garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-ix",5 "name": "Euclid's Elements, Book IX",6 "subject": "number_theory",7 "variant": "classical",8 "description": "Primes, perfect numbers, odd/even. 36 propositions. Depends on Books VII and VIII. Source: David E. Joyce.",9 "structure": {10 "books": 9,11 "propositions": 36,12 "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 IX Dependency Graph. Programming Framework.",27 "keywords": [28 "Euclid",29 "Elements",30 "Book IX",31 "prime",32 "perfect",33 "odd",34 "even"35 ]36 },37 "sources": [38 {39 "id": "joyce",40 "type": "digital",41 "authors": "Joyce, David E.",42 "title": "Euclid's Elements, Book IX",43 "year": "1996",44 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/bookIX.html",45 "notes": "Clark University; IX.20 infinitude of primes"46 }47 ],48 "nodes": [49 {50 "id": "BookVII",51 "type": "foundation",52 "label": "Book VII — Number theory",53 "shortLabel": "Book VII",54 "short": "Foundation",55 "book": 7,56 "colorClass": "foundation"57 },58 {59 "id": "BookVIII",60 "type": "foundation",61 "label": "Book VIII — Continued proportions",62 "shortLabel": "Book VIII",63 "short": "Foundation",64 "book": 8,65 "colorClass": "foundation"66 },67 {68 "id": "Prop1",69 "type": "proposition",70 "label": "Two similar plane numbers multiplied: product is square",71 "shortLabel": "Prop. IX.1",72 "short": "Similar plane product square",73 "book": 9,74 "number": 1,75 "colorClass": "proposition"76 },77 {78 "id": "Prop2",79 "type": "proposition",80 "label": "Two numbers product square: they are similar plane",81 "shortLabel": "Prop. IX.2",82 "short": "Product square: similar plane",83 "book": 9,84 "number": 2,85 "colorClass": "proposition"86 },87 {88 "id": "Prop3",89 "type": "proposition",90 "label": "Cubic number multiplied by itself: product is cube",91 "shortLabel": "Prop. IX.3",92 "short": "Cube times itself",93 "book": 9,94 "number": 3,95 "colorClass": "proposition"96 },97 {98 "id": "Prop4",99 "type": "proposition",100 "label": "Cubic times cubic: product is cube",101 "shortLabel": "Prop. IX.4",102 "short": "Cube times cube",103 "book": 9,104 "number": 4,105 "colorClass": "proposition"106 },107 {108 "id": "Prop5",109 "type": "proposition",110 "label": "If cube times any makes cube, the multiplied is cubic",111 "shortLabel": "Prop. IX.5",112 "short": "Cube times any makes cube",113 "book": 9,114 "number": 5,115 "colorClass": "proposition"116 },117 {118 "id": "Prop6",119 "type": "proposition",120 "label": "If number times itself makes cubic, it is cubic",121 "shortLabel": "Prop. IX.6",122 "short": "Number times itself cubic",123 "book": 9,124 "number": 6,125 "colorClass": "proposition"126 },127 {128 "id": "Prop7",129 "type": "proposition",130 "label": "Composite times any: product is solid",131 "shortLabel": "Prop. IX.7",132 "short": "Composite times any: solid",133 "book": 9,134 "number": 7,135 "colorClass": "proposition"136 },137 {138 "id": "Prop8",139 "type": "proposition",140 "label": "Numbers from unit in continued proportion: 3rd square, 4th cube, 7th both",141 "shortLabel": "Prop. IX.8",142 "short": "Continued proportion from unit",143 "book": 9,144 "number": 8,145 "colorClass": "proposition"146 },147 {148 "id": "Prop9",149 "type": "proposition",150 "label": "If second from unit square, all square; if cubic, all cubic",151 "shortLabel": "Prop. IX.9",152 "short": "Second square: all square",153 "book": 9,154 "number": 9,155 "colorClass": "proposition"156 },157 {158 "id": "Prop10",159 "type": "proposition",160 "label": "If second not square, only 3rd and every other square; similar for cube",161 "shortLabel": "Prop. IX.10",162 "short": "Second not square",163 "book": 9,164 "number": 10,165 "colorClass": "proposition"166 },167 {168 "id": "Prop11",169 "type": "proposition",170 "label": "Continued proportion from unit: less measures greater",171 "shortLabel": "Prop. IX.11",172 "short": "Less measures greater",173 "book": 9,174 "number": 11,175 "colorClass": "proposition"176 },177 {178 "id": "Prop12",179 "type": "proposition",180 "label": "If prime measures last, it measures second from unit",181 "shortLabel": "Prop. IX.12",182 "short": "Prime measures last",183 "book": 9,184 "number": 12,185 "colorClass": "proposition"186 },187 {188 "id": "Prop13",189 "type": "proposition",190 "label": "Continued proportion, second prime: only proportional numbers measure last",191 "shortLabel": "Prop. IX.13",192 "short": "Second prime: only those measure",193 "book": 9,194 "number": 13,195 "colorClass": "proposition"196 },197 {198 "id": "Prop14",199 "type": "proposition",200 "label": "Least measured by given primes: not measured by any other prime",201 "shortLabel": "Prop. IX.14",202 "short": "Least measured by primes",203 "book": 9,204 "number": 14,205 "colorClass": "proposition"206 },207 {208 "id": "Prop15",209 "type": "proposition",210 "label": "Three least in ratio: sum of any two prime to remainder",211 "shortLabel": "Prop. IX.15",212 "short": "Three in proportion",213 "book": 9,214 "number": 15,215 "colorClass": "proposition"216 },217 {218 "id": "Prop16",219 "type": "proposition",220 "label": "Two relatively prime: no third as first to second",221 "shortLabel": "Prop. IX.16",222 "short": "Relatively prime: no third",223 "book": 9,224 "number": 16,225 "colorClass": "proposition"226 },227 {228 "id": "Prop17",229 "type": "proposition",230 "label": "Continued proportion, extremes prime: last not to any as first to second",231 "shortLabel": "Prop. IX.17",232 "short": "Extremes prime: no extension",233 "book": 9,234 "number": 17,235 "colorClass": "proposition"236 },237 {238 "id": "Prop18",239 "type": "proposition",240 "label": "Given two numbers, investigate if third proportional exists",241 "shortLabel": "Prop. IX.18",242 "short": "Third proportional",243 "book": 9,244 "number": 18,245 "colorClass": "proposition"246 },247 {248 "id": "Prop19",249 "type": "proposition",250 "label": "Given three numbers, investigate when fourth proportional exists",251 "shortLabel": "Prop. IX.19",252 "short": "Fourth proportional",253 "book": 9,254 "number": 19,255 "colorClass": "proposition"256 },257 {258 "id": "Prop20",259 "type": "proposition",260 "label": "Prime numbers are more than any assigned multitude",261 "shortLabel": "Prop. IX.20",262 "short": "Infinitude of primes",263 "book": 9,264 "number": 20,265 "colorClass": "proposition"266 },267 {268 "id": "Prop21",269 "type": "proposition",270 "label": "Sum of even numbers is even",271 "shortLabel": "Prop. IX.21",272 "short": "Sum of evens even",273 "book": 9,274 "number": 21,275 "colorClass": "proposition"276 },277 {278 "id": "Prop22",279 "type": "proposition",280 "label": "Sum of odd numbers, even multitude: sum even",281 "shortLabel": "Prop. IX.22",282 "short": "Sum of odds (even count) even",283 "book": 9,284 "number": 22,285 "colorClass": "proposition"286 },287 {288 "id": "Prop23",289 "type": "proposition",290 "label": "Sum of odd numbers, odd multitude: sum odd",291 "shortLabel": "Prop. IX.23",292 "short": "Sum of odds (odd count) odd",293 "book": 9,294 "number": 23,295 "colorClass": "proposition"296 },297 {298 "id": "Prop24",299 "type": "proposition",300 "label": "Even minus even: remainder even",301 "shortLabel": "Prop. IX.24",302 "short": "Even minus even",303 "book": 9,304 "number": 24,305 "colorClass": "proposition"306 },307 {308 "id": "Prop25",309 "type": "proposition",310 "label": "Even minus odd: remainder odd",311 "shortLabel": "Prop. IX.25",312 "short": "Even minus odd",313 "book": 9,314 "number": 25,315 "colorClass": "proposition"316 },317 {318 "id": "Prop26",319 "type": "proposition",320 "label": "Odd minus odd: remainder even",321 "shortLabel": "Prop. IX.26",322 "short": "Odd minus odd",323 "book": 9,324 "number": 26,325 "colorClass": "proposition"326 },327 {328 "id": "Prop27",329 "type": "proposition",330 "label": "Odd minus even: remainder odd",331 "shortLabel": "Prop. IX.27",332 "short": "Odd minus even",333 "book": 9,334 "number": 27,335 "colorClass": "proposition"336 },337 {338 "id": "Prop28",339 "type": "proposition",340 "label": "Odd times even: product even",341 "shortLabel": "Prop. IX.28",342 "short": "Odd times even",343 "book": 9,344 "number": 28,345 "colorClass": "proposition"346 },347 {348 "id": "Prop29",349 "type": "proposition",350 "label": "Odd times odd: product odd",351 "shortLabel": "Prop. IX.29",352 "short": "Odd times odd",353 "book": 9,354 "number": 29,355 "colorClass": "proposition"356 },357 {358 "id": "Prop30",359 "type": "proposition",360 "label": "Odd measuring even: measures half",361 "shortLabel": "Prop. IX.30",362 "short": "Odd measures even",363 "book": 9,364 "number": 30,365 "colorClass": "proposition"366 },367 {368 "id": "Prop31",369 "type": "proposition",370 "label": "Odd relatively prime to any: also prime to its double",371 "shortLabel": "Prop. IX.31",372 "short": "Odd prime to double",373 "book": 9,374 "number": 31,375 "colorClass": "proposition"376 },377 {378 "id": "Prop32",379 "type": "proposition",380 "label": "Numbers doubled from 2: even-times even only",381 "shortLabel": "Prop. IX.32",382 "short": "Powers of 2",383 "book": 9,384 "number": 32,385 "colorClass": "proposition"386 },387 {388 "id": "Prop33",389 "type": "proposition",390 "label": "Number with half odd: even-times odd only",391 "shortLabel": "Prop. IX.33",392 "short": "Half odd",393 "book": 9,394 "number": 33,395 "colorClass": "proposition"396 },397 {398 "id": "Prop34",399 "type": "proposition",400 "label": "Number neither: both even-times even and even-times odd",401 "shortLabel": "Prop. IX.34",402 "short": "Neither",403 "book": 9,404 "number": 34,405 "colorClass": "proposition"406 },407 {408 "id": "Prop35",409 "type": "proposition",410 "label": "Continued proportion: (second−first):first = (last−first):sum of rest",411 "shortLabel": "Prop. IX.35",412 "short": "Geometric series",413 "book": 9,414 "number": 35,415 "colorClass": "proposition"416 },417 {418 "id": "Prop36",419 "type": "proposition",420 "label": "If sum of powers of 2 is prime, product with last is perfect",421 "shortLabel": "Prop. IX.36",422 "short": "Perfect numbers",423 "book": 9,424 "number": 36,425 "colorClass": "proposition"426 }427 ],428 "edges": [429 {430 "from": "BookVII",431 "to": "Prop1"432 },433 {434 "from": "BookVIII",435 "to": "Prop1"436 },437 {438 "from": "BookVII",439 "to": "Prop2"440 },441 {442 "from": "BookVIII",443 "to": "Prop2"444 },445 {446 "from": "BookVII",447 "to": "Prop3"448 },449 {450 "from": "BookVIII",451 "to": "Prop3"452 },453 {454 "from": "BookVII",455 "to": "Prop4"456 },457 {458 "from": "BookVIII",459 "to": "Prop4"460 },461 {462 "from": "BookVII",463 "to": "Prop5"464 },465 {466 "from": "BookVIII",467 "to": "Prop5"468 },469 {470 "from": "BookVII",471 "to": "Prop6"472 },473 {474 "from": "BookVIII",475 "to": "Prop6"476 },477 {478 "from": "BookVII",479 "to": "Prop7"480 },481 {482 "from": "BookVIII",483 "to": "Prop7"484 },485 {486 "from": "BookVII",487 "to": "Prop8"488 },489 {490 "from": "BookVIII",491 "to": "Prop8"492 },493 {494 "from": "BookVII",495 "to": "Prop9"496 },497 {498 "from": "BookVIII",499 "to": "Prop9"500 },501 {502 "from": "BookVII",503 "to": "Prop10"504 },505 {506 "from": "BookVIII",507 "to": "Prop10"508 },509 {510 "from": "BookVII",511 "to": "Prop11"512 },513 {514 "from": "BookVIII",515 "to": "Prop11"516 },517 {518 "from": "BookVII",519 "to": "Prop12"520 },521 {522 "from": "BookVIII",523 "to": "Prop12"524 },525 {526 "from": "BookVII",527 "to": "Prop13"528 },529 {530 "from": "BookVIII",531 "to": "Prop13"532 },533 {534 "from": "BookVII",535 "to": "Prop14"536 },537 {538 "from": "BookVIII",539 "to": "Prop14"540 },541 {542 "from": "BookVII",543 "to": "Prop15"544 },545 {546 "from": "BookVIII",547 "to": "Prop15"548 },549 {550 "from": "BookVII",551 "to": "Prop16"552 },553 {554 "from": "BookVIII",555 "to": "Prop16"556 },557 {558 "from": "BookVII",559 "to": "Prop17"560 },561 {562 "from": "BookVIII",563 "to": "Prop17"564 },565 {566 "from": "BookVII",567 "to": "Prop18"568 },569 {570 "from": "BookVIII",571 "to": "Prop18"572 },573 {574 "from": "BookVII",575 "to": "Prop19"576 },577 {578 "from": "BookVIII",579 "to": "Prop19"580 },581 {582 "from": "BookVII",583 "to": "Prop20"584 },585 {586 "from": "BookVIII",587 "to": "Prop20"588 },589 {590 "from": "BookVII",591 "to": "Prop21"592 },593 {594 "from": "BookVIII",595 "to": "Prop21"596 },597 {598 "from": "BookVII",599 "to": "Prop22"600 },601 {602 "from": "BookVIII",603 "to": "Prop22"604 },605 {606 "from": "BookVII",607 "to": "Prop23"608 },609 {610 "from": "BookVIII",611 "to": "Prop23"612 },613 {614 "from": "BookVII",615 "to": "Prop24"616 },617 {618 "from": "BookVIII",619 "to": "Prop24"620 },621 {622 "from": "BookVII",623 "to": "Prop25"624 },625 {626 "from": "BookVIII",627 "to": "Prop25"628 },629 {630 "from": "BookVII",631 "to": "Prop26"632 },633 {634 "from": "BookVIII",635 "to": "Prop26"636 },637 {638 "from": "BookVII",639 "to": "Prop27"640 },641 {642 "from": "BookVIII",643 "to": "Prop27"644 },645 {646 "from": "BookVII",647 "to": "Prop28"648 },649 {650 "from": "BookVIII",651 "to": "Prop28"652 },653 {654 "from": "BookVII",655 "to": "Prop29"656 },657 {658 "from": "BookVIII",659 "to": "Prop29"660 },661 {662 "from": "BookVII",663 "to": "Prop30"664 },665 {666 "from": "BookVIII",667 "to": "Prop30"668 },669 {670 "from": "BookVII",671 "to": "Prop31"672 },673 {674 "from": "BookVIII",675 "to": "Prop31"676 },677 {678 "from": "BookVII",679 "to": "Prop32"680 },681 {682 "from": "BookVIII",683 "to": "Prop32"684 },685 {686 "from": "BookVII",687 "to": "Prop33"688 },689 {690 "from": "BookVIII",691 "to": "Prop33"692 },693 {694 "from": "BookVII",695 "to": "Prop34"696 },697 {698 "from": "BookVIII",699 "to": "Prop34"700 },701 {702 "from": "BookVII",703 "to": "Prop35"704 },705 {706 "from": "BookVIII",707 "to": "Prop35"708 },709 {710 "from": "BookVII",711 "to": "Prop36"712 },713 {714 "from": "BookVIII",715 "to": "Prop36"716 }717 ],718 "colorScheme": {719 "foundation": {720 "fill": "#95a5a6",721 "stroke": "#7f8c8d"722 },723 "proposition": {724 "fill": "#1abc9c",725 "stroke": "#16a085"726 }727 }728}