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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 8v, page id 00067993:0016
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Schohum
Has propositiones quartam et quintam, etsi valde claras attente
consideranti adjiciendas duxi tamen, tum quod servient ad facilem admodum
per sola polygona inscripta sine circumscriptis demonstrationem apogogicam
qua prop. 7. utar, tum quod operae pretium videatur ipsius per se
differentiae proprietates considerare, quatenus ab excessu vel defectu animo
abstrahitur; cum scilicet non exprimitur quaenam ex differentibus quantitatio
altera major minorve sit
PROPOSITIO
V
Hujus propositionis lectio omitti potest, si quis
ot eindidebitium  in demonstranda
prop. 7. Summum rigorem non desideret. Ac itaque satius
erit eam praeteriri ab initio, ac reque tola intellecta
tum demum legi, ne ejus scrupulositas fatigatam
immature mentem a reliquis, longe amoenioribus,
absterreat. Hoc unum enim tantum conficit
duo spatia, quorum unum in alterum desinit si
in infinitum p; inscribendo progrediare;
etiam numero inscriptionum manente finito tantum
ad differentiam assignata quavis minorem
sibi appropinquare. Quod Plerumque etiam illi sumere
pro confesso solent, qui severas demonstrationes
afferre profitentur
Si a quolibet curvae cujusdam propositae, fpun
1C2C etc., 4C. (fig. 3.) puncto C. ad unum anguli cujusdum
recti TAB la,BB B in eodem cum ipsa plano
positi, latus AIB, B etc 4B. velut ad axem, ducantur ordinatae nom les
CB ad alterum latus A, T, T etc. +F. tangentes CT et ex punctis
occursus tangentium, F, agantur perpendiculares TD ad ordinatas respon ites AH; manifestum est triangulum ABC restanguli sub AH altitudine et BC
et curva nova 1D 2D etc. AD. per intersectiones harum perpendicular m
et ordinatarum transeat. Rursus si quaelibet in curva priore designat
puncta proxima, ut 1C et 2C,  vel etc. vel 3C et +C, aliabe quotemq
inter haec prima et ultima paria interjecta, eecti rectis inscriptis su
chordis 1C2C, etc. usque ad 3C+C jungantur, quae productae CM
eidem, cui tangentes Anguli illius recti FAB. lateri AF. occurrant in pncis
M. quae cadunt intra puncta d,  ut M inter 1Y et e2T,  5, et 3 M inter
3T et 4T. et ex his occursuum punctis M similiter perpendiculare
1M, N. D, aliaeque, usque ad 3 M3N3P, demittantur, quarum una quaelibet
duorum punctorum ad M pertinentium ordinatis occurrat, ut 1M, N, D.
om
(cujus punctum 1M, pertinet ad puncta curvae 1 C. 2C.) occurrere debet
ordinatae 1C,B in puncto. N, et ordinatae 2C.B in puncto 1P. id mu
in caeteris fieri debet; unde abtis hae perpendiculares ab ordinatis abcindem
portiones quasdam inde ab axe sumtas, ut 1N . B, aliave usque ad 3B N.
His ita praescriptis ac praeparatis, ajo in curvis puncta C. inter 16 2+C
et puncta D inter 1D et 4D tam sibi vicina tantoque numero assuma
intelligi posse, ut spatium rectilineum gradiforme 1N1B +B3 P3s Nt
compositum ex rectangulis 1 N1B2BDP, alisque usque ad 3 N3 B4B3 Pqba sub ordin un ABC, ut st diximus, et patet, ergo triangulum ABC etiam
31
d
reseno si opus est productarum nes s rescissis portionibus 1B.N, sue
usque ad 3B3 N, et sub intervallis 1B2 B, aliisve, usque ad 3B+B. ce
comprehenduntur; ab ipso spatio Qvadrilineo D, B+B4 D 3 D etc id
Faxe 1B4B, 0natis extromis 1B1D, 4.B4 D et curva nova 1D, De +D
 comprehenso) differ santitate minore quavis data. Et eadem demonstrtio
n locum habet in quovis alio spatio mixtilineo et gradiformi continua, aru
3)
3)
3
ad quendam arem applicatior formatis. Adeoque methodus indivisibium,
quae per summas linearum invenit areas spatiorum, pro demonstnta
haberi potest. Requiritur autem Curvas aut saltem partes in quas unt
97 sectae, esse ad easdem partes cavas, et carere punctis reversionum.