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 "book": {
  "id": "blackburn-elements-plane-trigonometry-1863",
  "title": "Elements of Plane Trigonometry",
  "authors": [
   "Hugh Blackburn"
  ],
  "year": 1863,
  "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
  "transcription": {
   "source": "Project Gutenberg eBook #32973",
   "url": "https://www.gutenberg.org/ebooks/32973",
   "released": "June 25, 2010",
   "licence_terms": "This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever.  You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org"
  },
  "file": "books/blackburn-elements-plane-trigonometry-1863.json"
 },
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   "id": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "number": "I",
   "title": "OF THE MENSURATION OF THE CIRCLE",
   "name": "Blackburn 1863, ch. I: OF THE MENSURATION OF THE CIRCLE",
   "pages": [
    "1",
    "11"
   ],
   "concepts": [
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    "concept/circular-sector",
    "concept/constant",
    "concept/geometrical-mean",
    "concept/limit",
    "concept/ratio",
    "concept/regular-polygon",
    "concept/similar-curves",
    "concept/variable",
    "method/approximating-pi-by-inscribed-and-circumscribed-polygons",
    "person/adrian-metius",
    "person/archimedes",
    "person/isaac-newton",
    "quantity/area",
    "quantity/circumference",
    "quantity/length-of-a-curve",
    "quantity/pi",
    "quantity/radius",
    "theorem/area-of-a-circular-sector",
    "theorem/areas-of-similar-segments-are-proportional-to-the-squares-on-their-chords",
    "theorem/convex-line-is-less-than-any-line-enveloping-it",
    "theorem/lengths-of-similar-arcs-are-proportional-to-their-chords",
    "theorem/limits-of-equal-variables-are-equal",
    "theorem/radii-of-inscribed-and-circumscribed-circles-of-doubled-polygons",
    "unit/radian"
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    "blackburn-elements-plane-trigonometry-1863/x-1f093d60aa",
    "blackburn-elements-plane-trigonometry-1863/x-7d65261339"
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    "blackburn-elements-plane-trigonometry-1863/eq-f24cdb358c",
    "blackburn-elements-plane-trigonometry-1863/eq-00af57c3af",
    "blackburn-elements-plane-trigonometry-1863/eq-ee246409b8",
    "blackburn-elements-plane-trigonometry-1863/eq-30d8edb714",
    "blackburn-elements-plane-trigonometry-1863/eq-a259018857",
    "blackburn-elements-plane-trigonometry-1863/eq-500b5b38e9",
    "blackburn-elements-plane-trigonometry-1863/eq-805c551708",
    "blackburn-elements-plane-trigonometry-1863/eq-61e3fce184",
    "blackburn-elements-plane-trigonometry-1863/eq-b35d2bcc52",
    "blackburn-elements-plane-trigonometry-1863/eq-51c08a8c4d",
    "blackburn-elements-plane-trigonometry-1863/eq-7bd8cbfc6a",
    "blackburn-elements-plane-trigonometry-1863/eq-ce2039ce2d",
    "blackburn-elements-plane-trigonometry-1863/eq-31d17b22a1"
   ],
   "exercise_sets": []
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "number": "II",
   "title": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "name": "Blackburn 1863, ch. II: OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "pages": [
    "11",
    "15"
   ],
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    "concept/angle-bisector",
    "concept/excircle",
    "concept/inscribed-circle",
    "concept/tangent",
    "quantity/area",
    "quantity/radius",
    "quantity/semi-perimeter",
    "theorem/heron-s-formula",
    "theorem/triangle-area-as-a-mean-proportional",
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    "theorem/triangle-area-as-semi-perimeter-times-inradius"
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    "blackburn-elements-plane-trigonometry-1863/eq-78d14cf27b",
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    "blackburn-elements-plane-trigonometry-1863/eq-cebebab2d5",
    "blackburn-elements-plane-trigonometry-1863/eq-4f0b26c1ab",
    "blackburn-elements-plane-trigonometry-1863/eq-e6a8b39f8d",
    "blackburn-elements-plane-trigonometry-1863/eq-8afd4f2418"
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  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "number": "III",
   "title": "OF SYMBOLS OF QUANTITY",
   "name": "Blackburn 1863, ch. III: OF SYMBOLS OF QUANTITY",
   "pages": [
    "15",
    "17"
   ],
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    "concept/negative-direction",
    "concept/origin",
    "concept/positive-direction",
    "person/euclid",
    "quantity/angle",
    "quantity/arc-of-a-circle"
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   "excerpts": [
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    "blackburn-elements-plane-trigonometry-1863/x-9a5ae2c8af",
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    "blackburn-elements-plane-trigonometry-1863/x-f2a9f1e5f7",
    "blackburn-elements-plane-trigonometry-1863/x-3566689306",
    "blackburn-elements-plane-trigonometry-1863/x-d59dc83555"
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    "blackburn-elements-plane-trigonometry-1863/eq-e03b6609e5"
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  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "number": "IV",
   "title": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "name": "Blackburn 1863, ch. IV: OF THE UNIT OF ANGULAR MAGNITUDE",
   "pages": [
    "17",
    "20"
   ],
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    "concept/complementary-angles",
    "concept/sexagesimal-system",
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    "quantity/pi",
    "quantity/right-angle",
    "unit/degree-of-angle",
    "unit/minute-of-arc",
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    "blackburn-elements-plane-trigonometry-1863/eq-081328cd8f",
    "blackburn-elements-plane-trigonometry-1863/eq-81e5fc28a5",
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    "blackburn-elements-plane-trigonometry-1863/eq-1777a0cfbf",
    "blackburn-elements-plane-trigonometry-1863/eq-ae67d7b780",
    "blackburn-elements-plane-trigonometry-1863/eq-710183bf7c",
    "blackburn-elements-plane-trigonometry-1863/eq-00ded65979",
    "blackburn-elements-plane-trigonometry-1863/eq-07b37c0fc2",
    "blackburn-elements-plane-trigonometry-1863/eq-174c0099f0",
    "blackburn-elements-plane-trigonometry-1863/eq-72c1018539",
    "blackburn-elements-plane-trigonometry-1863/eq-fb2ea2fe2a"
   ],
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  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "number": "V",
   "title": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "name": "Blackburn 1863, ch. V: CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "pages": [
    "20",
    "33"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/circular-measure",
    "concept/complementary-angles",
    "concept/cosecant",
    "concept/cosine",
    "concept/cotangent",
    "concept/function",
    "concept/periodic-function",
    "concept/regular-polygon",
    "concept/secant",
    "concept/sine",
    "concept/tangent-function",
    "concept/versine",
    "quantity/angle",
    "quantity/radius",
    "theorem/trigonometric-identity"
   ],
   "excerpts": [
    "blackburn-elements-plane-trigonometry-1863/x-5c3bea5050",
    "blackburn-elements-plane-trigonometry-1863/x-eef9e960ab",
    "blackburn-elements-plane-trigonometry-1863/x-9c7c821295",
    "blackburn-elements-plane-trigonometry-1863/x-4ab36fab10",
    "blackburn-elements-plane-trigonometry-1863/x-8f81ec33e7",
    "blackburn-elements-plane-trigonometry-1863/x-fdb188cb69",
    "blackburn-elements-plane-trigonometry-1863/x-527feb98f6"
   ],
   "equations": [
    "blackburn-elements-plane-trigonometry-1863/eq-2ee5101c5c",
    "blackburn-elements-plane-trigonometry-1863/eq-b78f2b353e",
    "blackburn-elements-plane-trigonometry-1863/eq-2b466fdea5",
    "blackburn-elements-plane-trigonometry-1863/eq-bcc06dae35",
    "blackburn-elements-plane-trigonometry-1863/eq-7f87858dba",
    "blackburn-elements-plane-trigonometry-1863/eq-7bdb332822",
    "blackburn-elements-plane-trigonometry-1863/eq-908c8bb301",
    "blackburn-elements-plane-trigonometry-1863/eq-787c1b8b8d",
    "blackburn-elements-plane-trigonometry-1863/eq-4a098e8ee6",
    "blackburn-elements-plane-trigonometry-1863/eq-9f3481315a",
    "blackburn-elements-plane-trigonometry-1863/eq-889fd6d24e",
    "blackburn-elements-plane-trigonometry-1863/eq-8ef7675e75",
    "blackburn-elements-plane-trigonometry-1863/eq-e9acdf5af5",
    "blackburn-elements-plane-trigonometry-1863/eq-75bf12b854",
    "blackburn-elements-plane-trigonometry-1863/eq-1206a8ee44",
    "blackburn-elements-plane-trigonometry-1863/eq-3b226912de",
    "blackburn-elements-plane-trigonometry-1863/eq-36881400d3",
    "blackburn-elements-plane-trigonometry-1863/eq-6f209ec7e7",
    "blackburn-elements-plane-trigonometry-1863/eq-5b753037f6",
    "blackburn-elements-plane-trigonometry-1863/eq-cf54a66496",
    "blackburn-elements-plane-trigonometry-1863/eq-472bd8a669",
    "blackburn-elements-plane-trigonometry-1863/eq-a3d11a0644",
    "blackburn-elements-plane-trigonometry-1863/eq-685f1d6298",
    "blackburn-elements-plane-trigonometry-1863/eq-8c0da28a70",
    "blackburn-elements-plane-trigonometry-1863/eq-fea2f16b85",
    "blackburn-elements-plane-trigonometry-1863/eq-9f83e647d4",
    "blackburn-elements-plane-trigonometry-1863/eq-d549076b45",
    "blackburn-elements-plane-trigonometry-1863/eq-c88ca72782"
   ],
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  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "number": "VI",
   "title": "OF LOGARITHMIC TABLES",
   "name": "Blackburn 1863, ch. VI: OF LOGARITHMIC TABLES",
   "pages": [
    "33",
    "37"
   ],
   "concepts": [
    "concept/base-of-a-logarithm-system",
    "concept/characteristic",
    "concept/common-logarithm",
    "concept/logarithm",
    "concept/mantissa",
    "method/interpolating-in-logarithmic-tables",
    "person/john-napier",
    "theorem/logarithm-of-1-is-zero",
    "theorem/logarithm-of-a-power",
    "theorem/logarithm-of-a-product",
    "theorem/logarithm-of-a-quotient"
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   "excerpts": [
    "blackburn-elements-plane-trigonometry-1863/x-27802618c1",
    "blackburn-elements-plane-trigonometry-1863/x-0a39a94bd9",
    "blackburn-elements-plane-trigonometry-1863/x-88d6482d60",
    "blackburn-elements-plane-trigonometry-1863/x-88af92efa9",
    "blackburn-elements-plane-trigonometry-1863/x-13177f43e0"
   ],
   "equations": [
    "blackburn-elements-plane-trigonometry-1863/eq-44fcb7bd2e",
    "blackburn-elements-plane-trigonometry-1863/eq-48c221fe6c",
    "blackburn-elements-plane-trigonometry-1863/eq-401be09736",
    "blackburn-elements-plane-trigonometry-1863/eq-cf316b3734",
    "blackburn-elements-plane-trigonometry-1863/eq-9535ca31a5",
    "blackburn-elements-plane-trigonometry-1863/eq-85b58e3c73",
    "blackburn-elements-plane-trigonometry-1863/eq-a3b4ebd3b9",
    "blackburn-elements-plane-trigonometry-1863/eq-566f2e71f9",
    "blackburn-elements-plane-trigonometry-1863/eq-928bc05ad1",
    "blackburn-elements-plane-trigonometry-1863/eq-e58183508e"
   ],
   "exercise_sets": []
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  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "number": "VII",
   "title": "SOLUTION OF TRIANGLES",
   "name": "Blackburn 1863, ch. VII: SOLUTION OF TRIANGLES",
   "pages": [
    "37",
    "44"
   ],
   "concepts": [
    "concept/ambiguous-case",
    "concept/triangle",
    "method/solving-a-right-angled-triangle-logarithmically",
    "method/solving-a-triangle",
    "method/solving-a-triangle-from-its-three-sides",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle",
    "theorem/law-of-sines",
    "theorem/radius-of-the-inscribed-circle-from-the-sides",
    "theorem/tangent-formula-for-two-sides-and-the-included-angle"
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    "blackburn-elements-plane-trigonometry-1863/eq-9fee30e497",
    "blackburn-elements-plane-trigonometry-1863/eq-2b468bec3a",
    "blackburn-elements-plane-trigonometry-1863/eq-953186424a",
    "blackburn-elements-plane-trigonometry-1863/eq-7209e8a3c6",
    "blackburn-elements-plane-trigonometry-1863/eq-0749ad0934",
    "blackburn-elements-plane-trigonometry-1863/eq-1ab0022818",
    "blackburn-elements-plane-trigonometry-1863/eq-ab6adceb96",
    "blackburn-elements-plane-trigonometry-1863/eq-0d0c5585cf",
    "blackburn-elements-plane-trigonometry-1863/eq-cb9f987031",
    "blackburn-elements-plane-trigonometry-1863/eq-01855a06d9",
    "blackburn-elements-plane-trigonometry-1863/eq-4e0bfebbb4",
    "blackburn-elements-plane-trigonometry-1863/eq-cfdf684cbd",
    "blackburn-elements-plane-trigonometry-1863/eq-932924e024",
    "blackburn-elements-plane-trigonometry-1863/eq-c5caf10392",
    "blackburn-elements-plane-trigonometry-1863/eq-ebbad58551",
    "blackburn-elements-plane-trigonometry-1863/eq-aff341bcef",
    "blackburn-elements-plane-trigonometry-1863/eq-bf2fd41781",
    "blackburn-elements-plane-trigonometry-1863/eq-faa20b0363",
    "blackburn-elements-plane-trigonometry-1863/eq-46d9a613a2",
    "blackburn-elements-plane-trigonometry-1863/eq-869d782d87",
    "blackburn-elements-plane-trigonometry-1863/eq-0d7047239a",
    "blackburn-elements-plane-trigonometry-1863/eq-bf08a87bf5",
    "blackburn-elements-plane-trigonometry-1863/eq-82d3448656",
    "blackburn-elements-plane-trigonometry-1863/eq-5a54a7477d",
    "blackburn-elements-plane-trigonometry-1863/eq-6716a62e33",
    "blackburn-elements-plane-trigonometry-1863/eq-cfbf497094",
    "blackburn-elements-plane-trigonometry-1863/eq-a20fb7a997",
    "blackburn-elements-plane-trigonometry-1863/eq-5609d4bce0",
    "blackburn-elements-plane-trigonometry-1863/eq-97f62021ec",
    "blackburn-elements-plane-trigonometry-1863/eq-514a2b1eae",
    "blackburn-elements-plane-trigonometry-1863/eq-3cd5988404",
    "blackburn-elements-plane-trigonometry-1863/eq-59ab9f2ca1",
    "blackburn-elements-plane-trigonometry-1863/eq-3e319aa0a6",
    "blackburn-elements-plane-trigonometry-1863/eq-d5062bba0e",
    "blackburn-elements-plane-trigonometry-1863/eq-c82095912b",
    "blackburn-elements-plane-trigonometry-1863/eq-923d7fa2af",
    "blackburn-elements-plane-trigonometry-1863/eq-6b23040ab0",
    "blackburn-elements-plane-trigonometry-1863/eq-a76e6b28a5",
    "blackburn-elements-plane-trigonometry-1863/eq-690664468e",
    "blackburn-elements-plane-trigonometry-1863/eq-94001971fb",
    "blackburn-elements-plane-trigonometry-1863/eq-505da7b5c9",
    "blackburn-elements-plane-trigonometry-1863/eq-abc3e60521",
    "blackburn-elements-plane-trigonometry-1863/eq-4e6d6434e1",
    "blackburn-elements-plane-trigonometry-1863/eq-50392e3c58",
    "blackburn-elements-plane-trigonometry-1863/eq-bea37ab88f",
    "blackburn-elements-plane-trigonometry-1863/eq-1e2ca23c5a",
    "blackburn-elements-plane-trigonometry-1863/eq-17dbaaaa58",
    "blackburn-elements-plane-trigonometry-1863/eq-79b401ec53"
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   "exercise_sets": []
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   "id": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "number": "VIII",
   "title": "OF TRIGONOMETRICAL SURVEYING",
   "name": "Blackburn 1863, ch. VIII: OF TRIGONOMETRICAL SURVEYING",
   "pages": [
    "44",
    "47"
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    "quantity/distance",
    "quantity/height"
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   "excerpts": [
    "blackburn-elements-plane-trigonometry-1863/x-ee412313e9",
    "blackburn-elements-plane-trigonometry-1863/x-c2d4ed49ec",
    "blackburn-elements-plane-trigonometry-1863/x-d47d9b8142",
    "blackburn-elements-plane-trigonometry-1863/x-cf907767c0",
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   "equations": [],
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  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "number": "IX",
   "title": "OF PROJECTIONS",
   "name": "Blackburn 1863, ch. IX: OF PROJECTIONS",
   "pages": [
    "47",
    "49"
   ],
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    "concept/broken-line",
    "concept/inclination-of-two-lines",
    "concept/line-of-projection",
    "concept/projection",
    "concept/projection-of-a-line",
    "theorem/projection-of-a-broken-line",
    "theorem/projection-of-a-line-on-a-line"
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   "excerpts": [
    "blackburn-elements-plane-trigonometry-1863/x-58ea15f04d",
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    "blackburn-elements-plane-trigonometry-1863/x-9275af9721",
    "blackburn-elements-plane-trigonometry-1863/x-112f3e089e"
   ],
   "equations": [
    "blackburn-elements-plane-trigonometry-1863/eq-043253d284",
    "blackburn-elements-plane-trigonometry-1863/eq-aec7cee42c"
   ],
   "exercise_sets": []
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "number": "X",
   "title": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "name": "Blackburn 1863, ch. X: THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "pages": [
    "49",
    "53"
   ],
   "concepts": [
    "concept/circular-measure",
    "concept/cosine",
    "concept/limit",
    "concept/projection",
    "concept/sine",
    "method/deriving-sum-and-difference-formulae-by-projection",
    "quantity/angle",
    "quantity/radius",
    "theorem/cosine-of-the-difference-of-two-angles",
    "theorem/cosine-of-the-sum-of-two-angles",
    "theorem/limiting-radius-of-inscribed-and-circumscribed-circles",
    "theorem/projection-of-a-broken-line",
    "theorem/sine-of-the-difference-of-two-angles",
    "theorem/sine-of-the-sum-of-two-angles"
   ],
   "excerpts": [
    "blackburn-elements-plane-trigonometry-1863/x-3caf928c41",
    "blackburn-elements-plane-trigonometry-1863/x-a829afd06a",
    "blackburn-elements-plane-trigonometry-1863/x-3ffa55dd66",
    "blackburn-elements-plane-trigonometry-1863/x-6d26e2b2db",
    "blackburn-elements-plane-trigonometry-1863/x-44e87e3591"
   ],
   "equations": [
    "blackburn-elements-plane-trigonometry-1863/eq-221cce7152",
    "blackburn-elements-plane-trigonometry-1863/eq-18342c1601",
    "blackburn-elements-plane-trigonometry-1863/eq-e35d999d91",
    "blackburn-elements-plane-trigonometry-1863/eq-a25882ff41",
    "blackburn-elements-plane-trigonometry-1863/eq-138592b15b",
    "blackburn-elements-plane-trigonometry-1863/eq-07d3a230a5",
    "blackburn-elements-plane-trigonometry-1863/eq-2cffb15855",
    "blackburn-elements-plane-trigonometry-1863/eq-6d0771fbad",
    "blackburn-elements-plane-trigonometry-1863/eq-fb0cb3d355",
    "blackburn-elements-plane-trigonometry-1863/eq-e1dd45729f"
   ],
   "exercise_sets": []
  }
 ],
 "excerpts": [
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-811984a21a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "1",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "A magnitude or ratio, which is fixed in value by the conditions of the question, is called a \\Emph{Constant}.",
   "markdown": "A magnitude or ratio, which is fixed in value by the conditions of the question, is called a Constant.",
   "why": "Gives the learner a plain definition of a constant before any limit argument.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/constant"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-0a3b94f17c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "2",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "If two variables are at every instant equal their limits are equal.",
   "markdown": "If two variables are at every instant equal their limits are equal.",
   "why": "States the lemma that the whole chapter's limit arguments rest on, in one sentence a learner can hold.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "theorem/limits-of-equal-variables-are-equal"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-70da0f39c1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "3",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "Then if the number of sides of the polygons increase these two ratios vary but remain always equal to each other, therefore (Lemma) their limits are equal.",
   "markdown": "Then if the number of sides of the polygons increase these two ratios vary but remain always equal to each other, therefore (Lemma) their limits are equal.",
   "why": "Shows the limiting argument in action: two ratios that stay equal must share a limit.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "theorem/lengths-of-similar-arcs-are-proportional-to-their-chords"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-3d7208f2fa",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "5",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "The area of any circular sector is half the rectangle contained by its arc and the radius of the circle.",
   "markdown": "The area of any circular sector is half the rectangle contained by its arc and the radius of the circle.",
   "why": "A memorable statement of the sector area rule that a learner can check against a simple circle.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/circular-sector",
    "theorem/area-of-a-circular-sector"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-7a9b9b5a41",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "2",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "Let a number of points be taken in a terminated curve line, and let straight lines be drawn from each point to the next, then if the number of points be conceived to increase and the distance between each two to diminish continually, the extremities remaining fixed, the limit of the sum of the straight lines is called the \\Emph{Length of the Curve}.",
   "markdown": "Let a number of points be taken in a terminated curve line, and let straight lines be drawn from each point to the next, then if the number of points be conceived to increase and the distance between each two to diminish continually, the extremities remaining fixed, the limit of the sum of the straight lines is called the Length of the Curve.",
   "why": "Defines arc length by a limit of inscribed polygonal lines, the idea a learner needs for curved lengths.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/limit",
    "quantity/length-of-a-curve"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-1f093d60aa",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "11",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "By the method of ``continued fractions'' it will be found that $\\dfrac{22}{7}$~and~$\\dfrac{355}{113}$ are nearer approximations to the value of~$\\pi$ than any simpler fractions.",
   "markdown": "By the method of “continued fractions” it will be found that $\\dfrac{22}{7}$ and $\\dfrac{355}{113}$ are nearer approximations to the value of $\\pi$ than any simpler fractions.",
   "why": "Gives learners a concrete sense of how close simple fractions come to pi.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-7d65261339",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "11",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "Of these $\\dfrac{22}{7}$ ($=3.14$) is the approximation discovered by Archimedes (killed, it is said, at the siege of Syracuse,",
   "markdown": "Of these $\\dfrac{22}{7}$ ($=3.14$) is the approximation discovered by Archimedes (killed, it is said, at the siege of Syracuse,",
   "why": "A historical remark placing the 22/7 approximation with Archimedes.",
   "use": [
    "history"
   ],
   "concepts": [
    "person/archimedes",
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-ced77fd644",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "11",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "A triangle is equal to the rectangle contained by its semi-perimeter and the radius of the inscribed circle.",
   "markdown": "A triangle is equal to the rectangle contained by its semi-perimeter and the radius of the inscribed circle.",
   "why": "States the chapter's first result in one line, so a learner sees the area formula in terms of half the perimeter and the inscribed radius.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/inscribed-circle",
    "quantity/area",
    "quantity/semi-perimeter",
    "theorem/triangle-area-as-semi-perimeter-times-inradius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-01975c8f91",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "The two tangents from each angle to the inscribed circle are equal: hence, if three tangents, one from each angle, be taken, their sum is the semi-perimeter, and therefore a tangent from one of the angles, together with the side opposite that angle, is equal to the semi-perimeter.",
   "markdown": "The two tangents from each angle to the inscribed circle are equal: hence, if three tangents, one from each angle, be taken, their sum is the semi-perimeter, and therefore a tangent from one of the angles, together with the side opposite that angle, is equal to the semi-perimeter.",
   "why": "Shows why a tangent plus the opposite side equals the semi-perimeter, a short argument a learner can follow step by step.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/inscribed-circle",
    "concept/tangent",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-8505dbbfc9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "Let two of the sides of the triangle $ABC$ be produced, and a circle described touching the two produced sides and the third side.",
   "markdown": "Let two of the sides of the triangle $ABC$ be produced, and a circle described touching the two produced sides and the third side.",
   "why": "Defines the excircle by construction, giving the learner a picture to draw before the excircle theorem.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/excircle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-83c76e1a90",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "This word is often spelled ``\\emph{escribed}'' improperly.",
   "markdown": "This word is often spelled “*escribed*” improperly.",
   "why": "A short aside on spelling that rewards a curious reader and explains why the word looks odd.",
   "use": [
    "website"
   ],
   "concepts": [
    "concept/excircle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-d38af63a9f",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "14",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "This most useful proposition was known to the Greeks of Alexandria, and by them communicated to the Arabians",
   "markdown": "This most useful proposition was known to the Greeks of Alexandria, and by them communicated to the Arabians",
   "why": "Places the mean-proportional result in history, showing that a formula taught today has a long lineage.",
   "use": [
    "history"
   ],
   "concepts": [
    "theorem/triangle-area-as-a-mean-proportional"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-e56315ac0d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "14",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "whence the area can be calculated in square units when the lengths of the sides are given numerically in units.",
   "markdown": "whence the area can be calculated in square units when the lengths of the sides are given numerically in units.",
   "why": "Tells the learner the practical purpose of the result: finding a triangle's area from its three side lengths.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/area",
    "theorem/heron-s-formula"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-a4e77e2c7a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "\\text{Then, numerically, the Area} = rs.",
   "markdown": "Then, numerically, the Area = rs.",
   "why": "Gives the compact numerical form of the area, linking the geometric statement to the symbols s and r a learner will compute with.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "quantity/area",
    "quantity/radius",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-5cd5566dd9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "15",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "In Euclid an angle is not defined as a magnitude but as the inclination of two lines, which never exceeds two right angles: and in most of the propositions in Euclid it is not necessary to treat an angle otherwise than as a change of direction of a line.",
   "markdown": "In Euclid an angle is not defined as a magnitude but as the inclination of two lines, which never exceeds two right angles: and in most of the propositions in Euclid it is not necessary to treat an angle otherwise than as a change of direction of a line.",
   "why": "It shows that the modern wide meaning of angle is a deliberate extension of Euclid's narrower definition, which helps a learner see why the word changes sense.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "person/euclid",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-9a5ae2c8af",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "15",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "So also in Trigonometry, where angular magnitude in general is treated numerically, it is desirable to use the term angle for the sum of a number of angles, which may be greater than two or than any number of right angles.",
   "markdown": "So also in Trigonometry, where angular magnitude in general is treated numerically, it is desirable to use the term angle for the sum of a number of angles, which may be greater than two or than any number of right angles.",
   "why": "It explains in plain words why a trigonometric angle may exceed a full turn, which is the idea learners most often find surprising.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-fbd11c40e2",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "15",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "For instance, if a distance $a$~miles be measured from a fixed point~$O$ along (or parallel to) a given line in a standard direction, say east, to~$A$; and a line~$AB$ be cut off from~$OA$ by measuring from~$A$ in the opposite direction, or westward, a distance $b$~miles, the distance~$OB$ may be said to be $= (a - b)$ miles east of~$O$ not only in the case where $a > b$, but also when $a < b$, if it be agreed to interpret the result as meaning $a - b$ east of~$O$ (or in the standard direction) if $a - b$ is a $+$~number, and $b - a$ miles west of~$O$ (or in the contrary direction), when $a - b$ is a $-$~number.",
   "markdown": "For instance, if a distance $a$ miles be measured from a fixed point $O$ along (or parallel to) a given line in a standard direction, say east, to $A$; and a line $AB$ be cut off from $OA$ by measuring from $A$ in the opposite direction, or westward, a distance $b$ miles, the distance $OB$ may be said to be $= (a - b)$ miles east of $O$ not only in the case where $a > b$, but also when $a < b$, if it be agreed to interpret the result as meaning $a - b$ east of $O$ (or in the standard direction) if $a - b$ is a $+$ number, and $b - a$ miles west of $O$ (or in the contrary direction), when $a - b$ is a $-$ number.",
   "why": "A worked distance example that shows how a signed result such as a minus b keeps one formula valid whichever quantity is larger.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/negative-number",
    "concept/origin",
    "concept/positive-number",
    "concept/variable"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-f2a9f1e5f7",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "16",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "Then the standard direction may be called the~$+$ (or positive) direction, and the contrary direction the~$-$ (or negative) direction.",
   "markdown": "Then the standard direction may be called the $+$ (or positive) direction, and the contrary direction the $-$ (or negative) direction.",
   "why": "It gives the plain naming of the two directions that a learner needs before any sign convention can be used.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/negative-direction",
    "concept/positive-direction",
    "concept/sign"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-3566689306",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "16",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "Again, with reference to angles, if the hand of a going clock be put back through an angle~$\\theta$, then, after the time during which the hand moves through an angle~$\\phi$, the hand will make an angle $\\theta - \\phi$ with its present position, the angle being~$+$ and measured in the opposite way to that in which the hand of the clock moves, if $\\theta > \\phi$; or $-$~and measured in the contrary direction, if $\\theta < \\phi$.",
   "markdown": "Again, with reference to angles, if the hand of a going clock be put back through an angle $\\theta$, then, after the time during which the hand moves through an angle $\\phi$, the hand will make an angle $\\theta - \\phi$ with its present position, the angle being $+$ and measured in the opposite way to that in which the hand of the clock moves, if $\\theta > \\phi$; or $-$ and measured in the contrary direction, if $\\theta < \\phi$.",
   "why": "A clock-hand picture makes a signed angle concrete and shows that the sign depends on which way the hand has turned.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/negative-number",
    "concept/positive-number",
    "concept/sign",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-d59dc83555",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "16",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "In what follows, an angle will be considered as if produced by the revolution of a radius of a circle, the direction of revolution from an initiatory position being considered as $-$~or~$+$ according as it takes place in the direction of the motion of the hand of a clock or the reverse.",
   "markdown": "In what follows, an angle will be considered as if produced by the revolution of a radius of a circle, the direction of revolution from an initiatory position being considered as $-$ or $+$ according as it takes place in the direction of the motion of the hand of a clock or the reverse.",
   "why": "It fixes the rotational meaning of angle used for the rest of the chapter, so learners know how the sign of a turn is chosen.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/sign",
    "quantity/angle",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-f8f91f6302",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "17",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "The angle of easiest construction is the angle of an equilateral triangle, which is also two-thirds of a right angle.",
   "markdown": "The angle of easiest construction is the angle of an equilateral triangle, which is also two-thirds of a right angle.",
   "why": "It gives learners a concrete angle to picture before any numerical scale is introduced.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "quantity/angle",
    "quantity/right-angle",
    "unit/degree-of-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-cc4998e7e4",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "18",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "It is called the \\emph{unit of circular measure}, and the ratio of any angle to this unit is called the \\emph{circular measure} of the angle.",
   "markdown": "It is called the *unit of circular measure*, and the ratio of any angle to this unit is called the *circular measure* of the angle.",
   "why": "It states plainly that angles can be measured as a ratio to a fixed standard angle, which is the core idea of the chapter.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circular-measure",
    "quantity/angle",
    "unit/radian"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-420d91dc0b",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "17",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "For such reasons perhaps the sexagesimal scale, which has prevailed since the time of Ptolemy\\footnotemark, was originally adopted.",
   "markdown": "For such reasons perhaps the sexagesimal scale, which has prevailed since the time of Ptolemy, was originally adopted.",
   "why": "It places the 60-part degree scale in its historical setting and shows that the chapter's conventions have a long history.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/sexagesimal-system",
    "person/ptolemy"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-6ab5f52916",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "Two angles are said to be complements, each of the other, when their sum is a right angle.",
   "markdown": "Two angles are said to be complements, each of the other, when their sum is a right angle.",
   "why": "It gives the defining condition for complementary angles in a form learners can test with any pair of angles.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/complementary-angles",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-5d22d0ea85",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "17",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "The $60$th part of a degree is a minute, denoted by~$1'$, $\\therefore 1° = 60'$.",
   "markdown": "The $60$th part of a degree is a minute, denoted by $1'$, $\\therefore 1° = 60'$.",
   "why": "It shows the subdivision rule that links minutes to degrees, which learners need before working with angles given in degrees, minutes and seconds.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "unit/degree-of-angle",
    "unit/minute-of-arc"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-5c3bea5050",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "20",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "When one magnitude or ratio is so connected with another that the former changes with the latter, but is determinable for any given value of the latter, the former is said to be a function of the latter.",
   "markdown": "When one magnitude or ratio is so connected with another that the former changes with the latter, but is determinable for any given value of the latter, the former is said to be a function of the latter.",
   "why": "Gives the learner the general idea of a function before any trigonometric ratio appears.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-eef9e960ab",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "The side of the hexagon inscribed is $= R$, and it subtends $\\dfrac{\\pi}{3}$ or~$60°$.",
   "markdown": "The side of the hexagon inscribed is $= R$, and it subtends $\\dfrac{\\pi}{3}$ or $60°$.",
   "why": "Shows how a known side of an inscribed regular polygon fixes a sine value, the method the chapter uses throughout.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/regular-polygon",
    "concept/sine",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-9c7c821295",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "23",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "It should be observed that, while the number~$\\theta$ continuously increases, the numbers $\\sin\\theta$, $\\cos\\theta$ pass through a series of values between $+1$~and~$-1$, and return to the same values again for every increase of~$2\\pi$ in the value of~$\\theta$.",
   "markdown": "It should be observed that, while the number $\\theta$ continuously increases, the numbers $\\sin\\theta$, $\\cos\\theta$ pass through a series of values between $+1$ and $-1$, and return to the same values again for every increase of $2\\pi$ in the value of $\\theta$.",
   "why": "Explains periodicity in plain terms, so the learner sees why angles beyond 2π reuse values.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cosine",
    "concept/periodic-function",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-4ab36fab10",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "25",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "If $AF$ be measured towards~$T$, it is to be considered~$+$, and $-$,~if in the contrary direction towards~$T'$.",
   "markdown": "If $AF$ be measured towards $T$, it is to be considered $+$, and $-$, if in the contrary direction towards $T'$.",
   "why": "Warns that a tangent's sign depends on the direction it is measured, a common source of sign errors.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-8f81ec33e7",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "Since the perpendicular from the centre of a circle on any chord bisects it at right angles, the ratio of the chord to the radius is twice the sine of half the angle subtended by the chord.",
   "markdown": "Since the perpendicular from the centre of a circle on any chord bisects it at right angles, the ratio of the chord to the radius is twice the sine of half the angle subtended by the chord.",
   "why": "Connects chords and sines in one clear step, and the reason is stated.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/sine",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-fdb188cb69",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "31",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "The seven ratios defined above are altogether independent of the size of the circle described, and depend only on the angle.",
   "markdown": "The seven ratios defined above are altogether independent of the size of the circle described, and depend only on the angle.",
   "why": "Tells the learner that a circular function is a property of the angle alone, which is the point of the definition.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/circular-function",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-527feb98f6",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "From these equations all the functions can be found, when one has been given.",
   "markdown": "From these equations all the functions can be found, when one has been given.",
   "why": "Shows that the identities let every circular function be recovered from any one of them.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circular-function",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-27802618c1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "36",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "But the inventor of Logarithms was not a Peer, and should not be styled Baron Napier as is often done.",
   "markdown": "But the inventor of Logarithms was not a Peer, and should not be styled Baron Napier as is often done.",
   "why": "A short historical correction that gives the learner a human story about who invented logarithms and why his title is misremembered.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/logarithm",
    "person/john-napier"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-0a39a94bd9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "33",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "Logarithms of ordinary numbers may be defined to be numbers, so calculated from the ordinary numbers, that the sum of the logarithms of two numbers is the logarithm of their product.",
   "markdown": "Logarithms of ordinary numbers may be defined to be numbers, so calculated from the ordinary numbers, that the sum of the logarithms of two numbers is the logarithm of their product.",
   "why": "Gives the defining property from which every other logarithm rule follows, so a learner sees why logarithms turn multiplication into addition.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/logarithm",
    "theorem/logarithm-of-a-product"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-88d6482d60",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "35",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "Thus, with tables, the result of multiplication is calculated by adding logarithms, of division by subtracting logarithms, of raising to a power by multiplying a logarithm by a number, of extracting a root by dividing a logarithm by a number.",
   "markdown": "Thus, with tables, the result of multiplication is calculated by adding logarithms, of division by subtracting logarithms, of raising to a power by multiplying a logarithm by a number, of extracting a root by dividing a logarithm by a number.",
   "why": "Summarises in one sentence the four arithmetic operations that the tables turn into simpler ones, which is the practical point of the chapter.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/logarithm",
    "method/division",
    "method/multiplication"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-88af92efa9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "Equation~\\Eq{5} shews that the logarithm increases with the number: therefore it appears that the common logarithm of a number of one integral digit is a proper fraction; that of a number of $2$~digits is $1 + \\text{a fraction}$; of $3$~digits $2 + \\text{a fraction}$, and so on.",
   "markdown": "Equation (5) shews that the logarithm increases with the number: therefore it appears that the common logarithm of a number of one integral digit is a proper fraction; that of a number of $2$ digits is $1 + \\text{a fraction}$; of $3$ digits $2 + \\text{a fraction}$, and so on.",
   "why": "Shows through the number of digits why the integer part of a common logarithm is what it is, a useful check a learner can apply by hand.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/characteristic",
    "concept/common-logarithm"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-13177f43e0",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "35",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "To avoid the use of negative numbers, the tabular logarithms of the circular functions are the common logarithms of these ratios increased by~$10$: which must be remembered in using the tables in calculations.",
   "markdown": "To avoid the use of negative numbers, the tabular logarithms of the circular functions are the common logarithms of these ratios increased by $10$: which must be remembered in using the tables in calculations.",
   "why": "Warns the learner that the printed values of trigonometric logarithms carry an offset of 10, a common source of error when using the tables.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/common-logarithm"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-0f54b14a5e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\First{A triangle} is said to be solved when the sides and angles are calculated from the data.",
   "markdown": "A triangle is said to be solved when the sides and angles are calculated from the data.",
   "why": "It states plainly what it means to solve a triangle, which frames every case that follows.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/triangle",
    "method/solving-a-triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-ed1933a036",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "Then if the side $AB$ be taken as radius, the ratio of $BC$ to $AB$ is the sine of~$A$; and of $AC$ to $AB$ is its cosine;",
   "markdown": "Then if the side $AB$ be taken as radius, the ratio of $BC$ to $AB$ is the sine of $A$; and of $AC$ to $AB$ is its cosine;",
   "why": "It defines sine and cosine as ratios of sides in a right-angled triangle, the basis for the right-angle formulas.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-adcd3e43d7",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "39",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "or \\emph{the sides are proportional to the sines of the opposite angles}.\nThis is true of all triangles.",
   "markdown": "or *the sides are proportional to the sines of the opposite angles*. This is true of all triangles.",
   "why": "It gives the law of sines in words, showing learners why the ratio of each side to the sine of its opposite angle is constant.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-c5971879f8",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "39",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "There will always be this ambiguity in determining the angle of a triangle from its sine, unless there be something to point out whether the angle is acute or obtuse: and therefore it is generally inconvenient to use a method involving the determination of the angle from its sine.",
   "markdown": "There will always be this ambiguity in determining the angle of a triangle from its sine, unless there be something to point out whether the angle is acute or obtuse: and therefore it is generally inconvenient to use a method involving the determination of the angle from its sine.",
   "why": "It warns that a sine alone does not fix an angle, since the sine of an angle and of its supplement are equal.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/ambiguous-case",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-f3aab8c4be",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "These relations give the simplest logarithmic solution of a triangle, when two sides and the contained angle are given",
   "markdown": "These relations give the simplest logarithmic solution of a triangle, when two sides and the contained angle are given",
   "why": "It names the method for two sides and the included angle as the simplest logarithmic route, showing learners how to choose a method.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/solving-a-triangle-from-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-4e4f4c5f69",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "The formulæ used here were discovered by William Purser of Dublin,\nin \\Date{1632}.",
   "markdown": "The formulæ used here were discovered by William Purser of Dublin, in 1632.",
   "why": "It credits the formulas for solving a triangle from its three sides to a named originator, giving the method a history.",
   "use": [
    "history"
   ],
   "concepts": [
    "method/solving-a-triangle-from-its-three-sides",
    "theorem/radius-of-the-inscribed-circle-from-the-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-947e75438f",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "There is also a simple verification of the process afforded by taking the sum of the three angles, which ought to be~$180°$",
   "markdown": "There is also a simple verification of the process afforded by taking the sum of the three angles, which ought to be $180°$",
   "why": "It gives learners a self-check for their answers: the three angles must total 180 degrees.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "method/verification",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-ee412313e9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "OF TRIGONOMETRICAL SURVEYING",
   "latex": "In Geodesy, or the application of this part of Trigonometry to surveying, a line, called \\emph{the Base}, is measured between two convenient stations; and the angles between lines from these stations to points visible from the stations are measured with appropriate instruments.",
   "markdown": "In Geodesy, or the application of this part of Trigonometry to surveying, a line, called *the Base*, is measured between two convenient stations; and the angles between lines from these stations to points visible from the stations are measured with appropriate instruments.",
   "why": "It states the whole method of a trigonometrical survey in one sentence: measure a base, then measure angles from its ends.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/base-line",
    "concept/geodesy",
    "concept/station"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-c2d4ed49ec",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "45",
   "location": "OF TRIGONOMETRICAL SURVEYING",
   "latex": "The horizontal angle between two objects is the angle between the vertical planes through the station and each object: it is also called the angle in azimuth.",
   "markdown": "The horizontal angle between two objects is the angle between the vertical planes through the station and each object: it is also called the angle in azimuth.",
   "why": "It separates the horizontal angle from the other angle classes, which is the first thing a learner must keep straight in surveying.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/horizontal-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-d47d9b8142",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "46",
   "location": "OF TRIGONOMETRICAL SURVEYING",
   "latex": "If the angle of elevation of $C$ be likewise measured at either station, the height of~$C$ above that station can be found by solving a right-angled triangle of which the hypotenuse and one of the acute angles are known.",
   "markdown": "If the angle of elevation of $C$ be likewise measured at either station, the height of $C$ above that station can be found by solving a right-angled triangle of which the hypotenuse and one of the acute angles are known.",
   "why": "It shows how a single measured angle of elevation and a right-angled triangle give a height that cannot be reached directly.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/angle-of-elevation",
    "method/solving-a-right-angled-triangle",
    "quantity/height"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-cf907767c0",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "45",
   "location": "OF TRIGONOMETRICAL SURVEYING",
   "latex": "For details of the measurement of a base, and a description of the instruments principally employed in practical surveying, the reader is referred to Professor Rankine's \\textit{Manual of Civil Engineering}, Part~\\osf{1}.",
   "markdown": "For details of the measurement of a base, and a description of the instruments principally employed in practical surveying, the reader is referred to Professor Rankine’s *Manual of Civil Engineering*, Part 1.",
   "why": "It shows the book sending the reader to a fuller practical source, which marks the chapter as a theoretical treatment of surveying rather than a manual.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/base-line",
    "concept/geodesy"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-7b534d5cff",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-viii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "47",
   "location": "OF TRIGONOMETRICAL SURVEYING",
   "latex": "Where the extent of the earth's surface surveyed is too great for the above supposition, the calculations have to be corrected for the sphericity of the earth (see books on Spherical Trigonometry), and in very great and very accurate surveys also for the earth's deviation from a perfect sphere.",
   "markdown": "Where the extent of the earth’s surface surveyed is too great for the above supposition, the calculations have to be corrected for the sphericity of the earth (see books on Spherical Trigonometry), and in very great and very accurate surveys also for the earth’s deviation from a perfect sphere.",
   "why": "It warns that flat-earth surveying breaks down over large areas, a limit a learner should know before trusting plane formulas.",
   "use": [
    "website",
    "lesson"
   ],
   "concepts": [
    "concept/geodesy"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-58ea15f04d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "47",
   "location": "OF PROJECTIONS",
   "latex": "\\First{The} point where the perpendicular from a given point on a given plane or a given line meets the plane or line is called the projection (or more precisely the orthogonal projection) of the point on the plane or line.",
   "markdown": "The point where the perpendicular from a given point on a given plane or a given line meets the plane or line is called the projection (or more precisely the orthogonal projection) of the point on the plane or line.",
   "why": "Gives the learner the precise definition of a projection as the foot of a perpendicular.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/projection"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-70ac2f58fe",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "47",
   "location": "OF PROJECTIONS",
   "latex": "It is convenient to take one direction of the line of projection (say from left to right) as the $+$ direction and the opposite as the $-$~direction;",
   "markdown": "It is convenient to take one direction of the line of projection (say from left to right) as the $+$ direction and the opposite as the $-$ direction;",
   "why": "Shows the sign convention that lets a projection carry a sign, which the later proofs depend on.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/line-of-projection",
    "concept/negative-direction",
    "concept/positive-direction"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-f881adea3e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "48",
   "location": "OF PROJECTIONS",
   "latex": "If $BAC$ be acute, $\\cos BAC$ is~$+$, and $A'B'$ is~$+$ and is measured from~$A'$ in the $+$~direction; but if $BAC$ be obtuse, $\\cos BAC$ is~$-$, and $A'B'$ is~$-$ and is measured from~$A'$ in the $-$~direction.",
   "markdown": "If $BAC$ be acute, $\\cos BAC$ is $+$, and $A'B'$ is $+$ and is measured from $A'$ in the $+$ direction; but if $BAC$ be obtuse, $\\cos BAC$ is $-$, and $A'B'$ is $-$ and is measured from $A'$ in the $-$ direction.",
   "why": "Explains why the sign of the cosine decides the direction of a projection, which is the point learners most often get wrong.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "concept/cosine",
    "concept/projection-of-a-line",
    "theorem/projection-of-a-line-on-a-line"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-9275af9721",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "48",
   "location": "OF PROJECTIONS",
   "latex": "The projection of a broken line is the algebraic sum of the projections of the parts of which it is made up.",
   "markdown": "The projection of a broken line is the algebraic sum of the projections of the parts of which it is made up.",
   "why": "States in one sentence why projections of pieces can be added with their signs, a result that simplifies many problems.",
   "use": [
    "lesson",
    "website"
   ],
   "concepts": [
    "concept/algebraic-sum",
    "concept/broken-line",
    "theorem/projection-of-a-broken-line"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-112f3e089e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "48",
   "location": "OF PROJECTIONS",
   "latex": "both in the case where all these projections are~$+$ and also where, as in the figure, $B'C'$~is~$-$.",
   "markdown": "both in the case where all these projections are $+$ and also where, as in the figure, $B'C'$ is $-$.",
   "why": "Warns that the sum rule still holds when one projection is negative, so learners should not assume all parts point the same way.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/negative-direction",
    "theorem/projection-of-a-broken-line"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-3caf928c41",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "50",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "The four expressions for $\\sin (\\theta ± \\phi)$ and $\\cos (\\theta ± \\phi)$ are true for all values of $\\theta$~and~$\\phi$, though the diagrams suppose the angles all acute. They form the fundamental formulæ of Analytical Trigonometry.",
   "markdown": "The four expressions for $\\sin (\\theta ± \\phi)$ and $\\cos (\\theta ± \\phi)$ are true for all values of $\\theta$ and $\\phi$, though the diagrams suppose the angles all acute. They form the fundamental formulæ of Analytical Trigonometry.",
   "why": "It tells the learner that the four formulae hold for all angles, not only acute ones, even though the diagrams show acute angles.",
   "use": [
    "lesson",
    "history"
   ],
   "concepts": [
    "theorem/cosine-of-the-difference-of-two-angles",
    "theorem/cosine-of-the-sum-of-two-angles",
    "theorem/sine-of-the-difference-of-two-angles",
    "theorem/sine-of-the-sum-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-a829afd06a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "49",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "Then $\\theta + \\phi$ is the circular measure of~$ACD$.",
   "markdown": "Then $\\theta + \\phi$ is the circular measure of $ACD$.",
   "why": "It shows that angles add by adding their circular measures, which is the step the whole derivation depends on.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circular-measure",
    "theorem/sine-of-the-sum-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-3ffa55dd66",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "50",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "Then $\\theta - \\phi =$ circular measure of $ACD$.",
   "markdown": "Then $\\theta - \\phi =$ circular measure of $ACD$.",
   "why": "It gives the matching step for the difference formulae, so a learner can see that subtraction of measures is handled the same way as addition.",
   "use": [
    "lesson"
   ],
   "concepts": [
    "concept/circular-measure",
    "theorem/sine-of-the-difference-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-6d26e2b2db",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "51",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "To shew that, when the difference between $R$~and~$r$ is small, the limit of each radius is very nearly $= \\dfrac{1}{3}(r + 2R)$.",
   "markdown": "To shew that, when the difference between $R$ and $r$ is small, the limit of each radius is very nearly $= \\dfrac{1}{3}(r + 2R)$.",
   "why": "It states the appendix's aim, a historical approximation for the common limit of inscribed and circumscribed radii, in the words of the period.",
   "use": [
    "history"
   ],
   "concepts": [
    "concept/limit",
    "quantity/radius",
    "theorem/limiting-radius-of-inscribed-and-circumscribed-circles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/x-44e87e3591",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "52",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "So that when, as in the text, $R - r < .00003$ the error in taking the ultimate radius $= \\dfrac{1}{3}(r + 2R)$ is $< \\dfrac{.0000000009}{45r}$ which does not affect the tenth decimal place.",
   "markdown": "So that when, as in the text, $R - r < .00003$ the error in taking the ultimate radius $= \\dfrac{1}{3}(r + 2R)$ is $< \\dfrac{.0000000009}{45r}$ which does not affect the tenth decimal place.",
   "why": "It shows how the error bound was used to reach ten decimal places by hand, a vivid picture of nineteenth-century computation.",
   "use": [
    "history",
    "website"
   ],
   "concepts": [
    "quantity/radius",
    "theorem/limiting-radius-of-inscribed-and-circumscribed-circles"
   ]
  }
 ],
 "equations": [
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-f24cdb358c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "3",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\text{the circumference $÷$ the diameter $= \\pi$,}",
   "name": null,
   "statement": "The ratio of the circumference of any circle to its diameter is one fixed number, denoted π.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference of a circle to its diameter"
    }
   ],
   "sympy": "Eq(C/D, pi)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/diameter",
    "quantity/circumference",
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-00af57c3af",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "3",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\dfrac{\\pi}{2}",
   "name": null,
   "statement": "The ratio of the semi-circumference to the diameter, common to all circles, is denoted π/2 as is customary.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference of a circle to its diameter"
    }
   ],
   "sympy": "Eq(S/D, pi/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "concept/diameter",
    "quantity/circumference",
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-ee246409b8",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "3",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\text{and the circumference $= 2\\pi R$.}",
   "name": null,
   "statement": "The circumference of a circle equals 2π times its radius.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "C",
     "meaning": "circumference of the circle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "Eq(C, 2*pi*R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "quantity/circumference",
    "quantity/pi",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-30d8edb714",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "4",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "AOC : AOB :: AC : AB",
   "name": null,
   "statement": "Angles at the centre of a circle are in the same ratio as the arcs they subtend (Euclid VI. 33).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AOC",
     "meaning": "angle at the centre O subtended by arc AC"
    },
    {
     "unit": null,
     "symbol": "AOB",
     "meaning": "angle at the centre O subtended by arc AB"
    },
    {
     "unit": null,
     "symbol": "AC",
     "meaning": "arc equal to the radius"
    },
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "arc subtending a right angle at the centre (a quarter of the circumference)"
    }
   ],
   "sympy": "Eq(AOC/AOB, AC/AB)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "concept/proportion",
    "quantity/angle",
    "quantity/arc-of-a-circle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a259018857",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "4",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "AOC = \\frac{2}{\\pi} × \\text{a right angle},",
   "name": null,
   "statement": "The angle subtended at the centre by an arc equal to the radius is 2/π of a right angle, the same fixed fraction for all circles.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AOC",
     "meaning": "angle at the centre subtended by an arc equal to the radius"
    },
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "Eq(AOC, 2/pi*RA)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/angle",
    "quantity/arc-of-a-circle",
    "quantity/pi",
    "quantity/right-angle",
    "unit/radian"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-500b5b38e9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "6",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\dfrac{1}{2}\\, R × \\text{circumference} = \\pi R^2",
   "name": null,
   "statement": "The area of a circle is half the radius times the circumference, which equals π times the square of the radius.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A",
     "meaning": "area of the circle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "Eq(A, pi*R**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-sector",
    "quantity/area",
    "quantity/circumference",
    "quantity/pi",
    "quantity/radius",
    "theorem/area-of-a-circle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-805c551708",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "6",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "= \\dfrac{\\pi}{4}",
   "name": null,
   "statement": "The ratio of the area of a circle to the square on its diameter is π/4.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "Eq(A/D**2, pi/4)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-ratio",
    "concept/diameter",
    "quantity/pi",
    "theorem/area-of-a-circle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-61e3fce184",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "8",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "r' = \\frac{R + r}{2}",
   "name": null,
   "statement": "The inscribed radius of the second polygon (with twice the sides) is the arithmetic mean of the inscribed and circumscribed radii of the first polygon.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "r'",
     "meaning": "radius of the circle inscribed in the second polygon"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle described about the first polygon"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle inscribed in the first polygon"
    }
   ],
   "sympy": "Eq(r_prime, (R + r)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/arithmetical-mean",
    "concept/inscribed-circle",
    "concept/regular-polygon",
    "method/approximating-pi-by-inscribed-and-circumscribed-polygons",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-b35d2bcc52",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "8",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "R' = \\sqrt{r' · R}",
   "name": null,
   "statement": "The circumscribed radius of the second polygon is the geometric mean of its inscribed radius and the circumscribed radius of the first polygon.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "R'",
     "meaning": "radius of the circle described about the second polygon"
    },
    {
     "unit": null,
     "symbol": "r'",
     "meaning": "radius of the circle inscribed in the second polygon"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle described about the first polygon"
    }
   ],
   "sympy": "Eq(R_prime, sqrt(r_prime*R))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circumscribed-circle",
    "concept/geometrical-mean",
    "concept/regular-polygon",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-51c08a8c4d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "8",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "EC: EF:: EF: EG",
   "name": null,
   "statement": "The circumscribed radius of the second polygon is a mean proportional between the circumscribed radius of the first polygon and the inscribed radius of the second.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "EC",
     "meaning": "radius of the circle described about the first polygon"
    },
    {
     "unit": null,
     "symbol": "EF",
     "meaning": "radius of the circle described about the second polygon"
    },
    {
     "unit": null,
     "symbol": "EG",
     "meaning": "radius of the circle inscribed in the second polygon"
    }
   ],
   "sympy": "Eq(EC/EF, EF/EG)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/geometrical-mean",
    "concept/proportion",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-7bd8cbfc6a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "10",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\dfrac{1}{3} (r + 2R)",
   "name": null,
   "statement": "When R and r nearly agree, (r + 2R)/3 is a very close approximation to the common limit of the two radii.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the circle inscribed in the polygon"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle described about the polygon"
    }
   ],
   "sympy": "Eq(L, (r + 2*R)/3)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circumscribed-circle",
    "concept/inscribed-circle",
    "concept/limit",
    "quantity/radius",
    "theorem/limiting-radius-of-inscribed-and-circumscribed-circles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-ce2039ce2d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "11",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\pi = \\frac{20000000000}{6366197723}",
   "name": null,
   "statement": "π may be taken as 20000000000/6366197723, which equals 3.141592654 to the stated accuracy.",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "Eq(pi, 20000000000/6366197723)",
   "physics": false,
   "states": [],
   "concepts": [
    "method/approximating-pi-by-inscribed-and-circumscribed-polygons",
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-31d17b22a1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-i",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "10",
   "location": "OF THE MENSURATION OF THE CIRCLE",
   "latex": "\\frac{2000000}{636621} < \\pi < \\frac{2000000}{636617}",
   "name": null,
   "statement": "Stopping at the 1024-sided polygon, π is bracketed between these two fractions (the transcription also carries a differing DPchg form, π < 2000000/636617 > 2000000/636621, which is flagged for review).",
   "kind": "approximation",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": "And(Lt(2000000/636621, pi), Lt(pi, 2000000/636617))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inequality",
    "method/approximating-pi-by-inscribed-and-circumscribed-polygons",
    "quantity/pi"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a4e77e2c7a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "\\text{Then, numerically, the Area} = rs.",
   "name": null,
   "statement": "The area of the triangle equals the radius of the inscribed circle times the semi-perimeter.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "Area",
     "meaning": "area of the triangle"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    }
   ],
   "sympy": "Eq(Area, r*s)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "quantity/area",
    "quantity/radius",
    "quantity/semi-perimeter",
    "theorem/triangle-area-as-semi-perimeter-times-inradius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-505da7b5c9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "Ab = Ac = s - a",
   "name": null,
   "statement": "The two tangent lengths from vertex A to the inscribed circle are equal, and each equals the semi-perimeter minus the side a opposite A.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "Ab",
     "meaning": "length of the tangent from vertex A to the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "Ac",
     "meaning": "length of the tangent from vertex A to the inscribed circle, the other tangent from A"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side opposite angle A"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "concept/side",
    "concept/tangent",
    "quantity/semi-perimeter"
   ],
   "pages": [
    "12",
    "43"
   ],
   "chapters": [
    "blackburn-elements-plane-trigonometry-1863/ch-ii",
    "blackburn-elements-plane-trigonometry-1863/ch-vii"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-78d14cf27b",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "Bc = Ba = s-b",
   "name": null,
   "statement": "The two tangent lengths from vertex B to the inscribed circle are equal, and each equals the semi-perimeter minus the side b opposite B.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "Bc",
     "meaning": "length of the tangent from vertex B to the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "Ba",
     "meaning": "length of the tangent from vertex B to the inscribed circle, the other tangent from B"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side opposite angle B"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "concept/side",
    "concept/tangent",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-20d227a15d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "12",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "Ca = Cb = s - c",
   "name": null,
   "statement": "The two tangent lengths from vertex C to the inscribed circle are equal, and each equals the semi-perimeter minus the side c opposite C.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "Ca",
     "meaning": "length of the tangent from vertex C to the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "Cb",
     "meaning": "length of the tangent from vertex C to the inscribed circle, the other tangent from C"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side opposite angle C"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "concept/side",
    "concept/tangent",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cebebab2d5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "13",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "(s - a) \\alpha = r s = \\text{the area}.",
   "name": null,
   "statement": "The area of the triangle equals both the excircle radius on side a times s minus a, and the inradius times the semi-perimeter.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\alpha",
     "meaning": "radius of the circle excribed on side BC"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "the area",
     "meaning": "area of the triangle"
    }
   ],
   "sympy": "Eq((s - a)*alpha, r*s)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/excircle",
    "concept/inscribed-circle",
    "quantity/area",
    "quantity/radius",
    "quantity/semi-perimeter",
    "theorem/triangle-area-as-excircle-radius-times-tangent"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-4f0b26c1ab",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "14",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "s(s-b) : \\Delta :: \\Delta : (s-a)(s-c)",
   "name": null,
   "statement": "The area is a mean proportional between s(s-b) and (s-a)(s-c).",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    },
    {
     "unit": "square units",
     "symbol": "\\Delta",
     "meaning": "area of the triangle in square units"
    }
   ],
   "sympy": "Eq(s*(s - b)/Delta, Delta/((s - a)*(s - c)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/geometrical-mean",
    "concept/proportion",
    "quantity/area",
    "quantity/semi-perimeter",
    "theorem/triangle-area-as-a-mean-proportional"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e6a8b39f8d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "14",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "\\Delta^2 = s(s-a)(s-b)(s-c)",
   "name": "Heron's formula",
   "statement": "The square of the area of a triangle equals the semi-perimeter times the three differences between the semi-perimeter and each side.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "square units",
     "symbol": "\\Delta",
     "meaning": "area of the triangle in square units"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side opposite angle A, BC"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side opposite angle B, CA"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side opposite angle C, AB"
    }
   ],
   "sympy": "Eq(Delta**2, s*(s - a)*(s - b)*(s - c))",
   "physics": false,
   "states": [
    "theorem/heron-s-formula"
   ],
   "concepts": [
    "concept/theorem-heron-s-formula",
    "concept/triangle",
    "quantity/area",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-8afd4f2418",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "14",
   "location": "OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE",
   "latex": "r^2 = \\dfrac{(s-a)(s-b)(s-c)}{s}",
   "name": null,
   "statement": "The square of the inscribed circle's radius equals the product of the three differences between the semi-perimeter and the sides, divided by the semi-perimeter.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "semi-perimeter of the triangle"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side opposite angle C"
    }
   ],
   "sympy": "Eq(r**2, (s - a)*(s - b)*(s - c)/s)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "quantity/radius",
    "quantity/semi-perimeter",
    "theorem/radius-of-the-inscribed-circle-from-the-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e03b6609e5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "15",
   "location": "OF SYMBOLS OF QUANTITY",
   "latex": "= (a - b)",
   "name": null,
   "statement": "The signed position of B relative to O equals a - b miles, read as eastward when a - b is positive and westward when it is negative, so one formula covers both cases of a > b and a < b.",
   "kind": "rule",
   "symbols": [
    {
     "unit": "mile",
     "symbol": "OB",
     "meaning": "signed distance of B from O along the standard (east) direction"
    },
    {
     "unit": "mile",
     "symbol": "a",
     "meaning": "distance from fixed point O to A along the standard direction"
    },
    {
     "unit": "mile",
     "symbol": "b",
     "meaning": "distance cut off from A in the opposite (westward) direction to reach B"
    }
   ],
   "sympy": "Eq(OB, a - b)",
   "physics": true,
   "states": [],
   "concepts": [
    "concept/algebraic-number",
    "concept/difference",
    "concept/negative-direction",
    "concept/origin",
    "concept/positive-direction",
    "quantity/distance"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-61c6a68755",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "17",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\therefore 1° = 60'",
   "name": null,
   "statement": "A degree is sixty minutes of arc.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "°",
     "meaning": "degree of angle"
    },
    {
     "unit": "minute",
     "symbol": "'",
     "meaning": "minute of arc"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sexagesimal-system",
    "unit/degree-of-angle",
    "unit/minute-of-arc"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-081328cd8f",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "17",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\therefore 1' =  60''",
   "name": null,
   "statement": "A minute of arc is sixty seconds of arc.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "minute",
     "symbol": "'",
     "meaning": "minute of arc"
    },
    {
     "unit": "second",
     "symbol": "''",
     "meaning": "second of arc"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sexagesimal-system",
    "unit/minute-of-arc",
    "unit/second-of-arc"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-81e5fc28a5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "18",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "1'' = 60'''",
   "name": null,
   "statement": "In older books a second of arc is divided sexagesimally into sixtieths, the third-order subdivision.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "second",
     "symbol": "''",
     "meaning": "second of arc"
    },
    {
     "unit": null,
     "symbol": "'''",
     "meaning": "third of arc (sixtieth of a second)"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sexagesimal-system",
    "unit/second-of-arc"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-3570ea0d60",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "18",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "1''' = 60^\\text{iv}",
   "name": null,
   "statement": "In older books the sixtieth of a third of arc is the fourth-order subdivision.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "'''",
     "meaning": "third of arc"
    },
    {
     "unit": null,
     "symbol": "^\\text{iv}",
     "meaning": "fourth-order sexagesimal subdivision of the angle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sexagesimal-system"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-1777a0cfbf",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "18",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "1^\\text{iv} = 60^\\text{v}",
   "name": null,
   "statement": "In older books the sixtieth of a fourth-order subdivision is the fifth-order subdivision.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "^\\text{iv}",
     "meaning": "fourth-order sexagesimal subdivision of the angle"
    },
    {
     "unit": null,
     "symbol": "^\\text{v}",
     "meaning": "fifth-order sexagesimal subdivision of the angle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/sexagesimal-system"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-ae67d7b780",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\theta = \\frac{AB}{AC} = \\frac{AB}{R}",
   "name": null,
   "statement": "The circular measure of an angle equals the arc subtending it at the centre divided by the radius, since the unit angle subtends an arc equal to the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "arc subtending the angle AOB at the centre"
    },
    {
     "unit": null,
     "symbol": "AC",
     "meaning": "arc subtending the unit angle AOC, equal to the radius"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius OA of the circle"
    }
   ],
   "sympy": "Eq(theta, AB/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/angle",
    "quantity/arc-of-a-circle",
    "quantity/radius",
    "unit/radian"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-710183bf7c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "AB = R\\theta",
   "name": null,
   "statement": "The arc subtending an angle at the centre equals the radius times the circular measure of the angle.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "arc subtending the angle AOB at the centre"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(AB, R*theta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/arc-of-a-circle",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-00ded65979",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\frac{AB}{R} = \\frac{A'B'}{R'}",
   "name": null,
   "statement": "The ratio of an arc to its radius is the same for any circle, because the same angle subtends both arcs.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "arc subtending an angle at the centre of the first circle"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the first circle"
    },
    {
     "unit": null,
     "symbol": "A'B'",
     "meaning": "arc subtending the same angle at the centre of the second circle"
    },
    {
     "unit": null,
     "symbol": "R'",
     "meaning": "radius of the second circle"
    }
   ],
   "sympy": "Eq(AB/R, ApBp/Rp)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/arc-of-a-circle",
    "quantity/radius"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-07b37c0fc2",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\frac{\\frac{1}{4} \\text{ circumference}}{R} = \\frac{\\pi}{2}",
   "name": null,
   "statement": "The circular measure of a right angle is a quarter of the circumference divided by the radius, which equals pi over two.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/circumference",
    "quantity/pi",
    "quantity/radius",
    "quantity/right-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-174c0099f0",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "= \\frac{180°}{\\pi}",
   "name": null,
   "statement": "The number of degrees in one unit of circular measure is 180 degrees divided by pi.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "π",
     "meaning": "ratio of the circumference to the diameter"
    },
    {
     "unit": "degree",
     "symbol": "°",
     "meaning": "degree of angle"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "quantity/pi",
    "unit/degree-of-angle",
    "unit/radian"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-72c1018539",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\theta > \\dfrac{\\pi}{2}",
   "name": null,
   "statement": "When the angle exceeds a right angle in circular measure, its complement is negative and measured in the negative direction.",
   "kind": "rule",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Gt(theta, pi/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "concept/complementary-angles",
    "concept/negative-direction",
    "quantity/right-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-fb2ea2fe2a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-iv",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "19",
   "location": "OF THE UNIT OF ANGULAR MAGNITUDE",
   "latex": "\\theta > \\pi",
   "name": null,
   "statement": "When the angle exceeds two right angles in circular measure, its supplement is negative and measured in the negative direction.",
   "kind": "rule",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "θ",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Gt(theta, pi)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-measure",
    "concept/negative-direction",
    "concept/supplementary-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-2ee5101c5c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "21",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin\\theta = \\dfrac{BD}{R}",
   "name": null,
   "statement": "The sine of the angle AOB is the perpendicular BD from the end of the arc onto the initial line OA, divided by the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "BD",
     "meaning": "perpendicular from B onto the initial line OA"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(sin(theta), BD/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/circular-measure",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-b78f2b353e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "21",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cos\\theta = \\sin\\left(\\frac{\\pi}{2} - \\theta\\right).",
   "name": null,
   "statement": "The cosine of an angle is the sine of its complement.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    }
   ],
   "sympy": "Eq(cos(theta), sin(pi/2 - theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/complementary-angles",
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-2b466fdea5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "21",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin\\theta = \\cos\\left(\\frac{\\pi}{2} - \\theta\\right).",
   "name": null,
   "statement": "The sine of an angle is the cosine of its complement.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    }
   ],
   "sympy": "Eq(sin(theta), cos(pi/2 - theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/complementary-angles",
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-bcc06dae35",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "21",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cos\\theta = \\sin COB = \\frac{BE}{R} = \\frac{OD}{R}.",
   "name": null,
   "statement": "The cosine of the angle AOB equals the sine of its complement COB, which in the diagram is the ratio OD to the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "OD",
     "meaning": "projection of OB onto the initial line OA"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(cos(theta), OD/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/complementary-angles",
    "concept/cosine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-7f87858dba",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "22",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin\\theta &= \\sin(2m\\pi + \\theta);",
   "name": null,
   "statement": "Adding a whole number multiple of 2π to the angle leaves the sine unchanged.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "any whole number"
    }
   ],
   "sympy": "Eq(sin(theta), sin(2*m*pi + theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/periodic-function",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-7bdb332822",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "22",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cos\\theta &= \\cos(2m\\pi + \\theta).",
   "name": null,
   "statement": "Adding a whole number multiple of 2π to the angle leaves the cosine unchanged.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    },
    {
     "unit": null,
     "symbol": "m",
     "meaning": "any whole number"
    }
   ],
   "sympy": "Eq(cos(theta), cos(2*m*pi + theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/periodic-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-908c8bb301",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "23",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin(-\\theta) &= -\\sin\\theta;",
   "name": null,
   "statement": "The sine of the negative of an angle is the negative of its sine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(sin(-theta), -sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/negative-direction",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-787c1b8b8d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "23",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cos(-\\theta) &= +\\cos\\theta.",
   "name": null,
   "statement": "The cosine of the negative of an angle equals the cosine of the angle.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(cos(-theta), cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/negative-direction"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-4a098e8ee6",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "23",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cos(\\pi - \\theta) &= -\\cos\\theta;",
   "name": null,
   "statement": "The cosine of π minus an angle is the negative of the cosine of the angle.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(cos(pi - theta), -cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/supplementary-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-9f3481315a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "25",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\tan\\theta = \\frac{AF}{R} \\text{ and } AF = R\\tan\\theta.",
   "name": null,
   "statement": "The tangent of an angle is the length AF cut off on the tangent at the initial end of the arc, divided by the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "AF",
     "meaning": "part of the tangent at A intercepted by the radius produced"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(tan(theta), AF/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/tangent",
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-889fd6d24e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "28",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\tan(-\\theta) = -\\tan\\theta; \\text{ and } \\cot(-\\theta) = -\\cot\\theta.",
   "name": null,
   "statement": "The tangent and cotangent of the negative of an angle are the negatives of those of the angle.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(tan(-theta), -tan(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cotangent",
    "concept/negative-direction",
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-8ef7675e75",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "28",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\tan(\\pi - \\theta) = -\\tan\\theta; \\quad \\cot(\\pi - \\theta) = -\\cot\\theta.",
   "name": null,
   "statement": "For supplementary angles the tangent and cotangent are equal in size and opposite in sign.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(tan(pi - theta), -tan(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cotangent",
    "concept/supplementary-angles",
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e9acdf5af5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "26",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cot\\theta = \\frac{CG}{R} \\text{ and } CG = R\\cot\\theta.",
   "name": null,
   "statement": "The cotangent of an angle is the length CG cut off on the tangent at C to the arc of the complement, divided by the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "CG",
     "meaning": "part of the tangent at C intercepted by the radius produced"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(cot(theta), CG/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/complementary-angles",
    "concept/cotangent"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-75bf12b854",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "29",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sec\\theta = \\frac{OS}{R};\\quad \\cosec\\theta = \\frac{OK}{R};",
   "name": null,
   "statement": "The secant and cosecant of an angle are the lengths OS and OK, cut off by the tangent at B, divided by the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "OS",
     "meaning": "segment of the initial radius produced to meet the tangent at B"
    },
    {
     "unit": null,
     "symbol": "OK",
     "meaning": "segment of the line COB produced to meet the tangent at B"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(sec(theta), OS/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosecant",
    "concept/secant"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-1206a8ee44",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "31",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\versin\\theta = \\frac{AD}{R}; \\text{ and } AD = R\\versin\\theta.",
   "name": null,
   "statement": "The versed sine of an angle is the segment AD cut off on the initial radius by the perpendicular from B, divided by the radius.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    },
    {
     "unit": null,
     "symbol": "AD",
     "meaning": "part of the initial radius between A and the perpendicular from B"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    }
   ],
   "sympy": "Eq(versin(theta), AD/R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/versine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-3b226912de",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "R^2 = R^2 \\sin^2\\theta + R^2 \\cos^2\\theta,",
   "name": null,
   "statement": "The square on the radius equals the sum of the squares on the sine and cosine lines, by the Pythagorean theorem applied to the right triangle BOD.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle"
    },
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle AOB"
    }
   ],
   "sympy": "Eq(R**2, R**2*sin(theta)**2 + R**2*cos(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "theorem/pythagorean-theorem"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-36881400d3",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin^2\\theta + \\cos^2\\theta = 1\\Add{.}",
   "name": "fundamental trigonometric identity",
   "statement": "The square of the sine plus the square of the cosine of any angle equals one.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(sin(theta)**2 + cos(theta)**2, 1)",
   "physics": false,
   "states": [
    "theorem/trigonometric-identity"
   ],
   "concepts": [
    "concept/cosine",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-6f209ec7e7",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sec^2\\theta = 1 + \\tan^2\\theta\\Add{.}",
   "name": null,
   "statement": "The square of the secant equals one plus the square of the tangent.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(sec(theta)**2, 1 + tan(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/secant",
    "concept/tangent-function",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-5b753037f6",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cosec^2\\theta = 1 + \\cot^2\\theta\\Add{.}",
   "name": null,
   "statement": "The square of the cosecant equals one plus the square of the cotangent.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(csc(theta)**2, 1 + cot(theta)**2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosecant",
    "concept/cotangent",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cf54a66496",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}",
   "name": null,
   "statement": "The tangent of an angle is its sine divided by its cosine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(tan(theta), sin(theta)/cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/sine",
    "concept/tangent-function",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-472bd8a669",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sec\\theta = \\frac{1}{\\cos\\theta}\\Add{.}",
   "name": null,
   "statement": "The secant of an angle is the reciprocal of its cosine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(sec(theta), 1/cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/reciprocal",
    "concept/secant"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a3d11a0644",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cotan\\theta = \\frac{\\cos\\theta}{\\sin\\theta} &= \\frac{1}{\\tan\\theta}",
   "name": null,
   "statement": "The cotangent of an angle is its cosine over its sine, and the reciprocal of its tangent.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(cot(theta), cos(theta)/sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cotangent",
    "concept/tangent-function",
    "theorem/trigonometric-identity"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-685f1d6298",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\cosec\\theta &= \\frac{1}{\\sin\\theta}",
   "name": null,
   "statement": "The cosecant of an angle is the reciprocal of its sine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(csc(theta), 1/sin(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosecant",
    "concept/reciprocal",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-8c0da28a70",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "32",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\versin\\theta = 1 - \\cos\\theta\\Add{.}",
   "name": null,
   "statement": "The versed sine of an angle equals one minus its cosine.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of the angle"
    }
   ],
   "sympy": "Eq(versin(theta), 1 - cos(theta))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/versine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-fea2f16b85",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin \\frac{\\pi}{4} = \\sin 45° = \\frac{1}{2}\\sqrt{2} = \\cos 45° = \\cos \\frac{\\pi}{4}.",
   "name": null,
   "statement": "The sine and cosine of 45° (π/4) both equal √2/2, from the side of the inscribed square.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\pi",
     "meaning": "ratio of circumference to diameter"
    }
   ],
   "sympy": "Eq(sin(pi/4), sqrt(2)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/regular-polygon",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-9f83e647d4",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin \\frac{\\pi}{6} = \\sin 30° = \\frac{1}{2} = \\cos 60° = \\cos \\frac{\\pi}{3}.",
   "name": null,
   "statement": "The sine of 30° (π/6) equals one half, from the side of the inscribed hexagon.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\pi",
     "meaning": "ratio of circumference to diameter"
    }
   ],
   "sympy": "Eq(sin(pi/6), Rational(1,2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/regular-polygon",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-d549076b45",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin \\frac{\\pi}{3} = \\sin 60° = \\frac{1}{2}\\sqrt{3} = \\cos 30° = \\cos \\frac{\\pi}{6}.",
   "name": null,
   "statement": "The sine of 60° (π/3) equals √3/2, from the side of the inscribed equilateral triangle.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\pi",
     "meaning": "ratio of circumference to diameter"
    }
   ],
   "sympy": "Eq(sin(pi/3), sqrt(3)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/regular-polygon",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-c88ca72782",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-v",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "24",
   "location": "CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS",
   "latex": "\\sin \\frac{\\pi}{10} = \\sin 18° = \\dfrac{\\sqrt{5} - 1}{4} = \\cos \\frac{4\\pi}{10} = \\cos 72°.",
   "name": null,
   "statement": "The sine of 18° (π/10) equals (√5 − 1)/4, from the side of the inscribed regular decagon.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\pi",
     "meaning": "ratio of circumference to diameter"
    }
   ],
   "sympy": "Eq(sin(pi/10), (sqrt(5) - 1)/4)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/regular-polygon",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-44fcb7bd2e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "33",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log m - \\log n = \\log(m ÷ n)",
   "name": null,
   "statement": "The difference of the logarithms of two numbers is the logarithm of their quotient.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    }
   ],
   "sympy": "Eq(log(m) - log(n), log(m/n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "theorem/logarithm-of-a-quotient"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-48c221fe6c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "33",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log m + \\log n = \\log (m × n)",
   "name": null,
   "statement": "The sum of the logarithms of two numbers is the logarithm of their product.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    }
   ],
   "sympy": "Eq(log(m) + log(n), log(m*n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "theorem/logarithm-of-a-product"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-401be09736",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log 1 = 0",
   "name": null,
   "statement": "The logarithm of 1 is zero.",
   "kind": "result",
   "symbols": [],
   "sympy": "Eq(log(1), 0)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/logarithm",
    "theorem/logarithm-of-1-is-zero"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cf316b3734",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log n^x = x \\log n",
   "name": null,
   "statement": "The logarithm of a power of n equals the exponent times the logarithm of n, for integer or fractional exponent x.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "an exponent, integer or fraction"
    }
   ],
   "sympy": "Eq(log(n**x), x*log(n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/logarithm",
    "theorem/logarithm-of-a-power"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-9535ca31a5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log n^p = p \\log n",
   "name": null,
   "statement": "For an integer p, the logarithm of n raised to the power p is p times the logarithm of n.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "an integer exponent"
    }
   ],
   "sympy": "Eq(log(n**p), p*log(n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/logarithm",
    "theorem/logarithm-of-a-power"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-85b58e3c73",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log n^{\\tfrac{p}{q}} = \\frac{p}{q} × \\log n",
   "name": null,
   "statement": "The logarithm of n raised to a fractional power p/q is p/q times the logarithm of n.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "p",
     "meaning": "numerator of the fractional exponent"
    },
    {
     "unit": null,
     "symbol": "q",
     "meaning": "denominator of the fractional exponent"
    }
   ],
   "sympy": "Eq(log(n**(p/q)), (p/q)*log(n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/exponent",
    "concept/logarithm",
    "theorem/logarithm-of-a-power"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a3b4ebd3b9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log_a n = x, \\text{ when } a^x = n",
   "name": null,
   "statement": "The logarithm of n to base a is the number x such that a raised to the power x equals n.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "the base of the system of logarithms"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    },
    {
     "unit": null,
     "symbol": "x",
     "meaning": "the logarithm of n to base a"
    }
   ],
   "sympy": "Eq(log(n, a), x)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/base-of-a-logarithm-system",
    "concept/logarithm"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-566f2e71f9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "34",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log 10 = 1",
   "name": null,
   "statement": "The common logarithm of 10 is 1, since common logarithms have base 10.",
   "kind": "result",
   "symbols": [],
   "sympy": "Eq(log(10), 1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/base-of-a-logarithm-system",
    "concept/common-logarithm",
    "concept/logarithm"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-928bc05ad1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "35",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log (10^m × n) &= m + \\log n",
   "name": null,
   "statement": "Multiplying a number by a power of 10 adds that integer power to its logarithm, changing only the characteristic.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "an integer power of 10"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    }
   ],
   "sympy": "Eq(log(10**m*n), m + log(n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/characteristic",
    "concept/common-logarithm",
    "theorem/logarithm-of-a-product"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e58183508e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vi",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "35",
   "location": "OF LOGARITHMIC TABLES",
   "latex": "\\log (n \\div 10^m) &= -m + \\log n",
   "name": null,
   "statement": "Dividing a number by a power of 10 subtracts that integer power from its logarithm, changing only the characteristic.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "m",
     "meaning": "an integer power of 10"
    },
    {
     "unit": null,
     "symbol": "n",
     "meaning": "a number"
    }
   ],
   "sympy": "Eq(log(n/10**m), -m + log(n))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/characteristic",
    "concept/common-logarithm",
    "theorem/logarithm-of-a-quotient"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-b60511c0fe",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "a &= c \\sin A",
   "name": null,
   "statement": "In a right triangle with hypotenuse c, the side a opposite A equals c times the sine of A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(a, c*sin(A))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/hypotenuse",
    "concept/right-triangle",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-9fee30e497",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "b &= c \\cos A",
   "name": null,
   "statement": "In a right triangle, the side b adjacent to A equals the hypotenuse c times the cosine of A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, adjacent to angle A"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(b, c*cos(A))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/right-triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-2b468bec3a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "b &= c \\sin B",
   "name": null,
   "statement": "In a right triangle, the side b opposite B equals the hypotenuse c times the sine of B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(b, c*sin(B))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/right-triangle",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-953186424a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "a &= c \\cos B",
   "name": null,
   "statement": "In a right triangle, the side a adjacent to B equals the hypotenuse c times the cosine of B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, adjacent to angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(a, c*cos(B))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/right-triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-7209e8a3c6",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "a &= b \\tan A",
   "name": null,
   "statement": "In a right triangle, the side a opposite A equals the side b adjacent to A times the tangent of A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, adjacent to angle A"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(a, b*tan(A))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/right-triangle",
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-0749ad0934",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "c &= b \\sec A",
   "name": null,
   "statement": "In a right triangle, the hypotenuse c equals the side b adjacent to A times the secant of A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, adjacent to angle A"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(c, b*sec(A))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/right-triangle",
    "concept/secant"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-1ab0022818",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "b &= a \\tan B",
   "name": null,
   "statement": "In a right triangle, the side b opposite B equals the side a adjacent to B times the tangent of B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, adjacent to angle B"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(b, a*tan(B))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/right-triangle",
    "concept/tangent-function"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-ab6adceb96",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "c &= a \\sec B",
   "name": null,
   "statement": "In a right triangle, the hypotenuse c equals the side a adjacent to B times the secant of B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, adjacent to angle B"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(c, a*sec(B))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/right-triangle",
    "concept/secant"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-0d0c5585cf",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log a &= \\log c + \\tab\\log \\sin A - 10",
   "name": null,
   "statement": "Common logarithm of side a equals the common logarithm of c plus the common logarithm of sin A, less 10, used to solve a right triangle given c and A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(log(a, 10), log(c, 10) + log(sin(A), 10) - 10)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/sine",
    "method/solving-a-right-angled-triangle-logarithmically"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cb9f987031",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log b &= \\log c + \\tab\\log \\cos A - 10",
   "name": null,
   "statement": "Common logarithm of side b equals the common logarithm of c plus the common logarithm of cos A, less 10, used to solve a right triangle given c and A.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, adjacent to angle A"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(log(b, 10), log(c, 10) + log(cos(A), 10) - 10)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/cosine",
    "method/solving-a-right-angled-triangle-logarithmically"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-01855a06d9",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log b &= \\log a + \\tab\\log \\tan B - 10",
   "name": null,
   "statement": "Common logarithm of side b equals the common logarithm of a plus the common logarithm of tan B, less 10, used to solve a right triangle given a and B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, adjacent to angle B"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(log(b, 10), log(a, 10) + log(tan(B), 10) - 10)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/tangent-function",
    "method/solving-a-right-angled-triangle-logarithmically"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-4e0bfebbb4",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log c &= \\log a + \\tab\\log \\sec B - 10",
   "name": null,
   "statement": "Common logarithm of the hypotenuse c equals the common logarithm of a plus the common logarithm of sec B, less 10, used to solve a right triangle given a and B.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, adjacent to angle B"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(log(c, 10), log(a, 10) + log(sec(B), 10) - 10)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/secant",
    "method/solving-a-right-angled-triangle-logarithmically"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cfdf684cbd",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "38",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log \\sin A &= 10 + \\log a - \\log c",
   "name": null,
   "statement": "Common logarithm of sin A equals 10 plus log a minus log c, used to find A when c and a are given.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, the hypotenuse"
    }
   ],
   "sympy": "Eq(log(sin(A), 10), 10 + log(a, 10) - log(c, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/sine",
    "method/solving-a-right-angled-triangle-logarithmically"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-932924e024",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "37",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "B = 90° - A",
   "name": null,
   "statement": "In a right triangle with C = 90°, the two acute angles A and B are complementary.",
   "kind": "definition",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(B, 90 - A)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/complementary-angles",
    "concept/right-triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-c5caf10392",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "38",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "AD = AB \\sin B = c \\sin B",
   "name": null,
   "statement": "The altitude AD from A to line BC equals AB times the sine of B, which equals c sin B.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AD",
     "meaning": "perpendicular from A to BC (produced if necessary)"
    },
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "side c, equal to c"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    }
   ],
   "sympy": "Eq(AD, AB*sin(B))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/angle-of-elevation",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-ebbad58551",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "38",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "AD = CA \\sin C = b \\sin C",
   "name": null,
   "statement": "The same altitude AD equals CA times the sine of C, which equals b sin C.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "AD",
     "meaning": "perpendicular from A to BC (produced if necessary)"
    },
    {
     "unit": null,
     "symbol": "CA",
     "meaning": "side b, equal to b"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    }
   ],
   "sympy": "Eq(AD, CA*sin(C))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/angle-of-elevation",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-aff341bcef",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "38",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}",
   "name": "law of sines",
   "statement": "In any triangle, each side is proportional to the sine of the opposite angle, with the same constant for all three sides.",
   "kind": "law",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    }
   ],
   "sympy": "Eq(a/sin(A), b/sin(B))",
   "physics": false,
   "states": [
    "theorem/law-of-sines"
   ],
   "concepts": [
    "concept/sine",
    "concept/triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-bf2fd41781",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "39",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\Tab\\log\\sin B = (\\tab\\log\\sin C - \\log c) + \\log b",
   "name": null,
   "statement": "The common logarithm of sin B equals the common logarithm of sin C minus log c plus log b, a logarithmic equation determining B from b, c and C.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    }
   ],
   "sympy": "Eq(log(sin(B), 10), (log(sin(C), 10) - log(c, 10)) + log(b, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/common-logarithm",
    "concept/sine",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-faa20b0363",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "39",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A = 180° - (B + C)",
   "name": null,
   "statement": "The third angle of a triangle is 180 degrees minus the sum of the other two.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    }
   ],
   "sympy": "Eq(A, 180 - (B + C))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/triangle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-46d9a613a2",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "39",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log a = \\tab\\log\\sin A - (\\tab\\log\\sin C - \\log c)",
   "name": null,
   "statement": "The common logarithm of side a equals log sin A minus (log sin C minus log c), giving the third side once the angles are known.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    }
   ],
   "sympy": "Eq(log(a, 10), log(sin(A), 10) - (log(sin(C), 10) - log(c, 10)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "method/solving-a-triangle",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-869d782d87",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log\\sin B_1 = \\log b + (\\tab\\log\\sin C - \\log c)",
   "name": null,
   "statement": "In the ambiguous case, the acute angle B_1 is determined by its common logarithm of sine, which equals log b plus (log sin C minus log c).",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "B_1",
     "meaning": "the acute value of angle B in the ambiguous case"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    }
   ],
   "sympy": "Eq(log(sin(B_1), 10), log(b, 10) + (log(sin(C), 10) - log(c, 10)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/common-logarithm",
    "concept/sine",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-0d7047239a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "B_2 = 180° - B_1",
   "name": null,
   "statement": "The second possible value of B in the ambiguous case is the supplement of the acute value B_1. The source text prints 'B_2 = 180° - B' in one place, which reads as a typo for B_1; flagged, not silently corrected in the record.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "B_2",
     "meaning": "the obtuse value of angle B in the ambiguous case"
    },
    {
     "unit": "degree",
     "symbol": "B_1",
     "meaning": "the acute value of angle B in the ambiguous case"
    }
   ],
   "sympy": "Eq(B_2, 180 - B_1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/supplementary-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-bf08a87bf5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A_1 = 180° - (B_1 + C)",
   "name": null,
   "statement": "With the acute value B_1, the third angle A_1 is 180 degrees minus B_1 and C.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A_1",
     "meaning": "third angle for the first solution of the ambiguous case"
    },
    {
     "unit": "degree",
     "symbol": "B_1",
     "meaning": "acute value of B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    }
   ],
   "sympy": "Eq(A_1, 180 - (B_1 + C))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-82d3448656",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A_2 = 180° - (B_2 + C)",
   "name": null,
   "statement": "With the obtuse value B_2, the third angle A_2 is 180 degrees minus B_2 and C.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A_2",
     "meaning": "third angle for the second solution of the ambiguous case"
    },
    {
     "unit": "degree",
     "symbol": "B_2",
     "meaning": "obtuse value of B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    }
   ],
   "sympy": "Eq(A_2, 180 - (B_2 + C))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/triangle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-5a54a7477d",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log a_1 = \\tab\\log\\sin A_1 - (\\tab\\log\\sin C - \\log c)",
   "name": null,
   "statement": "Common logarithm of the third side a_1 for the first solution of the ambiguous case.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a_1",
     "meaning": "third side for the first solution"
    },
    {
     "unit": "degree",
     "symbol": "A_1",
     "meaning": "third angle for the first solution"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    }
   ],
   "sympy": "Eq(log(a_1, 10), log(sin(A_1), 10) - (log(sin(C), 10) - log(c, 10)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/common-logarithm",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-6716a62e33",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log a_2 = \\tab\\log\\sin A_2 - (\\tab\\log\\sin C - \\log c)",
   "name": null,
   "statement": "Common logarithm of the third side a_2 for the second solution of the ambiguous case.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a_2",
     "meaning": "third side for the second solution"
    },
    {
     "unit": "degree",
     "symbol": "A_2",
     "meaning": "third angle for the second solution"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    }
   ],
   "sympy": "Eq(log(a_2, 10), log(sin(A_2), 10) - (log(sin(C), 10) - log(c, 10)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/ambiguous-case",
    "concept/common-logarithm",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-cfbf497094",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "DCB = A + B = 2BED",
   "name": null,
   "statement": "In the construction with C as centre, the angle DCB equals A + B and is twice the angle BED.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "DCB",
     "meaning": "angle at C in the construction"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "BED",
     "meaning": "angle at E in the construction"
    }
   ],
   "sympy": "Eq(DCB, A + B)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a20fb7a997",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A = ECF + CFA = ECF + B",
   "name": null,
   "statement": "Angle A decomposes into ECF plus CFA, and CFA equals B, so A equals ECF plus B.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "ECF",
     "meaning": "angle at C in the construction"
    },
    {
     "unit": "degree",
     "symbol": "CFA",
     "meaning": "angle at F in the construction"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(A, ECF + CFA)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-5609d4bce0",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "40",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "ECF = A - B = 2FBE",
   "name": null,
   "statement": "The angle ECF equals A minus B, and is twice the angle FBE.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "ECF",
     "meaning": "angle at C in the construction"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "FBE",
     "meaning": "angle at B in the construction"
    }
   ],
   "sympy": "Eq(ECF, A - B)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-97f62021ec",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "41",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "BED = \\dfrac{1}{2}(A + B)",
   "name": null,
   "statement": "The angle BED is half the sum of A and B.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "BED",
     "meaning": "angle at E in the construction"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(BED, (A + B)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/half-angle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-514a2b1eae",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "41",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "FBE = \\dfrac{1}{2}(A - B)",
   "name": null,
   "statement": "The angle FBE is half the difference of A and B.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "FBE",
     "meaning": "angle at B in the construction"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(FBE, (A - B)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/half-angle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-3cd5988404",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "41",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\frac{a+b}{\\cos\\dfrac{1}{2}(A - B)} = \\frac{c}{\\cos\\dfrac{1}{2}(A + B)}",
   "name": null,
   "statement": "The sum of two sides a+b over the cosine of half the difference of the opposite angles equals c over the cosine of half their sum; the relation used for the two-sides-and-included-angle solution.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq((a + b)/cos((A - B)/2), c/cos((A + B)/2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/half-angle",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-59ab9f2ca1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "41",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\frac{a-b}{\\sin\\dfrac{1}{2}(A - B)} = \\frac{c}{\\sin\\dfrac{1}{2}(A + B)}",
   "name": null,
   "statement": "The difference a-b over the sine of half the difference of the opposite angles equals c over the sine of half their sum.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq((a - b)/sin((A - B)/2), c/sin((A + B)/2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/half-angle",
    "concept/sine",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-3e319aa0a6",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "41",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\frac{\\tan\\dfrac{1}{2}(A - B)}{\\tan\\dfrac{1}{2}(A + B)} = \\frac{a - b}{a + b}",
   "name": null,
   "statement": "The ratio of the tangents of half the difference and half the sum of two angles equals (a-b)/(a+b); the tangent formula for two sides and the included angle.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(tan((A - B)/2)/tan((A + B)/2), (a - b)/(a + b))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/tangent-function",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle",
    "theorem/tangent-formula-for-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-d5062bba0e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A + B = 180° - C",
   "name": null,
   "statement": "The two angles A and B together make 180 degrees minus C.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    }
   ],
   "sympy": "Eq(A + B, 180 - C)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/triangle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-c82095912b",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log\\tan \\frac{1}{2} (A - B) = \\tab\\log\\tan \\frac{1}{2} (A + B) \\\\ + \\log (a - b) - \\log (a + b)",
   "name": null,
   "statement": "The common logarithm of tan of half the difference of A and B equals the common logarithm of tan of half their sum, plus log(a-b) minus log(a+b); determines the acute angle (A-B)/2.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    }
   ],
   "sympy": "Eq(log(tan((A - B)/2), 10), log(tan((A + B)/2), 10) + log(a - b, 10) - log(a + b, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle",
    "theorem/tangent-formula-for-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-923d7fa2af",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "A &= \\frac{1}{2} (A + B) + \\frac{1}{2} (A - B)",
   "name": null,
   "statement": "Angle A is half the sum of A and B plus half their difference.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(A, (A + B)/2 + (A - B)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/half-angle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-6b23040ab0",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "B &= \\frac{1}{2} (A + B) - \\frac{1}{2} (A - B)",
   "name": null,
   "statement": "Angle B is half the sum of A and B minus half their difference.",
   "kind": "result",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(B, (A + B)/2 - (A - B)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/half-angle",
    "quantity/angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a76e6b28a5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log c = \\tab\\log\\sin C - (\\tab\\log\\sin A - \\log a)",
   "name": null,
   "statement": "The common logarithm of c equals log sin C minus (log sin A minus log a), an alternative route to the third side.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    }
   ],
   "sympy": "Eq(log(c, 10), log(sin(C), 10) - (log(sin(A), 10) - log(a, 10)))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "method/solving-a-triangle",
    "theorem/law-of-sines"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-690664468e",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log c = \\tab\\log\\cos \\frac{1}{2} (A + B) - \\tab\\log\\cos \\frac{1}{2} (A - B) + \\log (a + b)",
   "name": null,
   "statement": "The common logarithm of the third side c equals log cos of half the sum minus log cos of half the difference, plus log(a+b).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(log(c, 10), log(cos((A + B)/2), 10) - log(cos((A - B)/2), 10) + log(a + b, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/cosine",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-94001971fb",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "42",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log c = \\tab\\log\\sin \\frac{1}{2} (A + B) - \\tab\\log\\sin \\frac{1}{2} (A - B) + \\log (a - b)",
   "name": null,
   "statement": "The common logarithm of the third side c equals log sin of half the sum minus log sin of half the difference, plus log(a-b).",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB, opposite angle C"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC, opposite angle A"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA, opposite angle B"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    }
   ],
   "sympy": "Eq(log(c, 10), log(sin((A + B)/2), 10) - log(sin((A - B)/2), 10) + log(a - b, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/sine",
    "method/solving-a-triangle-from-two-sides-and-the-included-angle"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-abc3e60521",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "43",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "a + b + c = 2s",
   "name": null,
   "statement": "The sum of the three sides is twice s, which is defined as half the perimeter.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    }
   ],
   "sympy": "Eq(a + b + c, 2*s)",
   "physics": false,
   "states": [],
   "concepts": [
    "quantity/perimeter",
    "quantity/semi-perimeter"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-4e6d6434e1",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "43",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "r = (s - a) \\tan \\frac{1}{2} A",
   "name": null,
   "statement": "The inscribed-circle radius r equals (s-a) times the tangent of half A; the same holds with B and C in place of A.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    },
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    }
   ],
   "sympy": "Eq(r, (s - a)*tan(A/2))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "concept/tangent-function",
    "quantity/semi-perimeter",
    "theorem/radius-of-the-inscribed-circle-from-the-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-50392e3c58",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "43",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "r^2 = \\frac{(s - a)(s - b)(s - c)}{s}",
   "name": null,
   "statement": "The square of the inscribed-circle radius equals (s-a)(s-b)(s-c) divided by s.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    }
   ],
   "sympy": "Eq(r**2, (s - a)*(s - b)*(s - c)/s)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/inscribed-circle",
    "quantity/semi-perimeter",
    "theorem/heron-s-formula",
    "theorem/radius-of-the-inscribed-circle-from-the-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-bea37ab88f",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "43",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\log r = \\frac{1}{2} \\bigl\\{ \\log (s - a) + \\log (s - b) + \\log (s - c) - \\log s \\bigr\\}",
   "name": null,
   "statement": "The common logarithm of r is half of log(s-a)+log(s-b)+log(s-c)-log s, the logarithmic form for solving a triangle from its three sides.",
   "kind": "formula",
   "symbols": [
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    }
   ],
   "sympy": "Eq(log(r, 10), (log(s - a, 10) + log(s - b, 10) + log(s - c, 10) - log(s, 10))/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "method/solving-a-triangle-from-its-three-sides",
    "theorem/radius-of-the-inscribed-circle-from-the-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-1e2ca23c5a",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log\\tan \\frac{1}{2} A &= 10 + \\log r - \\log (s - a)",
   "name": null,
   "statement": "The common logarithm of tan of half A equals 10 plus log r minus log(s-a), giving angle A from the three sides.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "A",
     "meaning": "angle at A"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "a",
     "meaning": "side BC"
    }
   ],
   "sympy": "Eq(log(tan(A/2), 10), 10 + log(r, 10) - log(s - a, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/tangent-function",
    "method/solving-a-triangle-from-its-three-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-17dbaaaa58",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log\\tan \\frac{1}{2} B &= 10 + \\log r - \\log (s - b)",
   "name": null,
   "statement": "The common logarithm of tan of half B equals 10 plus log r minus log(s-b), giving angle B from the three sides.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "B",
     "meaning": "angle at B"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "b",
     "meaning": "side CA"
    }
   ],
   "sympy": "Eq(log(tan(B/2), 10), 10 + log(r, 10) - log(s - b, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/tangent-function",
    "method/solving-a-triangle-from-its-three-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-79b401ec53",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-vii",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "44",
   "location": "SOLUTION OF TRIANGLES",
   "latex": "\\tab\\log\\tan \\frac{1}{2} C &= 10 + \\log r - \\log (s - c)",
   "name": null,
   "statement": "The common logarithm of tan of half C equals 10 plus log r minus log(s-c), giving angle C from the three sides.",
   "kind": "formula",
   "symbols": [
    {
     "unit": "degree",
     "symbol": "C",
     "meaning": "angle at C"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle"
    },
    {
     "unit": null,
     "symbol": "s",
     "meaning": "half the sum of the sides"
    },
    {
     "unit": null,
     "symbol": "c",
     "meaning": "side AB"
    }
   ],
   "sympy": "Eq(log(tan(C/2), 10), 10 + log(r, 10) - log(s - c, 10))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/common-logarithm",
    "concept/tangent-function",
    "method/solving-a-triangle-from-its-three-sides"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-043253d284",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "48",
   "location": "OF PROJECTIONS",
   "latex": "A'B'= AB \\cos BAC",
   "name": null,
   "statement": "The projection A'B' of a line AB on the line of projection equals the length of AB times the cosine of the angle BAC between the two lines, taken with the sign of the projection.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A'B'",
     "meaning": "projection of the line AB on the line of projection X'X, measured positive in the + direction when the angle is acute and negative when it is obtuse"
    },
    {
     "unit": null,
     "symbol": "AB",
     "meaning": "the projected line"
    },
    {
     "unit": null,
     "symbol": "BAC",
     "meaning": "the angle between the projected line AB and a line drawn from A parallel to the line of projection in the + direction"
    }
   ],
   "sympy": "Eq(ApBp, AB*cos(BAC))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/cosine",
    "concept/line-of-projection",
    "concept/projection-of-a-line",
    "quantity/angle",
    "theorem/projection-of-a-line-on-a-line"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-aec7cee42c",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-ix",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "48",
   "location": "OF PROJECTIONS",
   "latex": "A'D' = A'B' + B'C' + C'D'",
   "name": null,
   "statement": "The projection A'D' of the broken line ABCD on the line of projection equals the algebraic sum of the projections of its three parts AB, BC and CD.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "A'D'",
     "meaning": "projection of the broken line ABCD on the line of projection X'X"
    },
    {
     "unit": null,
     "symbol": "A'B'",
     "meaning": "projection of the part AB on X'X"
    },
    {
     "unit": null,
     "symbol": "B'C'",
     "meaning": "projection of the part BC on X'X, negative in the figure"
    },
    {
     "unit": null,
     "symbol": "C'D'",
     "meaning": "projection of the part CD on X'X"
    }
   ],
   "sympy": "Eq(ApDp, ApBp + BpCp + CpDp)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/algebraic-sum",
    "concept/line-of-projection",
    "concept/projection-of-a-line",
    "theorem/projection-of-a-broken-line"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-221cce7152",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "49",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "DE &= R \\sin (\\theta + \\phi)",
   "name": null,
   "statement": "The perpendicular DE from D to CA equals R times the sine of the sum of the two angles.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "DE",
     "meaning": "perpendicular from D to the line CA"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circle with centre C"
    },
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of angle ACB"
    },
    {
     "unit": "radian",
     "symbol": "\\phi",
     "meaning": "circular measure of angle BCD"
    }
   ],
   "sympy": "Eq(DE, R*sin(theta + phi))",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circular-function",
    "concept/circular-measure",
    "concept/perpendicular",
    "concept/sine"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-18342c1601",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "49",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\sin (\\theta + \\phi) &= \\sin\\theta \\cos\\phi + \\cos\\theta \\sin\\phi",
   "name": "sine addition formula",
   "statement": "The sine of the sum of two angles is the sine of the first times the cosine of the second plus the cosine of the first times the sine of the second.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of angle ACB"
    },
    {
     "unit": "radian",
     "symbol": "\\phi",
     "meaning": "circular measure of angle BCD"
    }
   ],
   "sympy": "Eq(sin(theta + phi), sin(theta)*cos(phi) + cos(theta)*sin(phi))",
   "physics": false,
   "states": [
    "theorem/sine-addition-formula"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/sine",
    "quantity/angle",
    "theorem/sine-of-the-sum-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e35d999d91",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "49",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\cos (\\theta + \\phi)   &= \\cos\\theta \\cos\\phi - \\sin\\theta \\sin\\phi",
   "name": "cosine addition formula",
   "statement": "The cosine of the sum of two angles is the product of the cosines minus the product of the sines.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of angle ACB"
    },
    {
     "unit": "radian",
     "symbol": "\\phi",
     "meaning": "circular measure of angle BCD"
    }
   ],
   "sympy": "Eq(cos(theta + phi), cos(theta)*cos(phi) - sin(theta)*sin(phi))",
   "physics": false,
   "states": [
    "theorem/cosine-addition-formula"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/sine",
    "quantity/angle",
    "theorem/cosine-of-the-sum-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-a25882ff41",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "50",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\sin (\\theta - \\phi) &= \\sin\\theta \\cos\\phi - \\cos\\theta \\sin\\phi",
   "name": "sine subtraction formula",
   "statement": "The sine of the difference of two angles is the sine of the first times the cosine of the second minus the cosine of the first times the sine of the second.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of angle ACB"
    },
    {
     "unit": "radian",
     "symbol": "\\phi",
     "meaning": "circular measure of angle BCD"
    }
   ],
   "sympy": "Eq(sin(theta - phi), sin(theta)*cos(phi) - cos(theta)*sin(phi))",
   "physics": false,
   "states": [
    "theorem/sine-subtraction-formula"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/sine",
    "quantity/angle",
    "theorem/sine-of-the-difference-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-138592b15b",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "50",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\cos (\\theta - \\phi) &= \\cos\\phi \\cos\\theta + \\sin\\phi \\sin\\theta",
   "name": "cosine subtraction formula",
   "statement": "The cosine of the difference of two angles is the product of the cosines plus the product of the sines.",
   "kind": "identity",
   "symbols": [
    {
     "unit": "radian",
     "symbol": "\\theta",
     "meaning": "circular measure of angle ACB"
    },
    {
     "unit": "radian",
     "symbol": "\\phi",
     "meaning": "circular measure of angle BCD"
    }
   ],
   "sympy": "Eq(cos(theta - phi), cos(phi)*cos(theta) + sin(phi)*sin(theta))",
   "physics": false,
   "states": [
    "theorem/cosine-subtraction-formula"
   ],
   "concepts": [
    "concept/circular-function",
    "concept/cosine",
    "concept/sine",
    "quantity/angle",
    "theorem/cosine-of-the-difference-of-two-angles"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-07d3a230a5",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "51",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "R - r = 2\\delta",
   "name": null,
   "statement": "The difference between the two radii R and r is written as twice the quantity δ.",
   "kind": "definition",
   "symbols": [
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circumscribed circle of the starting polygon"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle of the starting polygon"
    },
    {
     "unit": null,
     "symbol": "\\delta",
     "meaning": "half the difference R - r"
    }
   ],
   "sympy": "Eq(R - r, 2*delta)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circumscribed-circle",
    "concept/inscribed-circle",
    "method/approximating-pi-by-inscribed-and-circumscribed-polygons"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-2cffb15855",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "51",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "r_1 = \\dfrac{R + r}{2}",
   "name": null,
   "statement": "The inscribed radius of the first polygon is the average of R and r.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "r_1",
     "meaning": "radius of the circle inscribed in the first polygon"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circumscribed circle of the starting polygon"
    },
    {
     "unit": null,
     "symbol": "r",
     "meaning": "radius of the inscribed circle of the starting polygon"
    }
   ],
   "sympy": "Eq(r_1, (R + r)/2)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximating-sequence-of-polygons",
    "concept/inscribed-circle",
    "theorem/radii-of-inscribed-and-circumscribed-circles-of-doubled-polygons"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-6d0771fbad",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "51",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "R_1^2 &= r_1R",
   "name": null,
   "statement": "The square of the circumscribed radius of the first doubled polygon equals the product of r_1 and R.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "R_1",
     "meaning": "radius of the circle circumscribed about the first doubled polygon"
    },
    {
     "unit": null,
     "symbol": "r_1",
     "meaning": "radius of the circle inscribed in the first polygon"
    },
    {
     "unit": null,
     "symbol": "R",
     "meaning": "radius of the circumscribed circle of the starting polygon"
    }
   ],
   "sympy": "Eq(R_1**2, r_1*R)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/circumscribed-circle",
    "concept/inscribed-circle",
    "theorem/radii-of-inscribed-and-circumscribed-circles-of-doubled-polygons"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-fb0cb3d355",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "51",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\delta_1 = \\frac{\\delta}{4} - \\frac{\\delta_1^2}{r_1}",
   "name": null,
   "statement": "The first correction δ_1 equals one quarter of δ less a small quadratic term in δ_1 divided by r_1.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\delta_1",
     "meaning": "increase from the first inscribed radius to the second"
    },
    {
     "unit": null,
     "symbol": "\\delta",
     "meaning": "half the difference R - r"
    },
    {
     "unit": null,
     "symbol": "r_1",
     "meaning": "radius of the circle inscribed in the first polygon"
    }
   ],
   "sympy": "Eq(delta_1, delta/4 - delta_1**2/r_1)",
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximating-sequence-of-polygons",
    "concept/inscribed-circle",
    "concept/limit",
    "method/neglecting-higher-order-small-quantities"
   ]
  },
  {
   "id": "blackburn-elements-plane-trigonometry-1863/eq-e1dd45729f",
   "chapter": "blackburn-elements-plane-trigonometry-1863/ch-x",
   "book": "blackburn-elements-plane-trigonometry-1863",
   "edition": "Macmillan, 1863 (edition to be confirmed from the copy)",
   "page": "52",
   "location": "THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES",
   "latex": "\\delta + \\delta_1 + \\delta_2 + \\dots < \\frac{4}{3}\\, \\delta",
   "name": null,
   "statement": "The infinite sum of the successive corrections is less than four-thirds of δ.",
   "kind": "result",
   "symbols": [
    {
     "unit": null,
     "symbol": "\\delta",
     "meaning": "half the difference R - r"
    },
    {
     "unit": null,
     "symbol": "\\delta_1",
     "meaning": "first successive correction to the inscribed radius"
    },
    {
     "unit": null,
     "symbol": "\\delta_2",
     "meaning": "second successive correction to the inscribed radius"
    }
   ],
   "sympy": null,
   "physics": false,
   "states": [],
   "concepts": [
    "concept/approximating-sequence-of-polygons",
    "concept/infinite-sequence",
    "concept/inscribed-circle",
    "concept/limit"
   ]
  }
 ],
 "exercise_sets": [],
 "problems": [],
 "errata_statuses": [
  "transcriber_marked",
  "candidate",
  "probable",
  "confirmed",
  "dismissed",
  "note"
 ],
 "errata": []
}