= Solution
\b[The relevant historical route starts with phosphate-rich bone ash, converts it into a reducible phosphate preparation, and heats that preparation with <carbon> to obtain elemental <phosphorus>.] Bone contains <phosphorus> in <phosphate>, not as free pieces of <phosphorus> awaiting extraction.
Burning cleaned bones removes most organic matter and leaves <bone ash>, rich in calcium phosphates. Although natural bone mineral is not pure $\mathrm{Ca}_3(\mathrm{PO}_4)_2$, that formula is a useful historical reaction model. Treatment with <sulfuric acid> forms sparingly soluble <calcium sulfate> and a soluble acid phosphate fraction. With partial removal of <calcium>, an idealized balance is
$$
\boxed{\mathrm{Ca}_3(\mathrm{PO}_4)_2+2\mathrm H_2\mathrm{SO}_4\longrightarrow\mathrm{Ca(H}_2\mathrm{PO}_4)_2+2\mathrm{CaSO}_4.}
$$
Separate the sulfate-rich residue, concentrate the phosphate-bearing liquor, and heat the resulting material with <charcoal> in a retort. Dehydration and high-temperature <reduction> produce phosphorus vapour together with carbon oxides, while a calcium phosphate residue remains. One overall balance for the acid-phosphate route is
$$
\boxed{3\mathrm{Ca(H}_2\mathrm{PO}_4)_2+10\mathrm C\longrightarrow\mathrm{Ca}_3(\mathrm{PO}_4)_2+\mathrm P_4+10\mathrm{CO}+6\mathrm H_2\mathrm O.}
$$
This is a summed process balance; it does not imply that the hydrated phosphate survives unchanged until one elementary reaction suddenly occurs. It also makes clear that the partial-acid route leaves some of the original <phosphorus> in the calcium phosphate residue. Contemporary descriptions distinguish this route from treating the ash with enough <sulfuric acid> to give a more fully acidified phosphorus-bearing liquid.<Lessons in Elementary Chemistry by Henry Enfield Roscoe>[1868 printing, Lesson XV, printed pp. 131–133: bone ash, acid phosphate and charcoal reduction; later printing used as historical description, not a pharmacopoeia]{ref}
The retort outlet leads to a water-covered receiver: phosphorus vapour condenses and is protected from <air>. The collected product is principally <white phosphorus>, with molecular form $\mathrm P_4$. Its readiness to oxidize and ignite is why collection under <water> belongs to the explanation of the apparatus. This historical product must be distinguished from <red phosphorus>, obtained by a subsequent controlled transformation rather than directly identified with the freshly condensed product.
For comparison, the well-known silica-assisted phosphate-reduction balance is
$$
2\mathrm{Ca}_3(\mathrm{PO}_4)_2+6\mathrm{SiO}_2+10\mathrm C\longrightarrow\mathrm P_4+6\mathrm{CaSiO}_3+10\mathrm{CO}.
$$
Here <silica> binds <calcium> into a silicate-rich residue. Modern electric-furnace manufacture uses this principle; describing an electric furnace as the ordinary 1866 apparatus would be anachronistic. The historical answer is the sequence \b[<bone ash> → acid phosphate preparation → <carbon> reduction in a retort → condensation of <phosphorus> under <water>].
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