 0px; color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px; background-color: rgb(255, 255, 255);"></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><strong style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">A Bolyai-ki&aacute;ll&iacute;t&aacute;s plak&aacute;tja, alatta a ki&aacute;ll&iacute;t&aacute;s anyaga a Sapientia EMTE marosv&aacute;s&aacute;rhelyi kar&aacute;n. Jobbra: K&aacute;sa Zolt&aacute;n bemutatta a k&eacute;ziratos dokumentumokat a kolozsv&aacute;ri ki&aacute;ll&iacute;t&aacute;son,&nbsp; illetve n&eacute;h&aacute;ny Bolyaival kapcsolatos k&ouml;tet az 50 k&ouml;z&uuml;l. (Fot&oacute;k: K&aacute;sa Zolt&aacute;n, Rohonyi Iv&aacute;n)</strong></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Eml&iacute;ts&uuml;nk meg n&eacute;h&aacute;nyat a ki&aacute;ll&iacute;tott k&ouml;nyvek k&ouml;z&uuml;l. Mindenekel&#337;tt ott volt Bolyai J&aacute;nos f&#337; m&#369;ve, az <em>Appendix. A t&eacute;r tudom&aacute;nya</em> (Akad&eacute;miai Kiad&oacute;, Budapest, 1977) c&iacute;m&#369; k&ouml;tet, amely a geometria t&ouml;rt&eacute;net&eacute;nek &aacute;ttekint&eacute;s&eacute;vel kezd&#337;dik Euklid&eacute;szt&#337;l Bolyai m&#369;v&eacute;nek megjelen&eacute;s&eacute;ig, azut&aacute;n az <em>Appendix</em> eredeti, latin nyelv&#369; kiad&aacute;s&aacute;nak hasonm&aacute;sa k&ouml;vetkezik, majd ennek magyar ford&iacute;t&aacute;sa, v&eacute;g&uuml;l Bolyai eszm&eacute;inek m&aacute;ig &eacute;rezhet&#337; hat&aacute;s&aacute;t tekinti &aacute;t K&aacute;rteszi Ferenc matematikus, a szerkeszt&#337;. M&aacute;sik fontos m&#369;, a <em>Bolyai Farkas &eacute;s Carl Friedrich Gauss levelez&eacute;se</em> (Bolyai Kiad&oacute;, Budapest, 2015) is szerepelt a ki&aacute;ll&iacute;t&aacute;son.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Fellapozhattuk tov&aacute;bb&aacute; Benk&#337; Samu <em>Bolyai J&aacute;nos vallom&aacute;sai </em>(Irodalmi K&ouml;nyvkiad&oacute;, 1968) c&iacute;m&#369; munk&aacute;j&aacute;t. Benk&#337; Samu erd&eacute;lyi m&#369;vel&#337;d&eacute;st&ouml;rt&eacute;n&eacute;sz rendszerezte el&#337;sz&ouml;r a Bolyai-kutat&aacute;s &eacute;vsz&aacute;zados t&ouml;rt&eacute;net&eacute;ben &ndash; tizenhat &eacute;v munk&aacute;j&aacute;val sorba &aacute;ll&iacute;totta &eacute;s dosszi&eacute;kba helyezte a mintegy 12 ezer oldalt kitev&#337;, a marosv&aacute;s&aacute;rhelyi reform&aacute;tus gimn&aacute;ziumb&oacute;l a Teleki T&eacute;k&aacute;ba &aacute;tker&uuml;lt anyagot, Bolyai J&aacute;nos eg&eacute;sz k&eacute;ziratos hagyat&eacute;k&aacute;t. Ennek a k&ouml;tetnek &ndash; amelyben a t&ouml;rt&eacute;n&eacute;sz Bolyainak a nem matematikai hagyat&eacute;k&aacute;t dolgozta fel &ndash;, rom&aacute;n nyelv&#369; v&aacute;ltozata is l&aacute;that&oacute; volt a ki&aacute;ll&iacute;t&aacute;son (Benk&#337; Samu, <em>Confesiunile lui Bolyai J&aacute;nos, </em>Kriterion, 1976).</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">N&eacute;h&aacute;ny m&aacute;sik, igen jelent&#337;s munka: Weszely Tibor neves marosv&aacute;s&aacute;rhelyi Bolyai-kutat&oacute; k&ouml;tete, a <em>Bolyai J&aacute;nos matematikai munk&aacute;ss&aacute;ga</em> (Kriterion K&ouml;nyvkiad&oacute;, 1981), Kiss Elem&eacute;r matematikus, tudom&aacute;nyt&ouml;rt&eacute;n&eacute;sz egyik k&ouml;nyve, a <em>Matematikai kincsek Bolyai J&aacute;nos k&eacute;ziratos hagyat&eacute;k&aacute;b&oacute;l</em> (Typotex kiad&oacute;, 2005), Ol&aacute;h-G&aacute;l R&oacute;bert cs&iacute;kszeredai Bolyai-kutat&oacute; t&ouml;bb k&uuml;l&ouml;nb&ouml;z&#337; munk&aacute;ja, p&eacute;ld&aacute;ul a <em>Szemelv&eacute;nyek a Bolyaiak matematikai k&eacute;zirataib&oacute;l</em> (S&aacute;ndor J&oacute;zseffel egy&uuml;tt, 2023), <em>A Bolyaiak a mindennapok sodr&aacute;s&aacute;ban</em> (2016), vagy a <em>Gyerekeknek a Bolyaiakr&oacute;l</em> (2022) &ndash; mindh&aacute;rom a cs&iacute;kszeredai Pro Print Kiad&oacute; gondoz&aacute;s&aacute;ban jelent meg &ndash;, tov&aacute;bb&aacute; G&uuml;ndischn&eacute; Gajz&aacute;g&oacute; M&aacute;ria &ndash; Szenkovits Ferenc, <em>Bolyai Farkas fizik&aacute;ja &eacute;s csillag&aacute;szata. </em><em>M&aacute;sf&eacute;l &eacute;vsz&aacute;zada lappang&oacute; k&eacute;ziratok</em> (Budapest, 2013). Eml&iacute;t&eacute;sre m&eacute;lt&oacute; m&eacute;g az a k&uuml;l&ouml;nleges k&ouml;tet, amelyben Babits Mih&aacute;ly <em>Bolyai</em> c&iacute;m&#369; vers&eacute;t olvashatjuk 33 nyelven, valamint m&aacute;s k&ouml;lt&#337;k Bolyai t&eacute;m&aacute;j&uacute; verseit is.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A ki&aacute;ll&iacute;t&aacute;s Kolozsv&aacute;r ut&aacute;n a Sapientia EMTE marosv&aacute;s&aacute;rhelyi, majd pedig cs&iacute;kszeredai kar&aacute;nak &eacute;p&uuml;let&eacute;ben lesz l&aacute;that&oacute;, &eacute;s tervben van &aacute;lland&oacute; Bolyai-ki&aacute;ll&iacute;t&aacute;s l&eacute;trehoz&aacute;sa ugyancsak Kolozsv&aacute;ron.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><span style="font-family: nerisblack; font-size: 26px; line-height: 32px; margin-bottom: 26px; margin-top: 8px;"><strong>Kolozsv&aacute;r &ndash; Bolyai-eml&eacute;ksz&aacute;m</strong></span></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A Rad&oacute; Ferenc Matematikam&#369;vel&#337; T&aacute;rsas&aacute;g ifj&uacute;s&aacute;gi foly&oacute;irata, a <em>Matlap</em> az erd&eacute;lyi matematikai tehets&eacute;ggondoz&aacute;s egyik legjelent&#337;sebb &ndash; az id&#337;k sor&aacute;n m&aacute;s-m&aacute;s n&eacute;ven megjelen&#337;, de &ndash; 70 &eacute;ves m&uacute;ltra visszatekint&#337; kiadv&aacute;nya. A Bolyai-eml&eacute;k&eacute;v alkalm&aacute;b&oacute;l a lap szerkeszt&#337;s&eacute;ge eml&eacute;ksz&aacute;mot &aacute;ll&iacute;tott &ouml;ssze neves Bolyai-kutat&oacute;k &iacute;r&aacute;saib&oacute;l &ndash; Horv&aacute;th S&aacute;ndor docens (Marosv&aacute;s&aacute;rhely), K&aacute;sa Zolt&aacute;n egyetemi tan&aacute;r (Kolozsv&aacute;r), Ol&aacute;h-G&aacute;l R&oacute;bert adjunktus (Sapientia EMTE, Cs&iacute;kszereda), S&aacute;ndor J&oacute;zsef docens (BBTE, Kolozsv&aacute;r), Szab&oacute; P&eacute;ter G&aacute;bor &eacute;s Szilassi Lajos docensek (Szegedi Tudom&aacute;nyegyetem) cikkeib&#337;l. A k&uuml;l&ouml;nsz&aacute;mot a <em>Magyar tudom&aacute;ny napja Erd&eacute;lyben</em> konferencia matematikat&ouml;rt&eacute;neti szekci&oacute;j&aacute;nak &uuml;l&eacute;s&eacute;t is k&eacute;pez&#337; &iacute;r&oacute;-olvas&oacute; tal&aacute;lkoz&oacute; keret&eacute;ben mutatt&aacute;k be 2023. november 16-&aacute;n a B&aacute;thory Istv&aacute;n Elm&eacute;leti L&iacute;ceum d&iacute;szterm&eacute;ben.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A rendezv&eacute;nyen Schuller Hajnal, a l&iacute;ceum igazgat&oacute;ja k&ouml;sz&ouml;nt&ouml;tte az egybegy&#369;lteket, majd L&aacute;z&aacute;r Ir&eacute;nre, az iskola k&ouml;zelm&uacute;ltban elhunyt matematikatan&aacute;r&aacute;ra &eacute;s volt igazgat&oacute;j&aacute;ra eml&eacute;kezett, kiemelve L&aacute;z&aacute;r Ir&eacute;n igazgat&oacute;i k&eacute;pess&eacute;geit,&nbsp; j&oacute; gyakorlati, strat&eacute;giai &eacute;rz&eacute;k&eacute;t, illetve az 1989-es fordulat ut&aacute;ni b&aacute;tor &eacute;s gyors fell&eacute;p&eacute;s&eacute;t a magyar oktat&aacute;s vissza&aacute;ll&iacute;t&aacute;sa &eacute;rdek&eacute;ben. A r&eacute;sztvev&#337;k egyperces csenddel ad&oacute;ztak a tan&aacute;rn&#337; eml&eacute;k&eacute;nek. Ezut&aacute;n elhangzott Babits Mih&aacute;ly <em>Bolyai</em> c&iacute;m&#369; verse, majd a Gh. Dima Zeneakad&eacute;mia di&aacute;kjai muzsik&aacute;ltak (Gy&ouml;rgy Hunor heged&#369;n, Isabella Voropciuc zongor&aacute;n) &ndash; Bolyai k&ouml;zismert zenekedvel&#337; &eacute;s -m&#369;vel&#337; volt&aacute;ra eml&eacute;kezve egy-egy Bach-, illetve Paganini-darabot adtak el&#337;.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><img alt="" src="/sites/default/files/oldfiles/KVAR%2Bmatlap%2BUJ.jpg"></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="display: block; line-height: 20px; margin-bottom: 19px;"><strong>Kolozsv&aacute;ron a Matlap Bolyai-eml&eacute;ksz&aacute;m&aacute;nak bemutat&aacute;s&aacute;n Schuller Hajnal, a B&aacute;thory Istv&aacute;n Elm&eacute;leti L&iacute;ceum igazgat&oacute;ja k&ouml;sz&ouml;nt&ouml;tte a r&eacute;sztvev&#337;ket.&nbsp;Jobbra: Nagy L&aacute;szl&oacute; el&#337;ad&aacute;s&aacute;ban egy&uuml;tt volt a matematika &eacute;s a fizika. (Fot&oacute;k: facebook.com/matlap.matlap)&nbsp;</strong>&nbsp;</span></p><div><span style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">Az esem&eacute;ny h&aacute;rom el&#337;ad&aacute;ssal folytat&oacute;dott. Els&#337;k&eacute;nt Szenkovits Ferenc, a BBTE oktat&oacute;ja </span><em style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">200 &eacute;ves Bolyai J&aacute;nos hiperbolikus geometri&aacute;ja</em><span style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;"> c&iacute;m&#369; el&#337;ad&aacute;s&aacute;t hallhattuk, amelyb&#337;l cikk&uuml;nk elej&eacute;n m&aacute;r id&eacute;zt&uuml;nk. A tan&aacute;r &uacute;r el&#337;sz&ouml;r Bolyai &uacute;tj&aacute;t v&aacute;zolta iskolai tanulm&aacute;nyait&oacute;l kezdve a b&eacute;csi katonai akad&eacute;mi&aacute;n &aacute;t a katonai p&aacute;ly&aacute;t&oacute;l val&oacute; visszavonul&aacute;s&aacute;ig, majd matematikai munk&aacute;ss&aacute;g&aacute;nak jelent&#337;s&eacute;g&eacute;re t&eacute;rt &aacute;t. El&#337;ad&aacute;s&aacute;nak z&aacute;r&oacute;gondolat&aacute;b&oacute;l igen j&oacute;l &eacute;rz&eacute;kelhett&eacute;k a nem re&aacute;l k&eacute;pzetts&eacute;g&#369; hallgat&oacute;k is: Bolyai l&eacute;p&eacute;se olyan forradalminak sz&aacute;m&iacute;tott a matematik&aacute;ban, mint Kopernikusz felt&eacute;telez&eacute;se a vil&aacute;g heliocentrikus volt&aacute;r&oacute;l a csillag&aacute;szatban. Ez&eacute;rt is nevezik Bolyait a geometria Kopernikusz&aacute;nak.</span></div><div>&nbsp;</div><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">K&aacute;sa Zolt&aacute;n <em>A Bolyai-kutat&aacute;s m&aacute;sf&eacute;l &eacute;vsz&aacute;zada </em>c&iacute;m&#369; el&#337;ad&aacute;s&aacute;b&oacute;l megtudhattuk, hogy az els&#337; magyar Bolyai-kutat&oacute; Schmidt Ferenc temesv&aacute;ri sz&uuml;let&eacute;s&#369; &eacute;p&iacute;t&eacute;szm&eacute;rn&ouml;k volt, akinek az &eacute;desapja szem&eacute;lyesen ismerte Bolyait. Schmidtnek k&ouml;sz&ouml;nhet&#337;, hogy 1894-ben a Matematikai &eacute;s Fizikai T&aacute;rsas&aacute;g s&iacute;rk&#337;vel jel&ouml;ltette meg Bolyai J&aacute;nos marosv&aacute;s&aacute;rhelyi elfelejtett s&iacute;rj&aacute;t.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A m&aacute;sodik jeles Bolyai-kutat&oacute; George Bruce Halsted amerikai matematikus volt, aki leford&iacute;totta angolra az <em>Appendix</em>et &ndash; ezt 1891-ben adt&aacute;k ki el&#337;sz&ouml;r &ndash;, majd 1896-ban leutazott Kolozsv&aacute;rra &eacute;s Marosv&aacute;s&aacute;rhelyre, hogy jobban megismerhesse Bolyai &eacute;let&eacute;t &eacute;s munk&aacute;ss&aacute;g&aacute;t, &eacute;s biztatta Bed&#337;h&aacute;zi J&aacute;nos koll&eacute;giumi tan&aacute;rt, hogy &iacute;rja meg a k&eacute;t Bolyai &eacute;let&eacute;t, ami meg is t&ouml;rt&eacute;nt. Halstednek nem lehet el&eacute;g h&aacute;l&aacute;s a magyar tudom&aacute;nyoss&aacute;g &ndash; mondta K&aacute;sa Zolt&aacute;n.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Benk&#337; Samu m&#369;vel&#337;d&eacute;st&ouml;rt&eacute;n&eacute;sznek nagyon nagy szerepe volt a Bolyai-kutat&aacute;sok beindul&aacute;s&aacute;ban: a teljes hagyat&eacute;k rendszerez&eacute;se mellett feldolgozta annak nem matematikai r&eacute;sz&eacute;t. A h&aacute;romezer oldalnyi matematikai k&eacute;ziratot Kiss Elem&eacute;r marosv&aacute;s&aacute;rhelyi professzor olvasta v&eacute;gig az 1990-es &eacute;vekben: &eacute;let&eacute;nek utols&oacute; tizen&ouml;t &eacute;v&eacute;ben v&eacute;gzett munk&aacute;j&aacute;val kimutatta, hogy Bolyai a geometri&aacute;n k&iacute;v&uuml;l kor&aacute;nak m&aacute;s matematikai, p&eacute;ld&aacute;ul sz&aacute;melm&eacute;leti k&eacute;rd&eacute;seivel is foglalkozott.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Ol&aacute;h-G&aacute;l R&oacute;bert sokoldal&uacute; erd&eacute;lyi Bolyai-kutat&oacute; mintegy 15 k&ouml;nyvet &iacute;rt Bolyair&oacute;l. Szenkovits Ferenc a fizika, csillag&aacute;szat t&eacute;mak&ouml;r&eacute;be tartoz&oacute; Bolyai-munk&aacute;kkal foglalkozott. K&aacute;sa Zolt&aacute;n megeml&iacute;tette m&eacute;g Sz&eacute;n&aacute;si Barna, &Aacute;cs Tibor, Filep L&aacute;szl&oacute; &eacute;s Szab&oacute; P&eacute;ter G&aacute;bor magyarorsz&aacute;gi Bolyai-kutat&oacute;kat is.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">V&eacute;g&uuml;l Nagy L&aacute;szl&oacute; kolozsv&aacute;ri elm&eacute;leti fizikus tartotta meg <em>Bolyai J&aacute;nos &eacute;s a gravit&aacute;ci&oacute;, ahogy G&aacute;bos Zolt&aacute;n l&aacute;tta</em> c&iacute;m&#369; el&#337;ad&aacute;s&aacute;t, amelyet egykori tan&aacute;rn&#337;je, L&aacute;&shy;z&aacute;r Ir&eacute;n eml&eacute;k&eacute;nek aj&aacute;nlott.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">G&aacute;bos Zolt&aacute;n (1924&ndash;2018) az egyik legjelent&#337;sebb erd&eacute;lyi elm&eacute;leti fizikus volt, aki 1990 ut&aacute;n sokat foglalkozott Bolyai geometri&aacute;ja &eacute;s az &aacute;ltal&aacute;nos relativit&aacute;selm&eacute;let k&ouml;z&ouml;tti kapcsolattal. Bolyai szoros &ouml;sszef&uuml;gg&eacute;st t&eacute;telezett fel a t&eacute;r szerkezete &eacute;s a gravit&aacute;ci&oacute;s vonz&aacute;st&ouml;rv&eacute;ny k&ouml;z&ouml;tt. K&eacute;s&#337;bb Einstein &aacute;ltal&aacute;nos relativit&aacute;selm&eacute;let&eacute;ben bebizony&iacute;totta, hogy ez a szoros kapcsolat t&eacute;nyleg l&eacute;tezik: az anyag megg&ouml;rb&iacute;ti maga k&ouml;r&uuml;l a teret, ez hozza l&eacute;tre a gravit&aacute;ci&oacute;t, a gravit&aacute;ci&oacute; pedig &bdquo;megmondja&rdquo; az anyagnak, hogyan mozogjon. A gravit&aacute;ci&oacute; meg&eacute;rt&eacute;s&eacute;hez Einsteinnek nemeuklideszi geometri&aacute;ra volt sz&uuml;ks&eacute;ge, &#337; azonban nem Bolyai vagy Lobacsevszkij, hanem Riemann geometri&aacute;j&aacute;ra t&aacute;maszkodott elm&eacute;let&eacute;nek le&iacute;r&aacute;s&aacute;ban (Bernhard Riemann 19. sz&aacute;zadi n&eacute;met matematikus negyedsz&aacute;zaddal Bolyai &eacute;s Lobacsevszkij ut&aacute;n, m&aacute;s alapokra fektette le &ndash; szint&eacute;n nemeuklideszi &ndash; geometri&aacute;j&aacute;t, t&ouml;bb dimenzi&oacute;ra &aacute;ltal&aacute;nos&iacute;tva az addig h&aacute;romdimenzi&oacute;snak k&eacute;pzelt t&eacute;r elm&eacute;let&eacute;t). Mindenk&eacute;pp elmondhat&oacute; azonban, hogy Bolyai, Lobacsevszkij &eacute;s Riemann utat nyitott az &aacute;ltal&aacute;nos relativit&aacute;selm&eacute;let el&#337;tt.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><img alt="" src="/sites/default/files/oldfiles/Fizika%2BUJ.jpg"></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><strong style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">A Bolyai-eml&eacute;kt&aacute;bl&aacute;k Temesv&aacute;ron. (R&eacute;szlet&nbsp;S&aacute;ntha Imre G&eacute;za fot&oacute;j&aacute;b&oacute;l) Jobbra fent: Bolyai J&aacute;nos k&eacute;zirat&aacute;nak r&eacute;szlete: &bdquo;A neh&eacute;zked&eacute;s t&ouml;rv&eacute;nye is szoros &ouml;sszek&ouml;ttet&eacute;sben, folytat&aacute;sban tetszik (mutatkozik) az &#369;r term&eacute;szet&eacute;vel, val&oacute;j&aacute;val, milyens&eacute;g&eacute;vel&rdquo; &ndash;, azaz a t&eacute;rszerkezet &eacute;s a gravit&aacute;ci&oacute; szoros kapcsolatban van egym&aacute;ssal.&nbsp;</strong><strong style="color: rgb(72, 72, 74); font-family: nerislight; font-size: 16px;">Jobbra, lent: pozit&iacute;v, nulla, illetve negat&iacute;v g&ouml;rb&uuml;let&#369; fel&uuml;letek (&Eacute;rint&#337;, Elektronikus Matematikai Lapok / Szakk&ouml;r &ndash; Muzsnay Zolt&aacute;n)</strong></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Nagy L&aacute;szl&oacute; arra a k&eacute;rd&eacute;sre is kit&eacute;rt, hogy v&eacute;g&uuml;l is milyen a vil&aacute;gegyetem geometri&aacute;ja. Elm&eacute;letileg az univerzumunknak h&aacute;rom alakja lehets&eacute;ges, amelyek mindegyike a t&eacute;r g&ouml;rb&uuml;let&eacute;t&#337;l f&uuml;gg: nyereg alak&uacute; (negat&iacute;v g&ouml;rb&uuml;let), g&ouml;mb alak&uacute; (pozit&iacute;v g&ouml;rb&uuml;let) vagy lapos (g&ouml;rb&uuml;let n&eacute;lk&uuml;li). Az 1980-as &eacute;vek v&eacute;g&eacute;t&#337;l kezdve az &#369;robszervat&oacute;riumok egyre pontosabb m&eacute;r&eacute;seket v&eacute;geztek, amelyek azt mutatt&aacute;k, hogy a t&eacute;rnek egy&aacute;ltal&aacute;n nincs g&ouml;rb&uuml;lete &ndash; a csillag&aacute;szok &aacute;ltal m&eacute;rhet&#337; hat&aacute;rokig lapos. Ezek szerint nagy l&eacute;pt&eacute;kben vett geometri&aacute;ja euklideszi.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Elhangzott m&eacute;g, hogy Bolyai geometri&aacute;ja &uacute;gy viszonyul az euklideszi m&eacute;rtanhoz, mint a 20. sz&aacute;zadi fizika, a relativit&aacute;selm&eacute;let Newton tanaihoz. Nagy L&aacute;szl&oacute; eg&eacute;sz el&#337;ad&aacute;s&aacute;val hozz&aacute;tett kicsit mindannyiunk &aacute;ltal&aacute;nos m&#369;velts&eacute;g&eacute;hez.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><span style="font-family: nerisblack; font-size: 26px; line-height: 32px; margin-bottom: 26px; margin-top: 8px;"><strong>Kolozsv&aacute;r &ndash; a Bolyaiakr&oacute;l&nbsp;</strong><strong>a korabeli sajt&oacute;ban</strong></span></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A <em>Magyar tudom&aacute;ny napja Erd&eacute;lyben</em> konferencia matematika &eacute;s informatika szekci&oacute;j&aacute;nak november 18-i munk&aacute;latai keret&eacute;ben K&aacute;sa Zolt&aacute;n &uacute;jabb el&#337;ad&aacute;st tartott, <em>Bolyai-kultusz az erd&eacute;lyi magyar sajt&oacute;ban 1945-ig</em> c&iacute;mmel. Kutat&aacute;sai alapj&aacute;n &uacute;gy l&aacute;tja: &bdquo;A k&ouml;zv&eacute;leked&eacute;ssel ellent&eacute;tben, nem kev&eacute;s &iacute;r&aacute;s jelent meg a Bolyaiakr&oacute;l, f&#337;leg az 1902&ndash;1903-as megeml&eacute;kez&eacute;sek ut&aacute;n, &eacute;s nem csak kolozsv&aacute;ri &eacute;s marosv&aacute;s&aacute;rhelyi lapokban.&rdquo; N&eacute;h&aacute;ny p&eacute;lda k&ouml;vetkezik.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Magyar Polg&aacute;r</em>, Kolozsv&aacute;r, 1883. m&aacute;jus 29. </strong>&bdquo;<em>A Bolyai-eml&eacute;kk&#337;. Olvas&oacute;ink m&eacute;g eml&eacute;kezhetnek, hogy Bolyai Farkasnak, a marosv&aacute;s&aacute;rhelyi ev. ref. koll&eacute;gium egykori h&iacute;res mennyis&eacute;gtan-tan&aacute;r&aacute;nak, a marosv&aacute;s&aacute;rhelyi ref. k&ouml;ztemet&#337;ben jeltelen&uuml;l nyugv&oacute; porai megjel&ouml;l&eacute;s&eacute;re a koll&eacute;gium el&ouml;lj&aacute;r&oacute;s&aacute;g&aacute;ban 1879-ben mozgalom indulv&aacute;n meg, az el&ouml;lj&aacute;r&oacute;s&aacute;g e c&eacute;lra (...) egy bizotts&aacute;got k&uuml;ld&ouml;tt volt ki. A bizotts&aacute;g </em>&raquo;<em>&Aacute;ll&iacute;tsunk eml&eacute;kk&ouml;vet Bolyai Farkasnak!</em>&laquo;<em> c&iacute;m&#369; felh&iacute;v&aacute;st tett k&ouml;zz&eacute; a napisajt&oacute;ban, s egyszersmind a k&ouml;zszeretetben &aacute;llott n&eacute;hai tan&aacute;r m&eacute;g &eacute;l&#337; tan&iacute;tv&aacute;nyaihoz, ismer&#337;seihez, tisztel&#337;ihez gy&#369;jt&#337;&iacute;veket k&uuml;ld&ouml;tt (&hellip;)&rdquo;</em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Magyar Polg&aacute;r, </em>Kolozsv&aacute;r, 1886. febr. 25.</strong> <em>T&aacute;rcza. Bolyai Farkas.</em> (Brassai S&aacute;muel eml&eacute;kbesz&eacute;d&eacute;b&#337;l.)<em> &bdquo;A nagy matematikus Bolyai 1856-ban halt meg. A hal&aacute;l&aacute;t k&ouml;vet&#337; harmincadik esztend&#337;ben, h&eacute;tf&#337;n tartottak f&ouml;l&ouml;tte eml&eacute;kbesz&eacute;det a tud. akad&eacute;mi&aacute;n, melynek ama ritka tagjai k&ouml;z&eacute; tartozott, akik a k&uuml;lf&ouml;ld tudom&aacute;nyos vil&aacute;g&aacute;ban szint&eacute;n tekint&eacute;llyel b&iacute;rnak. Bolyait a francia &eacute;s n&eacute;met tud&oacute;sok s&#369;r&#369;bben emlegetik, mint n&aacute;lunk, pedig k&ouml;zt&uuml;nk &eacute;lt &eacute;s itthon halt meg. M&#369;veinek kiad&aacute;s&aacute;t most rendezi sajt&oacute; al&aacute; az akad&eacute;mia. &Eacute;letrajzi adatainak gy&#369;jt&eacute;se is megeredt v&eacute;gre, j&oacute; k&eacute;s&#337;n arra n&eacute;zve, hogy kort&aacute;rsai k&ouml;z&uuml;l min&eacute;l kevesebben tan&uacute;skodhassanak. (...)&rdquo; </em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Sz&eacute;kely H&iacute;rlap, </em>1896. november 19.</strong><em> &bdquo;A m&#369;szaki k&ouml;rr&#337;l. Felolvas&aacute;s. A marosv&aacute;s&aacute;rhelyi m&#369;szaki k&ouml;rben Bed&#337;h&aacute;zi J&aacute;nos ev. ref. koll&eacute;giumi professzor tartott felolvas&aacute;st az absolut geometri&aacute;r&oacute;l &eacute;s Bolyai J&aacute;nosr&oacute;l. El&#337;adta az Euklides tizenegyedik axi&oacute;m&aacute;ja &aacute;ltal felvetett k&eacute;rd&eacute;s t&ouml;rt&eacute;net&eacute;t, amely k&eacute;rd&eacute;s t&ouml;bb mint k&eacute;tezer &eacute;ven &aacute;t annyi fejt&ouml;r&eacute;st szerzett a tud&oacute;soknak, a sikertelen t&ouml;rekv&eacute;st, hogy ez axi&oacute;ma, amely tulajdonk&eacute;ppen nem az, bebizony&iacute;ttass&eacute;k; Gaussnak a dolog m&eacute;ly&eacute;re val&oacute; n&eacute;zeteit a k&eacute;rd&eacute;sr&#337;l s Bolyai J&aacute;nosnak igazi l&aacute;ng&eacute;szre vall&oacute; megold&aacute;s&aacute;t, mely &aacute;ltal egyik&eacute;v&eacute; v&aacute;lt amaz els&#337;knek, akik egy r&eacute;git&#337;l elt&eacute;r&#337; &uacute;j geometri&aacute;nak vetett&eacute;k meg az alapj&aacute;t. (...)&rdquo;</em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Magyar Polg&aacute;r</em>, Kolozsv&aacute;r, 1900. m&aacute;rcius 20.</strong> <em>&bdquo;Bolyai J&aacute;nos sz&uuml;l&#337;h&aacute;za. A kolozsv&aacute;ri tudom&aacute;nyegyetem matematikai &eacute;s term&eacute;szettudom&aacute;nyi kara Bolyai J&aacute;nos sz&uuml;let&eacute;s&eacute;nek 100-ik &eacute;vfordul&oacute;j&aacute;t meg&uuml;nnepelni s sz&uuml;let&eacute;s-h&aacute;z&aacute;t eml&eacute;kt&aacute;bl&aacute;val megjel&ouml;lni sz&aacute;nd&eacute;kozik. Irodalmi adatokb&oacute;l tudjuk, hogy Bolyai Farkasnak &eacute;s nej&eacute;nek Benk&#337; Zsuzs&aacute;nn&aacute;nak J&aacute;nos nev&#369; fia 1802. dec. 15-&eacute;n sz&uuml;letett Kolozsv&aacute;rtt a Bels&#337;k&ouml;z&eacute;p-utc&aacute;ban, Bodor P&aacute;l h&aacute;z&aacute;ban, Benk&#337; Zsuzs&aacute;nna sz&uuml;leinek h&aacute;z&aacute;ban. E h&aacute;znak hiteles megjel&ouml;l&eacute;se c&eacute;lj&aacute;b&oacute;l a tan&aacute;cs f&ouml;lh&iacute;vja &eacute;s f&ouml;lk&eacute;ri mindazokat, kik ama Bodor P&aacute;l-f&eacute;le h&aacute;zr&oacute;l &eacute;s hely&eacute;r&#337;l tudom&aacute;ssal b&iacute;rnak, hogy ezt tan&aacute;csunknak, a h&aacute;z megjel&ouml;l&eacute;s&eacute;vel, bejelenteni sz&iacute;veskedjenek.&rdquo;</em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Aranyosvid&eacute;k</em>, 1913. okt&oacute;ber 11. </strong><em>&bdquo;Nagy magyarok (...). Volt id&#337;, mid&#337;n kor&aacute;nak egyik legels&#337;, haz&aacute;j&aacute;nak eddigel&eacute; legnagyobb matematikusa Bolyai Farkas hirdette a marosv&aacute;s&aacute;rhelyi ev. ref. gimn&aacute;ziumban a tudom&aacute;ny ig&eacute;it. Milyen id&#337;k is voltak azok (1804&ndash;1851), mid&#337;n a tudom&aacute;ny fejedelmei kerest&eacute;k fel lev&eacute;lben a kis Maros-V&aacute;s&aacute;rhely szer&eacute;ny koll&eacute;gium&aacute;nak nagy professzor&aacute;t, hogy tudom&aacute;nyos k&eacute;rd&eacute;sekben n&eacute;zeteit hallj&aacute;k. H&iacute;res munk&aacute;j&aacute;t, a Tentament latin nyelven &iacute;rta; ehhez hozz&aacute;k&ouml;tve jelent meg fia &eacute;s matematikai l&aacute;ngesz&#369;nek &ouml;r&ouml;k&ouml;se: Bolyai J&aacute;nos neve alatt a vil&aacute;gh&iacute;r&#369; Appendix. Ez a munka tette a Bolyai nevet vil&aacute;gh&iacute;r&#369;v&eacute;. (...)&rdquo;</em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Keleti &Uacute;js&aacute;g, </em>1941. &aacute;prilis 8. </strong>&bdquo;<em>Megkoszor&uacute;zt&aacute;k Bolyai J&aacute;nos s&iacute;rj&aacute;t a katonai m&#369;szaki akad&eacute;mia vezet&#337;i. Tud&oacute;s&iacute;t&oacute;nk jelenti: Csipk&eacute;s Ern&#337; ezredes &eacute;s Gh&eacute;czy Gy&ouml;rgy sz&aacute;zados, (...) a marosv&aacute;s&aacute;rhelyi reform&aacute;tus koll&eacute;gium tan&aacute;ri kar&aacute;nak k&iacute;s&eacute;ret&eacute;ben vas&aacute;rnap a reform&aacute;tus temet&#337;ben megkoszor&uacute;zt&aacute;k Bolyai J&aacute;nos s&iacute;rj&aacute;t. A katonai m&#369;szaki akad&eacute;mia vezet&#337;i ezzel akart&aacute;k kegyelet&uuml;ket ler&oacute;ni a halhatatlan magyar tud&oacute;s ir&aacute;ni, akir&#337;l az int&eacute;zetet elnevezt&eacute;k. (...).</em>&rdquo;</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong><em>Sz&eacute;kely N&eacute;p</em>, 1944. m&aacute;rcius 18.</strong> &bdquo;<em>Marosv&aacute;s&aacute;rhelyen &aacute;ll&iacute;tj&aacute;k fel B&aacute;thori Istv&aacute;n szobr&aacute;t P&aacute;ly&aacute;zatot hirdetnek a k&eacute;t Bolyai szobr&aacute;nak elk&eacute;sz&iacute;t&eacute;s&eacute;re. Marosv&aacute;s&aacute;rhely, m&aacute;rc. 17. Marosv&aacute;s&aacute;rhely t&ouml;rv&eacute;nyhat&oacute;s&aacute;gi bizotts&aacute;g&aacute;nak p&eacute;nteki k&ouml;zgy&#369;l&eacute;s&eacute;n Papp K&aacute;roly dr. polg&aacute;rmesterhelyettes k&ouml;z&ouml;lte, hogy a budapesti </em>&raquo;<em>B&aacute;thori Istv&aacute;n eml&eacute;kbizotts&aacute;g</em>&laquo;<em> Marosv&aacute;s&aacute;rhelynek aj&aacute;nd&eacute;kozta a nagy erd&eacute;lyi fejedelem, k&eacute;s&#337;bbi lengyel kir&aacute;ly bronzszobr&aacute;t. (...) Ugyancsak Papp K&aacute;roly dr. ismertette a k&eacute;t Bolyainak, Marosv&aacute;s&aacute;rhely nagy fiainak eml&eacute;k&eacute;t meg&ouml;r&ouml;k&iacute;t&#337; szobor fel&aacute;ll&iacute;t&aacute;s&aacute;r&oacute;l sz&oacute;l&oacute; javaslatot. (...)</em>&rdquo;</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A fenti &uacute;js&aacute;gcikkek teljes v&aacute;ltozata, illetve m&aacute;s lapok Bolyai-cikkei az Erd&eacute;lyi Magyar M&#369;szaki Tudom&aacute;nyos T&aacute;rsas&aacute;g publik&aacute;ci&oacute;j&aacute;ban, a Historia Sci&shy;en&shy;ti&shy;a&shy;rum 2023/21. sz&aacute;m&aacute;ban olvashat&oacute;k az interneten (https://ojs.emt.ro/hs/article/view/1491). V&eacute;gezet&uuml;l olvassuk el Bolyai J&aacute;nos gy&aacute;szjelent&eacute;s&eacute;t, amelyet <strong>1860. febru&aacute;r 5-&eacute;n k&ouml;z&ouml;lt a <em>Kolozsv&aacute;ri K&ouml;zl&ouml;ny</em>:</strong></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">&bdquo;<em>Necrolog. Korunk kit&#369;n&#337; f&eacute;rfiainak egyike, a l&aacute;ngesz&#369; sz&aacute;mtud&oacute;s cs. k. nyugalmazott m&eacute;rn&ouml;kkapit&aacute;ny bolyai Bolyai J&aacute;nos foly&oacute; h&oacute; 26-&aacute;n, &eacute;lete 58. &eacute;v&eacute;ben jobb haz&aacute;ba k&ouml;lt&ouml;z&ouml;tt. Nemcsak kis Erd&eacute;ly&uuml;nk, hanem az &ouml;sszes tud&oacute;s vil&aacute;g, mely el&#337;tt egy r&ouml;vid sz&aacute;mtani m&#369;v&eacute;vel honunknak f&eacute;nyt, dics&#337;s&eacute;get szerzett, sokat vesztett kora hal&aacute;la &aacute;ltal, mert nagybecs&#369; k&eacute;ziratait nem adhat&aacute;, elhat&aacute;rozott c&eacute;lja szerint, sajt&oacute; al&aacute;, s k&eacute;rd&eacute;s vajon siker&uuml;lend-e avatott kezeknek &uacute;gy rendbeszedni s vil&aacute;g elibe bocs&aacute;tani, hogy magas &eacute;rt&eacute;k&ouml;k szerint m&eacute;lt&oacute; elismer&eacute;st v&iacute;vjanak ki. Mint nyelv&eacute;sz &eacute;s heged&#369;m&#369;v&eacute;sz is rendk&iacute;v&uuml;li egy&eacute;nis&eacute;g volt. Nagy vesztes&eacute;g re&aacute;nk n&eacute;zve, hogy l&aacute;ng&eacute;sz &eacute;s m&eacute;ly tudom&aacute;ny tulajdonos&aacute;nak &eacute;lete szinte minden haszon n&eacute;lk&uuml;l zajlott le k&ouml;r&uuml;nkben, &eacute;s k&uuml;l&ouml;nck&ouml;d&#337;, emberker&uuml;l&#337; term&eacute;szet&eacute;n&eacute;l fogva minden t&aacute;rsas&aacute;gt&oacute;l f&eacute;lrevonultan kirekeszt&#337;leg nagy eszm&eacute;ivel foglalkozva &eacute;lt. B&eacute;ke hamvainak!&rdquo;</em></span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><img alt="" src="/sites/default/files/oldfiles/Ujsag%2BUJ.jpg"></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><strong>Bolyai J&aacute;nos gy&aacute;szjelent&eacute;se a Kolozsv&aacute;ri K&ouml;zl&ouml;ny 1860. febru&aacute;r 5-i sz&aacute;m&aacute;ban</strong></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">A Bolyai<em>-</em>eml&eacute;k&eacute;v temesv&aacute;ri, kolozsv&aacute;ri &ndash; &eacute;s megel&#337;legezhet&#337;en a november 20-&aacute;n kezd&#337;d&#337;, t&ouml;bb h&eacute;ten &aacute;t tart&oacute; v&aacute;ltozatos marosv&aacute;s&aacute;rhelyi &ndash; programsorozata minden k&eacute;ts&eacute;get kiz&aacute;r&oacute;an bizony&iacute;tja: Bolyai nagybecs&#369; k&eacute;ziratainak rendbeszed&eacute;s&eacute;t &eacute;s vil&aacute;g el&eacute;be bocs&aacute;t&aacute;s&aacute;t avatott kezek napjainkban is folytatj&aacute;k, magas &eacute;rt&eacute;k&uuml;k szerint pedig a m&eacute;lt&oacute; elismer&eacute;st r&eacute;ges r&eacute;gen kiv&iacute;vt&aacute;k.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><strong>Bor&iacute;t&oacute;k&eacute;p</strong>:&nbsp;</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;"><em>A</em><em>ppendix. A</em><em> t&eacute;r absolut igaz tudom&aacute;nya, a XI. Euklides-f&eacute;le axioma helyes vagy t&eacute;ves volt&aacute;t&oacute;l (melyet a priori nem d&ouml;nthet&uuml;nk el) f&uuml;ggetlen&uuml;l t&aacute;rgyalva, a XI. axioma t&eacute;ves volt&aacute;nak elfogad&aacute;s&aacute;hoz hozz&aacute;csatolv&aacute;n a k&ouml;r geometriai n&eacute;gysz&ouml;ges&iacute;t&eacute;s&eacute;t</em> / &iacute;rta bolyai Bolyai J&aacute;nos, az Osztr&aacute;k cs. k. Hadseregben t&aacute;bori m&eacute;rn&ouml;kkapit&aacute;ny; ford&iacute;totta Sut&aacute;k J&oacute;zsef. &ndash; Budapest: Horny&aacute;nszky, 1897.</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;"><span style="color:#48484a;display:block;font-family:nerislight;font-size:16px;line-height:20px;margin-bottom:19px;text-align:left;">Az <em>Appendix </em>itt l&aacute;that&oacute; p&eacute;ld&aacute;nya Bolyai J&aacute;nos saj&aacute;t munkap&eacute;ld&aacute;nya volt. Erre utalnak saj&aacute;t kez&#369; bejegyz&eacute;sei &eacute;s ceruz&aacute;s jav&iacute;t&aacute;sai. A p&eacute;ld&aacute;nyt 1901-ben Schmidt Ferenc hagyat&eacute;k&aacute;b&oacute;l v&aacute;s&aacute;rolta meg a Magyar Tudom&aacute;nyos Akad&eacute;mia. A lap tetej&eacute;n Schmidt Ferenc piros tint&aacute;s autogr&aacute;f megjegyz&eacute;se: Handschrift von Johan Bolyai. (K&eacute;p forr&aacute;sa: Wikip&eacute;dia)</span></p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;">&nbsp;</p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;">&nbsp;</p><p style="font-family:nerislight;color:#48484a;font-size:16px;line-height:20px;text-align:left;margin-bottom:19px;display:block;">&nbsp;</p></p>\n"
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