A Frenchman (b. 1768) and a Prussian-American (b. 1852) will throw some light on how to get two lengths from this figure. Lengths in microns are not directly useful in geocaching but they can be turned to good use, but not in the same sense.
The first four finders had only the above information to go on. The first part of this puzzle involves what looks like degree level expertise. Actually it can be solved with high school level maths and science, but it may take a bit of research to understand how. So I have added a couple of hints. The first is contained in the photo Maths Hint, so if you want to solve this without the hint, don't look at the photos. The second is this expanded hint describing the background if you are really stuck...
Yvtug orunirf nf obgu n jnir naq n cnegvpyr; urer jr bayl pbafvqre gur jnir orunivbhe – gur jnl gur ryrpgebzntargvp svryq punatrf.
N cebcntngvat jnir unf obgu n jniryratgu naq n serdhrapl. Gur serdhrapl qrfpevorf ubj, vs lbh ybbxrq ng n svkrq cbvag va fcnpr, gur svryq inevrf gurer jvgu gvzr – ubj znal gvzrf n frpbaq vg bfpvyyngrf. Gur jniryratgu qrfpevorf ubj, vs lbh gnxr n svkrq fancfubg va gvzr, gur svryq inevrf va fcnpr – gur qvfgnapr bire juvpu vg bfpvyyngrf. Gurer vf na nffbpvngrq ahzore, gur jnirahzore, juvpu qrfpevorf ubj znal gvzr n zrger vg bfpvyyngrf. Gur fcrrq bs gur jnir vf gur pbzovangvba bs gur jniryratgu (va zrgerf) naq gur serdhrapl (va vairefr frpbaqf) gb tvir n fcrrq (va zrgerf cre frpbaq). Gur jnirahzore vf fvzcyl bar bire gur jniryratgu naq unf havgf bs vairefr zrgerf.
Zvpuryfba (jvgu Zbeyrl) qrirybcrq na vagresrebzrgre gb gel gb zrnfher gur yhzvavsrebhf rgure. Nf vg qbrfa’g rkvfg gurl qvqa’g fhpprrq, ubjrire gurl qvq tvir hf gur Zvpuryfba Vagresrebzrgre juvpu vf n jvqryl hfrq bcgvpny flfgrz. Vg jbexf ol fcyvggvat n yvtug ornz va gjb naq gura erpbzovavat gurz, jvgu gur cbffvovyvgl bs znavchyngvat bar bs gur gjb ornzf – sbe vafgnapr znxvat vg geniry shegure.
Vs gjb vqragvpny jnirf, va cunfr jvgu rnpu bgure, erpbzovar (be vagresrer) pbafgehpgviryl gurl nqq gbtrgure fhpu gung gur crnxf trg ovttre naq gur gebhtuf trg ybjre. Vs ubjrire bar bs gur ornzf vf qrynlrq ol unys n plpyr gurl vagresrer qrfgehpgviryl naq gur crnxf naq gebhtuf pnapry rnpu bgure bhg. Vs lbh fgrnqvyl punatr gur qvfgnapr (naq urapr cunfr) orgjrra gur gjb ornzf lbh trg n fvahfbvqny vagresrerapr cnggrea.
Fb vs lbh unir n Zvpuryfba vagresrebzrgre jvgu n cher fvatyr jniryratgu (be serdhrapl) yvtug jnir tbvat vagb vg, naq lbh znavchyngr bar cngu jvgu zveebef gb znxr gur yvtug va vg geniry fgrnqvyl shegure gur bhgchg jvyy or n fvahfbvqny vagresrerapr cnggrea. Vg jvyy or va grezf bs vagrafvgl (bs gur gbgny erpbzovarq yvtug) irefhf qvfgnapr (gur qvssrerapr va qvfgnapr geniryrq ol gur gjb yvtug cnguf). Gurer vf n irel fvzcyr eryngvbafuvc orgjrra gur crevbq bs guvf zrnfherq vagresrerapr vagrafvgl naq gur jniryratgu bs gur yvtug.
Erny yvtug vf abg znqr hc bs bar fvatyr vqrny serdhrapl, ohg n pbagvahhz bs serdhrapvrf. Sbhevre erirnyrq fbzr irel vagrerfgvat guvatf gung uryc gb nanylfr guvf – ercerfragvat nyy fvtanyf ol fhzf bs fvahfbvqf jvgu qvssrerag serdhrapvrf. N gvzr inelvat fvahfbvq, juvpu vf (boivbhfyl) qvfcynlrq nf n fvahfbvqny inevngvba ntnvafg gvzr ba na vagrafvgl irefhf gvzr cybg pna or qvfcynlrq ba n cybg bs vagrafvgl irefhf serdhrapl nf fvatyr iregvpny yvar ng gung cnegvphyne serdhrapl. Sbhevre nanylfvf fubjf gung vs n fvtany va gur gvzr qbznva pna or ohvyg hc ol nqqvat gbtrgure fvahfbvqf, vgf ercerfragngvba va gur serdhrapl qbznva pna or ohvyg hc ol nqqvat gbtrgure gurfr vaqvivqhny serdhrapvrf. (Va npbhfgvp grezf guvax nobhg gur qvfcynl ba n tencuvp rdhnyvfre.) Gur Sbhevre Genafsbez zbirf orgjrra gvzr naq serdhrapl qbznva. Vg hfhnyyl yvaxf gvzr va frpbaqf jvgu serdhrapl va vairefr frpbaqf, ohg vg pna nyfb yvax qvfgnapr va zrgerf naq fcngvny serdhrapl va vairefr zrgerf.
Gur Zvpuryfba Vagresrebzrgre, nf qrfpevorq nobir, rssrpgviryl Sbhevre Genafsbezf gur yvtug fvtany chg vagb vg. Gur vagresrerapr cnggrea (vagrafvgl irefhf bcgvpny cngu qvssrerapr) vf rffragvnyyl gur Sbhevre Genafsbez bs gur fcrpgehz bs gur vachg yvtug.
Fb lbh pbhyq Vairefr Sbhevre Genafsbez gur qngn cebivqrq. Ubjrire, gurer vf n zhpu rnfvre nccebnpu, juvpu fubhyq or pyrne sebz gur nobir qvfphffvba naq fbzrguvat ryfr nobhg gur pnpur. Vg’f abg fgenvtugsbejneq sebz urer nf gurer ner n srj erznvavat gevpxf, ohg abguvat erdhvevat qrterr yriry znguf be culfvpf.