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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 232r, page id 00067993:0465
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ex basi quadiantis
HF EN = ACCAE) AE seu AE _. sive momenta tangentium et langentium complementi simul, aequantum quadratis
Jam Momunta tung Coroll.d.Hinc sequitur summam ex tangente et tangente
Comb. t.ito bregiu demonstres. in Ho radii. Ergo sunt quadrabilia.
AHF altiludo AE= altitudui FZ. diimidio complementi, aequari secanti, seu HF = AH. quia AH =  AED et HF= AEH. Coroll. 2.
3.35. Ina
Conl 3 Et
item
Hinc differentia inter conchoeidem et Hyperbolam, quadrari potest aquatur Tangentibus semicomplementi
Coroll. 3. item tangens arcus dimidii auctus tangente complementi aequalis secanti complementi.
ull. q opomentae tangentia sermiusptti ex basi, quadbtia . Torelles Ergo, um. 7. prop. 25. dud. Hyprb
C propib
Dla similia: AEF et ESF HG. = HE. Eitang. sunitompl. quadrabil. Error
ADCEF= DF= FK(FK = AE) ob Dla ADF. ergo secans (complementi) in tangentem arcus dimidus (complementi)
aequalis radio in differentiam Tangento, et Tangsentis auus dimidis Ccomp) Coroll. Hinc haberi possunt et tan
gentes auu dimidii in Tangiarcus dimid. compl. Si a Tang. arc. dimid. in sec. anferatur Tanus. arc.
dimid. in tangs. quia sec = tanus + tanus. compl. arc. dimid.
DED = DF DG. seu quadratum  differentiae secantis et radi, est aequale differentiae inter tangentem
et tangentem arcus dimidii, ductae in disserentiam inter tangentem et Anum. Esto secans d. radius a.
langens c. tangens arcs dimidi . 1. sinus b. fiet: d + a  syda = b- 1,  6- b. =  + bi -ei- He bc
iam d= 7 c + a. prop. 11.
pone
sceoitimus ad Ia simuilia intermissa, de quibus ante prop. ab.
HC = xe = xH habulims HE EG= KECIF,
restant:
op. 13.
da = bc + af adde prop. 20.
ubi idem aliter demonstratum, (insignicalculi
confirmaine
HEGt = XHEF. tangens in differentiam Tangentis dd semiusplementi et sius aequatur diffeentiae secantis et sunus
Ergo ce. seu tangentes seincomplementi in tangentes vel tangentes complementi in
complementi in tangentem semicomplement.1.
c. ce = de- fe
tt
qu rconta)
quadrabi
A. 4. corill. 3.
tangentes arcus dimidii pendent ext da seu alindra Aperbolic, velquadia.
tura hyperbo circuli, con rocidectus
(orll. 2. cumque A sit d+ e ete = d. sit d = c + e+ . 2cC.
ideoque e seu  summa Qvadratorum Tangentium
semicomplementi, vel summa quadrata Tangentium
a per crp 11.
arcus dimidii ad basin pendent ex quadratura bo. Circuli, cunharda tysp
He GF = KH EG seu sinus in diffcrentiam sinus e completmenti, et tangentio semicomplementi aequantur dise
rentrae intersecantem et sinum complementi in tangentem semicomplementi, differentiam sinus et radii
bof. be. = d - f5 a - h. = da + fb - db - sa :, b.
= ZF (21) EN. ob Plum ENF in quo posita basi EN. altitudo est ZF posita basi EF altelydo est NI
Ni
AE rad
ead. circerttyp.
quadrat
 u hoeid
A  AR  ANsinus
AR. Ad. sui opsecans qui
investiganda jam quantitas ipsius NI. Satet X1. parallelam AE. Ergu
licios ut ante substituamus. At =  d=b. seu Ni = 8a ba. Alqui g. a. Ergo M = a - ba - a- ba ()a- b
Ergo cylinder tangentium semicomplementi, additis demis Ractangulis ex Tangentibus senicomplementi 1 et Tangenti
applicatis semmparabolae axi paralletis, aequatur momentis differentiarum inter sinum, et radium utiue quadrabitibus. Nam
posito sinu b et fradio a erit b applicata parabolee Circularis axi paralleta Quia Anus est media proportionatis inte
applicalai parabolae sircularis, det radium. Pr parabola autem circularis est, cijus altitudo, et basis aequa
cadde iussign. 41. F Aeda EI. e his focile haberi potest est enim A4   Hq f- a- + + 2b adde 
Tanus
ACDF= AD EF cylinder tangeantis, nto (complementi) demto cylindio semiauu (conchocu fu diund cat ae
condioen
Inaps. 4n. c3.
(complementi) = secans (compl) in Tungs. semiarcus (compl) seu Tongens + Tangis semiarus, in Tang. semiareu
seu longens in lunu simiareus + Quadu Janus semiauu. Coroll.1 Ergo quadrata Tangentum seniamus pendent ex G vadratura
oroll. 2 Cylender tangentium cplementi (quadrabilis) dento cyludios tangenttum semius plenem
aequatur Tangentibus complementi in otangentes semicomplementi (id est intindio secantum complementi demto cytindro radu
vdditis Quadratis tangentum semicomplementi (qua vndent ex Z. Euc. prop au q (mo ex peric. ht dineus. Tamus cyl. turg. semigat.)
Conchocidis et Hyperbolae.
Ca = EN. quia in Dlo AEC.. AC EN = AE ( = AC)Ca.
AaC AN. Na. V At Nrmile Vlo VENA simile Ylo Aa C. quia add ha-ACve
aequalia sunt EN = Ca, et AC= AE ergo et tertia seu Aa= A N. Egi Ea = AC
portionis circuloris
AMEC = AC EN. Chorda arcue in sipum complementi arcus dimidiis, aequatur cylindio sinuum arcusdati. vel. Chorda
arius complementi ad quadrantem (non ad semicirculum) in sinum arcus dati= cylindio simmum complementi (quadrabiti)
AD Ca = AC CD. Momentum Secantium complementi = cylindio tangentium complementi quadrabili. aode prop. 18:
EC MF = EFEa, vel FAFC EG. Chada arcus dims in differentian secantus acus dimidii a siuu complementi
dangentium
De) semiareus in sinum complementi, seu moneulum sermarus
arcus dimidii  Tansenle arcus damidii in cinum versum arcus dati.
radio in differentiam sujus, et tang. pu ia semiareu. Coroll. Tangens amicomplementi in sinum cyli  radio
Tancens aus dimisti (
in sinus complementi, dento cylindio Tanus semiuspl. At per prop. 46 junct. 20. Cylinder Conthoeid - Cyl. Hyp. + had. in sing gpl.
AD HZ. EFEEK = XZAC
AE. Momientum Juns compl = Ratio in diff. rad. X sinus adve p. 47. cort et. Ain. H tanus gpl. in
lonus. compl. in sin. Ergo res eodem redit
Quadrg torad. in arcu prop. 50
152.
semi
prop. 47
prop. 48.
prop. g. Repetitio prop. 14.
prop. 49.
Per PH. da af.
inuig ant
ui
prop. 54
prop. 55
prop. 5b
prop 3
prop.bo.
prop. b1.
E EN = NC (d CG) A
sin quadruli