garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-viii",5 "name": "Euclid's Elements, Book VIII",6 "subject": "number_theory",7 "variant": "classical",8 "description": "Continued proportions. 27 propositions. Depends on Book VII. Source: David E. Joyce.",9 "structure": {10 "books": 8,11 "propositions": 27,12 "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 VIII Dependency Graph. Programming Framework.",27 "keywords": [28 "Euclid",29 "Elements",30 "Book VIII",31 "continued proportion",32 "plane",33 "solid"34 ]35 },36 "sources": [37 {38 "id": "joyce",39 "type": "digital",40 "authors": "Joyce, David E.",41 "title": "Euclid's Elements, Book VIII",42 "year": "1996",43 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookVIII/bookVIII.html",44 "notes": "Clark University"45 }46 ],47 "nodes": [48 {49 "id": "BookVII",50 "type": "foundation",51 "label": "Book VII — Number theory",52 "shortLabel": "Book VII",53 "short": "Foundation",54 "book": 7,55 "colorClass": "foundation"56 },57 {58 "id": "Prop1",59 "type": "proposition",60 "label": "Continued proportion, extremes relatively prime: least in ratio",61 "shortLabel": "Prop. VIII.1",62 "short": "Continued proportion, extremes prime",63 "book": 8,64 "number": 1,65 "colorClass": "proposition"66 },67 {68 "id": "Prop2",69 "type": "proposition",70 "label": "To find numbers in continued proportion, least in given ratio",71 "shortLabel": "Prop. VIII.2",72 "short": "Find numbers in continued proportion",73 "book": 8,74 "number": 2,75 "colorClass": "proposition"76 },77 {78 "id": "Prop3",79 "type": "proposition",80 "label": "Least in continued proportion: extremes relatively prime",81 "shortLabel": "Prop. VIII.3",82 "short": "Least: extremes prime",83 "book": 8,84 "number": 3,85 "colorClass": "proposition"86 },87 {88 "id": "Prop4",89 "type": "proposition",90 "label": "Given ratios in least numbers, find least in continued proportion",91 "shortLabel": "Prop. VIII.4",92 "short": "Find continued proportion from ratios",93 "book": 8,94 "number": 4,95 "colorClass": "proposition"96 },97 {98 "id": "Prop5",99 "type": "proposition",100 "label": "Plane numbers have ratio compounded of ratios of sides",101 "shortLabel": "Prop. VIII.5",102 "short": "Plane numbers: compound ratio",103 "book": 8,104 "number": 5,105 "colorClass": "proposition"106 },107 {108 "id": "Prop6",109 "type": "proposition",110 "label": "Continued proportion: if first does not measure second, none measures another",111 "shortLabel": "Prop. VIII.6",112 "short": "First does not measure second",113 "book": 8,114 "number": 6,115 "colorClass": "proposition"116 },117 {118 "id": "Prop7",119 "type": "proposition",120 "label": "Continued proportion: if first measures last, it measures second",121 "shortLabel": "Prop. VIII.7",122 "short": "First measures last",123 "book": 8,124 "number": 7,125 "colorClass": "proposition"126 },127 {128 "id": "Prop8",129 "type": "proposition",130 "label": "Numbers between two in continued proportion correspond to ratios",131 "shortLabel": "Prop. VIII.8",132 "short": "Numbers between in proportion",133 "book": 8,134 "number": 8,135 "colorClass": "proposition"136 },137 {138 "id": "Prop9",139 "type": "proposition",140 "label": "Two relatively prime: numbers between them as between each and unit",141 "shortLabel": "Prop. VIII.9",142 "short": "Relatively prime: numbers to unit",143 "book": 8,144 "number": 9,145 "colorClass": "proposition"146 },147 {148 "id": "Prop10",149 "type": "proposition",150 "label": "Numbers between number and unit correspond to between two numbers",151 "shortLabel": "Prop. VIII.10",152 "short": "Numbers from unit",153 "book": 8,154 "number": 10,155 "colorClass": "proposition"156 },157 {158 "id": "Prop11",159 "type": "proposition",160 "label": "Between two squares one mean proportional; duplicate ratio",161 "shortLabel": "Prop. VIII.11",162 "short": "Mean proportional of squares",163 "book": 8,164 "number": 11,165 "colorClass": "proposition"166 },167 {168 "id": "Prop12",169 "type": "proposition",170 "label": "Between two cubes two mean proportionals; triplicate ratio",171 "shortLabel": "Prop. VIII.12",172 "short": "Two means between cubes",173 "book": 8,174 "number": 12,175 "colorClass": "proposition"176 },177 {178 "id": "Prop13",179 "type": "proposition",180 "label": "Continued proportion: products proportional; products of products",181 "shortLabel": "Prop. VIII.13",182 "short": "Products proportional",183 "book": 8,184 "number": 13,185 "colorClass": "proposition"186 },187 {188 "id": "Prop14",189 "type": "proposition",190 "label": "Square measures square iff side measures side",191 "shortLabel": "Prop. VIII.14",192 "short": "Square measures square",193 "book": 8,194 "number": 14,195 "colorClass": "proposition"196 },197 {198 "id": "Prop15",199 "type": "proposition",200 "label": "Cube measures cube iff side measures side",201 "shortLabel": "Prop. VIII.15",202 "short": "Cube measures cube",203 "book": 8,204 "number": 15,205 "colorClass": "proposition"206 },207 {208 "id": "Prop16",209 "type": "proposition",210 "label": "Square does not measure square iff side does not measure side",211 "shortLabel": "Prop. VIII.16",212 "short": "Square does not measure square",213 "book": 8,214 "number": 16,215 "colorClass": "proposition"216 },217 {218 "id": "Prop17",219 "type": "proposition",220 "label": "Cube does not measure cube iff side does not measure side",221 "shortLabel": "Prop. VIII.17",222 "short": "Cube does not measure cube",223 "book": 8,224 "number": 17,225 "colorClass": "proposition"226 },227 {228 "id": "Prop18",229 "type": "proposition",230 "label": "Between similar plane numbers one mean proportional; duplicate ratio",231 "shortLabel": "Prop. VIII.18",232 "short": "Similar plane: mean proportional",233 "book": 8,234 "number": 18,235 "colorClass": "proposition"236 },237 {238 "id": "Prop19",239 "type": "proposition",240 "label": "Between similar solid numbers two mean proportionals; triplicate ratio",241 "shortLabel": "Prop. VIII.19",242 "short": "Similar solid: two means",243 "book": 8,244 "number": 19,245 "colorClass": "proposition"246 },247 {248 "id": "Prop20",249 "type": "proposition",250 "label": "If one mean between two numbers, they are similar plane",251 "shortLabel": "Prop. VIII.20",252 "short": "One mean: similar plane",253 "book": 8,254 "number": 20,255 "colorClass": "proposition"256 },257 {258 "id": "Prop21",259 "type": "proposition",260 "label": "If two means between two numbers, they are similar solid",261 "shortLabel": "Prop. VIII.21",262 "short": "Two means: similar solid",263 "book": 8,264 "number": 21,265 "colorClass": "proposition"266 },267 {268 "id": "Prop22",269 "type": "proposition",270 "label": "Three in continued proportion, first square: third square",271 "shortLabel": "Prop. VIII.22",272 "short": "Three in proportion: first square",273 "book": 8,274 "number": 22,275 "colorClass": "proposition"276 },277 {278 "id": "Prop23",279 "type": "proposition",280 "label": "Four in continued proportion, first cube: fourth cube",281 "shortLabel": "Prop. VIII.23",282 "short": "Four in proportion: first cube",283 "book": 8,284 "number": 23,285 "colorClass": "proposition"286 },287 {288 "id": "Prop24",289 "type": "proposition",290 "label": "If ratio as square to square and first square, second square",291 "shortLabel": "Prop. VIII.24",292 "short": "Ratio as square to square",293 "book": 8,294 "number": 24,295 "colorClass": "proposition"296 },297 {298 "id": "Prop25",299 "type": "proposition",300 "label": "If ratio as cube to cube and first cube, second cube",301 "shortLabel": "Prop. VIII.25",302 "short": "Ratio as cube to cube",303 "book": 8,304 "number": 25,305 "colorClass": "proposition"306 },307 {308 "id": "Prop26",309 "type": "proposition",310 "label": "Similar plane numbers have ratio of square to square",311 "shortLabel": "Prop. VIII.26",312 "short": "Similar plane: square ratio",313 "book": 8,314 "number": 26,315 "colorClass": "proposition"316 },317 {318 "id": "Prop27",319 "type": "proposition",320 "label": "Similar solid numbers have ratio of cube to cube",321 "shortLabel": "Prop. VIII.27",322 "short": "Similar solid: cube ratio",323 "book": 8,324 "number": 27,325 "colorClass": "proposition"326 }327 ],328 "edges": [329 {330 "from": "BookVII",331 "to": "Prop1"332 },333 {334 "from": "BookVII",335 "to": "Prop2"336 },337 {338 "from": "Prop1",339 "to": "Prop2"340 },341 {342 "from": "BookVII",343 "to": "Prop3"344 },345 {346 "from": "Prop2",347 "to": "Prop3"348 },349 {350 "from": "BookVII",351 "to": "Prop4"352 },353 {354 "from": "Prop2",355 "to": "Prop4"356 },357 {358 "from": "BookVII",359 "to": "Prop5"360 },361 {362 "from": "BookVII",363 "to": "Prop6"364 },365 {366 "from": "Prop1",367 "to": "Prop6"368 },369 {370 "from": "BookVII",371 "to": "Prop7"372 },373 {374 "from": "Prop1",375 "to": "Prop7"376 },377 {378 "from": "BookVII",379 "to": "Prop8"380 },381 {382 "from": "BookVII",383 "to": "Prop9"384 },385 {386 "from": "Prop8",387 "to": "Prop9"388 },389 {390 "from": "BookVII",391 "to": "Prop10"392 },393 {394 "from": "Prop9",395 "to": "Prop10"396 },397 {398 "from": "BookVII",399 "to": "Prop11"400 },401 {402 "from": "Prop8",403 "to": "Prop11"404 },405 {406 "from": "BookVII",407 "to": "Prop12"408 },409 {410 "from": "Prop8",411 "to": "Prop12"412 },413 {414 "from": "BookVII",415 "to": "Prop13"416 },417 {418 "from": "BookVII",419 "to": "Prop14"420 },421 {422 "from": "Prop11",423 "to": "Prop14"424 },425 {426 "from": "BookVII",427 "to": "Prop15"428 },429 {430 "from": "Prop12",431 "to": "Prop15"432 },433 {434 "from": "BookVII",435 "to": "Prop16"436 },437 {438 "from": "Prop14",439 "to": "Prop16"440 },441 {442 "from": "BookVII",443 "to": "Prop17"444 },445 {446 "from": "Prop15",447 "to": "Prop17"448 },449 {450 "from": "BookVII",451 "to": "Prop18"452 },453 {454 "from": "Prop8",455 "to": "Prop18"456 },457 {458 "from": "BookVII",459 "to": "Prop19"460 },461 {462 "from": "Prop8",463 "to": "Prop19"464 },465 {466 "from": "BookVII",467 "to": "Prop20"468 },469 {470 "from": "Prop18",471 "to": "Prop20"472 },473 {474 "from": "BookVII",475 "to": "Prop21"476 },477 {478 "from": "Prop19",479 "to": "Prop21"480 },481 {482 "from": "BookVII",483 "to": "Prop22"484 },485 {486 "from": "Prop1",487 "to": "Prop22"488 },489 {490 "from": "BookVII",491 "to": "Prop23"492 },493 {494 "from": "Prop1",495 "to": "Prop23"496 },497 {498 "from": "BookVII",499 "to": "Prop24"500 },501 {502 "from": "Prop22",503 "to": "Prop24"504 },505 {506 "from": "BookVII",507 "to": "Prop25"508 },509 {510 "from": "Prop23",511 "to": "Prop25"512 },513 {514 "from": "BookVII",515 "to": "Prop26"516 },517 {518 "from": "Prop18",519 "to": "Prop26"520 },521 {522 "from": "BookVII",523 "to": "Prop27"524 },525 {526 "from": "Prop19",527 "to": "Prop27"528 }529 ],530 "colorScheme": {531 "foundation": {532 "fill": "#95a5a6",533 "stroke": "#7f8c8d"534 },535 "proposition": {536 "fill": "#1abc9c",537 "stroke": "#16a085"538 }539 }540}