garywelz/programming_framework
0
1{2 "schemaVersion": "1.0",3 "discourse": {4 "id": "euclid-elements-book-x",5 "name": "Euclid's Elements, Book X",6 "subject": "incommensurables",7 "variant": "classical",8 "description": "Incommensurables, binomials, apotomes. 16 definitions, 115 propositions. Depends on Books I, V, VI. Source: David E. Joyce.",9 "structure": {10 "books": 10,11 "definitions": 16,12 "propositions": 115,13 "foundationTypes": [14 "foundation"15 ]16 }17 },18 "metadata": {19 "created": "2026-03-18",20 "lastUpdated": "2026-03-18",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 X Dependency Graph. Programming Framework.",28 "keywords": [29 "Euclid",30 "Elements",31 "Book X",32 "incommensurable",33 "binomial",34 "apotome",35 "medial"36 ]37 },38 "sources": [39 {40 "id": "joyce",41 "type": "digital",42 "authors": "Joyce, David E.",43 "title": "Euclid's Elements, Book X",44 "year": "1996",45 "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookX/bookX.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": "Prop1",79 "type": "proposition",80 "label": "Two unequal magnitudes: subtract more than half repeatedly; remainder less than lesser",81 "shortLabel": "Prop. X.1",82 "short": "Antenaresis for magnitudes",83 "book": 10,84 "number": 1,85 "colorClass": "proposition"86 },87 {88 "id": "Prop2",89 "type": "proposition",90 "label": "If less subtracted from greater repeatedly and remainder never measures prior: incommensurable",91 "shortLabel": "Prop. X.2",92 "short": "Incommensurable criterion",93 "book": 10,94 "number": 2,95 "colorClass": "proposition"96 },97 {98 "id": "Prop3",99 "type": "proposition",100 "label": "To find greatest common measure of two commensurable magnitudes",101 "shortLabel": "Prop. X.3",102 "short": "GCM of two commensurable",103 "book": 10,104 "number": 3,105 "colorClass": "proposition"106 },107 {108 "id": "Prop4",109 "type": "proposition",110 "label": "To find greatest common measure of three commensurable magnitudes",111 "shortLabel": "Prop. X.4",112 "short": "GCM of three commensurable",113 "book": 10,114 "number": 4,115 "colorClass": "proposition"116 },117 {118 "id": "Prop5",119 "type": "proposition",120 "label": "Commensurable magnitudes have ratio which a number has to a number",121 "shortLabel": "Prop. X.5",122 "short": "Commensurable: number ratio",123 "book": 10,124 "number": 5,125 "colorClass": "proposition"126 },127 {128 "id": "Prop6",129 "type": "proposition",130 "label": "If two magnitudes have number-to-number ratio, they are commensurable",131 "shortLabel": "Prop. X.6",132 "short": "Number ratio: commensurable",133 "book": 10,134 "number": 6,135 "colorClass": "proposition"136 },137 {138 "id": "Prop7",139 "type": "proposition",140 "label": "Incommensurable magnitudes do not have number-to-number ratio",141 "shortLabel": "Prop. X.7",142 "short": "Incommensurable: no number ratio",143 "book": 10,144 "number": 7,145 "colorClass": "proposition"146 },147 {148 "id": "Prop8",149 "type": "proposition",150 "label": "If magnitudes lack number-to-number ratio, they are incommensurable",151 "shortLabel": "Prop. X.8",152 "short": "No number ratio: incommensurable",153 "book": 10,154 "number": 8,155 "colorClass": "proposition"156 },157 {158 "id": "Prop9",159 "type": "proposition",160 "label": "Squares on commensurable lines have square-to-square ratio; converse",161 "shortLabel": "Prop. X.9",162 "short": "Squares: commensurable iff square ratio",163 "book": 10,164 "number": 9,165 "colorClass": "proposition"166 },167 {168 "id": "Prop10",169 "type": "proposition",170 "label": "To find two lines incommensurable with assigned: one in length, one in square",171 "shortLabel": "Prop. X.10",172 "short": "Find incommensurable lines",173 "book": 10,174 "number": 10,175 "colorClass": "proposition"176 },177 {178 "id": "Prop11",179 "type": "proposition",180 "label": "Four proportional: first commensurable with second iff third with fourth",181 "shortLabel": "Prop. X.11",182 "short": "Proportion preserves commensurability",183 "book": 10,184 "number": 11,185 "colorClass": "proposition"186 },187 {188 "id": "Prop12",189 "type": "proposition",190 "label": "Magnitudes commensurable with same magnitude are commensurable",191 "shortLabel": "Prop. X.12",192 "short": "Commensurable with same",193 "book": 10,194 "number": 12,195 "colorClass": "proposition"196 },197 {198 "id": "Prop13",199 "type": "proposition",200 "label": "If two commensurable, one incommensurable with third, so is the other",201 "shortLabel": "Prop. X.13",202 "short": "Commensurable: incommensurable partner",203 "book": 10,204 "number": 13,205 "colorClass": "proposition"206 },207 {208 "id": "Prop14",209 "type": "proposition",210 "label": "Given two lines: find square by which greater square exceeds less; find line whose square equals sum",211 "shortLabel": "Prop. X.14",212 "short": "Square difference, sum of squares",213 "book": 10,214 "number": 14,215 "colorClass": "proposition"216 },217 {218 "id": "Prop15",219 "type": "proposition",220 "label": "Two commensurable added: whole commensurable with each",221 "shortLabel": "Prop. X.15",222 "short": "Commensurable sum",223 "book": 10,224 "number": 15,225 "colorClass": "proposition"226 },227 {228 "id": "Prop16",229 "type": "proposition",230 "label": "Two incommensurable added: sum incommensurable with each",231 "shortLabel": "Prop. X.16",232 "short": "Incommensurable sum",233 "book": 10,234 "number": 16,235 "colorClass": "proposition"236 },237 {238 "id": "Prop17",239 "type": "proposition",240 "label": "Applied parallelogram falling short by square: parts commensurable iff square difference commensurable",241 "shortLabel": "Prop. X.17",242 "short": "Parallelogram falling short",243 "book": 10,244 "number": 17,245 "colorClass": "proposition"246 },247 {248 "id": "Prop18",249 "type": "proposition",250 "label": "Applied parallelogram: parts incommensurable iff square difference incommensurable",251 "shortLabel": "Prop. X.18",252 "short": "Parallelogram: incommensurable parts",253 "book": 10,254 "number": 18,255 "colorClass": "proposition"256 },257 {258 "id": "Prop19",259 "type": "proposition",260 "label": "Rectangle by rational lines commensurable in length is rational",261 "shortLabel": "Prop. X.19",262 "short": "Rational rectangle from rational",263 "book": 10,264 "number": 19,265 "colorClass": "proposition"266 },267 {268 "id": "Prop20",269 "type": "proposition",270 "label": "Rational area applied to rational line: breadth rational and commensurable",271 "shortLabel": "Prop. X.20",272 "short": "Rational area on rational",273 "book": 10,274 "number": 20,275 "colorClass": "proposition"276 },277 {278 "id": "Prop21",279 "type": "proposition",280 "label": "Rectangle by rational in square only is irrational; side called medial",281 "shortLabel": "Prop. X.21",282 "short": "Medial: rational in square only",283 "book": 10,284 "number": 21,285 "colorClass": "proposition"286 },287 {288 "id": "Prop22",289 "type": "proposition",290 "label": "Square on medial applied to rational: breadth rational, incommensurable in length",291 "shortLabel": "Prop. X.22",292 "short": "Medial applied to rational",293 "book": 10,294 "number": 22,295 "colorClass": "proposition"296 },297 {298 "id": "Prop23",299 "type": "proposition",300 "label": "Line commensurable with medial is medial",301 "shortLabel": "Prop. X.23",302 "short": "Commensurable with medial",303 "book": 10,304 "number": 23,305 "colorClass": "proposition"306 },307 {308 "id": "Prop24",309 "type": "proposition",310 "label": "Rectangle by medial lines commensurable in length is medial",311 "shortLabel": "Prop. X.24",312 "short": "Medial rectangle: commensurable length",313 "book": 10,314 "number": 24,315 "colorClass": "proposition"316 },317 {318 "id": "Prop25",319 "type": "proposition",320 "label": "Rectangle by medial in square only: rational or medial",321 "shortLabel": "Prop. X.25",322 "short": "Medial: square only",323 "book": 10,324 "number": 25,325 "colorClass": "proposition"326 },327 {328 "id": "Prop26",329 "type": "proposition",330 "label": "Medial area does not exceed medial by rational area",331 "shortLabel": "Prop. X.26",332 "short": "Medial minus medial",333 "book": 10,334 "number": 26,335 "colorClass": "proposition"336 },337 {338 "id": "Prop27",339 "type": "proposition",340 "label": "To find medial lines commensurable in square only containing rational rectangle",341 "shortLabel": "Prop. X.27",342 "short": "Find medial in square only, rational rect",343 "book": 10,344 "number": 27,345 "colorClass": "proposition"346 },347 {348 "id": "Prop28",349 "type": "proposition",350 "label": "To find medial lines commensurable in square only containing medial rectangle",351 "shortLabel": "Prop. X.28",352 "short": "Find medial in square only, medial rect",353 "book": 10,354 "number": 28,355 "colorClass": "proposition"356 },357 {358 "id": "Prop29",359 "type": "proposition",360 "label": "To find two rational in square only: square diff commensurable with greater",361 "shortLabel": "Prop. X.29",362 "short": "Find rational in square only, commensurable diff",363 "book": 10,364 "number": 29,365 "colorClass": "proposition"366 },367 {368 "id": "Prop30",369 "type": "proposition",370 "label": "To find two rational in square only: square diff incommensurable with greater",371 "shortLabel": "Prop. X.30",372 "short": "Find rational in square only, incommensurable diff",373 "book": 10,374 "number": 30,375 "colorClass": "proposition"376 },377 {378 "id": "Prop31",379 "type": "proposition",380 "label": "To find two medial in square only, rational rect: diff commensurable with greater",381 "shortLabel": "Prop. X.31",382 "short": "Find medial, rational rect, commensurable diff",383 "book": 10,384 "number": 31,385 "colorClass": "proposition"386 },387 {388 "id": "Prop32",389 "type": "proposition",390 "label": "To find two medial in square only, medial rect: diff commensurable with greater",391 "shortLabel": "Prop. X.32",392 "short": "Find medial, medial rect, commensurable diff",393 "book": 10,394 "number": 32,395 "colorClass": "proposition"396 },397 {398 "id": "Prop33",399 "type": "proposition",400 "label": "To find two incommensurable in square: sum of squares rational, rectangle medial",401 "shortLabel": "Prop. X.33",402 "short": "Find incommensurable: rational sum, medial rect",403 "book": 10,404 "number": 33,405 "colorClass": "proposition"406 },407 {408 "id": "Prop34",409 "type": "proposition",410 "label": "To find two incommensurable in square: sum medial, rectangle rational",411 "shortLabel": "Prop. X.34",412 "short": "Find incommensurable: medial sum, rational rect",413 "book": 10,414 "number": 34,415 "colorClass": "proposition"416 },417 {418 "id": "Prop35",419 "type": "proposition",420 "label": "To find two incommensurable in square: sum and rect medial, incommensurable",421 "shortLabel": "Prop. X.35",422 "short": "Find incommensurable: both medial",423 "book": 10,424 "number": 35,425 "colorClass": "proposition"426 },427 {428 "id": "Prop36",429 "type": "proposition",430 "label": "Two rational in square only added: whole irrational, called binomial",431 "shortLabel": "Prop. X.36",432 "short": "Binomial defined",433 "book": 10,434 "number": 36,435 "colorClass": "proposition"436 },437 {438 "id": "Prop37",439 "type": "proposition",440 "label": "Two medial in square only, rational rect added: first bimedial",441 "shortLabel": "Prop. X.37",442 "short": "First bimedial defined",443 "book": 10,444 "number": 37,445 "colorClass": "proposition"446 },447 {448 "id": "Prop38",449 "type": "proposition",450 "label": "Two medial in square only, medial rect added: second bimedial",451 "shortLabel": "Prop. X.38",452 "short": "Second bimedial defined",453 "book": 10,454 "number": 38,455 "colorClass": "proposition"456 },457 {458 "id": "Prop39",459 "type": "proposition",460 "label": "Two incommensurable in square, rational sum, medial rect added: major",461 "shortLabel": "Prop. X.39",462 "short": "Major defined",463 "book": 10,464 "number": 39,465 "colorClass": "proposition"466 },467 {468 "id": "Prop40",469 "type": "proposition",470 "label": "Two incommensurable in square, medial sum, rational rect added: side of rational plus medial",471 "shortLabel": "Prop. X.40",472 "short": "Side of rational plus medial",473 "book": 10,474 "number": 40,475 "colorClass": "proposition"476 },477 {478 "id": "Prop41",479 "type": "proposition",480 "label": "Two incommensurable in square, both medial added: side of sum of two medial",481 "shortLabel": "Prop. X.41",482 "short": "Side of sum of two medial",483 "book": 10,484 "number": 41,485 "colorClass": "proposition"486 },487 {488 "id": "Prop42",489 "type": "proposition",490 "label": "Binomial divided into terms at one point only",491 "shortLabel": "Prop. X.42",492 "short": "Binomial: unique division",493 "book": 10,494 "number": 42,495 "colorClass": "proposition"496 },497 {498 "id": "Prop43",499 "type": "proposition",500 "label": "First bimedial divided at one point only",501 "shortLabel": "Prop. X.43",502 "short": "First bimedial: unique division",503 "book": 10,504 "number": 43,505 "colorClass": "proposition"506 },507 {508 "id": "Prop44",509 "type": "proposition",510 "label": "Second bimedial divided at one point only",511 "shortLabel": "Prop. X.44",512 "short": "Second bimedial: unique division",513 "book": 10,514 "number": 44,515 "colorClass": "proposition"516 },517 {518 "id": "Prop45",519 "type": "proposition",520 "label": "Major divided at one point only",521 "shortLabel": "Prop. X.45",522 "short": "Major: unique division",523 "book": 10,524 "number": 45,525 "colorClass": "proposition"526 },527 {528 "id": "Prop46",529 "type": "proposition",530 "label": "Side of rational plus medial divided at one point only",531 "shortLabel": "Prop. X.46",532 "short": "Side rational+medial: unique division",533 "book": 10,534 "number": 46,535 "colorClass": "proposition"536 },537 {538 "id": "Prop47",539 "type": "proposition",540 "label": "Side of sum of two medial divided at one point only",541 "shortLabel": "Prop. X.47",542 "short": "Side two medial: unique division",543 "book": 10,544 "number": 47,545 "colorClass": "proposition"546 },547 {548 "id": "Prop48",549 "type": "proposition",550 "label": "To find the first binomial line",551 "shortLabel": "Prop. X.48",552 "short": "Find first binomial",553 "book": 10,554 "number": 48,555 "colorClass": "proposition"556 },557 {558 "id": "Prop49",559 "type": "proposition",560 "label": "To find the second binomial line",561 "shortLabel": "Prop. X.49",562 "short": "Find second binomial",563 "book": 10,564 "number": 49,565 "colorClass": "proposition"566 },567 {568 "id": "Prop50",569 "type": "proposition",570 "label": "To find the third binomial line",571 "shortLabel": "Prop. X.50",572 "short": "Find third binomial",573 "book": 10,574 "number": 50,575 "colorClass": "proposition"576 },577 {578 "id": "Prop51",579 "type": "proposition",580 "label": "To find the fourth binomial line",581 "shortLabel": "Prop. X.51",582 "short": "Find fourth binomial",583 "book": 10,584 "number": 51,585 "colorClass": "proposition"586 },587 {588 "id": "Prop52",589 "type": "proposition",590 "label": "To find the fifth binomial line",591 "shortLabel": "Prop. X.52",592 "short": "Find fifth binomial",593 "book": 10,594 "number": 52,595 "colorClass": "proposition"596 },597 {598 "id": "Prop53",599 "type": "proposition",600 "label": "To find the sixth binomial line",601 "shortLabel": "Prop. X.53",602 "short": "Find sixth binomial",603 "book": 10,604 "number": 53,605 "colorClass": "proposition"606 },607 {608 "id": "Prop54",609 "type": "proposition",610 "label": "Area by rational and first binomial: side is binomial",611 "shortLabel": "Prop. X.54",612 "short": "Rational × first binomial",613 "book": 10,614 "number": 54,615 "colorClass": "proposition"616 },617 {618 "id": "Prop55",619 "type": "proposition",620 "label": "Area by rational and second binomial: side is first bimedial",621 "shortLabel": "Prop. X.55",622 "short": "Rational × second binomial",623 "book": 10,624 "number": 55,625 "colorClass": "proposition"626 },627 {628 "id": "Prop56",629 "type": "proposition",630 "label": "Area by rational and third binomial: side is second bimedial",631 "shortLabel": "Prop. X.56",632 "short": "Rational × third binomial",633 "book": 10,634 "number": 56,635 "colorClass": "proposition"636 },637 {638 "id": "Prop57",639 "type": "proposition",640 "label": "Area by rational and fourth binomial: side is major",641 "shortLabel": "Prop. X.57",642 "short": "Rational × fourth binomial",643 "book": 10,644 "number": 57,645 "colorClass": "proposition"646 },647 {648 "id": "Prop58",649 "type": "proposition",650 "label": "Area by rational and fifth binomial: side is rational plus medial",651 "shortLabel": "Prop. X.58",652 "short": "Rational × fifth binomial",653 "book": 10,654 "number": 58,655 "colorClass": "proposition"656 },657 {658 "id": "Prop59",659 "type": "proposition",660 "label": "Area by rational and sixth binomial: side is sum of two medial",661 "shortLabel": "Prop. X.59",662 "short": "Rational × sixth binomial",663 "book": 10,664 "number": 59,665 "colorClass": "proposition"666 },667 {668 "id": "Prop60",669 "type": "proposition",670 "label": "Square on binomial applied to rational: breadth first binomial",671 "shortLabel": "Prop. X.60",672 "short": "Square on binomial",673 "book": 10,674 "number": 60,675 "colorClass": "proposition"676 },677 {678 "id": "Prop61",679 "type": "proposition",680 "label": "Square on first bimedial applied to rational: breadth second binomial",681 "shortLabel": "Prop. X.61",682 "short": "Square on first bimedial",683 "book": 10,684 "number": 61,685 "colorClass": "proposition"686 },687 {688 "id": "Prop62",689 "type": "proposition",690 "label": "Square on second bimedial applied to rational: breadth third binomial",691 "shortLabel": "Prop. X.62",692 "short": "Square on second bimedial",693 "book": 10,694 "number": 62,695 "colorClass": "proposition"696 },697 {698 "id": "Prop63",699 "type": "proposition",700 "label": "Square on major applied to rational: breadth fourth binomial",701 "shortLabel": "Prop. X.63",702 "short": "Square on major",703 "book": 10,704 "number": 63,705 "colorClass": "proposition"706 },707 {708 "id": "Prop64",709 "type": "proposition",710 "label": "Square on rational+medial applied to rational: breadth fifth binomial",711 "shortLabel": "Prop. X.64",712 "short": "Square on rational+medial",713 "book": 10,714 "number": 64,715 "colorClass": "proposition"716 },717 {718 "id": "Prop65",719 "type": "proposition",720 "label": "Square on sum of two medial applied to rational: breadth sixth binomial",721 "shortLabel": "Prop. X.65",722 "short": "Square on two medial",723 "book": 10,724 "number": 65,725 "colorClass": "proposition"726 },727 {728 "id": "Prop66",729 "type": "proposition",730 "label": "Line commensurable with binomial is binomial, same order",731 "shortLabel": "Prop. X.66",732 "short": "Commensurable with binomial",733 "book": 10,734 "number": 66,735 "colorClass": "proposition"736 },737 {738 "id": "Prop67",739 "type": "proposition",740 "label": "Line commensurable with bimedial is bimedial, same order",741 "shortLabel": "Prop. X.67",742 "short": "Commensurable with bimedial",743 "book": 10,744 "number": 67,745 "colorClass": "proposition"746 },747 {748 "id": "Prop68",749 "type": "proposition",750 "label": "Line commensurable with major is major",751 "shortLabel": "Prop. X.68",752 "short": "Commensurable with major",753 "book": 10,754 "number": 68,755 "colorClass": "proposition"756 },757 {758 "id": "Prop69",759 "type": "proposition",760 "label": "Line commensurable with rational+medial is rational+medial",761 "shortLabel": "Prop. X.69",762 "short": "Commensurable with rational+medial",763 "book": 10,764 "number": 69,765 "colorClass": "proposition"766 },767 {768 "id": "Prop70",769 "type": "proposition",770 "label": "Line commensurable with sum of two medial is sum of two medial",771 "shortLabel": "Prop. X.70",772 "short": "Commensurable with two medial",773 "book": 10,774 "number": 70,775 "colorClass": "proposition"776 },777 {778 "id": "Prop71",779 "type": "proposition",780 "label": "Rational and medial added: four irrationals arise",781 "shortLabel": "Prop. X.71",782 "short": "Rational + medial: four irrationals",783 "book": 10,784 "number": 71,785 "colorClass": "proposition"786 },787 {788 "id": "Prop72",789 "type": "proposition",790 "label": "Two medial incommensurable added: two irrationals arise",791 "shortLabel": "Prop. X.72",792 "short": "Two medial: two irrationals",793 "book": 10,794 "number": 72,795 "colorClass": "proposition"796 },797 {798 "id": "Prop73",799 "type": "proposition",800 "label": "Rational minus rational in square only: remainder irrational, apotome",801 "shortLabel": "Prop. X.73",802 "short": "Apotome defined",803 "book": 10,804 "number": 73,805 "colorClass": "proposition"806 },807 {808 "id": "Prop74",809 "type": "proposition",810 "label": "Medial minus medial, rational rect: first apotome of medial",811 "shortLabel": "Prop. X.74",812 "short": "First apotome of medial",813 "book": 10,814 "number": 74,815 "colorClass": "proposition"816 },817 {818 "id": "Prop75",819 "type": "proposition",820 "label": "Medial minus medial, medial rect: second apotome of medial",821 "shortLabel": "Prop. X.75",822 "short": "Second apotome of medial",823 "book": 10,824 "number": 75,825 "colorClass": "proposition"826 },827 {828 "id": "Prop76",829 "type": "proposition",830 "label": "Line minus incommensurable: sum rational, rect medial: remainder minor",831 "shortLabel": "Prop. X.76",832 "short": "Minor defined",833 "book": 10,834 "number": 76,835 "colorClass": "proposition"836 },837 {838 "id": "Prop77",839 "type": "proposition",840 "label": "Line minus incommensurable: sum medial, rect rational: produces rational+medial",841 "shortLabel": "Prop. X.77",842 "short": "Produces rational+medial",843 "book": 10,844 "number": 77,845 "colorClass": "proposition"846 },847 {848 "id": "Prop78",849 "type": "proposition",850 "label": "Line minus incommensurable: both medial, incommensurable: produces medial+medial",851 "shortLabel": "Prop. X.78",852 "short": "Produces medial+medial",853 "book": 10,854 "number": 78,855 "colorClass": "proposition"856 },857 {858 "id": "Prop79",859 "type": "proposition",860 "label": "To apotome only one rational can be annexed in square only",861 "shortLabel": "Prop. X.79",862 "short": "Apotome: unique annex",863 "book": 10,864 "number": 79,865 "colorClass": "proposition"866 },867 {868 "id": "Prop80",869 "type": "proposition",870 "label": "To first apotome of medial: unique medial annex, rational rect",871 "shortLabel": "Prop. X.80",872 "short": "First apotome medial: unique annex",873 "book": 10,874 "number": 80,875 "colorClass": "proposition"876 },877 {878 "id": "Prop81",879 "type": "proposition",880 "label": "To second apotome of medial: unique medial annex, medial rect",881 "shortLabel": "Prop. X.81",882 "short": "Second apotome medial: unique annex",883 "book": 10,884 "number": 81,885 "colorClass": "proposition"886 },887 {888 "id": "Prop82",889 "type": "proposition",890 "label": "To minor: unique annex incommensurable in square",891 "shortLabel": "Prop. X.82",892 "short": "Minor: unique annex",893 "book": 10,894 "number": 82,895 "colorClass": "proposition"896 },897 {898 "id": "Prop83",899 "type": "proposition",900 "label": "To produces rational+medial: unique annex",901 "shortLabel": "Prop. X.83",902 "short": "Rational+medial: unique annex",903 "book": 10,904 "number": 83,905 "colorClass": "proposition"906 },907 {908 "id": "Prop84",909 "type": "proposition",910 "label": "To produces medial+medial: unique annex",911 "shortLabel": "Prop. X.84",912 "short": "Medial+medial: unique annex",913 "book": 10,914 "number": 84,915 "colorClass": "proposition"916 },917 {918 "id": "Prop85",919 "type": "proposition",920 "label": "To find the first apotome",921 "shortLabel": "Prop. X.85",922 "short": "Find first apotome",923 "book": 10,924 "number": 85,925 "colorClass": "proposition"926 },927 {928 "id": "Prop86",929 "type": "proposition",930 "label": "To find the second apotome",931 "shortLabel": "Prop. X.86",932 "short": "Find second apotome",933 "book": 10,934 "number": 86,935 "colorClass": "proposition"936 },937 {938 "id": "Prop87",939 "type": "proposition",940 "label": "To find the third apotome",941 "shortLabel": "Prop. X.87",942 "short": "Find third apotome",943 "book": 10,944 "number": 87,945 "colorClass": "proposition"946 },947 {948 "id": "Prop88",949 "type": "proposition",950 "label": "To find the fourth apotome",951 "shortLabel": "Prop. X.88",952 "short": "Find fourth apotome",953 "book": 10,954 "number": 88,955 "colorClass": "proposition"956 },957 {958 "id": "Prop89",959 "type": "proposition",960 "label": "To find the fifth apotome",961 "shortLabel": "Prop. X.89",962 "short": "Find fifth apotome",963 "book": 10,964 "number": 89,965 "colorClass": "proposition"966 },967 {968 "id": "Prop90",969 "type": "proposition",970 "label": "To find the sixth apotome",971 "shortLabel": "Prop. X.90",972 "short": "Find sixth apotome",973 "book": 10,974 "number": 90,975 "colorClass": "proposition"976 },977 {978 "id": "Prop91",979 "type": "proposition",980 "label": "Area by rational and first apotome: side is apotome",981 "shortLabel": "Prop. X.91",982 "short": "Rational × first apotome",983 "book": 10,984 "number": 91,985 "colorClass": "proposition"986 },987 {988 "id": "Prop92",989 "type": "proposition",990 "label": "Area by rational and second apotome: side is first apotome of medial",991 "shortLabel": "Prop. X.92",992 "short": "Rational × second apotome",993 "book": 10,994 "number": 92,995 "colorClass": "proposition"996 },997 {998 "id": "Prop93",999 "type": "proposition",1000 "label": "Area by rational and third apotome: side is second apotome of medial",1001 "shortLabel": "Prop. X.93",1002 "short": "Rational × third apotome",1003 "book": 10,1004 "number": 93,1005 "colorClass": "proposition"1006 },1007 {1008 "id": "Prop94",1009 "type": "proposition",1010 "label": "Area by rational and fourth apotome: side is minor",1011 "shortLabel": "Prop. X.94",1012 "short": "Rational × fourth apotome",1013 "book": 10,1014 "number": 94,1015 "colorClass": "proposition"1016 },1017 {1018 "id": "Prop95",1019 "type": "proposition",1020 "label": "Area by rational and fifth apotome: side produces rational+medial",1021 "shortLabel": "Prop. X.95",1022 "short": "Rational × fifth apotome",1023 "book": 10,1024 "number": 95,1025 "colorClass": "proposition"1026 },1027 {1028 "id": "Prop96",1029 "type": "proposition",1030 "label": "Area by rational and sixth apotome: side produces medial+medial",1031 "shortLabel": "Prop. X.96",1032 "short": "Rational × sixth apotome",1033 "book": 10,1034 "number": 96,1035 "colorClass": "proposition"1036 },1037 {1038 "id": "Prop97",1039 "type": "proposition",1040 "label": "Square on apotome of medial applied to rational: breadth first apotome",1041 "shortLabel": "Prop. X.97",1042 "short": "Square on apotome medial",1043 "book": 10,1044 "number": 97,1045 "colorClass": "proposition"1046 },1047 {1048 "id": "Prop98",1049 "type": "proposition",1050 "label": "Square on first apotome of medial: breadth second apotome",1051 "shortLabel": "Prop. X.98",1052 "short": "Square on first apotome medial",1053 "book": 10,1054 "number": 98,1055 "colorClass": "proposition"1056 },1057 {1058 "id": "Prop99",1059 "type": "proposition",1060 "label": "Square on second apotome of medial: breadth third apotome",1061 "shortLabel": "Prop. X.99",1062 "short": "Square on second apotome medial",1063 "book": 10,1064 "number": 99,1065 "colorClass": "proposition"1066 },1067 {1068 "id": "Prop100",1069 "type": "proposition",1070 "label": "Square on minor applied to rational: breadth fourth apotome",1071 "shortLabel": "Prop. X.100",1072 "short": "Square on minor",1073 "book": 10,1074 "number": 100,1075 "colorClass": "proposition"1076 },1077 {1078 "id": "Prop101",1079 "type": "proposition",1080 "label": "Square on produces rational+medial: breadth fifth apotome",1081 "shortLabel": "Prop. X.101",1082 "short": "Square on rational+medial",1083 "book": 10,1084 "number": 101,1085 "colorClass": "proposition"1086 },1087 {1088 "id": "Prop102",1089 "type": "proposition",1090 "label": "Square on produces medial+medial: breadth sixth apotome",1091 "shortLabel": "Prop. X.102",1092 "short": "Square on medial+medial",1093 "book": 10,1094 "number": 102,1095 "colorClass": "proposition"1096 },1097 {1098 "id": "Prop103",1099 "type": "proposition",1100 "label": "Line commensurable with apotome is apotome, same order",1101 "shortLabel": "Prop. X.103",1102 "short": "Commensurable with apotome",1103 "book": 10,1104 "number": 103,1105 "colorClass": "proposition"1106 },1107 {1108 "id": "Prop104",1109 "type": "proposition",1110 "label": "Line commensurable with apotome of medial is apotome of medial",1111 "shortLabel": "Prop. X.104",1112 "short": "Commensurable with apotome medial",1113 "book": 10,1114 "number": 104,1115 "colorClass": "proposition"1116 },1117 {1118 "id": "Prop105",1119 "type": "proposition",1120 "label": "Line commensurable with minor is minor",1121 "shortLabel": "Prop. X.105",1122 "short": "Commensurable with minor",1123 "book": 10,1124 "number": 105,1125 "colorClass": "proposition"1126 },1127 {1128 "id": "Prop106",1129 "type": "proposition",1130 "label": "Line commensurable with produces rational+medial is same",1131 "shortLabel": "Prop. X.106",1132 "short": "Commensurable with rational+medial",1133 "book": 10,1134 "number": 106,1135 "colorClass": "proposition"1136 },1137 {1138 "id": "Prop107",1139 "type": "proposition",1140 "label": "Line commensurable with produces medial+medial is same",1141 "shortLabel": "Prop. X.107",1142 "short": "Commensurable with medial+medial",1143 "book": 10,1144 "number": 107,1145 "colorClass": "proposition"1146 },1147 {1148 "id": "Prop108",1149 "type": "proposition",1150 "label": "From rational area subtract medial: side is apotome or minor",1151 "shortLabel": "Prop. X.108",1152 "short": "Rational minus medial",1153 "book": 10,1154 "number": 108,1155 "colorClass": "proposition"1156 },1157 {1158 "id": "Prop109",1159 "type": "proposition",1160 "label": "From medial subtract rational: first apotome of medial or rational+medial",1161 "shortLabel": "Prop. X.109",1162 "short": "Medial minus rational",1163 "book": 10,1164 "number": 109,1165 "colorClass": "proposition"1166 },1167 {1168 "id": "Prop110",1169 "type": "proposition",1170 "label": "From medial subtract medial incommensurable: second apotome or medial+medial",1171 "shortLabel": "Prop. X.110",1172 "short": "Medial minus medial",1173 "book": 10,1174 "number": 110,1175 "colorClass": "proposition"1176 },1177 {1178 "id": "Prop111",1179 "type": "proposition",1180 "label": "Apotome is not the same as binomial",1181 "shortLabel": "Prop. X.111",1182 "short": "Apotome ≠ binomial",1183 "book": 10,1184 "number": 111,1185 "colorClass": "proposition"1186 },1187 {1188 "id": "Prop112",1189 "type": "proposition",1190 "label": "Square on rational applied to binomial: breadth apotome, same order",1191 "shortLabel": "Prop. X.112",1192 "short": "Rational on binomial",1193 "book": 10,1194 "number": 112,1195 "colorClass": "proposition"1196 },1197 {1198 "id": "Prop113",1199 "type": "proposition",1200 "label": "Square on rational applied to apotome: breadth binomial, same order",