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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
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Reductio circuli ad Figuram Rationalem aequivalentum
definit figuram Rationalem voco, cujus abscissae sunt rationales ad ordinatas, vel ordinatae ad absissas
Ut sinus versus est ad rectum, ita ntangens arcus dimidii ad radium MB.
centro B
Estorurcus crculi AB quem tangant rectae,
AD et DD,  jugatur CD, potet
AD tangentes Comonicae, ccuentro C, sit AF
arcus circuli descriptus AFB. quem contingunt
recta AD, BD qum cujus tangens cujus dimidius
AF, ejusque tangens AD, jungatur BD, in
quam ec A pp si opus est productam,
agatur perpendicularis AO, ponr  Arcus AB
sinus versus este AE, et sinus rectus EB ajo esse
AD ad CA, uti AE ad EB. Nam cum
AD et EB, item AD, et CB sint paralle la
eunt Triangula OD et EBC similia erunt adeoque
AD ad EB, ut AD ad  CB. jam AO aequalis ipsi AE,
(: est enim AO aequalis ipsi BG et BG ipsi AE.) Ergo
Ergo I. AE ad EB ut AD ad CB, vel ut AD ad AC. Quod erat ostendendum
 Analyticis, sina verso posito X, et radio a, constat
Sp t. u redtum ope san p set hagetemus dimti aspellata
Stop t.  d La C Sim. no, sut VB) scan
Si Tangens arcus ddmidi, appelletur Y,  sius veriae picus dati, x et tadius a
i
fiet: X 1
2ay
Prop. II. aue dae vel
Nam simat verso posito X, eriti sinus rectus 2aX - X i. verit ergo.
a +y
2aX - x a Ergo raCy = Xy  a X sive 2ay = XV B a X
1X
Hy et denique X 
cay
ar+ y
Unde corollurium sequitur sutis memorabile:
nempe si Tangens
 nempe si Tangens arcus sit radio commensurabilis, simmm
versum arcus dupli, commensurabilem quoqe ipisis fore. sive dat
in numeris atangente arcus circuli  et radio, etiame
sinum versum arcus aco proinde et sinum complementi usd
dupli
in nuimeris rationalibus haberi posse, Idque o non est minus
memorabile, ac illud a Tarissimo Pellio observatum, quod tangenote
aliqua posita rationali, etiam Tangens arcus dupli sit rationalis, nam
3a2V
ic sangente appellata Y, et tangente arcus dupli 2, fier Z A a- y.
unde jungendo haec duo theoremata inter se, patet Tangente posita rationali arcu
versum sore rationales
dupli tam tangentem quam sagittam sive sinum si paniter ipsi, ac monide inter
se commensurabiles, sive.
Sed quod de sinu verso invenimus tanto
ab eo inprimis aestrnandum est, quod viam nobis aperuit ad figuram
rationalem auculo aequivetentem inveniendam
syntomos
de id est quae aequatione exprimi possunt, in quibus unjus incognitae valot purus simplexque est
defir figuras aquintentes voco; quarum potiones portionibus contime aequantur
A sctanscns auq
idi sive ys sia
a7- 727247.
ery p2 saea2
vel y +24y+  
Ttuorema prop. 1.