garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-ii",5 "name": "Euclid's Elements, Book II",6 "subject": "geometry",7 "variant": "classical",8 "description": "The 14 propositions of Book II on geometric algebra (rectangles, squares). Props 1-10 are logically independent within Book II; 11-14 depend on 6, 4, 7, 5. All depend on Book I. Source: David E. Joyce, Clark University.",9 "structure": {10 "books": 2,11 "propositions": 14,12 "foundationTypes": [13 "definition"14 ]15 }16 },17 "metadata": {18 "created": "2026-03-15",19 "lastUpdated": "2026-03-15",20 "version": "1.0.0",21 "license": "CC BY 4.0",22 "authors": [23 "Welz, G."24 ],25 "methodology": "Programming Framework",26 "citation": "Welz, G. (2026). Euclid's Elements Book II Dependency Graph. Programming Framework.",27 "keywords": [28 "Euclid",29 "Elements",30 "Book II",31 "geometric algebra",32 "rectangles",33 "squares",34 "golden section",35 "quadrature"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book II",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookII/bookII.html",46 "notes": "Clark University; dependency table from Logical structure"47 },48 {49 "id": "euclid-heath",50 "type": "primary",51 "authors": "Heath, T.L.",52 "title": "The Thirteen Books of Euclid's Elements",53 "year": "1908",54 "edition": "2nd",55 "publisher": "Cambridge University Press",56 "url": "https://archive.org/details/euclidheath00heatiala",57 "notes": "Standard English translation"58 }59 ],60 "nodes": [61 {62 "id": "BookI",63 "type": "foundation",64 "label": "Book I — Fundamentals of plane geometry",65 "shortLabel": "Book I",66 "short": "Foundation",67 "book": 1,68 "colorClass": "foundation"69 },70 {71 "id": "Def1",72 "type": "definition",73 "label": "Rectangle contained by two straight lines containing the right angle",74 "shortLabel": "Def. II.1",75 "book": 2,76 "number": 1,77 "colorClass": "definition"78 },79 {80 "id": "Def2",81 "type": "definition",82 "label": "Gnomon: parallelogram about diameter with two complements",83 "shortLabel": "Def. II.2",84 "book": 2,85 "number": 2,86 "colorClass": "definition"87 },88 {89 "id": "Prop1",90 "type": "proposition",91 "label": "If one line is cut into segments, rectangle by whole equals sum of rectangles by each segment",92 "shortLabel": "Prop. II.1",93 "short": "Rectangle = sum of rectangles",94 "book": 2,95 "number": 1,96 "colorClass": "proposition"97 },98 {99 "id": "Prop2",100 "type": "proposition",101 "label": "If a line is cut at random, sum of rectangles by whole and each segment equals square on whole",102 "shortLabel": "Prop. II.2",103 "short": "Sum of rectangles = square on whole",104 "book": 2,105 "number": 2,106 "colorClass": "proposition"107 },108 {109 "id": "Prop3",110 "type": "proposition",111 "label": "If a line is cut at random, rectangle by whole and one segment equals rectangle by segments plus square on that segment",112 "shortLabel": "Prop. II.3",113 "short": "Rectangle = rectangle + square",114 "book": 2,115 "number": 3,116 "colorClass": "proposition"117 },118 {119 "id": "Prop4",120 "type": "proposition",121 "label": "If a line is cut at random, square on whole equals squares on segments plus twice rectangle contained by segments",122 "shortLabel": "Prop. II.4",123 "short": "Square on whole = squares + 2×rectangle",124 "book": 2,125 "number": 4,126 "colorClass": "proposition"127 },128 {129 "id": "Prop5",130 "type": "proposition",131 "label": "If a line cut into equal and unequal segments, rectangle by unequal segments plus square on difference equals square on half",132 "shortLabel": "Prop. II.5",133 "short": "Unequal segments: rectangle + square = square on half",134 "book": 2,135 "number": 5,136 "colorClass": "proposition"137 },138 {139 "id": "Prop6",140 "type": "proposition",141 "label": "If a line bisected and added to, rectangle by whole-with-added and added plus square on half equals square on half-plus-added",142 "shortLabel": "Prop. II.6",143 "short": "Bisected + added: rectangle + square = square",144 "book": 2,145 "number": 6,146 "colorClass": "proposition"147 },148 {149 "id": "Prop7",150 "type": "proposition",151 "label": "If a line cut at random, square on whole plus square on one segment equals twice rectangle by whole and segment plus square on remainder",152 "shortLabel": "Prop. II.7",153 "short": "Square on whole + square on segment",154 "book": 2,155 "number": 7,156 "colorClass": "proposition"157 },158 {159 "id": "Prop8",160 "type": "proposition",161 "label": "If a line cut at random, four times rectangle by whole and one segment plus square on remainder equals square on whole-plus-segment",162 "shortLabel": "Prop. II.8",163 "short": "Four times rectangle + square",164 "book": 2,165 "number": 8,166 "colorClass": "proposition"167 },168 {169 "id": "Prop9",170 "type": "proposition",171 "label": "If a line cut into equal and unequal segments, sum of squares on unequal segments is double sum of square on half and square on difference",172 "shortLabel": "Prop. II.9",173 "short": "Unequal segments: sum of squares",174 "book": 2,175 "number": 9,176 "colorClass": "proposition"177 },178 {179 "id": "Prop10",180 "type": "proposition",181 "label": "If a line bisected and added to, square on whole-with-added plus square on added equals double sum of square on half and square on half-plus-added",182 "shortLabel": "Prop. II.10",183 "short": "Bisected + added: sum of squares",184 "book": 2,185 "number": 10,186 "colorClass": "proposition"187 },188 {189 "id": "Prop11",190 "type": "proposition",191 "label": "To cut a given line so that rectangle by whole and one segment equals square on remaining segment",192 "shortLabel": "Prop. II.11",193 "short": "Cut line: rectangle = square (golden section)",194 "book": 2,195 "number": 11,196 "colorClass": "proposition"197 },198 {199 "id": "Prop12",200 "type": "proposition",201 "label": "In obtuse-angled triangles, square on side opposite obtuse angle greater than sum of squares on sides containing it",202 "shortLabel": "Prop. II.12",203 "short": "Obtuse triangle: law of cosines",204 "book": 2,205 "number": 12,206 "colorClass": "proposition"207 },208 {209 "id": "Prop13",210 "type": "proposition",211 "label": "In acute-angled triangles, square on side opposite acute angle less than sum of squares on sides containing it",212 "shortLabel": "Prop. II.13",213 "short": "Acute triangle: law of cosines",214 "book": 2,215 "number": 13,216 "colorClass": "proposition"217 },218 {219 "id": "Prop14",220 "type": "proposition",221 "label": "To construct a square equal to a given rectilinear figure",222 "shortLabel": "Prop. II.14",223 "short": "Construct square = rectilinear figure",224 "book": 2,225 "number": 14,226 "colorClass": "proposition"227 }228 ],229 "edges": [230 {231 "from": "BookI",232 "to": "Prop1"233 },234 {235 "from": "Def1",236 "to": "Prop1"237 },238 {239 "from": "BookI",240 "to": "Prop2"241 },242 {243 "from": "Def1",244 "to": "Prop2"245 },246 {247 "from": "BookI",248 "to": "Prop3"249 },250 {251 "from": "Def1",252 "to": "Prop3"253 },254 {255 "from": "BookI",256 "to": "Prop4"257 },258 {259 "from": "Def1",260 "to": "Prop4"261 },262 {263 "from": "BookI",264 "to": "Prop5"265 },266 {267 "from": "Def1",268 "to": "Prop5"269 },270 {271 "from": "Def2",272 "to": "Prop5"273 },274 {275 "from": "BookI",276 "to": "Prop6"277 },278 {279 "from": "Def1",280 "to": "Prop6"281 },282 {283 "from": "Def2",284 "to": "Prop6"285 },286 {287 "from": "BookI",288 "to": "Prop7"289 },290 {291 "from": "Def1",292 "to": "Prop7"293 },294 {295 "from": "BookI",296 "to": "Prop8"297 },298 {299 "from": "Def1",300 "to": "Prop8"301 },302 {303 "from": "BookI",304 "to": "Prop9"305 },306 {307 "from": "Def1",308 "to": "Prop9"309 },310 {311 "from": "BookI",312 "to": "Prop10"313 },314 {315 "from": "Def1",316 "to": "Prop10"317 },318 {319 "from": "BookI",320 "to": "Prop11"321 },322 {323 "from": "Prop6",324 "to": "Prop11"325 },326 {327 "from": "BookI",328 "to": "Prop12"329 },330 {331 "from": "Prop4",332 "to": "Prop12"333 },334 {335 "from": "BookI",336 "to": "Prop13"337 },338 {339 "from": "Prop7",340 "to": "Prop13"341 },342 {343 "from": "BookI",344 "to": "Prop14"345 },346 {347 "from": "Prop5",348 "to": "Prop14"349 }350 ],351 "colorScheme": {352 "foundation": {353 "fill": "#95a5a6",354 "stroke": "#7f8c8d"355 },356 "definition": {357 "fill": "#3498db",358 "stroke": "#2980b9"359 },360 "proposition": {361 "fill": "#1abc9c",362 "stroke": "#16a085"363 }364 }365}