garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-v",5 "name": "Euclid's Elements, Book V",6 "subject": "geometry",7 "variant": "classical",8 "description": "Theory of ratio and proportion (Eudoxus). 18 definitions, 25 propositions. Does not depend on previous books. Source: David E. Joyce.",9 "structure": {10 "books": 5,11 "definitions": 18,12 "propositions": 25,13 "foundationTypes": [14 "definition"15 ]16 }17 },18 "metadata": {19 "created": "2026-03-15",20 "lastUpdated": "2026-03-15",21 "version": "1.0.0",22 "license": "CC BY 4.0",23 "authors": [24 "Welz, G."25 ],26 "methodology": "Programming Framework",27 "citation": "Welz, G. (2026). Euclid's Elements Book V Dependency Graph. Programming Framework.",28 "keywords": [29 "Euclid",30 "Elements",31 "Book V",32 "proportion",33 "ratio",34 "Eudoxus"35 ]36 },37 "sources": [38 {39 "id": "joyce",40 "type": "digital",41 "authors": "Joyce, David E.",42 "title": "Euclid's Elements, Book V",43 "year": "1996",44 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookV/bookV.html",45 "notes": "Clark University; Logical structure"46 }47 ],48 "nodes": [49 {50 "id": "Def1",51 "type": "definition",52 "label": "A magnitude is a part of a magnitude when it measures it",53 "shortLabel": "Def. V.1",54 "short": "Part",55 "book": 5,56 "number": 1,57 "colorClass": "definition"58 },59 {60 "id": "Def2",61 "type": "definition",62 "label": "The greater is a multiple of the less when it is measured by the less",63 "shortLabel": "Def. V.2",64 "short": "Multiple",65 "book": 5,66 "number": 2,67 "colorClass": "definition"68 },69 {70 "id": "Def3",71 "type": "definition",72 "label": "A ratio is a sort of relation in respect of size between two magnitudes",73 "shortLabel": "Def. V.3",74 "short": "Ratio",75 "book": 5,76 "number": 3,77 "colorClass": "definition"78 },79 {80 "id": "Def4",81 "type": "definition",82 "label": "Magnitudes have a ratio when the less can be multiplied to exceed the greater",83 "shortLabel": "Def. V.4",84 "short": "Same ratio",85 "book": 5,86 "number": 4,87 "colorClass": "definition"88 },89 {90 "id": "Def5",91 "type": "definition",92 "label": "Magnitudes in same ratio when equimultiples alike exceed, equal, or fall short",93 "shortLabel": "Def. V.5",94 "short": "In same ratio (Eudoxus)",95 "book": 5,96 "number": 5,97 "colorClass": "definition"98 },99 {100 "id": "Def6",101 "type": "definition",102 "label": "Magnitudes which have the same ratio are proportional",103 "shortLabel": "Def. V.6",104 "short": "Proportional",105 "book": 5,106 "number": 6,107 "colorClass": "definition"108 },109 {110 "id": "Def7",111 "type": "definition",112 "label": "When of equimultiples first exceeds second, third does not exceed fourth",113 "shortLabel": "Def. V.7",114 "short": "Greater ratio",115 "book": 5,116 "number": 7,117 "colorClass": "definition"118 },119 {120 "id": "Def8",121 "type": "definition",122 "label": "Compound ratio is the ratio of the products of corresponding terms",123 "shortLabel": "Def. V.8",124 "short": "Compound ratio",125 "book": 5,126 "number": 8,127 "colorClass": "definition"128 },129 {130 "id": "Def9",131 "type": "definition",132 "label": "Duplicate ratio is the ratio of the squares",133 "shortLabel": "Def. V.9",134 "short": "Duplicate ratio",135 "book": 5,136 "number": 9,137 "colorClass": "definition"138 },139 {140 "id": "Def10",141 "type": "definition",142 "label": "Triplicate ratio is the ratio of the cubes",143 "shortLabel": "Def. V.10",144 "short": "Triplicate ratio",145 "book": 5,146 "number": 10,147 "colorClass": "definition"148 },149 {150 "id": "Def11",151 "type": "definition",152 "label": "Corresponding magnitudes in proportion",153 "shortLabel": "Def. V.11",154 "short": "Corresponding magnitudes",155 "book": 5,156 "number": 11,157 "colorClass": "definition"158 },159 {160 "id": "Def12",161 "type": "definition",162 "label": "Alternate: first to third as second to fourth",163 "shortLabel": "Def. V.12",164 "short": "Alternate ratio",165 "book": 5,166 "number": 12,167 "colorClass": "definition"168 },169 {170 "id": "Def13",171 "type": "definition",172 "label": "Inverse: second to first as fourth to third",173 "shortLabel": "Def. V.13",174 "short": "Inverse ratio",175 "book": 5,176 "number": 13,177 "colorClass": "definition"178 },179 {180 "id": "Def14",181 "type": "definition",182 "label": "Composition: first+second to second as third+fourth to fourth",183 "shortLabel": "Def. V.14",184 "short": "Composition of ratio",185 "book": 5,186 "number": 14,187 "colorClass": "definition"188 },189 {190 "id": "Def15",191 "type": "definition",192 "label": "Separation: first−second to second as third−fourth to fourth",193 "shortLabel": "Def. V.15",194 "short": "Separation of ratio",195 "book": 5,196 "number": 15,197 "colorClass": "definition"198 },199 {200 "id": "Def16",201 "type": "definition",202 "label": "Conversion: first to first−second as third to third−fourth",203 "shortLabel": "Def. V.16",204 "short": "Conversion of ratio",205 "book": 5,206 "number": 16,207 "colorClass": "definition"208 },209 {210 "id": "Def17",211 "type": "definition",212 "label": "Ex aequali: when first to second as second to third",213 "shortLabel": "Def. V.17",214 "short": "Ex aequali",215 "book": 5,216 "number": 17,217 "colorClass": "definition"218 },219 {220 "id": "Def18",221 "type": "definition",222 "label": "Ex aequali perturbed: when ratios are in perturbed order",223 "shortLabel": "Def. V.18",224 "short": "Ex aequali perturbed",225 "book": 5,226 "number": 18,227 "colorClass": "definition"228 },229 {230 "id": "Prop1",231 "type": "proposition",232 "label": "If magnitudes each same multiple of others, sum is that multiple of sum",233 "shortLabel": "Prop. V.1",234 "short": "Sum of multiples",235 "book": 5,236 "number": 1,237 "colorClass": "proposition"238 },239 {240 "id": "Prop2",241 "type": "proposition",242 "label": "If first:second as third:fourth, sum of first and fifth as sum of third and sixth",243 "shortLabel": "Prop. V.2",244 "short": "Equimultiples sum",245 "book": 5,246 "number": 2,247 "colorClass": "proposition"248 },249 {250 "id": "Prop3",251 "type": "proposition",252 "label": "Equimultiples of equimultiples are equimultiples",253 "shortLabel": "Prop. V.3",254 "short": "Equimultiples of equimultiples",255 "book": 5,256 "number": 3,257 "colorClass": "proposition"258 },259 {260 "id": "Prop4",261 "type": "proposition",262 "label": "If a:b = c:d, then ma:nb = mc:nd",263 "shortLabel": "Prop. V.4",264 "short": "Equimultiples preserve ratio",265 "book": 5,266 "number": 4,267 "colorClass": "proposition"268 },269 {270 "id": "Prop5",271 "type": "proposition",272 "label": "Multiple of whole minus multiple of part = multiple of remainder",273 "shortLabel": "Prop. V.5",274 "short": "Multiple of difference",275 "book": 5,276 "number": 5,277 "colorClass": "proposition"278 },279 {280 "id": "Prop6",281 "type": "proposition",282 "label": "Equimultiples minus equimultiples equal or equimultiples",283 "shortLabel": "Prop. V.6",284 "short": "Equimultiples minus equimultiples",285 "book": 5,286 "number": 6,287 "colorClass": "proposition"288 },289 {290 "id": "Prop7",291 "type": "proposition",292 "label": "Equal magnitudes have same ratio to same; same to equals",293 "shortLabel": "Prop. V.7",294 "short": "Equals in ratio",295 "book": 5,296 "number": 7,297 "colorClass": "proposition"298 },299 {300 "id": "Prop8",301 "type": "proposition",302 "label": "Of unequal magnitudes, greater has greater ratio to same",303 "shortLabel": "Prop. V.8",304 "short": "Greater has greater ratio",305 "book": 5,306 "number": 8,307 "colorClass": "proposition"308 },309 {310 "id": "Prop9",311 "type": "proposition",312 "label": "Magnitudes with same ratio to same are equal",313 "shortLabel": "Prop. V.9",314 "short": "Same ratio implies equal",315 "book": 5,316 "number": 9,317 "colorClass": "proposition"318 },319 {320 "id": "Prop10",321 "type": "proposition",322 "label": "Of magnitudes with ratio to same, greater ratio implies greater",323 "shortLabel": "Prop. V.10",324 "short": "Greater ratio implies greater",325 "book": 5,326 "number": 10,327 "colorClass": "proposition"328 },329 {330 "id": "Prop11",331 "type": "proposition",332 "label": "Ratios same with same ratio are same with one another",333 "shortLabel": "Prop. V.11",334 "short": "Transitivity of ratios",335 "book": 5,336 "number": 11,337 "colorClass": "proposition"338 },339 {340 "id": "Prop12",341 "type": "proposition",342 "label": "Proportional: one antecedent to one consequent as sum to sum",343 "shortLabel": "Prop. V.12",344 "short": "Sum of antecedents/consequents",345 "book": 5,346 "number": 12,347 "colorClass": "proposition"348 },349 {350 "id": "Prop13",351 "type": "proposition",352 "label": "If a:b = c:d and c:d > e:f, then a:b > e:f",353 "shortLabel": "Prop. V.13",354 "short": "Substitution in ratio inequality",355 "book": 5,356 "number": 13,357 "colorClass": "proposition"358 },359 {360 "id": "Prop14",361 "type": "proposition",362 "label": "If a:b = c:d and a>c, then b>d",363 "shortLabel": "Prop. V.14",364 "short": "Equal ratios, equal magnitudes",365 "book": 5,366 "number": 14,367 "colorClass": "proposition"368 },369 {370 "id": "Prop15",371 "type": "proposition",372 "label": "Parts have same ratio as their equimultiples",373 "shortLabel": "Prop. V.15",374 "short": "Parts as equimultiples",375 "book": 5,376 "number": 15,377 "colorClass": "proposition"378 },379 {380 "id": "Prop16",381 "type": "proposition",382 "label": "If a:b = c:d, then a:c = b:d",383 "shortLabel": "Prop. V.16",384 "short": "Alternate proportion",385 "book": 5,386 "number": 16,387 "colorClass": "proposition"388 },389 {390 "id": "Prop17",391 "type": "proposition",392 "label": "If (a+b):b = (c+d):d, then a:b = c:d",393 "shortLabel": "Prop. V.17",394 "short": "Jointly implies separately",395 "book": 5,396 "number": 17,397 "colorClass": "proposition"398 },399 {400 "id": "Prop18",401 "type": "proposition",402 "label": "If a:b = c:d, then (a+b):b = (c+d):d",403 "shortLabel": "Prop. V.18",404 "short": "Separately implies jointly",405 "book": 5,406 "number": 18,407 "colorClass": "proposition"408 },409 {410 "id": "Prop19",411 "type": "proposition",412 "label": "If (a+b):(c+d) = a:c, then also = b:d",413 "shortLabel": "Prop. V.19",414 "short": "Whole to whole as part to part",415 "book": 5,416 "number": 19,417 "colorClass": "proposition"418 },419 {420 "id": "Prop20",421 "type": "proposition",422 "label": "If a:b = d:e and b:c = e:f and a>c, then d>f",423 "shortLabel": "Prop. V.20",424 "short": "Ex aequali (direct)",425 "book": 5,426 "number": 20,427 "colorClass": "proposition"428 },429 {430 "id": "Prop21",431 "type": "proposition",432 "label": "If a:b = e:f and b:c = d:e and a>c, then d>f",433 "shortLabel": "Prop. V.21",434 "short": "Ex aequali (perturbed)",435 "book": 5,436 "number": 21,437 "colorClass": "proposition"438 },439 {440 "id": "Prop22",441 "type": "proposition",442 "label": "If a1:a2 = b1:b2, a2:a3 = b2:b3, ..., then a1:an = b1:bn",443 "shortLabel": "Prop. V.22",444 "short": "Ex aequali chain",445 "book": 5,446 "number": 22,447 "colorClass": "proposition"448 },449 {450 "id": "Prop23",451 "type": "proposition",452 "label": "If a:b = y:z and b:c = x:y, then a:c = x:z",453 "shortLabel": "Prop. V.23",454 "short": "Ex aequali perturbed chain",455 "book": 5,456 "number": 23,457 "colorClass": "proposition"458 },459 {460 "id": "Prop24",461 "type": "proposition",462 "label": "If a:b = c:d and e:b = f:d, then (a+e):b = (c+f):d",463 "shortLabel": "Prop. V.24",464 "short": "Sum of ratios",465 "book": 5,466 "number": 24,467 "colorClass": "proposition"468 },469 {470 "id": "Prop25",471 "type": "proposition",472 "label": "If a:b = c:d and a greatest, d least, then a+d > b+c",473 "shortLabel": "Prop. V.25",474 "short": "Sum of extremes > sum of means",475 "book": 5,476 "number": 25,477 "colorClass": "proposition"478 }479 ],480 "edges": [481 {482 "from": "Prop2",483 "to": "Prop3"484 },485 {486 "from": "Prop3",487 "to": "Prop4"488 },489 {490 "from": "Prop1",491 "to": "Prop5"492 },493 {494 "from": "Prop2",495 "to": "Prop6"496 },497 {498 "from": "Prop1",499 "to": "Prop8"500 },501 {502 "from": "Prop8",503 "to": "Prop9"504 },505 {506 "from": "Prop7",507 "to": "Prop10"508 },509 {510 "from": "Prop8",511 "to": "Prop10"512 },513 {514 "from": "Prop1",515 "to": "Prop12"516 },517 {518 "from": "Prop8",519 "to": "Prop14"520 },521 {522 "from": "Prop10",523 "to": "Prop14"524 },525 {526 "from": "Prop13",527 "to": "Prop14"528 },529 {530 "from": "Prop7",531 "to": "Prop15"532 },533 {534 "from": "Prop12",535 "to": "Prop15"536 },537 {538 "from": "Prop11",539 "to": "Prop16"540 },541 {542 "from": "Prop14",543 "to": "Prop16"544 },545 {546 "from": "Prop15",547 "to": "Prop16"548 },549 {550 "from": "Prop1",551 "to": "Prop17"552 },553 {554 "from": "Prop2",555 "to": "Prop17"556 },557 {558 "from": "Prop11",559 "to": "Prop18"560 },561 {562 "from": "Prop14",563 "to": "Prop18"564 },565 {566 "from": "Prop17",567 "to": "Prop18"568 },569 {570 "from": "Prop11",571 "to": "Prop19"572 },573 {574 "from": "Prop16",575 "to": "Prop19"576 },577 {578 "from": "Prop17",579 "to": "Prop19"580 },581 {582 "from": "Prop7",583 "to": "Prop20"584 },585 {586 "from": "Prop8",587 "to": "Prop20"588 },589 {590 "from": "Prop10",591 "to": "Prop20"592 },593 {594 "from": "Prop13",595 "to": "Prop20"596 },597 {598 "from": "Prop7",599 "to": "Prop21"600 },601 {602 "from": "Prop8",603 "to": "Prop21"604 },605 {606 "from": "Prop10",607 "to": "Prop21"608 },609 {610 "from": "Prop13",611 "to": "Prop21"612 },613 {614 "from": "Prop4",615 "to": "Prop22"616 },617 {618 "from": "Prop20",619 "to": "Prop22"620 },621 {622 "from": "Prop11",623 "to": "Prop23"624 },625 {626 "from": "Prop15",627 "to": "Prop23"628 },629 {630 "from": "Prop16",631 "to": "Prop23"632 },633 {634 "from": "Prop21",635 "to": "Prop23"636 },637 {638 "from": "Prop7",639 "to": "Prop24"640 },641 {642 "from": "Prop18",643 "to": "Prop24"644 },645 {646 "from": "Prop22",647 "to": "Prop24"648 },649 {650 "from": "Prop7",651 "to": "Prop25"652 },653 {654 "from": "Prop11",655 "to": "Prop25"656 },657 {658 "from": "Prop14",659 "to": "Prop25"660 },661 {662 "from": "Prop19",663 "to": "Prop25"664 }665 ],666 "colorScheme": {667 "definition": {668 "fill": "#3498db",669 "stroke": "#2980b9"670 },671 "proposition": {672 "fill": "#1abc9c",673 "stroke": "#16a085"674 }675 }676}