Wikisource lawikisource https://la.wikisource.org/wiki/Pagina_prima MediaWiki 1.47.0-wmf.19 first-letter Media Specialis Disputatio Usor Disputatio Usoris Vicifons Disputatio Vicifontis Fasciculus Disputatio Fasciculi MediaWiki Disputatio MediaWiki Formula Disputatio Formulae Auxilium Disputatio Auxilii Categoria Disputatio Categoriae Scriptor Disputatio Scriptoris Pagina Disputatio Paginae Liber Disputatio Libri TimedText TimedText talk Modulus Disputatio Moduli Event Event talk 0 1421 280901 257103 2026-09-14T14:32:31Z Saumache 27923 280901 wikitext text/x-wiki l4ev5xgp5td5rr97q2ppoe3c9jcsb6d 102 2126 280915 280759 2026-09-14T19:50:30Z Alpaga2 25642 /* Opera */ De gloria beatorum confessorum 280915 wikitext text/x-wiki 6zhhqv6fhkfxkw4uatinbwsqm78j7xe 280916 280915 2026-09-14T19:52:44Z Alpaga2 25642 /* Opera */ doubles 280916 wikitext text/x-wiki 0u6173a7ba2ckc7sz804nlecs0h2n2a 0 2536 280918 128236 2026-09-14T20:23:05Z Demetrius Talpa 13304 280918 wikitext text/x-wiki b7j5y4fwmn6ysk0qjkw6jhzajzv06j3 280919 280918 2026-09-14T20:24:08Z Demetrius Talpa 13304 280919 wikitext text/x-wiki 76lmbsak5g8ogz1s7fybjxk64th6b8t 0 2549 280893 280841 2026-09-14T13:28:15Z Saumache 27923 280893 wikitext text/x-wiki sr6u82rm8btmg1zl2z5hyg54getssl7 0 2552 280887 280835 2026-09-14T13:09:54Z Saumache 27923 /* 98. */ 280887 wikitext text/x-wiki kdkkiioo0tyn5jptm3fagtxxlxsqk4v 280894 280887 2026-09-14T13:35:33Z Saumache 27923 280894 wikitext text/x-wiki dpse5wtu475zeeewpu678ik6thc7ywz 0 2553 280888 280834 2026-09-14T13:10:12Z Saumache 27923 280888 wikitext text/x-wiki mli1xxae3yelxjw4k6uhv924js1xnls 0 2554 280889 280833 2026-09-14T13:10:28Z Saumache 27923 280889 wikitext text/x-wiki nousv3k3pj1nndmcgsfjh91oddlk3wl 104 82521 280909 274668 2026-09-14T15:23:20Z Saumache 27923 280909 proofread-page text/x-wiki ojz676xzu0d4o3gxaweoxxd4m7bd3eu 104 82542 280897 271449 2026-09-14T13:52:33Z Saumache 27923 /* Emendata */ 280897 proofread-page text/x-wiki 02zke8duchu53g852r4p9fe6rbu9468 280898 280897 2026-09-14T13:52:49Z Saumache 27923 280898 proofread-page text/x-wiki c06y71bvevczyu3q4e48brodexmllv6 104 82543 280899 271415 2026-09-14T14:05:01Z Saumache 27923 /* Emendata */ 280899 proofread-page text/x-wiki jx7w79f7gtw09fclo64rswl1h63thr4 104 82544 280902 271501 2026-09-14T14:40:43Z Saumache 27923 /* Emendata */ 280902 proofread-page text/x-wiki 3d3yyd4io4uk55984cf63w9i2hjou9i 280903 280902 2026-09-14T14:41:33Z Saumache 27923 280903 proofread-page text/x-wiki 6aqhhgnfuws6mnx8ak42gmua0sazd5k 104 82545 280904 271507 2026-09-14T14:49:03Z Saumache 27923 /* Emendata */ 280904 proofread-page text/x-wiki 151aokp8547msssoci3fn36w2x857nd 104 82546 280905 271416 2026-09-14T15:00:48Z Saumache 27923 /* Emendata */ notha? 280905 proofread-page text/x-wiki 20poie4i40antd3rw6uvyfoof7mx7le 280906 280905 2026-09-14T15:01:03Z Saumache 27923 280906 proofread-page text/x-wiki s3h25qvb80ujo21iugd17a1vr30nxd6 104 82547 280907 266749 2026-09-14T15:10:29Z Saumache 27923 /* Emendata */ 280907 proofread-page text/x-wiki 5bbu7v7gik87ev8mba6nnal53xn9z8m 104 82548 280908 271492 2026-09-14T15:20:46Z Saumache 27923 /* Emendata */ 280908 proofread-page text/x-wiki 3lc6mmiy2eebdhoqw9wmwwddnlczmr8 0 85608 280886 280737 2026-09-14T13:08:10Z Coronus Tenebrosius 31709 /* 15. Chroati in Nestoris Chronica */ 280886 wikitext text/x-wiki hfzkdg7l55j1x8nobbdtbqgssa4q7b8 280890 280886 2026-09-14T13:11:05Z Coronus Tenebrosius 31709 /* Archaeologica investigationes in Ucraina occidentali */ 280890 wikitext text/x-wiki g1xgwbhxz5kinmm7p87odnloyhtbrkv 280891 280890 2026-09-14T13:26:07Z Coronus Tenebrosius 31709 /* Archaeologica investigationes in Ucraina occidentali */ 280891 wikitext text/x-wiki 01n6ojehr00tcm6t3u7kv2veqonzel9 280892 280891 2026-09-14T13:26:28Z Coronus Tenebrosius 31709 /* Archaeologica investigationes in Ucraina occidentali */ 280892 wikitext text/x-wiki 54gd39xstdngwvcpe3557fhksvb0szq 280895 280892 2026-09-14T13:38:25Z Coronus Tenebrosius 31709 /* 16. Archaeologica investigationes in Ucraina occidentali */ 280895 wikitext text/x-wiki ietn5lxmm87b0nrn04utd2mk618kjsc 280896 280895 2026-09-14T13:45:21Z Coronus Tenebrosius 31709 /* 16. Archaeologica investigationes in Ucraina occidentali */ 280896 wikitext text/x-wiki 1a0uf2wncr4392nyrh7er0xg7d875fq 280900 280896 2026-09-14T14:13:32Z Coronus Tenebrosius 31709 /* 16. Archaeologica investigationes in Ucraina occidentali */ 280900 wikitext text/x-wiki 96rb8lx6uqkkj29c83s69zf7riyi4m0 280917 280900 2026-09-14T20:18:39Z Coronus Tenebrosius 31709 /* 15. Chroati in Nestoris Chronica */ 280917 wikitext text/x-wiki k397lxni4ip6t7pro49wqimig2cf2sz 280920 280917 2026-09-14T20:25:19Z Coronus Tenebrosius 31709 /* 15. Chroati in Nestoris Chronica */ 280920 wikitext text/x-wiki dzpbjhpdh7wvoeky1ibcje0sdr8n0kc 280994 280920 2026-09-15T08:37:45Z Coronus Tenebrosius 31709 /* 4. Thesis de Chroatia Alba in territorio Carinthiae mediaevalis, id est provinciae hodiernae Austriae Kärnten */ 280994 wikitext text/x-wiki q5en4x195h59ei2kuwrtst4jvphel3w 104 86051 280910 280864 2026-09-14T15:45:54Z Nederlandse Leeuw 5519 280910 proofread-page text/x-wiki 2erzo8t5qywdsh71wczh6a3u8l6qta5 104 86052 280911 280867 2026-09-14T16:17:58Z Nederlandse Leeuw 5519 280911 proofread-page text/x-wiki rbz6u1n1oju675va10bxvfahojgbu4i 104 86053 280912 280871 2026-09-14T16:39:28Z Nederlandse Leeuw 5519 280912 proofread-page text/x-wiki b3yyoy2lkf3lfk0qqstznwia8wfbv6f 104 86054 280913 280873 2026-09-14T16:57:09Z Nederlandse Leeuw 5519 280913 proofread-page text/x-wiki gifcapwrluwj6a71vw1ser5o6iixyq3 280914 280913 2026-09-14T17:01:57Z Nederlandse Leeuw 5519 280914 proofread-page text/x-wiki 62bj21wcd7ma6hoi2x22tdhba35cfiw 104 86063 280921 2026-09-15T01:26:17Z Sjgallagher2 29663 /* Nondum emendata */ 280921 proofread-page text/x-wiki 46sq1tedv6fe3kkaawocwpbcxu2ejlx 280924 280921 2026-09-15T01:32:53Z Sjgallagher2 29663 Add missing part of word 280924 proofread-page text/x-wiki oz3qcwebqwl2r09p6l5hzh39mxtd751 280925 280924 2026-09-15T01:34:12Z Sjgallagher2 29663 Spacing 280925 proofread-page text/x-wiki 9yascatog0zahkxxl17u4fitwzkbbv0 280989 280925 2026-09-15T01:58:57Z Sjgallagher2 29663 /* Emendata */ 280989 proofread-page text/x-wiki 62h8sm4rkf16lghksu9l86te9ddzgy0 280990 280989 2026-09-15T02:00:13Z Sjgallagher2 29663 Missing comma 280990 proofread-page text/x-wiki rjs1pncdm9w6vxkhgkszdcl8yv8ntsx 104 86064 280922 2026-09-15T01:28:12Z Sjgallagher2 29663 /* Nondum emendata */ 280922 proofread-page text/x-wiki 377cj5rbs7rk8w91zactf7ym5701klm 280923 280922 2026-09-15T01:31:27Z Sjgallagher2 29663 /* Nondum emendata */ 280923 proofread-page text/x-wiki m9hs4tgtcb8m4kgsqvnahg5fxvr6kjp 104 86065 280926 2026-09-15T01:36:43Z Sjgallagher2 29663 /* Nondum emendata */ 280926 proofread-page text/x-wiki r9v8hkhj2dlmb8kx2qd6c1961t4ud3q 280987 280926 2026-09-15T01:50:26Z Sjgallagher2 29663 /* Emendata */ 280987 proofread-page text/x-wiki sk09yetwufv8uaw3mdmwxniq5pus000 280988 280987 2026-09-15T01:52:23Z Sjgallagher2 29663 Fix typo 280988 proofread-page text/x-wiki 4pak6eo2hzztwmdhiodgbl3y1tm4fk4 104 86066 280927 2026-09-15T01:37:22Z Sjgallagher2 29663 /* Nondum emendata */ 280927 proofread-page text/x-wiki bt1u9qe7rrl733hh8qkhukxw96hz8yu 104 86067 280928 2026-09-15T01:37:29Z Sjgallagher2 29663 /* Nondum emendata */ 280928 proofread-page text/x-wiki ibdgiuvj8ihgawd3hj1z7fh630i7org 104 86068 280929 2026-09-15T01:37:35Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRAGT ATA. 269 Designando itaque multitudinem quantitatum a, 6, c, d . .. . per m, erunt S S S . . 8""-^ nee non T T TK . . T'""' omnes = 0: porro ^^_i ^ rj^m-x ^ 1 . ßm summa quantitatum E'"", E' E' E''^ , . . . 8""+' summa productorum omnium e binis diversis seu identicis; S^'^^ summa pro- ductorum e ternis etc. : et perinde valores summarum T™, T^+^^ j,m+2 ^^^^ ^ quantitatibus £-*'«, E-' E'"', E'... 280929 proofread-page text/x-wiki fe5yttm22o00ox91rb3xxairpvipd45 104 86069 280930 2026-09-15T01:37:39Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '2 70 NACHLASS. _|__£'-l(n+i-Jm)a-^*) ( 2 i )'"-' sin A (a — 6) sin - {a — c) sin {a — d). ubi Signum superius vel inferius valet , prout m impar est vel par. Quam cum parte prima summae S'^ addendo, sive ab eadem subtrahendo, concludimus fieri primo pro valore impari ipsius m partem primam summae i(S"+ T^) cos((n-|-l — ^m)a — s) {2iY*~^s,in{a — ft)sinj(a — c)sinA(a— <?) . . . ß n rpn partem primam summae — —. — sin ( (w...' 280930 proofread-page text/x-wiki 4v79w8p97wilkee73bvybsodqtq0o9y 104 86070 280931 2026-09-15T01:37:43Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA mTEEPOLATIONIS METHODO NOVA TßACTATA. 271 sin{Xa4-^) sin^(a — 6)sin4-(a — c)äin^{a — d) . . . sin {b --k) sin ^ (ö — a) sia 4 (i — c) sin ^ (i — d) . . . sin (kc --k) sin«-(c — a)sin-j(c — 5) sin *- (c — d) . . . + sin ^{d — a) sin A(d — b) sin ^{d — c) Casus primus , si m est impar. In hoc casu X debet esse non numerus integer sed fractione ^ affectus sive numeri imparis semissis. Ad valores negativos ipsiu...' 280931 proofread-page text/x-wiki dsigejgkt46826qopqb6jbvbtfije5b 104 86071 280932 2026-09-15T01:37:48Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '272 NACHLASS. -|-sin(^5 + A'4-^)+ etc.j yirn+i ^ _2'^'U'^~^{cos{is-{-k--a)-{-cos{is--k--b)-^cos[is--k--c) --cos{is--k--d)-- etc.j ^4»*+2^ jjpn+6 ^^^^ aequalis producto ex 2^~^ i^~'^ in summam sinuum omnium angulorum, qui oriuntur addendo s-{-k cum binis, ternis etc. angulorum a, 6, c diversis seu identicis; 17^^+ ^ C^'""''^ etc. aequalis producto ex — 2*""" ^"^~ in summam cosinuum omnium angulorum, qui oriuntur addendo s--k cum b... 280932 proofread-page text/x-wiki 045wgf03g16wxmhwhtb1djl5keuq728 104 86072 280933 2026-09-15T01:37:52Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. Multiplicentur hae aeqiiationes resp. per 273 {a — b){a — c) {a — d) . . . . (« — t) 1 {b~a){b — c){b — d). . . ,{b~ t) 1 (c — a){c — b){c — d) (c — 1 {d — a){d — b){d — c) {d-t) etc. 1 {t — a) {t — b){i—c) prodeatque inde per productorum additionem A {a-^b){a — c){a — d) . . . (a — t) ^^ {b — a){b — c){b — d) ,-. , (b — i) '^ic — a){c...' 280933 proofread-page text/x-wiki hsh9vfwjstqkufze6t2ncg8g4b03a0l 104 86073 280934 2026-09-15T01:37:56Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '274 NACHLASS. cus tantum valor ipsius X e valoribus datis computandus est. Quando plures sunt eriiendi, transformatio sequens formulae nostrae nonnunquam praeferri poterit + ('—")(' *)((„_ j)(<,jr^)+(j_aj(i^+(„_„,(„j:jj) A B + [t—a) [t— b) {t — c) ((^_j)(^_g)(^_-7) H- (ö_«)(j_c)(ft_7) + etc. (quae ex formula art. 3 facillime derivatur, si in hac multitudo quantitatum a,b,c,d... ab una a ad duas «, b, inde ad tres a, b, c etc. s... 280934 proofread-page text/x-wiki chkw3644k7no1v1q52rc3u4gpb2dfzi 104 86074 280935 2026-09-15T01:38:01Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOKIA INTEKPOLATIONIS METHODO NOVA TRACTATA. 275 quam m — 1*^ ordinis; atque 31. ^, (S, 2) . . . . 2" eius valores pro x = a, b, c, d. . .t resp. Tunc per art. 1 erit 31 + (a — b){a — c) [a — d) , . . {a — t) ^ {b-a){b — c){b-d). ..(b — t) l^(c — a)(c — Ä)(c — rf) . . .{c — t) ^^ {d — a){d — b){d — c) . . . {d-t) + etc. , 1 S %^ , "T" {t-a) {t-b) {t-c){t-d) . . . Hinc facile deducitur I (t-b)it-c)it-d).....' 280935 proofread-page text/x-wiki 7r9sm1od2c6u6ncfcgrhbukei9o1bw6 104 86075 280936 2026-09-15T01:38:05Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '276 NACHLASS. sint termini in X post potestatem a?'"~* hi fx<2?"'+va?"'+^-j-etc. Erit itaquein art. 3 non W=^0, sed ut ex art. 1 sequitur, TF= (i-f-v(a4-i-j-c-f-</-|-. ..-|-^)-j-etc. Hinc valori ipsius T illic tradito adhuc adiici debet {t—a){t—b){t—c){t—d).. . x!(x+v(a+6 + c+<?4- ... 4-^)4- etc.| 7. Casus in praxi maxpe frequens est, ubi a,' b, c, d . . . progressionem arith- meticam constituunt. Ponendo intervallum r= i , ita u... 280936 proofread-page text/x-wiki tvxlwkcio2h358cnyz4jjra0fqovc7v 104 86076 280937 2026-09-15T01:38:12Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 277 Manifesto hoc ita est decidendum, ut error a differentia m}^ ordinis oriundus fiat quam minimus ; hie vero error fit = {t—a)(t—a~)[t^a-1) .... {t—a—m^)j-^—^ si adhibentur terraini ad x --= a, a+l, a---2, . . . a--m — 1 pertinentes, vel = [t—a—)(t-~a — 2)(t—a — ^)...,{t—a—m)-^^- si adhibentur termini ad w = a--, a-{-1, a--^, . . . a--7n pertinentes, vel A == (^ — « + 1...' 280937 proofread-page text/x-wiki 153m848x8s5uki0e96tpu7wryikqzdq 104 86077 280938 2026-09-15T01:38:19Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '278 NACHLASS. • cuius valor numericus ex A = 9,5918780 B = 9,6093133 C = 9,6259483 D = 9,6418420 eruitur 9,6177271.5; tabulae dant 9,6177270. Quodsi hie loco log. sin 2 3^ logarithmum sin 27° = ^= 9,6570468 ad- hibuissemus , formula H-tV^ + H^— tV^+tV£ dedisset 9,6177267.375. Formula art. praec. pro hoc casu dat errorem ex neglecta differentia quarta A oriundum = ^f-g-^. si Ä, B, C, D, contra = — rf^^. si -B. C, D, E ad- h... 280938 proofread-page text/x-wiki 1et7nydbyrnhln7qs003mxg0542kkqn 104 86078 280939 2026-09-15T01:38:23Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TEACTATA. 279 «=—29® 19' 57" =—105596" A = —2 6=— 16 6 47 =— 58007 B=—l c = — 2 33 56 =— 9236 C= d = --ll 13 2 =+40382 i) = -|-l e = 4-25 7 6 == + 90426 £=+2 t = Hinc computatur /'-?'"-t"~,n'M = +0,0149084 (a — 6)(a — c)(a — d){a — e) ' ' lt-a){t-c){t-d){t-e) _ ........ (ö_«)(6_c)(5_d)(5_.) = — 0,10o0660 it-ant-l^it-ä)it-e) _ ^0,9624567 (c — a)(c — o)(c — cZ)(c...' 280939 proofread-page text/x-wiki 9vt8u2spjqg28vrhyi9mdwkqqv3h0lp 104 86079 280940 2026-09-15T01:38:28Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '280 . NACHLASS. A = a--a cos a-{-a" cos 2 a --a" cos da -- etc. -j- ö ' sin a -f- 0" sin 2 a -(- Ö '" sin 3 a-}- etc. ^ = a-{-a cos ö -j- a" cos 2b-- a!" cos 3 6 -|- etc. -f-ö'sin6-fö"sin26-|-ö^'sin36-f etc. C = a -f- a' cos c -|- a" cos 2c~-a"' cosdc~- etc., -hÖ'sinc+€"sin2c-f-ö"'sin3c4- etc. Z) =:: a-f-a'cost?-}-«" cos 2 (?-}-«"' COS 3rf-|- etc. -{-ß'smd-ir^"sm2d-{-ß"'smZd-{- etc.' etc. T = a-|-a'cos^-j-a"cos2^-f-a"'cos3^-... 280940 proofread-page text/x-wiki g7na7gh9sli4hgf8ye1yxn1n2zauuqq 104 86080 280941 2026-09-15T01:38:34Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOßlA INTERPOL ATIONI 8 METHODO NOVA TRACTATA. 281 Tunc ex art. 2, ubi m idem erat, quod hie est 1m--2, adeoque casus se- cundus locum habet , fit TF = ; quamobrem multiplicando per sin^(^ — a)sinf (# — 6)sinf (? — c)sin|-(? — d) prodit sin ^ {t — b) sin ^{t — c) sin *- [t — d) sin ^ (a — b) sin -*- (a — c) sin ^{a — d) . , sin - {t — a) sin | {t — c) sin {t — d) . sin 4^ (b — a) sin ^{b — c) sin-*- {b — d) ....' 280941 proofread-page text/x-wiki ls3dphdp0h9pbmkshur6uazcue7corq 104 86081 280942 2026-09-15T01:38:39Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '282 " NACHLASS. %cosit-{-'ä'cosft--Wcosft-{- etc. -{-^smit-{-^'smit-{-^"smft-- etc. Quamobrem pro hoc casu viam aliam ingrodi oportebit. Multiplicatur X per cos[4ra;--k), proditque, si termini ultimi in X sta- tuuntur a"^cosmji!-{-^^smmcc acos{^x--k)---^ a cos{fx--k)-{- ^a" cos{'^x-j-k)-- .... 4- f a*" cos ((m 4- ^) a? + /f) -|-ia'cos(^a7 — k) -- -l a" cos (f 0! — k]-{-^a" cos{fx — k)--, . . . 4--^a"*cos((m — i)a; — k) ■ -f-... 280942 proofread-page text/x-wiki 6dqwy6ahazdkodize40yaoi8db3atyi 104 86082 280943 2026-09-15T01:38:44Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 283 W = A cos (a a + ä) 8ini(a — 6)sinJ-(a — c)sin^(a — d) .... sin 4^ (a — t) + BcoB{^b + k) sinA(i — a) sin | (6 — c) sini(6 — d) ,. . . sin|-(ö — Ccos (A c + A;) ' sin| (c—a) sin -J(c — 6) sin L(c—c;) 8in-i-(c— iS) _l Dcos(^-t;+Z :) ""'" sin I (cZ— a) sin J (d — &) sin a (ü!— c) . .. .sinJ((Z— ^) + etc. "•" sin 4- (i! — a) sin {t — b) sin i{t —...' 280943 proofread-page text/x-wiki r1qsgtew6hsy339hmsx1it5khhx7168 104 86083 280944 2026-09-15T01:38:49Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '284 NACHLASS. IV. Quodsi autem tantummodo expressio eiusdem formae ut a + a' cos t -j- cd' cos 2t-- etc. -j- ö ' sin ^ -j- ö " sin 2 ^ -j- etc. quaeritur (sive cum hac identica sit , sive non) , per quam valores propositi A, B, C,D etc. repraesententur, si pro t resp. a, b, c, detc. substituitur : quantitas a"^cos^s--ß"^sm^s ex incognita arbitraria evadit, quam si placet etiam = statuere licebit; reveraque formula nostra pro Tita co... 280944 proofread-page text/x-wiki ql8gsli8ewhzo9xhyzhkw6neufcx48q 104 86084 280945 2026-09-15T01:38:56Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'Ä' = THEORIA INTERPOL ATIONIS METHODO NOVA TRACTATA. 285 A , B sin ^ (a — b) ' sin ^ (6 — a) A„ ^cos^(a — c) , ^0084^(6— c) , C Ä" sin^(a — 6)8in|(a — c) ' sin4^(6 — c()sin|(S — c) ' sin4-(c — a)8inf(c — b) A I B sin * (a — 6)sin-*-(« — c)sin *-(a — d) "' sinA(6 — a)sin-|-(6 — c)sin|(6 — d) H L_l^ R ' sin|^(c — a)sin^(c — 6)sin^(c — d) ' sin|^((^ — a)sm^{d — &)sin|(d — c) A titr Acos^{a —...' 280945 proofread-page text/x-wiki h0en417i0tjej21657s33npoi9hmehp 104 86085 280946 2026-09-15T01:39:01Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '286 NACHIiASS. a, Ä, b, B, c, C, d, D etc. resp. mutantur in vel in Tunc erit c, C, d, D, e, E, f, F etc. d, D, e, E, f, F, g, G etc. etc. A — ^-^ B' — ^-^ C — ^-^ 9ic sm^(a-^6) sin-^(6 — c) sin-|(c — d) .„ __ A<to%{a-c)- B' j^„ _ B'co8i{b-d)-C' . ^ — sin^(a-c) ' -^ — sin^ib-d) ^^' .,„ A" cos^ {a — c) + A' sin j {a — c) — I{" „w B" cos^{b—d)-- B' &m^{b—d) — C" . -^ — ain^{a — d) ' ^ ainl-{b — e)...' 280946 proofread-page text/x-wiki ttw0knmmd2n7z1zg5ysxmmwg7lirja7 104 86086 280947 2026-09-15T01:39:06Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 287 denique pro a, a — a sive , manifestum est , partes determinatas hoc modo ad- iunctas , mutuo se tollere adeoque functionem X hoc modo non variari. Singu- lae vero partes ipsius X nunc per sin^(^ — a) divisibiles erunt, puta cos na; — cos na n ' tr ^ ^ ^ — ^-TT r- = — 2sin((w — TJ-^ + i«) — 2sin((w — f)^H-fa) — 2sin((w— |-).^ + |a) — etc. — 2 sin ((1"^+ (^ — 4") <^)...' 280947 proofread-page text/x-wiki tb8fp45p03hkupj5j1qz5nukcdam67v 104 86087 280948 2026-09-15T01:39:11Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '288 NACHLASS. ut per multiplicationem facile confirmatur. Uterque quotiens est formae F. Quam- obrem functio tota X per sm^[oc — a) divisibilis, quotiensque ad formam F reducibilis erit. Q. E. D. 14. Theorema. Si functio X formae F vel G pro pliirihus valorihus diversis*) ipsius X, puta a, 6, c, d etc. ßt = , per productum sxi^[x — a)sm(x — h)^xi{x — c)&m{x — d) .... divisibilis , quotiensque vel eiusdem formae erit ut X vel diver... 280948 proofread-page text/x-wiki 5yezsa42zt8av80zlmawfep917svql2 104 86088 280949 2026-09-15T01:39:15Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTEßPOLATlONIS METHODO NOVA TEACTATA. 289 G, prout multitiMdo valorum a, b, c, d etc. , quam per {x designabimus , par est vel impar. lam observamus : I. In artt. 10, 11, 12 eruere docuimus functionem X ordinis m^^ (i.e. ultra cosmx et sinm^ non progredientem), quae (x valoribus propositis A,B,C,DQic. satisfaciat, ita ut m sit =i-jJ. — ^, vel ^(jl, prout [x impar est vel par. Pro- ductum P manifeste est ordinis y^^' Quando it... 280949 proofread-page text/x-wiki mvw7lb2phtk4o3086j1eoydf1gx9bqw 104 86089 280950 2026-09-15T01:43:02Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '290 NACHLASS. kcos ^s + /sin^ s K' l—L'k sin is ÄCOS^ s + ^sin-^s K' l—L'k cos ±s kcQB^ s-- /sin-^s esse 1 cos 4 5 . 1 = ± sin^s _L 2."-' 2/<-i = ponitur, oriunda, attamen ab hac diversa esse potest. Sit functio illa X", haec X'"; adeoque X'"= X"--hP, designante h quantitatem definitam. Po- namus terminos summos in X" esse K' cos 'l-^{v-{-L' sin -^iaj; in X'" autem Ä'"cos-|-[x<2?-|-jL"sin^[x<... 280950 proofread-page text/x-wiki m8r29kh1pzymcbeoh5c8xy7m3f10ck4 104 86090 280951 2026-09-15T01:43:06Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTEEPOLATIONIS METHODO NOVA TRACTATA. 291 les habere debent. Quibus ita intellectis sint m-j-1 valores dati functionis X hi A, B, C, D etc. , valoribus ipsius cv bis a, b, c, d etc. respondentes , quaeri- turque expressio generalis pro valore T, quem functio pro <^^ = adipiscitur. Quum functionem X in casu nostro etiam sub formam y-|-7'^os,2? + 7"(cos<r)^ + y"'(^os<2?j^-f- etc. — (-y'"(cos<2?)"* reducere liceat, habebimus per... 280951 proofread-page text/x-wiki g30a2g6mi5fhz6px5jvimh3rt526u9k 104 86091 280952 2026-09-15T01:43:11Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '292 NACHLASS. reduci posse , adeoque, quum valores functionis - — pro x = a,h c d = sina' "^^' ■= — ' —-1 haberi per eundem art 3 sina' sjjjö smc sina ^ . fiant sin t (cos t — cos b) (cos t — cos c) (cos t — cosd) . sin a (cos a — cos 6) (cos a — cos c) (cos a — cos d ) . sin t {cost — cosa) {cost — cosc) (cos t — cos d) . sinb (cos b — cosa) (cos 6 — cosc) (cos ö — cosö!) . sin t (cos t~cosa) (cos t — cos 6...' 280952 proofread-page text/x-wiki 0wtizleesvb7ln45g90fkexkhssfx24 104 86092 280953 2026-09-15T01:43:16Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOEIA INTERPOLATIONIS METHODO NOVA TRACTATA. 293 ^tum deprimatur, uncle necessario functio X ipsa, quae e valoribus Ä,B, C^D etc. per methodum art. praec. eruitur, prodire debebit. Quum methodus terminum summum functionis Y atque inde deinceps inferiores evolvendi, tamquam casus specialis e raethodo in art. 15,11 tradita peti possit, non opus est, huic rei hie diutius inhaerere. ■ 18. Ömnes disquisitiones praecedentes superstructa... 280953 proofread-page text/x-wiki 2h45o0e0dlxs5lbcj7i0tat8m4zucdz 104 86093 280954 2026-09-15T01:43:22Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '294 NACHLASS. sin4-(^ — a)sin4-(if — b)sm^{t — c)sin-|-(^ — d) etc. ßt = ^^ — ^Sn^— ) w&^ Signum superius vel inferius valet, prout jjl par est vel impar. Demonstr. Potest quidem hoc theorema facile ex alio ab ill. Eülero in In- trod. in Anal. Infin. I, §. 240 tradito derivari; ne quid vero hie desiderari vi- deatur, demonstrationem aliam paucis hie explicabimus . I. Si produetum P primo ponitur sub formam eos-^(^ — ö! — 180^)eosJ...' 280954 proofread-page text/x-wiki 1uqnf5lhvlua06mdolcphavtektmm31 104 86094 280955 2026-09-15T01:43:26Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOEIA mTERPOLATIONlS METHODO NOVA TRACTATÄ. 295 cosif — cosa = 2sin-|-(^ — a)sin4-( — t — a) cost — cosb = 2sin^(^ — &)sin-|-( — t- — b) cost — cosc = 2sm-^{t — c)sin^( — t — c) ■ cost — cosd = 2sm^it — ^)sin^( — t — d) etc. productum ex his factoribus fit, per lemma primum sinA[i.(^ — «)sin^fji( — t — a) cos[).t — costAa f r;^ t = g^z — ^]Sri ^. -Ci. JJ. Quum habeatur sin-5-{j(,(^ — a) = — sin^...' 280955 proofread-page text/x-wiki a30d8l4yfv3fkyh9qz31fz9vip5cy3x 104 86095 280956 2026-09-15T01:43:30Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '296 NACHLASS. Hinc in formula art. 1 pro T fit coefficiens ipsius A — l(i_|_2cos(^ — a)H-2cos2(f — a) + 2cos3(# — a)+ . . . + 2cosm(^ — a)) Simili modo fit ^m^{t—a)&m^{t—c)&m^(t—d) . . . = — 2^--sinx(^-6) == _-i-^(l-f-2cos(^ — 6) + 2cos2(^— 6) + 2cos3(^ — 6)-j- .'. . +2 cosm(^— 6)) atque sin|-(6 — a)sin-^(& — c)sin-f (6 — d) . . . = — —-, Hinc coefficiens ipsius B in formula pro T fit — l(l-|-2cos(^— 6) +...' 280956 proofread-page text/x-wiki 1zlz07x7nqlg7aw4qm1ptupltcknut2 104 86096 280957 2026-09-15T01:43:38Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOKIA INTEßPOLATIONIS METHODO NOVA TEACTATA. 297 pletus erit, itemque valores singulorum coeificientium a, a! ß a", ö" etc. ibi traditi correctione opus habebunt. Haec autem commodius per methodum sequen- tem quam per summationem art. 2 determinatur. ijite omnia observamus , esse cos n a -- cos 11 b-{- cos nc-- cos nd-{- . . . = (xcoswa sinn a -- sin 7tb-{- sinne-}- sin 71 d-- . . . = isinna quoties n est integer per [Jt divisibili... 280957 proofread-page text/x-wiki ju8zwml7qanr4wdky2opkonxmcuoouh 104 86097 280958 2026-09-15T01:43:43Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '298 , NACHLASS. y-|-7'cos^ + 7"cos2#-|-Y"'cos3^+ . . . +y"*cosm? ita ut habeatur j = —{A--B-- C--D-{- etc.) etc. Substituantur pro Ä, B, C, D etc. valores sui, puta A = a -|- a' cos öj + a" cos 2a-[- . . . -f- a^" cos we « + a"*"^' cos {m--l)a-- . . + ö'sin« + €"sin2rt+. . . 4-ö'"sin?wa + ö^+' sin(m + l)a+ . . etc. fietque (per summationem modo traditam) ■y = a--a^cosi).a-{-a^^cos2ii.a-{-a^^cosdia-{- etc. --ß^ &m lia -{-ß^^ sin... 280958 proofread-page text/x-wiki 6m4iphrn95y8uoqv7la4ro9yehdkts3 104 86098 280959 2026-09-15T01:43:48Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOEIA INTERPOLATIONIS METHODO NOVA TKACTATA. 299 cosi-{t—a) smi-{t ■^b)smi{t— c) sin i-{t — d) . . . sm{-[i.(t—a)cos-L{t — a) 2f^-' sin i{t — a) == ^ir7-^(cos4-(# — a) + cos| {t— a]-i- cos f{t — a) -{-... --cos^'{t — a)) == — ^1(1+ 2 cos(^— ö) 4- 2 cos 2 (?— a) + 2 cos 3 (f— «)-f . . . -f 2 cos(m— l)(A-a) + cos m{t — aj) Quare in formula art. J 1 pro T coefficiens ipsius Ä fit = ■^(l+2cos(#— a) + 2cos2(...' 280959 proofread-page text/x-wiki i6l0p4ob91htjh3bhr0hm1nho9u4lnv 104 86099 280960 2026-09-15T01:43:56Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '300 NACHLASS. exhibetur, omnes coefRcientes y, y, ß', y", 8" etc. praecise eosdem valores ha- bere patet, ut in art. praec. , ubi erat [i =■ '2m-}-l, exceptis duobus ulti- mis y"*, ^^, qui illic duplo erant maiores. Porro liquet, omnes coefRcientes a, a a" . . . (X^- 6', Ö" . . . ^'""^ Ulis y, y, f... j"^-', ^',^"... ^^-' resp. aequa- les adeoque cognitos fieri, inter Ultimos a"*, ß"^ autem haberi aequationes ^m __ '^'^ --(^a^ sin ma... 280960 proofread-page text/x-wiki dps5qznaohl87ecfxxrckrloe27j7ah 104 86100 280961 2026-09-15T01:44:01Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 301 In hoc itaque casu coefficientes y» T' ^'' T"» ^" • • • 7"*~ 8"^~^ pro veris a, d, €', a' ö" . . . a'"~', ö™"~* adoptare licet, quatenus sequentes «"^"'" ö"*+' etc. negligi possunt; ad Ultimos vero y"*, ^'" nisi ipsi pro negligendis haberi possunt, haec conclusio non est extendenda. 24. Per disquisitionem inde ab art. 2 traditam eo pervenimus, ut propositis |x valoribus functionis X... 280961 proofread-page text/x-wiki he2ed3k65je4eh0ezm52oed57fjjac7 104 86101 280962 2026-09-15T01:44:06Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '302 NACHLASS. sive quod eodem redit per Lsm{x--l) — Xsin((X — (J-j-^ + ^+f^a) designante X integrum. Quocirca in X" pro quavis parte Lsm(lx-{-l) sub- stituere licebit hanc iysin((X — i)x--l-{-iia) et simili ratione pro hac rarsus jLsin((X — 2|x)a?+/+2{jia) porroque -Lsin((X— 3 [ji)<2?-j-/-[-3 [xa) etc. Ut similis conclusio ad cosinus extendatur, sufficit observatio, cosinus pro sinubus prodire, si modo pro / scribatur /-|-90". Hinc... 280962 proofread-page text/x-wiki blkhf5ul2ud8bxfm1qh7n0soo21z4z0 104 86102 280963 2026-09-15T01:44:10Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 303 25. Si functio X cum terminis cosmx, smmx non abrumpitur , sed ulterius excurrit, coefficientesque terminorum sequentium adhuc nimis considerabiles sunt, ita ut ipsos negligere non liceat, coefficientes 7, 7', ^' etc. in functione X', quae [i, valoribus functionis X periodu^ completam formantibus, pro oj = a, b,c,d etc. satisfacit, notabiliter a coefficientibus respondentibus a, a, ö'... 280963 proofread-page text/x-wiki 3bzm3aimkuq2b8q246rr9efrcj75g2t 104 86103 280964 2026-09-15T01:44:15Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '304 NACHLASS. X' evolvitur, nanciscimur v valores singulorum coefficientium y, y', d' etc., qui hisce deinceps valoribus ipsius ^ respondent ^ = ^a, ^ r= |i,a' = (i«-f--360°, 1/ = {la = ixa-{- — ^QO^, etc. adeoque periodos completas v terminorum formant. Quocirca, per methodum no- stram, singuli coefficientes y, y', 8' etc. per expressionem talem -f- C'siny -}-C"sin 2 j/ -}-.... -f-C'^ sin wj/ exhiberi poterunt, ita iit sit n ve... 280964 proofread-page text/x-wiki b6ge3vi7qx28ic8g0v1z0o13crfxh4s 104 86104 280965 2026-09-15T01:44:22Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOKIA INTERPOLATIONIS METHODO NOVA TRACTATA. 305 dem ordinis ut X'". Sufficit itaque utriusque functionis terminos Ultimos com- parare. Haud difficile perspicitiir , hos terminos Ultimos in X" prodire unice ex terminis -^"^ cosmx -{-^"^ sinmcc , et quidem ex iis terminis coefficientium y"*, S"*, qui continent cosni/ et sinni/ sive cos[ina^ et siB.inx. Sint hi termini in y"^ .... Ä cos [1/^.2? + /sin {xw<^ in ^"* . . . . k' cos inaj-... 280965 proofread-page text/x-wiki kobk46bdshuwvh8s46yqwcl2mm6rkug 104 86105 280966 2026-09-15T01:44:27Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '306 NACHI.ASS. Quocirca erit -^{k — V) = hcos^iza = h cos ^{^n--m)a i-{l-^k') = hshi^Tza = h sin ^{nn'--m)a Q. E. D. Haec vero aequatio ipsissimum erit criterium coefficientium ultimorum functionis X" ut in art. 25 ostentum est: quare in hoc quoque casu functio X" cum X'" prorsus identica erit. III. Quando v par est, fit iin--m = -^Tz-{-m, adeoque X" ordinis altio- ris quam X'". Facile autem perspicitur, omnes terminos in X", qui o... 280966 proofread-page text/x-wiki 8qdzpa150bsl5y5k9erwrovfv5pnv7l 104 86106 280967 2026-09-15T01:44:32Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRAGTATA. 307 Sed ex aequatione ksin^iza = /cosfica, quae in utroque casu inter k et / lo- cum habere debet (art. 24) , nullo negotio deducitur ^ [k -{- k cos iz a-{-l sin TZ a) = k {l--ksmT^a — /cos TT a) = l Hinc tandem coUigitur , pro iis partibus in X", quae ex terminis k cos ny--l sin ny cuiusvis coefficientis y^ vel h^ producerentur , designante 'k indicem maiorem quam 0, statim subst... 280967 proofread-page text/x-wiki 6nnb5kr8lwzs8yaimi1w44pv6xsjkoj 104 86107 280968 2026-09-15T01:44:38Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '308 NACHLASS, rursus esset compositus , in quo casu manifesto quaevis periodus /x terminorum in plures periodos minores subdividi potest. Ceterum ex pluribus aliis annota- tionibus applicationi practicae inservientibus hie sequentes tantummodo attinge- mus. Valores talium aggregatorum Acos'ka--Bcos'kb--Ccos'Kc--Dcos'kd-{- etc. =p Asm'ka--Bsm'kb-{- C sinXc -- D siiiXä -- etc. = q (ubi ut in praecedentibus, supponimus b — a= c — b =... 280968 proofread-page text/x-wiki s7q5v9pr1b5fzuqoh3qp8kx9ed6x3s3 104 86108 280969 2026-09-15T01:44:45Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTEßPOLATIONIS METHODO NOVA TRACTATA. 309 Distribuamus hanc periodum primo in tres periodos quaternorum terminorum + 408 -- 10 4-1238 + 1462 a= 0« Ä b = 90« B c = 180« C d = 270« D In formula a = 30« Ä =^ 89 d' = 60« A' = — 66 y = 120« J5' = -j- 338 h" = 150« B" =: + 807 1 c = 210« C' = +1511 c = 240« C" = +1583 d' = 300« D' = +1183 d" = 330« D" = +... 280969 proofread-page text/x-wiki 5n07jy3swtjo07fud4nx7kev95xyzwl 104 86109 280970 2026-09-15T01:44:48Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '310 NACHLASS. + 780,6 -|-411,0cosa? — 720,2sm.3? -- 43,4cos2a? — 2,2sin2a? — 4,3cos3<2?-|- 5,5sin3<2? — 1,1 cos 4 a? — 1,0 sin 4 <2? -f- 0,3cos5a7 — 0,3sin5.3? -|- 0,1 cos 6 .3? Distribuamus secundo eandem periodum in quatuor periodos ternorum termino- rum , quibus singulis itaque per formulam primi ordinis X.' = Y-l-y'cos.2? + ^'sin^ satisfaciendum erit. Hie invenitur Pro periodo ubi ^ = dcc T 7 8' prima 0«... 280970 proofread-page text/x-wiki t18n2uup33pqq43ugp2yahp0v8r4dkt 104 86110 280971 2026-09-15T01:44:53Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 311 atque a-{--^QO^--a = /x 360^ sive a = il — k)—. Quare in hoc casu a per — divisibilis erit, adeoque smfia=0; porro habetur b = (ixl — ^+2) — , atque sinfxö = 0, et perinde de sequentibus c, d etc. Supponendo itaque, a esse minorem quam —7- , sed non negativum (quod permissum est, quum ab omni- bus a, b, c, d etc. multiplum proxime minus totius peripheriae subtrahere et ex residuis mini... 280971 proofread-page text/x-wiki iic8j8d9ivw43gho5febv5furup932v 104 86111 280972 2026-09-15T01:44:57Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '312 NACHLASS. terminus secundus = sinj(xa4~sin(ju, — 2)a ultimus = sin j(xa — sin {fj, — 2)a porro terminus tertius = sin fx« + sin (f^ — 4) a terminus penultimus = sin/x« — sin(|üt — 4)a etc. Hinc colligitur , coefficientem ipsius A in formula art. 1 6 pro T fieri r^^ — I sin nx Ö5 -}■ 2 sin (ju, — l)a cos ^ -|- 2 sin (ju, — 2) a cos 2t-- etc. -j- 2 sin a sin (ju, — 1 )f j |jisin| Prorsus similes expressiones proveni...' 280972 proofread-page text/x-wiki j7krfeuibe8pew9xr6fizpb47cg9iaq 104 86112 280973 2026-09-15T01:45:02Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 313 comparare, quam per X." exprimemus. Comparando valores coefficientium y, y, S y", ^"etc. in artt. 20, 22 traditos cum valoribus coefficientium e, e', e'etc. modo inventis , invenitur s = y ... E = Y — 8' cotB,ng fjba, ^^"^ = ^'cosecjU/ö s" = y" — ^" cotang fJL a , s^'~~^ = ^" cosec [la £= y — cotangfxa, £' =0 cosecjii,« etc. usque ad (4-)e*'*r=y"^— g^VotangjLi«, (i)£f^-'"= a'^... 280973 proofread-page text/x-wiki ayedbrn5pe0ui8klunm1arz2s6phlc2 104 86113 280974 2026-09-15T01:45:07Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '314 NACHLASS. £ z:s=i a--a^' COS iJia-{-a^^cos2 ^a-- etc. ^' = cTt:::; | a sin fxa 4- a^-^^ sin 2aa-{- a^'"'+^ sin 3 ua + etc. — a~f^~^ sin^a — a^'"~^ sin 2 jix a — a^f^"^ sin 3 ju, a — etc. j e" ^= ^-r^ j a" sin fia-j- (x'^+" sin 2 jix a -}- oc^f^+^ sin 3 |(x + etc. — a^^~^sm^a — a^f^~^sin2]u,a — a^f^'^sinSjua — etc.} etc. ^^~^ == ^r^ j a^~^ sin |ii a + a^^'^'^ sin 2 jix a + a^^"^ sin 3 fi« -j- etc. — a^~^^smixa — a^^'^^ sin...' 280974 proofread-page text/x-wiki hxfolzq6eyims4kh58waloefsb6z6x0 104 86114 280975 2026-09-15T01:45:14Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOKIA INTEIiPOLATIONIS METHODO NOVA TRACTATA. 315 in casu tertio Lcos^^acos{k-{-^)^acos-^lJiX — Lsm-^{xacos[k--^)fjiacotg^acos^^a? + L sin^|u,6« cos {k-{- 1-) |[xa cosec jixa cos^ |(jt<a? quae expressio in omnibus tribus casibus reducitur ad L sin (k--i)iJ.a cos "k'x — L sin kixa cos (a — ')x sin[A a Quoties autem X'= 0, habemus in X' simpliciter Lcosk^ia, qui termi- nus sine variatione in X" retinetur. 33. Sit secundo X fu... 280975 proofread-page text/x-wiki tb0t6n86et482rrtttgqbu7r6manzr9 104 86115 280976 2026-09-15T01:45:18Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens ' 316 NACHLÄSS. cum X identica erit, mutando t in cv, unde coefficientes C', C", C" etc. ipsis ö', ^", ß'" etc. resp, aequales erunt. 34. Si ut in art. 3 1 omnibus valoribus propositis per functionem talem X' = 'y + T'cosa:'-j--)'"cos2<37+ etc. -{-y^^^cosmw -- ^' sin 0) -{- ^" sin 2 Ol -- etc. -{-^"^sinmaj satisfactum est, existente m = {(jl — ^ vel = ^-jix, prout fx impar vel par est, comparatio huius functionis cum hac X" = C... 280976 proofread-page text/x-wiki o83ondfznju4mtpxggbudh9b51j5xx4 104 86116 280977 2026-09-15T01:45:23Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOKIA INTERPOLATIONIS METHODO NOVA TRACTATA. 317 -, . , > - . / s , . y'- cos u. a sin X a; 4- Tf^' sin (ix — X)x 7^cotg[Aö^sinX^-l-7^cosec|x«sin(|x — Ä),2? sive ^ ^^^^^ — quod praeceptum lectores cum iis, quae in art. 24 tradidimus , ipsi comparent. 35. Si functio X cum sinnx non abrumpitur, sed ulterius excurrit, terminis sequentibus per ^f^+*sin((jL-|-l)^H-^'''+^sin([x+ 2)<2? etc. expressis , habebimus: C = -J_!a'sinaa + gf'... 280977 proofread-page text/x-wiki qzm9koxjrd0frmh21bpg7x1vwi0l14l 104 86117 280978 2026-09-15T01:45:28Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '318 NACHLASS. 36. Transformationes in art. praec. atque in art. 32 traditae concinnius ex theo- remate quodam generali deduci possunt , quod quum per se quoque satis elegans sit , paucis hie adhuc attingemus. Theorema. Designantihus X, V, X" numeros integros qiioscunque , jx numerum integrum, qui differentias inter illos , X' — X, X" — X', X — X" metitur [e. g. unitatem), X arcum indeßtiitum, a arcum deßnitum: functiones P = sin (... 280978 proofread-page text/x-wiki 3medpglyapb2n62bxd0z088pln85xw3 104 86118 280979 2026-09-15T01:45:33Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEOEIA INTERPOLATIONIS METHODO NOVA TEACTATA. 319 P, Q ex partibus per cos[Jt<2? — cosfia divisibiles compositas , ideoque ipsas quo- que per hunc divisorem divisibiles esse. II. Quando X per (i non est divisibilis , sit l numerus integer arbitrarius per [1 divisibilis, ponaturque X = l-{-b , X'=/'-|-6, X"=/"-|-Q, unde etiam /', /" per (X divisibiles erunt. lam patet, si ponatur sin(/' — l")acosla!-{-sm{r — l)acosra;-{-sm{l — l')aco... 280979 proofread-page text/x-wiki qp0wvw2nlkuizmui1ea9htybpnhs2s4 104 86119 280980 2026-09-15T01:45:39Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '^20 NACHLASS. Sit primo X functio formae y -- Y cos a^ -- y" cos 2 a^ -}- etc. -[-y"coswa? adeoque multitudo coeiRcientium incognitorum = l-f-w, lam scimus, ex Omni- bus valoribus propositis functionis X eos reiici debere , qui respondent valori ipsius maiori quam 180^: quatuor itaque casus hie sunt distinguendi: 1) quando |jl par, atque a = 0, erit 180^ valor i-[x-l-l*"^ ipsius o?; quare quum sequentes reiici debeant, remanent val... 280980 proofread-page text/x-wiki efbz1c0377kyxmfy1egofsuodq37g88 104 86120 280981 2026-09-15T01:45:44Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TKACT ATA. 321 in qua coefficientes y, y', y" etc. determinantur per formulas y = -1(^ + 1^+04-1)+ etc.) y'=-(^cosa --Bcosb +Ccosc --Dcosd + etc.) y"= - (^ COS 2 öf + J5 COS 25+ Ccos 2 C + -D COS 2^+ etc.) etc. , ultimus autem , quando i par est atque adeo m =■ -^- [x , per hanc y"^=: — (J.cosma + ^cos?w&+ Ccosmc + J)cosm^+ etc.) necessario cum functione X identica erit, siquidem haec non est gr... 280981 proofread-page text/x-wiki f4imtknp6yuw4qfb9eu9zx0gx8vphs8 104 86121 280982 2026-09-15T01:45:50Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '322 NACHLASS. in casu 2 et 4, superius in casu 1 et 2 ; denique pro ultimo habebitur in casu primo y"^ = a"^4- a'"^ + a'"^ + etc. in casu secundo autem y"* = 0. Hae aequationes ostendunt, quatenus differen- tiam inter functiones X et X' negligere permissum esse possit. Haec posterior functio inter omnes , quae jn valoribus propositis satisfaciunt, simplicissima erit, quae omnes sub forma X.'--Ysm({-fjLaj — if^a) contenti erunt, quae... 280982 proofread-page text/x-wiki 7tydi1qgmgwlukva7jqdmavp7av2i6g 104 86122 280983 2026-09-15T01:45:56Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 323 datorum subductis superfluis , esse debebit = n , ut ad coefficientium determi- nationem completam sufficiant. lam quum ut superflui in hoc casu reiiciendi sint valores functionis X ii, qui respondent valori ipsius <2? maiori quam 180^ nee non valori et 180°, habebimus pro quatuor casibus supra distinctis : 1. quando ju, par, a= 0, erit n = ifj^ — 1 1 8 0*^ 2. quando fx par, a = —... 280983 proofread-page text/x-wiki hlcqvr0zzm8i4yzb2r1xqjxpzhgu7gc 104 86123 280984 2026-09-15T01:46:01Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '324 NACHLASS. dinem j^fi — 1 tantum ascendit , quum fiat 8"'= -(^sin04-5sinl80''4-Csin360"4-Dsin540"+ etc.) = 40. Si functio X non est ordinis w^ ut supposuimus, sed ulterius excurrit, ae- quationes sequentes docebunt, quomodo differentia inter X' et X a coefficienti- bus sequentibus pendeat (v. artt. 21, 23) g' = g' Zjl ßi'-' ± ßi>+' _ g2[^-i + b''^+' + etc. • g"= 6"Ijl€!^-2 + öi-+2_€2!^-2 + ^2f-+2q: etc. et sie porro usque ad... 280984 proofread-page text/x-wiki 3csyfnxtv46w63yit3oxk8iuw0icphw 104 86124 280985 2026-09-15T01:46:06Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens 'THEORIA INTERPOLATTONIS METHODO NOVA TRACTATA. 325 primo i. e. pro a= 0, et valore pari ipsius ^i evadit '=-^fji, terminus Lsin'Kaj omDino destruatur. 41. Quum omnes casus speciales in artt. 29 — 40 considerati ad casum genera- lem in artt. 20 — 28 absolute reducti sint, omnia artificia, per quae in hoc casu calculus abbreviatur, qualia in artt. 25, 26, 27 explicavimus , etiam ad illos applicari poterunt. Quamobrem non opus erit ,... 280985 proofread-page text/x-wiki q20otk7rgif8zn4f97q159hwggbu1rx 104 86125 280986 2026-09-15T01:46:10Z Sjgallagher2 29663 /* Nondum emendata */ Paginam instituit, scribens '326 NACHLASS. ubi pro coefficientibus y, y', ^' etc. valores sequentes invenimus : Periodus y=^x Y t' prima secunda 0« 60 -(- 56"205 — 217"757 tertia 120 + 56,115 — 217,467 quarta quinta sexta 180 240 300 — 56,115 — 56, 205 -217,467 — 217,757 t" 4 761"671 ! + 760, 805 — 760 805 — 761, 671 — 1528,825 — 1527,397 + 1527,397 -j-1528,825 -103830,105 -103 9 37,... 280986 proofread-page text/x-wiki r583p54bkotul3e9biuq2fpadgmuda8 104 86126 280991 2026-09-15T05:11:36Z Trooper57 22870 /* Problematic */ 280991 proofread-page text/x-wiki 5ouov2vcf5n3ysseo7kly8brep135u4 280992 280991 2026-09-15T06:21:49Z Trooper57 22870 /* Proofread */ 280992 proofread-page text/x-wiki p6xlhmlfi8m3jnhmavt5bwhxjbq8l0z 280993 280992 2026-09-15T06:22:40Z Trooper57 22870 280993 proofread-page text/x-wiki rkgxp435lhlb3fztbpnurbqt6zcao3j