<50v>

Lemma.

\1/[Editorial Note 1] [1] If one body move from a to b in ye same time in wch another moves from a to c & a 3d body move from a wth motion compoundee|d| of those two it shall (completeing ye parallelogram abcd ) move to d in ye same time. For those motion would severally {carry} it ye one from a to c ye other from c to d &c

[2] \2/ In ye description of any Mechanicall line what ever, {illeg} there may bee found s|t|wo such motions wch compound or make up ye motion of ye point describeing it, whose ∼ motion being by them found \by ye Lemma/, its determinacon shall bee in a tangent to ye mechanicall line.

[3] Example ye 1st. If abe is an helix, {illeg} wch {illeg} described by ye point b move {sic}ing uniformely in ye line abc from ye point center a about wch ye line abc circulates uniformely. {illeg} |the line ab increasing uniformely whilest it also circulates uniformely about ye center a| Let ye radius of ye circle dmbd bee ab. & let {dm $\mathrm{b}$  {sic}} measure ye quantity of the giration of ab (viz ad touching ye helix at ye center) let bf be a tangent to ye circle dmbd . yn is ye motion of ye line point b towards c to its motion towards f , as ab , to dmb {illeg}. therefore make $bc=fg:bf=cg\colon\colon ab:dmb$. & (by ye Lemma) ye diagonall bg shall touch ye helix in b. Or make $bc=fg=ab$. & $bf=cg=dmb$. the diagonall bg shall touch ye helix. (ye length of bf may be thus found viz; $ae:bc=ad=ab\colon\colon dmbd:dmb=bf$.)

[4] Example ye 2d. If ye center $\left(a\right)$ of a globe \$\left(bae\right)$/ moves uniformely in a streight line parallel to eh, whilest ye Globe uniformely girates. Each point $\left(b\right)$ in ye Globe will describe a Trochoides: to wch a ye point b I thus draw a tangent. Draw ye radi {sic} ab & bc perpendicular to it yn is ye circular motion of {illeg}|th|e point b determined in ye line bc , & its progesive {sic} in bf . If therefore I make $bc=fg$ to $bf=cg$ as ye circular motion of ye point b to its progressive ye Diagonall (by ye Lemma) bg shall {illeg}g touch ye Trochoides in b . As if ye Globe roule upon ye pl{illeg}|ain|e eh , & I make $bc=fg=ab$. & $bf=cg=ae$. yn do{illeg}|th| ye Diagonall bg touch ye Trochoides. (Or be , {illeg} passing through ye point in wch ye globe & plaine touch, is a perpendicular to ye {Tro}.

[5] Example ye 3d. If ye line $bh\perp ak$ moves uniformely ye length of ah whilest { ab girate} uniformely from ak to am about ye center a , ye point of their intersection $\left(b\right)$ will descibe {sic} ye Quadratrix kbn . Draw $bc\perp abp\perp pq$. & $bf\perp am$ yn \$an:am=ak\colon\colon$/ $ka:kpm\colon\colon$ motion of $\left(b\right)$ to f ∶ motion of $\left(p\right)$ to q , {(sup)}. And motion of $\left(p\right)$ to q ∶ motion of $\left(b\right)$ to c ∷ \ $am=$/ $ap:ab\colon\colon kpm:sbt$ . Therefore motion of $\left(b\right)$ to f ∶ motion of $\left(b\right)$ to c $\colon\colon ka:sbt\colon\colon an:ab$. Therefore makeing $ak:sbt\colon\colon$ $an:ab\colon\colon bf=cg:bc=fg$ . (Or wch is makeing $bf=an=cg$ , & $bc=ba=fg$ ) ye Diagonall bg shall touch ye Quadratrix at b .

[6] [Scholium. The tangents of Geometricall \lines/ may be found by their descriptions {after} ye same manner. As the Ellipsis (whose foci are a & f ) being described by ye thred abf ye thred ab lengthens so much as ye thred bf shortens, or the point b moves equally from a & to f. Therefore I take $bc=bf$. $cg=fg$ \& $fg\perp bf$./ & ye diagonall bg will touch the Ellipsis in b ]. (This should follow ye 3d Example's substitute) See fol 57.

[7] Although ye nature of a Mechanicall line is not knowne from its description but from some other principle yet may a tangnt be drawne to it by ye same method.

As if be is an Hyperbola. tad its asymptote & $cf\parallel tad$ . & $gp:ph\colon\colon cadf:adeb$. {illeg} to draw a tangent to ye tangent {illeg} line gh {illeg} ghm , I consider yt, $df:de\colon\colon$ increasing of acdf ∶ increasing of abde ∷ increase of gp ∶ increase of ph ∷ motion of ye point h towards k ∶ motion of h towards r , if $hk\parallel gp$. Therefore I make {$rh=sk:rs=hk\colon\colon$} $\colon\colon de:df\colon\colon hp:pw$. & ye diagonall hs or wh shall touch ye line ghm . Or if $vg=ta=ab=xy$ . $gp=ad$. $ph=vxy\parallel phr$. \$vp\parallel yh$/. yn doth xhs touch ye line ghm at h .

Tangents to mechanichanicall lines may sometimes bee found by finding such a point wch is immoveable in respect of ye line described & also doth {{illeg}vary} in distance from ye describing point. Then in ye for ye {Sicunf{illeg}} through yt point. Thus in ye Trochoides when ye point $\left(e\right)$ toucheth ye plaine eh tis immoveable, & tis ever equidistant from ye describing point b ({illeg} both of ym fin{illeg} points in ye Globe). Therefore ye line {illeg} drawne {fro=} ye describeing point to ye touch point of ye Globe & plaine $\left(eh\right)$ is perpendicular to {ye} trochoides. But in ye spirall though ye point {illeg} is {illeg} from yt {illeg}.

[8] Instead {in ye} third example {illeg} {illeg} {illeg}. <51r> Therefore $af:ab\colon\colon ak:gbl\colon\colon$ {illeg}|abs|olute & whole motion of b towards c (or acf ) ∶ whole motion of b towards d (or ed ). Soe yt makeing $bc:bd\colon\colon af:ab$. {$ab\parallel ed\perp ed$} & $hb\parallel ce\perp cb$. The point b will be moved {illeg} to ye line {illeg} ce & ed in same times wch cannot bee unlesse it move to e (their common intersection). The point b therefore move in ye line be wch doth therefore touch ye Quadratrix at b : (The same is done by makeing $bd=$ $bl=bd\perp de\parallel ab$ . & drawing \ye tangent/ be through ye common intersection of ed & aem .)

To resolve these and such like Problems these following propositions may bee very usefull.

May 14. 1666.

[9] Prop 1. If ye body a being in a circumference of ye circle \or sphære/ adce doth move towards its center b its acceleration \motion or velocity/ to|wards| each point \ d , c , e / of ye circ{illeg}umference is yn as ye cordes ad , ac , ae , drawn from yt body to those points are. This may be Demonstrated by Te|h|eorem R pag 57.

[10] Prop 2d. If \ ∠  {sic}$adc$ {illeg} is a parallelogram \ sim $△aec$ although they bee not in yesame plane &// three bodys move \uniformely from a /. ye first from a to b d ye 2d from a to e , ye 3d from a to c their motions being \each to other/ as ye {illeg} directing lines ad , ae , ac , are in ye same time, & adce is a parallelogram then is ye motion of ye third body compounded of ye other two. Demonstration. For \makeing $df\perp ac$f & $eb\perp${adba}/ |{illeg} makeing $df\parallel eb\perp ac$| ye motion of ye first body towards d is to its motion towards f as ad = {illeg} is to af (prop 1); \{illeg}/ & ye \{illeg}/ motion of ye second body towards be is to its motion towards db as ae $=dc$ is to b ba (prop 1). Therefore But $af+ab=ac$. Therefore &c

Prop 3d. If a moveing line keepe parallel to it selfe all its pts have equall motion.

Prop 4th. If a line move in plano, so yt all its points keepe equidistant from some common point center the motions of those points are as their distances from yt center.

Prop 5t. If ye motion of a line in plano bee mixed of parallell & circular motion, ye motion of all its points are compound (see prop 2) of that motion which they would have, had ye line onely its centrall parallel motion, & of yt wch they would have, had ye line onely its circular motion.

Schol: All motion in plano is reducible to one of these three cases, & in ye 3d case any point in yt plaine may bee taken for a center to ye circular motion.

[11] Prop 6t. If ye \streight/ line ea doth rest & da doth move: soe yt ye point a fixed in ye line da moveth towards b : Then from ye moveing line da drawing $de\parallel ab$ , & ye same way wch ye point a moveth; These motions, viz of ye \fixed/ point a towards b , of ye intersection point a in ye line ad towards d , & of ye intersection point a towar in ye line ae towards e , shall bee one to another, as their correspondent lines de , ad , & ae are.

[12] Prop 7th. If ye \streight/ lines adm , ane , doe move, soe yt ye point a fixed in ye line amd moveth towards b , & ye point a fixed in ye line ae moveth in ye towards c : Then from ye line each line to ye other draw two lines de , nm parallell to the mo the line amd , draw ∼ $de\parallel ab$ & ye same way: & from ye line ae draw $nm\parallel ac$, & ye contrary way, to make up ye Trapezium denm . And if any two of these foure lines de , {illeg} mn , md , ne , bee to any ∼ correspondent two of these foure motions, viz: of ye point a (fixed in ye line dma ) towards b , of ye point a (fixed in ye line ane ) towards c , of ye intersection point a moveing in ye line dma according to ye order of ye letters {illeg} m , d & of ye intersection point a in ye line ane according to ye order of ye letters {illeg} n , e : Also all ye foure lines shall be one to another as those foure motions are.

Note yt \in ye two last propositions/ if ye moveing lines \may/ bee crooked {illeg} \so yt/ amd , ane , bee tangents to them in ye point a .

Note also yt by ye place of a body is meant its center of gravity.

To resolve Problems by motion ye 6 following prop: are necessary & suffcient.

May 16. 1666.

[13] If ye body a in ye perimeter of ye circle or sphære adce moveth towards its center b . its velocity to each point \$d.c.e.$ / of yt circumference is as ye cordes ad , ac , ae , drawne from yt body to those points are.

Prop 2. If ye △s adc , aec are alike though in diverse planes; & 3 bodys move from ye point a uniformely & in equall times, ye first to d , ye 2d to e , ye 3d to c : yn is ye 3d's {sic} motion compounded of ye motion of ye 1st & {2d.}

Note yt by a body is meant its center of gravity.

Prop. 3. All ye points of a body keeping parallel to it selfe are in equall motion.

Prop. 4. If a body onely move circularly about some axis, ye motion of its points are as their distances from {illeg}|th|at axis.                     Call these 2 simple motions

Prop. 5. If ye motion of a body is considered as comp mixed of simple motions: ye motions of all its points are compounded of their simple motions, so as ye motion towards c (in prop 2d) is compounded of ye motion towards d & e .

Note yt all motion is reducible to one of these 3 cases: & in ye 3d case any line may bee taken for {the} axis (or if a line or superficies {move} in plano any point of yt plaine may bee taken for ye center) of motion.

[14] Prop. 6. If ye lines {illeg} ah being moved doe continually intersect; I describe ye Trapezium abcd {illeg} its diagonall ac : & say yt ye proportion & position of these five lines ab , ad , ac , cb , cd being determined by {requisite data} they shall designe ye proportion & position of these 5 motions: {illeg} of ye point a fixed in ye {illeg} moveing towards b ; of ye point a fixed in ye line ah {illeg} moveing towards d ; of ye intersection point a moveing in ye plaine abcd towards c (for those 5 lines are {illeg} in ye same plaine though {illeg} & ah may only touch ye plaine in their intersection point): of ye intersection point a moveing in ye line ae parallely to {illeg} cb & according to ye order of ye letters c , b : & of ye {inter}section point a moveing in ye line ah parallelly to cd & according to ye order of those {illeg}

[15] Note yt a streight line is said to designe ye position of curved motion in any point {illeg} {illeg} if toucheth ye line described by ye motion in yt point, or when tis (as ab , ad , ac ), or {illeg} tis parallell to such a {illeg} (as ad , {illeg}). Note also yt one line ah resting (as in Fig 3 & 4) ye points d & a are coincident & ye point c shall bee in ye line ah if {illeg} bee streight {illeg} (fig 3), otherwise in its tangent ac (fig 4) {illeg}. Haveing an equation expressing ye relation of two lines x & y described by two bodys A & B whose motions {illeg} q ; Translate {illeg} ye termes to one side & multiply ym, being ordered according to x {illeg} {illeg} progression {$\frac{3p}{x}.\frac{2p}{x}.\frac{p}{x}.0.\frac{-p}{x}.\frac{-pp}{x}.$ } &c: & being ordered by ye dimensions of y multiply those by {illeg} {illeg} &c. ye summa of those products {illeg} equation expressing ye relation {illeg} {illeg} motions p & q .

<51v>

To draw a tangent to ye Ellipsis

[16] Suppose ye Ellipsis to be described by ye thred acb , & yt ce is its tangent. Since ye thred ac is diminished with ye same proportion velocity yt be increased|t|h, yt is, yt ye point c hath ye same motion towards a & d , ye angles dce , ace , must bee equall, by prop 1. I And, so of ye othe {sic} {conicks}.

<94v>

Quæstionum solutio Geometrica.

1 Angulum datum DAB recta datæ longitudinis CB subtendere quæ ad datum punctum P converget Cape PQ=CD et Q erit in circulo cujus centrum P radius PQ. Age QR{}∥AD et {Q}PRD∥AD|B| et erit PD.DC=AD−QR∷ ∷PR.QR. Ergo Q{illeg} in conica sectione est. Pone QR infinitū et erit $PR=\frac{PD×QR}{AD-QR}$ AD−P|Q|R.PQ|QR|∷PD.QPQ|R|. seu PR =−PD. Pone PR infinitus et erit PD+PR.PR∷AD.QR ergo AD=QR. et AB Asymptotos. alteras Cap{er}|{e}| ergo PS=PD et per S parallelam AD age alteram Asymtoton & {his} Asymptotis per punctus P describe Hyperb{o}lam secantem circulum prædictus in Q.

2 Inter circulum PDF et rectam DF ponere rectam datæ longitudinis BC quæ ad punctum P in circumferentia circuli datum converget. Biseca DF in E. Age PD,PE,PF. Cape PQ=BC. Age QR ∥DC & occurrentem PE in R. Et erit PR.{illeg}PQ∷PE.PC. PR.PE∷PQ(BC).|PC∷RQ.EC.| {illeg}Et PQ.|(|BC).FC∷DC.PC Et {PR×PQ} Ergo PQ×PC=FC×DC =ECq−EFq PR.PQ Et $BC.\frac{PE×RQ}{PR}-EF\left(FC\right)\colon\colon$ $\frac{PE×RQ}{PR}+EF\left(DC\right).\frac{PE×BC}{PR}\left(PC\right).$ Seu BQ|C|,PR.PE,PR|RQ|−PR,EF∷ $\frac{PE×RQ}{}+PR,EF.PE,BC.{BC}^{\mathrm{q}},PE,PR={PE}^{\mathrm{q}},{RQ}^{\mathrm{q}}-{PR}^{\mathrm{q}},{EF}^{\mathrm{q}}$ Si PR infinitum{Q} Ergo Q locatur in Conica sectione cujus diameter PR, ordinata RP|{PR}|. Sit RQ=0, erit PR=0 et $\frac{-{BC}^{\mathrm{q}}PE}{{EF}^{\mathrm{q}}}$ In EP producta cape ergo $PS.\frac{1}{2}PE\colon\colon {BC}^{\mathrm{q}}.{EF}^{\mathrm{q}}$ et erit S centrum et P vertex figuræ. Pone PR infinitus et erit PEq,RQq=PRq,EFq, seu PE{:|,|}RQ=±EF,PR. Quare per S ipsis PD,PS age parallelas et hæ erunt Asymptoti figuræ His igitur Asymptotis per punctum P describe Hyperbolam, ut et cent{illeg}|ro| P radio PQ circulum & per eorum intersectionem Q age rectam PC.

Corol. si ang. PEC {illeg}|re|ctus est Problema planus erit. Nam circuli centrum incidit in {illeg}|ax|em figuræ.

3 A dato puncto P rectam PC ducere cujus pars BC inter circulum et recta productam diametrum æquab DF æquabitur \semi/ diametro EF. Age EF ac demitte ⊥ PG,BH. Est EH. HB∷GC(GE+2EH).GP. Ergo punctum B in Hyperbola est. Pone {illeg} EH=0 et erit HB×GE=0 adeo HB=0. Quare Hyperbola transit per punctum E. Pone EH infinitus et erit EH.HB∷2EH.GP. Ergo $\frac{1}{2}GP=HB$. Pone HB infinitus ergo et erit EH.GE∷HB.\/GP−2HB∷HB.−2HB Ergo $\frac{1}{2}GE=-EH$. Quare biseca PE in S et per S age Asymtotos parallelas EH et HB et per punctum E vel P describe Hyperbolam secantem circul{u}m in B|{D}|. Et per B age PC.

Corol. Hinc si ang PEG semirectus erit PE axis Hyper{illeg}|bola| adeo Problema in eo casu planum.

<95v>

Quæstionum solutio Geometrica

1 Datis trianguli cujuvis angulo latere et summa vel differentia {re}liquorum laterum datur triangulum{,} Detur latus AB reliquorum laterum AC+BC summa {illeg}vel differentia AD. Si detur angulus datus dato lateri conterminus est, sit iste A. Et dabitur triangulum DAB. Angulorum vero |C|DB ADBD differentia in priori casu summa in posteriori est ang ABC.

Si{t} angulus datus dato lateri oponitur, sit iste C {illeg} {illeg}datum{illeg} ang DCB dabuntur ang CDB CBD est dabitur tr dabuntur anguli dabitur triangulum CDB specie. In triangulo autem ADB datis lateribus AB AD et ang D datur ang ABD. Unde datur Ang ABC ut ante.

2 Data differentia segmentorum basis uno angulo summa vel differentia laterum et uno angulorum datur triangulum. Nam si datur summa laterum dabitur ratio differentiæ laterum ad basin si differentia debitur ratio summæ laterum ad basin. In utra casu \Ex ratione utravis & uno angulorum {illeg}/ per problema superius datur triangulus specie. Deinde ex data differentia segmentorum basis et ex data ratione differentiæ segmentorum \basis/ ad latera dantur latera.

3 Data summa {illeg}|vel| differentia laterum uno angulorum et ratione basis ad perpendiculum: ex duobus posterioribus dabitur triangulus specie ex priori dabitur etiam magnitudine.

4 Data summa vel differentia laterum uno angulorum et area: \ex area/ rectangulum laterum datum angulum comprehendent{ur}. Si istorum summa \vel diff./ datur ad|a| quadrat{us}|o| summæ aufer duplum rectangulus vel ad quadratum differentiæ \{laterus}/ adde {qu}duplus rectangululum {sic} et habebitur priori casu quadratus {illeg},|differenti{æ}| posteriori quamdratus differentiæ s summæ laterum: Ex dat{illeg}|is| autem summa ac differentia laterum dantur latera. Si angulus datus basi conterminus est problema erit solidum.

5 Datis angulo \{illeg}/ /A\ latera \AC vel BC/ et differentia segmentorum basi|s| AD dabitur triangulum ADC |ut| et angulus B \quod est/ complementus est anguli ADC.

Si detur angulus verticalis C laterum alterutra AC vel BC et segmento basis AC: quiescant BC,AC et punctum D in {circulo} erit radio CB centro C descripto. [Et et AB.$\frac{{AC}^{\mathrm{q}}-{BC}^{\mathrm{q}}}{AB}=AD$ ergo D determinatur per Geometriam planam simplicicem. Sed et ubi punctum D incidit in B positio rectæ AD|B| determinatur. Ergo locus puncti D conica est sectio] {illeg}|Ut| et in Chonchoide {illeg}|Po|lo {B} {illeg}|asy|mptoto AC intervallo AD descrip{illeg}|ta|.

Vel sic. Dato angulo AC{D|B|} datur summa ang: A+B. seu A+CDB Aufer hoc de duobus rectis ac dabitur differentia ang ADC−A. Unde datur triang. per sequ. Prob.

7 Datis \basi &/ differentia angulorum ad basin una cum basi et \una cum/ latere al{illeg}|tera|tro vel summa differentia ratione \laterum/ aut summa vel differentia {lo} laterum aut area, perpendiculo vel segm{en}{illeg}|ta|to basis aut summa vel differentia lateris alterutrius et perpendicula vel segment{i} basis. &c Datur triang. Nam data basi et angulorum ad {illeg}|Basem| differentia, {illeg} C erit ad Hyperbolam; et ex dato tertio, punctum C erit ad recta aut circulus aut conican aliquā sectionem.

8 Ubi datar angulus verticalis et differentia segmentorum basis et tertium aliquod, habebitur aliud triangulum ADC ubi datur {illeg} differentia angulorum ad ba{illeg}|se|m, et tertium aliquod.

9 Dat{illeg} basi ratione laterum et tertio quovis ut ⊥ segment{illeg} basis{,} angulo aliquo{,} ratione ⊥ ad lat{eri} {peh} {illeg} ad segmentum d|D|{tis} {illeg}. Nam {illeg} data {illeg} lat. Dat{ur} circulos {illeg} {illeg}

<96r>

Quæstionum solutio Geometrica. Prob 1

Circulum \ABE/ per data duo puncta \A, B/ describere quæ rectam FG    positione datam continget.

Junge A{,}B. D biseca eam in D. Erige normalem DF occurrentem FG in F. {illeg} Produc AB donec occurrat FG i{n}

|Sit| E punctum contactus. Produc AB donec occurrat FG in G et erit EG medium proprortionale inter datas AG,BG.

Prob. 2

Circulum \ABE/ per datum punctum A describere qui recta duas FE, FH continget. {illeg}

Center F, radi{s} FA describe Recta FD biseca{illeg} angulum HFE. Ad FD demitte normalem AD et produc donec occurrat FE in G. Cape D ad B ut sit DB=AD et per puncta A, D describe circulum ut prius qui contingat rectam FE.

Prob. 3

Circulum \ABE/ per data duo puncta \A, B/ describere qui alium circulum positione datum \EKL/ continget.

\Puta factum/Sit punctum contactus E. Linea contingens EM. et erit AM×BM=EMq=MK×ML. Divide ergo BK in M ut sit AM.MK∷ML.MB. Cape ME medium proportionale inter AM et BM et centro M radio ME describe circulum. Hic secabit circulum EKL in puncto contactus E. Recta autem BK sic secatur in M. Est AM= AB+MB,|.| MK=BK−BMB.ML=BL−{B}MB. ergo AB+MB.BK−MB∷BL−MB. MB. Et componendo AB+BK(AK).BK−MB∷BL.MB. et inverse AK. BL∷BK−MB.MB. et rersus componendo AK+BL.BL∷BK.MB.

Si AB non secat EKL {illeg} pro MK×ML scribe MGq+FGq−KF erit AM×BM{} AG−MG×BG−{illeg}G(AM×BM)=MGq+FGq−KFq {illeg} est AG×BG−FGq+KFq=AG+BG×MG. Seu 2AG.BL∷BK:MB. {illeg} \{Unde}/ cum sit BL.BO∷M{illeg}BN.BK, erit 2AG.BN∷BN.BM. Quæ solutio versalis est.

Prob 4

Circulum \BDE/ per datum punctum \B/ describere qui datum circulus {illeg} & rectam lineam |AD| postione datam continget.

AB est 2CD−AH. NF est \2/CQ−NS posito CQ=CF=CS. Ergo HF est 2CD−{illeg} AB−NF est 2CD−AH−2CQ+NS seu. AB−NF+DQ est Adde 2DQ, erit AB+DQ−HF=NS−AH=$\frac{{AD}^{\mathrm{q}}}{AB}$ $=\frac{{NQ}^{\mathrm{q}}}{NF}-\frac{{AD}^{\mathrm{q}}}{AB}$. Dividendo|a| est ita \data/ AH in D ita ut $\frac{{DH}^{\mathrm{q}}}{NF}-\frac{{AD}^{\mathrm{q}}}{AB}$ dato{illeg} æquale sit, nempo dato AB(Hb)+DQ−HF, seu bk. {ADq}DHq×AB AD =AB, HK, bL. DH=A{D}{illeg} AH−AD.DHq=AHq−2DAH+ADq.AD×{illeg}Kq{−2A}{illeg}{AH} −2DAH+ADq{illeg}$\frac{HK×{AD}^{\mathrm{q}}}{AB}=HK,bk$. [A{B}{illeg}−HK(bK).AH∷A{illeg}|{B}|.AV.AHq−2DA{H} $\frac{{AD}^{\mathrm{q}},AH}{AV}=HK×BL$] $AH-2DA+\frac{AB-HK}{BAH}{AD}^{\mathrm{q}}=\frac{HK,bk}{AH}$. \Fact/ AB−HK.{illeg}AH∷BA.AV. & AH. $HK\colon\colon bk.Hp.AH-2DA+\frac{{AD}^{\mathrm{q}}}{AV}=Hp$. ADq−DA{illeg}V+D{illeg} AVq=AV×PV{illeg}\{P}/AD{}AV {illeg}PV={DV.}AD {illeg}AV{illeg}PV{illeg}−AD+AV{illeg}=AD{illeg}|{×}|PV =DV. Age ergo BK occurre{nt}em AH in VHK ad {H}A versus A si {HK} {illeg} versus {b} aliter {illeg} A {illeg} ad bk {illeg} {illeg}

<96v>

Nota etiam quod Problematis quatuor sunt sunt {sic} casus quorum duo sunt impossibiles ubi circulus datus et recta data se mutuò secant. Casus impossibiles sunt ubi punctum v cadit inter A et P.

Vel \in {a}ng GED/ agatur GD datam per A transiens posito AE quadrato, quære summam radicum Fd, FD Ad AD erige ⊥ DK age erit CK {illeg} F{illeg} Ergo AK summa illa, et CDq+CKq(DKq)+GDq=GKq Aufer BGq seu CKq et restabit CDq+GDq=BKq Datur ergo {illeg} summa AK. Quare cum ang ADK rectus; super diametro AK describe circulus secantem FE in D, d

Super datis rectis tribus AB, CD, EF, tria constituere triangula quorum vertices erunt ad \idem/ punctum G et anguli ad vertices AGB, CGD, EGF æquales.Super Junge AD, BC. {illeg}|Bi|seca eas in r, s. Produ AB {illeg} secent AB, CD se mutuo in t. Age tG∥sr. Idem fac in lineis CD FE.

Vel sic. super lineis AB, CD, EF describe similia segmenta quorumvis circulorum satis magnorum \ita ut se mutuo secent compl{illeg} segm. ad circulos {illeg}/. Per intersectionem circulorum AB, CD age rectam, ut et aliam rectam per intersectionem circulorum CD, FE: nam hæ rectæ se secabunt in puncto G

The Problem {illeg} \in Schooten/ de tribus baculis may be solved more easily by supposing ye Ellipsis to be a circle first & then reducing it to ye desired circle.

In triangulo DEF \dato ABC/ {illeg} aliud triangulum DEF {illeg} dato def simile inscribere cujus latus EF transibit per datum punctum G. Nemper vi|e|rticis trianguli DEF locus {illeg} est linea recta.

In data conica sectione ABCDE, trapezium ACDB inscribere cujus anguli op duo oppositi CAD CBD dantur et data puncta A et B consistunt. Vizt si locus puncti D est conica sectio locus punctis c erit linea recta.

<97r>

[77] Ex observationibus proprijs Cometæ anni 168\0/.

A stella major et orientalior duarum in orientali \australi/ pede Persei, B stella minor earundem. AB stellarum distantia 1gr.46′6″. α, β, γ &c loca Comet{æ}

$\begin{array}{l}\text{Feb 25 hora}\phantom{\rule{0.3em}{0ex}}8⁤\frac{1}{2}\phantom{\rule{1em}{0ex}}\text{vesp.}\text{Dist}\text{Aα}\phantom{\rule{0.3em}{0ex}}{2}^{gr}\text{.17′.42″. Ang BAα.}\\ \text{F}\end{array}$

$\begin{array}{lll}& \text{Dist.}\phantom{\rule{0.3em}{0ex}}\text{C}\text{ometæ}\phantom{\rule{0.3em}{0ex}}\text{stella A.}& \text{Angulus}\\ \text{Die}\phantom{\rule{0.3em}{0ex}}\text{Feb 25 hora}\phantom{\rule{0.3em}{0ex}}8⁤\frac{1}{2}\phantom{\rule{0.3em}{0ex}}\text{vesp}& A\alpha =104,\frac{7}{12}\phantom{\rule{0.3em}{0ex}}\text{part}={2}^{gr}.17\prime .42″& \text{BAα}\phantom{\rule{0.3em}{0ex}}{9}^{gr}.17\prime \\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Feb 25 hora}\phantom{\rule{0.3em}{0ex}}9⁤\frac{1}{2}& A\beta =103⁤\frac{3}{4}\phantom{\rule{1em}{0ex}}\phantom{\text{part}}={2}^{\phantom{gr}}.16\prime .36″& \text{BAβ}\phantom{\rule{0.3em}{0ex}}{9}^{\phantom{gr}}.27\prime \\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Feb 27 hora}\phantom{\rule{0.3em}{0ex}}8⁤\frac{1}{4}\phantom{\rule{0.3em}{0ex}}& A\gamma =77-& \text{BAγ}\\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Mart 1 hora}\phantom{\rule{0.3em}{0ex}}11& A\delta =52\phantom{\rule{3em}{0ex}}\phantom{\text{part}}={1}^{\phantom{gr}}.8\prime .30″& \text{BAδ}\phantom{\rule{0.3em}{0ex}}{31}^{\phantom{gr}}.40\prime \\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Mart 2 hora}\phantom{\rule{0.3em}{0ex}}8& A\epsilon =43\phantom{\rule{3em}{0ex}}\phantom{\text{part}}={0}^{\phantom{gr}}.56\prime .37″& \text{BAε}\phantom{\rule{0.3em}{0ex}}{43}^{\phantom{gr}}.8\prime \\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Mart 5 hora}\phantom{\rule{0.3em}{0ex}}11⁤\frac{1}{2}& A\zeta =& \text{BAζ}\\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Mart 7 hora}\phantom{\rule{0.3em}{0ex}}9⁤\frac{1}{2}& A\eta =& \text{BAη}\\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\text{Mart 9 h}\text{o}\text{ra}\phantom{\rule{0.3em}{0ex}}8⁤\frac{1}{2}& A\theta =78⁤\frac{7}{12}\phantom{\rule{1em}{0ex}}\phantom{\text{part}}={1}^{\phantom{gr}}.43\prime .30″& \text{BAθ}\phantom{\rule{0.3em}{0ex}}{143}^{\phantom{gr}}.00\prime \\ \phantom{\text{Die}}\phantom{\rule{0.3em}{0ex}}\phantom{}\text{Mart 9 hora}\phantom{\rule{0.3em}{0ex}}12& A\iota =80⁤\frac{1}{3}\phantom{\rule{1.7em}{0ex}}\phantom{\text{part}}={1}^{\phantom{gr}}.45\prime .46″& \text{BAι}\end{array}$

<98r>

Observationes Comet{illeg}|æ| habitæ ab Academia Physicomathematica Romana anno 1680 et 1681, a Ponthæo æditæ.

$\begin{array}{cc}\begin{array}{cccccc}\text{Stylo vel.}& \text{Stylo novo}& \text{Hora. mat}& \text{Cometæ Long.}& \text{Lat. Aust}& \text{L}\text{ong.}\phantom{\rule{0.3em}{0ex}}\text{candæ}\\ \text{Novemb 17}& \text{Novem.}\phantom{\rule{0.3em}{0ex}}27& 6\phantom{⁤\frac{1}{2}}& \phantom{0}8.30\prime & 0.40& {15}^{gr}\phantom{\rule{0.3em}{0ex}}\text{et ultra}& \\ & \phantom{\text{Novem.}}\phantom{\rule{0.3em}{0ex}}28& 6⁤\frac{1}{2}& \phantom{}13.\phantom{0}0\phantom{\prime }& 1.20& & \\ & \text{Decem.}\phantom{\rule{0.3em}{0ex}}\phantom{0}1& 7⁤\frac{1}{4}& \phantom{}27.50\phantom{\prime }& 1.16& & \\ & \phantom{\text{Decem.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}4& 7\phantom{⁤\frac{1}{2}}& 12.\phantom{0}0\phantom{\prime }& 0.40& & \\ & \phantom{\text{Decem.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}5& 7⁤\frac{1}{4}& \phantom{}15.50\phantom{\prime }& 0.40& & \\ \\ & \phantom{\text{Decem.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}6& 7⁤\frac{1}{2}& 20.\phantom{0}0\phantom{\prime }& 0.35& & \\ & \phantom{\text{Decem.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}7& 7⁤\frac{1}{2}& 24.\phantom{0}0\phantom{\prime }& 0.30& & \end{array}& \begin{array}{l}\text{Caput Nov 27}\\ \\ \text{aquabat prim}\\ \text{notæ stellas mag}\\ \text{nitudine at lu}\\ \text{mine ab eis mul}\\ \text{tum deficiebat}\end{array}\end{array}$

Observationes ejusdem cometæ habitæ a R. P Ango in Fleche

Novem 28 \hor. 5 matitin./ in medio erat inter stellas duas exiguas quarum una est minima {illeg} trium quæ sunt in manu australi Virginis altera est in extremitate alce: Adeo longitudo Cometæ jam erat ≏ 13, Latitudo australis 50′.

Decem 1. hora 5 matutina. erat in Libra 27. 45′

Observationes Venetijs habitæ a M. Montenaro.

Novem 30 hora {illeg} post occasum solis duodecima |Cometa| erat in ≏ 23gr cum lat Aust. 1gr 30′

Decem 1, Erat in hor 5 matutina erat in 27. 51 ≏.

Decem 2 erat in {illeg}|| 2. 33

Decem 4 erat in 12. 52

Credidit M. Montenari latitudinem {illeg} ad us finem harum observationem augæri.

Observationes Hevelij destituti instrumentis

Anno 1680 Decem. 2 Cometa erat in ≏ 25 cum lat. Austr. 5gr

Decem 3 {illeg}|{Arc}|te ortum is hora sexta erat in 4 cum lat. austr 4gr.

Decem 4 mane hor 6 20′ erat in 10 cum lat austr. 3gr.

<98v>

Observationes Cometæ Mense Novembri anni 1680

Canterburiæ per Artificem quendam nomine Hill, instrumento cujus radius erat 4 pedum Die Veneris Novemb 11 tempore matutino, |C|ometa inventus est in 12gr cum lat. b|B|oreali 2gr. Locus is 29gr. 53′

Romæ per Marcum Antonium Cellium observationes hæ factæ sunt.

$\begin{array}{ccccc}\text{Stylo veter.}& \text{Stylo novo}& \text{Cometæ Longit}& \text{Lat austral}& \text{Locus}\phantom{\rule{0.3em}{0ex}}{☉}^{\text{is}}\\ \text{Novemb}\phantom{\rule{0.3em}{0ex}}17& \text{Nov}\phantom{\text{em}}\phantom{\rule{0.3em}{0ex}}27\phantom{\rule{0.3em}{0ex}}\text{mane}& \text{≏}\phantom{0}{8\phantom{.}}^{gr}30\prime & {0}^{gr}\phantom{.}\phantom{+}30\prime \phantom{,\text{circiter}}& \text{♐}{5.}^{gr}55\prime \\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}18& \phantom{\text{Nov}\phantom{\text{em}}\phantom{\rule{0.3em}{0ex}}}28\phantom{\rule{0.3em}{0ex}}\text{mane}& \phantom{\text{≏}}{13.}^{\phantom{gr}}30\phantom{\prime }& {1}^{\phantom{gr}}.\phantom{+}00\phantom{\prime },\text{circiter}& \phantom{\text{♐}}{6.}^{\phantom{gr}}56\phantom{\prime }\\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}21& \text{Decem.}\phantom{\rule{0.3em}{0ex}}\phantom{0}1\phantom{\rule{0.3em}{0ex}}\text{mane}& \phantom{\text{≏}}{28.}^{\phantom{gr}}\phantom{0}0\phantom{\prime }& {1}^{\phantom{gr}}\phantom{.}+00\phantom{\prime },\text{circiter}& \phantom{\text{♐}}{9.}^{\phantom{gr}}58\phantom{\prime }\\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}24& \phantom{\text{Decem.}\phantom{\rule{0.3em}{0ex}}0}4\phantom{\rule{0.3em}{0ex}}\text{mane}& \text{♏}{11.}^{\phantom{gr}}40\phantom{\prime }& {1}^{\phantom{gr}}\phantom{.+}00\phantom{\prime },\text{circiter}& \phantom{\text{♐}}{13.}^{\phantom{gr}}\phantom{0}1\phantom{\prime }\\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}25& \phantom{\text{Decem.}\phantom{\rule{0.3em}{0ex}}0}5\phantom{\rule{0.3em}{0ex}}\text{mane}& \phantom{\text{♏}}{15.}^{\phantom{gr}}47\phantom{\prime }& {1}^{\phantom{gr}}\phantom{.+}00\phantom{\prime ,}\text{circiter}& \phantom{\text{♐}}{14.}^{\phantom{gr}}\phantom{0}2\phantom{\prime }\\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}26& \phantom{\text{Decem.}\phantom{\rule{0.3em}{0ex}}0}6\phantom{\rule{0.3em}{0ex}}\text{mane}& \phantom{\text{♏}}{19.}^{\phantom{gr}}45\phantom{\prime }& {1}^{\phantom{gr}}\phantom{.+}00\phantom{\prime ,}\text{circiter}& \phantom{\text{♐}}{15.}^{\phantom{gr}}\phantom{0}3\phantom{\prime }\\ \phantom{\text{Novemb}\phantom{\rule{0.3em}{0ex}}}27& \phantom{\text{Decem.}\phantom{\rule{0.3em}{0ex}}0}7\phantom{\rule{0.3em}{0ex}}\text{mane}& \phantom{\text{♏}}{23.}^{\phantom{gr}}35\phantom{\prime }& {1}^{\phantom{gr}}\phantom{.+}00\phantom{\prime ,}\text{circiter}& \phantom{\text{♐}}{16.}^{\phantom{gr}}\phantom{0}4\phantom{\prime }\end{array}$

Romeæ per Galletium hæ

$\begin{array}{cccc}\text{Stylo vet}& \text{St. Novo}& \text{Comet. Long}& \text{Lat. austral.}\\ \text{Novem}\phantom{\rule{0.3em}{0ex}}17& \text{Novem}\phantom{.}\phantom{\rule{0.3em}{0ex}}27\phantom{\rule{0.3em}{0ex}}\text{Hora}\phantom{\rule{0.3em}{0ex}}18\phantom{⁤\frac{1}{2}}& \phantom{0}8.\phantom{0}0& 0.00\\ \phantom{\text{Novem}}\phantom{\rule{0.3em}{0ex}}18& \phantom{\text{Novem}.}\phantom{\rule{0.3em}{0ex}}28\phantom{\rule{0.3em}{0ex}}\phantom{\text{Hora}}\phantom{\rule{0.3em}{0ex}}17⁤\frac{1}{2}& 13.\phantom{0}0& 1.00\\ \phantom{\text{Novem}}\phantom{\rule{0.3em}{0ex}}21& \text{Dec.}\phantom{\text{em}}\phantom{\rule{0.3em}{0ex}}\phantom{0}1\phantom{\rule{0.3em}{0ex}}\phantom{\text{Hora}}\phantom{\rule{0.3em}{0ex}}17⁤\frac{1}{2}& \phantom{0}0.\phantom{0}0& 4.00\\ \phantom{\text{Novem}}\phantom{\rule{0.3em}{0ex}}23& \phantom{\text{Dec.}\text{em}}\phantom{\rule{0.3em}{0ex}}\phantom{0}3\phantom{\rule{0.3em}{0ex}}\phantom{\text{Hora}}\phantom{\rule{0.3em}{0ex}}17\phantom{⁤\frac{1}{2}}& \phantom{0}9.\phantom{0}0& 3.00\\ \phantom{\text{Novem}}\phantom{\rule{0.3em}{0ex}}25& \phantom{\text{Dec.}\text{em}}\phantom{\rule{0.3em}{0ex}}\phantom{0}5\phantom{\rule{0.3em}{0ex}}\phantom{\text{Hora}}\phantom{\rule{0.3em}{0ex}}18\phantom{⁤\frac{1}{2}}& 16.15& 2.00\\ \phantom{\text{Novem}}\phantom{\rule{0.3em}{0ex}}26& \phantom{\text{Dec.}\text{em}}\phantom{\rule{0.3em}{0ex}}\phantom{0}6\phantom{\rule{0.3em}{0ex}}\phantom{\text{Hora}}\phantom{\rule{0.3em}{0ex}}18\phantom{⁤\frac{1}{2}}& 19.30& 1.00\end{array}$

Cantabrigiæ per juvenem quendam Cometa observatus est Novemb 19 juxta spicam Virginis, quasi duobus gradibus |supra| stellam illam, ad sive ad boream, circa horam quartam vel quintam matutinam. Et cauda extendebatur ad us stellam illam primæ magnitudinis quæ cauda Leonis dicitur.

Observationes Parisijs habitæ Cometæ subsequentis 1680 & 1681.

[78]$\begin{array}{cclllll}& \text{Longitudo}& \text{Lat. bor}& \text{A}\text{s}\text{cbn. f.}& \text{Decl}& \text{Long. caud.}& \text{Decl. caudæ ab op.}\phantom{\rule{0.3em}{0ex}}\text{O}\\ \phantom{\rule{4em}{0ex}}\text{St. vet.}\phantom{\rule{0.3}{0ex}}\text{Dec.}\phantom{\rule{0.3em}{0ex}}19\phantom{.}\phantom{\rule{0.3em}{0ex}}\text{hor}\phantom{\rule{0.3em}{0ex}}6.30\text{p.m.}& {27.}^{gr}\phantom{0}4\phantom{⁤\frac{1}{2}}\prime & 18.53& & & {62}^{gr}& {2}^{gr}.48\prime .\\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}\phantom{\rule{0.3}{0ex}}\text{Dec.}}\phantom{\rule{0.3em}{0ex}}24\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}5.30\phantom{\text{p.m.}}& {18.}^{\phantom{gr}}36⁤\frac{1}{2}\phantom{\prime }& 25.26& & & & {3}^{gr}.43\prime \phantom{.}\\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}\phantom{\rule{0.3}{0ex}}\text{Dec.}}\phantom{\rule{0.3em}{0ex}}28\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.00\phantom{\text{p.m.}}& {\phantom{0}8.}^{\phantom{gr}}12⁤\frac{1}{2}\phantom{\prime }& 28.\phantom{0}3& & & {62}^{gr}& \\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}\phantom{\rule{0.3}{0ex}}\text{Dec.}}\phantom{\rule{0.3em}{0ex}}29\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.\phantom{0}6\phantom{\text{p.m.}}& \phantom{}{12.}^{\phantom{gr}}55\phantom{⁤\frac{1}{2}\prime }& 28.16& & & & \\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}}\phantom{\rule{0.3em}{0ex}}\text{Jan}\phantom{\text{.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}4.\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.\phantom{0}6\phantom{\text{p.m.}}& \phantom{}{\phantom{0}5.}^{\phantom{gr}}53\phantom{⁤\frac{1}{2}\prime }& 26.38& & & & \\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}}\phantom{\rule{0.3em}{0ex}}\phantom{\text{Jan}\text{.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}6.\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.30\phantom{\text{p.m.}}& \phantom{}{11.}^{\phantom{gr}}33\phantom{⁤\frac{1}{2}\prime }& 25.42& & & & {5}^{gr}.13\prime \phantom{.}\\ \phantom{\rule{4em}{0ex}}\phantom{\text{St. vet.}}\phantom{\rule{0.3em}{0ex}}\phantom{\text{Jan}\text{.}}\phantom{\rule{0.3em}{0ex}}\phantom{0}8.\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.50\phantom{\text{p.m.}}& \phantom{}{16.}^{\phantom{gr}}36\phantom{⁤\frac{1}{2}\prime }& 24.44⁤\frac{1}{2}& \phantom{0}4.40& 29.\phantom{0}7& & \\ \text{Observatio crassa}\phantom{\rule{0.3em}{0ex}}13\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.20\phantom{\text{p.m.}}& \phantom{}{26.}^{\phantom{gr}}\phantom{0}7⁤\frac{1}{2}\phantom{\prime }& 22.20.& & & & \\ \phantom{\text{Observatio crassa}}\phantom{\rule{0.3em}{0ex}}23\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.40\phantom{\text{p.m.}}& & & 28.27& 31.35& & \\ \phantom{\text{Observatio crassa}}\phantom{\rule{0.3em}{0ex}}24\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.20\phantom{\text{p.m.}}& & & 29.30⁤\frac{1}{2}& 31.36& & \\ \phantom{\text{Observatio crassa}}\phantom{\rule{0.3em}{0ex}}23\phantom{.}\phantom{\rule{0.3em}{0ex}}\phantom{\text{hor}}\phantom{\rule{0.3em}{0ex}}6.40\phantom{\text{p.m.}}& & & 30.30& 31.37& & \end{array}$

Ejusdem posterioris Cometæ Observationes Grenovici habitæ

$\begin{array}{c}\begin{array}{cc}\phantom{\rule{59.5em}{0ex}}& \begin{array}{l}\text{Hæ æquan}\\ \text{tur}\phantom{\rule{0.3em}{0ex}}\text{Perism}\\ \text{addendo vel}\\ \text{auferado}\\ \end{array}\end{array}\\ \begin{array}{ccccccccc}\text{Loca}\phantom{\rule{0.3em}{0ex}}{}^{\text{is}}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}\phantom{\rule{0.3em}{0ex}}12}& \text{Tempus verus}& \text{Ascentio recta}& \text{Declinatio}& \text{Longitudo}& \text{Latitudo}& \text{Longit}& \text{in}\hfill \\ & & & & \text{borealis}& & \text{borealis}& \text{caudæ}& \text{long. lat}\hfill \\ \phantom{0}1.53\phantom{⁤\frac{1}{2}}& \text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}\phantom{\rule{0.3em}{0ex}}12& {4}^{\mathrm{h}}46\prime \text{p. m.}& \text{————}& \text{————}& \phantom{0}{6}^{gr}.33\prime \phantom{⁤\frac{1}{2}}& \phantom{0}{8}^{gr}.26\prime \phantom{⁤\frac{1}{2}}& 35+& \\ \phantom{}11.\phantom{0}8\phantom{⁤\frac{1}{2}}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}}\phantom{\rule{0.3em}{0ex}}21& {6}^{\phantom{\mathrm{h}}}31\phantom{\prime \text{p. m.}}& 302.20⁤\frac{1}{2}& \phantom{0}2\phantom{.}\phantom{0}5⁤\frac{1}{3}& \phantom{0}{5}^{\phantom{gr}}.08\phantom{\prime }⁤\frac{1}{5}& {21}^{\phantom{gr}}.42\phantom{\prime }⁤\frac{1}{6}& 70\phantom{+}& \\ \phantom{}14.11⁤\frac{1}{3}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}}\phantom{\rule{0.3em}{0ex}}24& {6}^{\phantom{\mathrm{h}}}24\phantom{\prime \text{p. m.}}& 313\phantom{.}33\phantom{⁤\frac{1}{2}}& \phantom{0}9\phantom{.}\phantom{0}0\phantom{⁤\frac{1}{3}}& \phantom{}{58}^{\phantom{gr}}.52\phantom{\prime ⁤\frac{1}{5}}& {25}^{\phantom{gr}}.26\phantom{\prime ⁤\frac{1}{6}}& 65\phantom{+}& -\phantom{0}3.\phantom{\frac{1}{2}}+2\\ \phantom{}16.10⁤\frac{2}{3}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}}\phantom{\rule{0.3em}{0ex}}26& {5}^{\phantom{\mathrm{h}}}16\phantom{\prime \text{p. m.}}& 321\phantom{.}15\phantom{⁤\frac{1}{2}}& 13\phantom{.}19⁤\frac{2}{3}& \phantom{}{28}^{\phantom{gr}}.28\phantom{\prime }⁤\frac{1}{2}& {27}^{\phantom{gr}}.\phantom{0}5\phantom{\prime }⁤\frac{1}{2}& 60\phantom{+}& \\ \phantom{}19.21\phantom{⁤\frac{2}{3}}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}}\phantom{\rule{0.3em}{0ex}}29& {8}^{\phantom{\mathrm{h}}}\phantom{0}0\phantom{\prime \text{p. m.}}& 333\phantom{.}27\phantom{⁤\frac{1}{2}}& 19\phantom{.}21\phantom{⁤\frac{2}{3}}& \phantom{}{13}^{\phantom{gr}}.12\phantom{\prime ⁤\frac{1}{2}}& {28}^{\phantom{gr}}.10\phantom{\prime ⁤\frac{1}{2}}& 50\phantom{+}& +\phantom{0}6.\phantom{\frac{1}{2}}+8\\ \phantom{}20.22⁤\frac{1}{2}& \phantom{\text{St. vet.}\phantom{\rule{0.3em}{0ex}}\text{Decem}}\phantom{\rule{0.3em}{0ex}}30& {8}^{\phantom{\mathrm{h}}}\phantom{0}4\phantom{\prime \text{p. m.}}& 337\phantom{.}14\phantom{⁤\frac{1}{2}}& 21\phantom{.}00\phantom{⁤\frac{2}{3}}& \phantom{}{17}^{\phantom{gr}}.39\phantom{\prime ⁤\frac{1}{2}}& {28}^{\phantom{gr}}.\phantom{}12\phantom{\prime ⁤\frac{1}{2}}& 25\phantom{+}& \\ \phantom{}26.23⁤\frac{1}{3}& \phantom{\text{St. vet.}}\phantom{\rule{2.5em}{0ex}}\text{Jan}\phantom{\rule{0.3em}{0ex}}\phantom{0}5& {5}^{\phantom{\mathrm{h}}}53\phantom{\prime \text{p. m.}}& 356\phantom{.}45⁤\frac{1}{2}& 27\phantom{.}25⁤\frac{1}{2}& \phantom{0}{8}^{\phantom{gr}}.49\phantom{\prime }⁤\frac{1}{6}& {26}^{\phantom{gr}}.\phantom{}15\phantom{\prime }⁤\frac{1}{2}& 15\phantom{+}& +\phantom{0}1\phantom{.}\frac{1}{2}-4\\ \phantom{0}0.30\phantom{⁤\frac{1}{3}}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Jan}}\phantom{\rule{0.3em}{0ex}}\phantom{0}9& {6}^{\phantom{\mathrm{h}}}50\phantom{\prime \text{p. m.}}& \phantom{0}6\phantom{.}57\phantom{⁤\frac{1}{2}}& 29\phantom{.}31\phantom{⁤\frac{1}{2}}& \phantom{}{18}^{\phantom{gr}}.44\phantom{\prime ⁤\frac{1}{6}}& {24}^{\phantom{gr}}.\phantom{}12\phantom{\prime }⁤\frac{1}{10}& & +\phantom{0}9.\phantom{\frac{1}{2}-}0\\ \phantom{}\phantom{0}1.28⁤\frac{1}{2}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Jan}}\phantom{\rule{0.3em}{0ex}}10& {5}^{\phantom{\mathrm{h}}}56\phantom{\prime \text{p. m.}}& \phantom{0}9\phantom{.}\phantom{0}2\phantom{⁤\frac{1}{2}}& 29\phantom{.}51\phantom{⁤\frac{1}{2}}& \phantom{}{20}^{\phantom{gr}}.41\phantom{\prime }⁤\frac{1}{2}& {23}^{\phantom{gr}}.\phantom{}44\phantom{\prime }⁤\frac{1}{2}& 10& +18.\phantom{\frac{1}{2}-0}\\ \phantom{}\phantom{0}4.34\phantom{⁤\frac{1}{2}}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Jan}}\phantom{\rule{0.3em}{0ex}}13& {6}^{\phantom{\mathrm{h}}}55\phantom{\prime \text{p. m.}}& 14\phantom{.}52\phantom{⁤\frac{1}{2}}& 30\phantom{.}37\phantom{⁤\frac{1}{2}}& \phantom{}{25}^{\phantom{gr}}.59\phantom{\prime }⁤\frac{1}{2}& {22}^{\phantom{gr}}.\phantom{}17\phantom{\prime }⁤\frac{1}{2}& \phantom{0}5& +12.\phantom{\frac{1}{2}}+3\\ \phantom{}16.45⁤\frac{2}{3}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Jan}}\phantom{\rule{0.3em}{0ex}}25& {7}^{\phantom{\mathrm{h}}}44\phantom{\prime \text{p. m.}}& 30\phantom{.}35\phantom{⁤\frac{1}{2}}& 31\phantom{.}37\phantom{⁤\frac{1}{2}}& \phantom{0}{9}^{\phantom{gr}}.36\phantom{\prime ⁤\frac{1}{2}}& {17}^{\phantom{gr}}.\phantom{}57\phantom{\prime ⁤\frac{1}{2}}& & \\ \phantom{}21.50\phantom{⁤\frac{2}{3}}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Jan}}\phantom{\rule{0.3em}{0ex}}30& {8}^{\phantom{\mathrm{h}}}\phantom{0}7\phantom{\prime \text{p. m.}}& 35\phantom{.}\phantom{0}2\phantom{⁤\frac{1}{2}}& 31\phantom{.}41\phantom{⁤\frac{1}{2}}& \phantom{}{13}^{\phantom{gr}}.20\phantom{\prime ⁤\frac{1}{2}}& {16}^{\phantom{gr}}.\phantom{}41\phantom{\prime ⁤\frac{1}{2}}& & \\ \phantom{}24.43⁤\frac{1}{5}& \phantom{\text{St. vet.}}\phantom{\rule{2.5em}{0ex}}\text{Feb}\phantom{\rule{0.3em}{0ex}}\phantom{0}2& {6}^{\phantom{\mathrm{h}}}20\phantom{\prime \text{p. m.}}& 37\phantom{.}18\phantom{⁤\frac{1}{2}}& 31\phantom{.}41\phantom{⁤\frac{1}{2}}& \phantom{}{15}^{\phantom{gr}}.14\phantom{\prime ⁤\frac{1}{2}}& {16}^{\phantom{gr}}.\phantom{}02\phantom{\prime ⁤\frac{1}{2}}& & \\ \phantom{}27.50⁤\frac{1}{2}& \phantom{\text{St. vet.}\phantom{\rule{2.5em}{0ex}}\text{Feb}}\phantom{\rule{0.3em}{0ex}}\phantom{0}5& {7}^{\phantom{\mathrm{h}}}\phantom{0}8\phantom{\prime \text{p. m.}}& 39\phantom{.}26\phantom{⁤\frac{1}{2}}& 31\phantom{.}41\phantom{⁤\frac{1}{2}}& \phantom{}{17}^{\phantom{gr}}.00\phantom{\prime }⁤\frac{1}{2}& {15}^{\phantom{gr}}.\phantom{}27\phantom{\prime ⁤\frac{1}{2}}& & \end{array}\end{array}$

<99r>

Observationes de Cauda Cometæ prioris

Novemb 19 Cometa juxta spicam virginis existens caudam projiciebat ad us caudam Leonis, spectante juvenes quodam.

Postea caudam versu{illeg} per meride|i|em versus occidenti|e|m projici {illeg} longam satis & ad horizontem obliquam capite {illeg} vel sub horizonte vel pone ædificia delitescente vidit Humf. Bab. S. T. D.

De cauda Cometæ posterioris

Decemb 8 stylo veteris Hallius noster tempore matutino Parisias versus iter faciens prope Bolonian ante ortum solis Caudam vidit Cometæ quasi perpendiculariter ex horizonte surgentem, ut ipse retulit in epistola quadam cit{illeg}|an|te Flamstedio. Unde Cometa inquit Flamstedius tunc borealem habeb{a}t latitudinem & cum solenondum conjunctus fuerat. |Apparebat autem cauda lat{illeg}|{æ}| divergens et {illeg} ex corpore {illeg} eg{illeg}|re|ssa aer prius {illeg} quam {illeg} {illeg} \{illeg}/ {illeg} {\{illeg}/} {illeg}|

Decemb 10. duabus horis post occasum Solis, {illeg}bat cauda per medium distantiæ inter caudam serpentis Ophi{illeg}{illeg}|cha| et stellam (Bayero δ) in ala austrina Aquilæ. Desinebat vero ad stellas tres exiguas (Bayero Awb) in dorso Aquilæ juxta caudam, eductione caudæ Aquilæ ejusdem, id est in linea jungente stellam|s| lucida{illeg} secundæ magnitudinis in eductione colli Aquilæ, et estellam tertiæ quæ penultima est in cauda ejus, ac stellæ illi penultim{æ} [duplo quidem] propior existebat qu{à}m alteri in eductione colli. {illeg}|F|lamstedius in Epistolis ad nos datis. Desinebat igitur cauda in $19\frac{1}{2}$ cum lat. bor. $34\frac{1}{4}$ circiter

Decemb 11 {Cau} post occasum Solis cauda instar jubaris apparuit ab horizonte erecti et lunâ latioris. Post crepusculi cessationem ex tendebat ad us stellas \duas/ quartæ m{æ}gnitudinis (Bayero α, β,) in capite \seu glyphidæ/ Sagittæ. (Flam{st}. ib.) adeo desinebat in 26gr 43′ cum lat bor. 38gr 34′.

Decemb. 12. Quamprimum non obscura facta est, cauda transibat per medium sagittæ, ne ultra medium longè extendebat. (Flam{st} ib) L{in}quebat igitur stellas 5 et 6 magnitudinis, δ et ζ in tribulo sagittæ, quasi 40′ ad occidentem, et ultra per 3grad circiter vel fort{æ} 4 extendens desinebat in {} 4 cum lat bor $42\frac{1}{2}$ circiter vel 42 43 /34\ 43. Desinebat uti e regione superior{is} duarum informium 4 magnitudinis quæ supra sagittam sunt {illeg} non et ultra extendebat{illeg}. Nam cauda ensiformis nobis visa {illeg} sagittam \paul{o}/ longius superare quam Flamstedio, in viam lacteam {illeg} nihil extendens & termino acuto paulatim languescens. Ca|e|ter {illeg} in A{s}trola{b}io Flamstedij, cauda hac nocte {illeg} desinit accurat ad stellas duas exiguas prædictas in tribulo sagittæ.

Decemb 15 hor $5\frac{3}{4}$ lucida Aquilæ erat in medio caudæ fere {Pergebat vero cauda} Ancon item austrinus Aquilæ erat {illeg} {illeg} \{illeg}/la parte caudæ prope terminum ejus \{medio} caudæ fere prope terminū ejus ad latus australe vergens./ (Ipse \ego & {Bainbro} et ellis/ part{im} ex observa{tione} partim ex circu{s} {stansijs}) Erat autem cauda 50 grad. longa ({illeg} {steed} epist. 1) {nec tutren} \tenuem/ extremitatem ejus propter Lunæ \novel/ splendorem oliquam apparuise probabila est.

< insertion from the left margin of f 99r >

Decemb. 16 hor 5 P.M. Cometa existente in 17 cum lat. bor. 15gr circiter(|]| cauda lucidam Aquilæ (quæ nocte superiori erat in medio ejus) latere suo boreali \{illeg}/ tangebat, \aut quasi;/ ut et lucidam in {ancone} austrino cygni tangebat eodem latere |{illeg} aut quasi| Tota Caudæ longitudo erat 60 grad: feré, latitudo 2 gradus. (Observator quidam Scotus.) Unde Cauda terminabatur in long. 10 vel 12 circiter 9 lat. bor. 53.

Decem. 19 Hor $5⁤\frac{1}{2}$ P.M. Transibat cauda per Delphini caput dein latere suo boreali stellam penultimam in austrina ala{illeg} Cygni stringebat, tendans inde versus lucidam in Cassiopeiæ cathedra et quasi 60 gradus longa existens ({illeg} observator Scotus) vel potius 63 aut 64 grad ut ex alijs colligo, si non et paullo ultra. Desinebat igitur in 6 cum lat. bor. 52 |vel {illeg} $81⁤\frac{1}{2}$|.

Decem 17 cauda inferiùs duos gradus lata