garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-xii",5 "name": "Euclid's Elements, Book XII",6 "subject": "measurement",7 "variant": "classical",8 "description": "Measurement of figures: circles, pyramids, cones, cylinders, spheres. 18 propositions. Depends on Books I, V, VI, XI. Source: David E. Joyce.",9 "structure": {10 "books": 12,11 "propositions": 18,12 "foundationTypes": [13 "foundation"14 ]15 }16 },17 "metadata": {18 "created": "2026-03-18",19 "lastUpdated": "2026-03-18",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 XII Dependency Graph. Programming Framework.",27 "keywords": [28 "Euclid",29 "Elements",30 "Book XII",31 "measurement",32 "pyramid",33 "cone",34 "cylinder",35 "sphere"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book XII",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXII/bookXII.html",46 "notes": "Clark University"47 }48 ],49 "nodes": [50 {51 "id": "BookI",52 "type": "foundation",53 "label": "Book I — Plane geometry",54 "shortLabel": "Book I",55 "short": "Foundation",56 "book": 1,57 "colorClass": "foundation"58 },59 {60 "id": "BookV",61 "type": "foundation",62 "label": "Book V — Proportions",63 "shortLabel": "Book V",64 "short": "Foundation",65 "book": 5,66 "colorClass": "foundation"67 },68 {69 "id": "BookVI",70 "type": "foundation",71 "label": "Book VI — Similar figures",72 "shortLabel": "Book VI",73 "short": "Foundation",74 "book": 6,75 "colorClass": "foundation"76 },77 {78 "id": "BookXI",79 "type": "foundation",80 "label": "Book XI — Solid geometry",81 "shortLabel": "Book XI",82 "short": "Foundation",83 "book": 11,84 "colorClass": "foundation"85 },86 {87 "id": "Prop1",88 "type": "proposition",89 "label": "Similar polygons in circles: to one another as squares on diameters",90 "shortLabel": "Prop. XII.1",91 "short": "Similar polygons: as squares on diameters",92 "book": 12,93 "number": 1,94 "colorClass": "proposition"95 },96 {97 "id": "Prop2",98 "type": "proposition",99 "label": "Circles are to one another as the squares on their diameters",100 "shortLabel": "Prop. XII.2",101 "short": "Circles: as squares on diameters",102 "book": 12,103 "number": 2,104 "colorClass": "proposition"105 },106 {107 "id": "Prop3",108 "type": "proposition",109 "label": "Pyramid with triangular base: divided into two pyramids, two prisms; prisms greater than half",110 "shortLabel": "Prop. XII.3",111 "short": "Pyramid divided",112 "book": 12,113 "number": 3,114 "colorClass": "proposition"115 },116 {117 "id": "Prop4",118 "type": "proposition",119 "label": "Two pyramids same height, triangular bases, divided: base to base as all prisms",120 "shortLabel": "Prop. XII.4",121 "short": "Pyramids: base as prisms",122 "book": 12,123 "number": 4,124 "colorClass": "proposition"125 },126 {127 "id": "Prop5",128 "type": "proposition",129 "label": "Pyramids same height, triangular bases: to one another as bases",130 "shortLabel": "Prop. XII.5",131 "short": "Pyramids: as bases",132 "book": 12,133 "number": 5,134 "colorClass": "proposition"135 },136 {137 "id": "Prop6",138 "type": "proposition",139 "label": "Pyramids same height, polygonal bases: to one another as bases",140 "shortLabel": "Prop. XII.6",141 "short": "Pyramids polygonal: as bases",142 "book": 12,143 "number": 6,144 "colorClass": "proposition"145 },146 {147 "id": "Prop7",148 "type": "proposition",149 "label": "Prism with triangular base: divided into three equal pyramids",150 "shortLabel": "Prop. XII.7",151 "short": "Prism into three pyramids",152 "book": 12,153 "number": 7,154 "colorClass": "proposition"155 },156 {157 "id": "Prop8",158 "type": "proposition",159 "label": "Similar pyramids triangular bases: in triplicate ratio of corresponding sides",160 "shortLabel": "Prop. XII.8",161 "short": "Similar pyramids: triplicate ratio",162 "book": 12,163 "number": 8,164 "colorClass": "proposition"165 },166 {167 "id": "Prop9",168 "type": "proposition",169 "label": "Equal pyramids triangular bases: bases reciprocally proportional to heights",170 "shortLabel": "Prop. XII.9",171 "short": "Equal pyramids: bases reciprocally proportional",172 "book": 12,173 "number": 9,174 "colorClass": "proposition"175 },176 {177 "id": "Prop10",178 "type": "proposition",179 "label": "Any cone is third part of cylinder same base and equal height",180 "shortLabel": "Prop. XII.10",181 "short": "Cone third of cylinder",182 "book": 12,183 "number": 10,184 "colorClass": "proposition"185 },186 {187 "id": "Prop11",188 "type": "proposition",189 "label": "Cones and cylinders same height: to one another as bases",190 "shortLabel": "Prop. XII.11",191 "short": "Cones, cylinders: as bases",192 "book": 12,193 "number": 11,194 "colorClass": "proposition"195 },196 {197 "id": "Prop12",198 "type": "proposition",199 "label": "Similar cones and cylinders: in triplicate ratio of diameters of bases",200 "shortLabel": "Prop. XII.12",201 "short": "Similar cones, cylinders: triplicate",202 "book": 12,203 "number": 12,204 "colorClass": "proposition"205 },206 {207 "id": "Prop13",208 "type": "proposition",209 "label": "Cylinder cut by plane parallel to opposite: cylinder to cylinder as axis to axis",210 "shortLabel": "Prop. XII.13",211 "short": "Cylinder cut: as axes",212 "book": 12,213 "number": 13,214 "colorClass": "proposition"215 },216 {217 "id": "Prop14",218 "type": "proposition",219 "label": "Cones and cylinders on equal bases: to one another as heights",220 "shortLabel": "Prop. XII.14",221 "short": "Cones, cylinders equal bases: as heights",222 "book": 12,223 "number": 14,224 "colorClass": "proposition"225 },226 {227 "id": "Prop15",228 "type": "proposition",229 "label": "Equal cones and cylinders: bases reciprocally proportional to heights",230 "shortLabel": "Prop. XII.15",231 "short": "Equal cones, cylinders: reciprocally proportional",232 "book": 12,233 "number": 15,234 "colorClass": "proposition"235 },236 {237 "id": "Prop16",238 "type": "proposition",239 "label": "Given two circles same center: inscribe in greater equilateral polygon even sides not touching lesser",240 "shortLabel": "Prop. XII.16",241 "short": "Inscribe polygon in greater circle",242 "book": 12,243 "number": 16,244 "colorClass": "proposition"245 },246 {247 "id": "Prop17",248 "type": "proposition",249 "label": "Given two spheres same center: inscribe in greater polyhedral solid not touching lesser",250 "shortLabel": "Prop. XII.17",251 "short": "Inscribe polyhedron in greater sphere",252 "book": 12,253 "number": 17,254 "colorClass": "proposition"255 },256 {257 "id": "Prop18",258 "type": "proposition",259 "label": "Spheres are to one another in triplicate ratio of their diameters",260 "shortLabel": "Prop. XII.18",261 "short": "Spheres: triplicate ratio",262 "book": 12,263 "number": 18,264 "colorClass": "proposition"265 }266 ],267 "edges": [268 {269 "from": "BookI",270 "to": "Prop1"271 },272 {273 "from": "BookV",274 "to": "Prop1"275 },276 {277 "from": "BookVI",278 "to": "Prop1"279 },280 {281 "from": "BookXI",282 "to": "Prop1"283 },284 {285 "from": "BookI",286 "to": "Prop2"287 },288 {289 "from": "BookV",290 "to": "Prop2"291 },292 {293 "from": "BookVI",294 "to": "Prop2"295 },296 {297 "from": "BookXI",298 "to": "Prop2"299 },300 {301 "from": "BookI",302 "to": "Prop3"303 },304 {305 "from": "BookV",306 "to": "Prop3"307 },308 {309 "from": "BookVI",310 "to": "Prop3"311 },312 {313 "from": "BookXI",314 "to": "Prop3"315 },316 {317 "from": "BookI",318 "to": "Prop4"319 },320 {321 "from": "BookV",322 "to": "Prop4"323 },324 {325 "from": "BookVI",326 "to": "Prop4"327 },328 {329 "from": "BookXI",330 "to": "Prop4"331 },332 {333 "from": "BookI",334 "to": "Prop5"335 },336 {337 "from": "BookV",338 "to": "Prop5"339 },340 {341 "from": "BookVI",342 "to": "Prop5"343 },344 {345 "from": "BookXI",346 "to": "Prop5"347 },348 {349 "from": "BookI",350 "to": "Prop6"351 },352 {353 "from": "BookV",354 "to": "Prop6"355 },356 {357 "from": "BookVI",358 "to": "Prop6"359 },360 {361 "from": "BookXI",362 "to": "Prop6"363 },364 {365 "from": "BookI",366 "to": "Prop7"367 },368 {369 "from": "BookV",370 "to": "Prop7"371 },372 {373 "from": "BookVI",374 "to": "Prop7"375 },376 {377 "from": "BookXI",378 "to": "Prop7"379 },380 {381 "from": "BookI",382 "to": "Prop8"383 },384 {385 "from": "BookV",386 "to": "Prop8"387 },388 {389 "from": "BookVI",390 "to": "Prop8"391 },392 {393 "from": "BookXI",394 "to": "Prop8"395 },396 {397 "from": "BookI",398 "to": "Prop9"399 },400 {401 "from": "BookV",402 "to": "Prop9"403 },404 {405 "from": "BookVI",406 "to": "Prop9"407 },408 {409 "from": "BookXI",410 "to": "Prop9"411 },412 {413 "from": "BookI",414 "to": "Prop10"415 },416 {417 "from": "BookV",418 "to": "Prop10"419 },420 {421 "from": "BookVI",422 "to": "Prop10"423 },424 {425 "from": "BookXI",426 "to": "Prop10"427 },428 {429 "from": "BookI",430 "to": "Prop11"431 },432 {433 "from": "BookV",434 "to": "Prop11"435 },436 {437 "from": "BookVI",438 "to": "Prop11"439 },440 {441 "from": "BookXI",442 "to": "Prop11"443 },444 {445 "from": "BookI",446 "to": "Prop12"447 },448 {449 "from": "BookV",450 "to": "Prop12"451 },452 {453 "from": "BookVI",454 "to": "Prop12"455 },456 {457 "from": "BookXI",458 "to": "Prop12"459 },460 {461 "from": "BookI",462 "to": "Prop13"463 },464 {465 "from": "BookV",466 "to": "Prop13"467 },468 {469 "from": "BookVI",470 "to": "Prop13"471 },472 {473 "from": "BookXI",474 "to": "Prop13"475 },476 {477 "from": "BookI",478 "to": "Prop14"479 },480 {481 "from": "BookV",482 "to": "Prop14"483 },484 {485 "from": "BookVI",486 "to": "Prop14"487 },488 {489 "from": "BookXI",490 "to": "Prop14"491 },492 {493 "from": "BookI",494 "to": "Prop15"495 },496 {497 "from": "BookV",498 "to": "Prop15"499 },500 {501 "from": "BookVI",502 "to": "Prop15"503 },504 {505 "from": "BookXI",506 "to": "Prop15"507 },508 {509 "from": "BookI",510 "to": "Prop16"511 },512 {513 "from": "BookV",514 "to": "Prop16"515 },516 {517 "from": "BookVI",518 "to": "Prop16"519 },520 {521 "from": "BookXI",522 "to": "Prop16"523 },524 {525 "from": "BookI",526 "to": "Prop17"527 },528 {529 "from": "BookV",530 "to": "Prop17"531 },532 {533 "from": "BookVI",534 "to": "Prop17"535 },536 {537 "from": "BookXI",538 "to": "Prop17"539 },540 {541 "from": "BookI",542 "to": "Prop18"543 },544 {545 "from": "BookV",546 "to": "Prop18"547 },548 {549 "from": "BookVI",550 "to": "Prop18"551 },552 {553 "from": "BookXI",554 "to": "Prop18"555 }556 ],557 "colorScheme": {558 "foundation": {559 "fill": "#95a5a6",560 "stroke": "#7f8c8d"561 },562 "proposition": {563 "fill": "#1abc9c",564 "stroke": "#16a085"565 }566 }567}