De loco lunae ex ecHpsihis lunaribus determinando. 367 



bita, quem tabulae lunares indicare debent, neque ad hoc reductione loci lunae ad eclipticam 

 lus habebimus, uti vera oppositio postulat. 



§ 7. Repraesentet erg-o arcus 6% non veram oppositionem, sed eum luminarium situm, in quo 

 cus QX aequalis sit arcui Qo, hocque oveniat elapsis post initium SL horis t. Sit ergo hoc 

 omento tam longitudo umbrae, quam longitudo lunae in orbita a nodo computata Qa = QX=^ x^ 

 it ob So={m-+-k)t", LX =\fi-+-k) t'\ arcus QS=x — {m-v-k)t" et arcus QL=x — {n-\-k)t" . 

 ?inde vero erit arcus Qs = x-\-{in-^k) {h. — t)" et Ql = x -\- {n -+ k) {h — t)" . Ponatur 

 eulus XQg=iq, erit ex natura triangulorum sphaericorum 



cos SL = cos Q sin QL sin QiS'-+- cos QL cos QS^ cos sl = cos q sin Ql sin Qs -h cos Ql cos Qs. 



lare cum sit SL = sl, ent 



K cos QL cos QS — cos Ql cos Qs = cos q (sin Ql sin Qs — sin QL sin QS). 



^k 8. Est vero per compositionera-angulorum ut sequitur 



cos QL = cos X cos {n -f- k) t'' h- sin x sin {n -t- k) t" 

 eos S^iS* = cos X cos (w -i- /c) f" h- sin a; sin {m -t-k)t" 

 Qos Ql = cos X cos (« -H k) {h — t)" — sin x sin {n -+- k) {h — t)" 

 cos Qs = cos X cos {m~\-k) {h — t)" — sin x sin {m-+-k) {h — t)" 

 sin Q / = sin x cos {n -+- k) {h — t)" -h cos x sin {n -t-k) (h — t)" 

 sie 9 * = sin x cos (m -+k){h — e)'' -\- cos a; sin (m -h /c) (/j — f)" 

 sin QL = sin£C cos {/i-i- A:) <" — cos cc sin (/i -*- /cj i" 

 sin QS =smxcos{m-\-k)t" — cosxsm{m-i-k)t" 



H per alias sinuum proprietates 



cosQLcosQ^^^-f cos(Q^--QL)-f--^cos(Q5'-i-QL) , 



cos Q /. cos Q 5 = -|- cos (Q / — Q 5 ) -f- 4- cos (Q 5 -H Q /) 



sin Q/ sin Q5 = -|-cos (Q/ — Qs) — ^cos(Q5-h Ql) 



sin QL sin QS= \ cos (Q^* — QL) — 4-cos {QS h- QL). 



l 9. Cum igitur sit 



QS—QL = {n^m)t" QS-+QL = 2x-^{m-^n-+2k)t" 



Ql~^Qs ={n — m){h^ t)" Q^ n- Q Z = 2a; -*- (iw -4- w -h 2/c) (/i — «)". 



h ergo valoribus substitutls reperiemus 



'ios{n-m)t"-^cos(2x-{m-^n-+2k)t") ) ( cos(/i-m)(/i-<)"-cos(2a;-»-(in-^-/i-^2/c)(/i-t)") 



~08(/i-m)(/i-«)"-cos(2£C-H(m-4-/i-f-2/c)(/i-<)")j l-cos(/i-m)«"-*-cos(2aj-(m-«-/i-*-2/f)(") 



reducta abit in 

 (Vcos^^^cosl/i-m^e^-cos^/i-m)^/^— t)'')^^!— cos())(cos(2a;-H(m-*-/i-4-2/0(/i-<)'0-cos(2aj-(m-^/i-^2/c)f'0. 



