Newton's Waste Book (Part 3)Isaac Newtonc.42,677 wordsThe Newton ProjectFalmer2013Newton Project, University of Sussex
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c. 1664 - c. 1685, c. 44,926 words, 187 pp.187 pp.MS Add. 4004Newton's Waste Book (Part 2) [MS Add. 4004, ff. 15v-50r]MS Add. 4004, ff. 50v-198v, Cambridge University Library, Cambridge, UKUKCambridgeCambridge University LibraryPortsmouth CollectionMS Add. 4004, ff. 50v-198vc. 1664 - c. 1685EnglandEnglishLatinHolographBarnabas SmithUnknown HandUnknown Hand (1)Newton's NotebooksScienceMathematicsCatalogue information compiled from CUL Janus Catalogue by Michael HawkinsTranscription and encoding of transcription Isaac Newton's text from front to ff. 63r by Daniele CassisaYvonne Santacreu tagged transcription of Barnabas Smith's notesTranscription and encoding of Isaac Newton's text from ff 64v-198v by Margarita Fernandez-ChasProofed by Robert IliffePreliminary audit of XML by Michael Hawkins
How to Draw Tangents to Mechanicall Lines
Lemma.
1Newton numbered the first two paragraphs to indicate their order within the text. These paragraphs have been moved in our transcription according to Newton's numbering. If one body move from to in yethe same time in wchwhich another moves from to & a 3d body move from wthwith motion compoundeed of those two it shall (completeing yetheparallelogram ) move to in yethe same time. For those motion would severally carryit yethe one from to yethe other from to &c
November 8th 1665.2 In yethe description of any Mechanicall line what ever, there may bee found stwo such motions wchwhich compound or make up yethe motion of yethe point describeing it, whose ∼ motion being by them found by yethe Lemma, its determinacon shall bee in a tangent to yethe mechanicall line. Example yethe 1st. If is an helix, wchwhich described by yethe point movemoveing uniformely in yethe line from yethe point center about wchwhichyethe line circulates uniformely. the line increasing uniformely whilest it also circulates uniformely about yethe center Let yethe radius of yethe circle bee . & let measure yethe quantity of the giration of (viz touching yethehelix at yethe center) let be a tangent to yethe circle . ynthen is yethe motion of yetheline point towards to its motion towards , as , to . therefore make . & (by yethe Lemma) yethe diagonall shall touch yethe helix in . Or make . & . the diagonall shall touch yethe helix. (yethe length of may be thus found viz; .) Example yethe 2d. If yethe center of a globe moves uniformely in a streight line parallel to , whilest yethe Globe uniformely girates. Each point in yethe Globe will describe a Trochoides: to wchwhich a yethe point I thus draw a tangent. Draw yetheradiradii & perpendicular to it ynthen is yethe circular motion of the point determined in yethe line , & its progesiveprogressive in . If therefore I make to as yethecircular motion of yethe point to its progressive yethe Diagonall (by yethe Lemma) shall g touch yethe Trochoides in . As if yethe Globe roule upon yethe plaine , & I make . & . ynthen dothyethe Diagonall touch yethe Trochoides. (Or , passing through yethe point in wchwhichyethe globe & plaine touch, is a perpendicular to yetheTroTrochoides. Example yethe 3d. If yethe line moves uniformely yethe length of whilest girateuniformely from to about yethe center , yethe point of their intersection will descibedescribeyethe Quadratrix . Draw . & ynthen motion of to motion of to , (sup). And motion of to motion of to . Therefore motion of to motion of to . Therefore makeing . (Or wchwhich is makeing , & ) yethe Diagonall shall touch yethe Quadratrix at . [Scholium. The tangents of Geometricall lines may be found by their descriptions afteryethe same manner. As the Ellipsis (whose foci are & ) being described by yethe thred yethe thred lengthens so much as yethe thred shortens, or the point moves equally from & to . Therefore I take . & . & yethe diagonall will touch the Ellipsis in ]. (This should follow yethe 3d Example's substitute) See folfolium 57. Although yethe nature of a Mechanicall line is not knowne from its description but from some other principle yet may a tangnttangent be drawne to it by yethe same method.As if is an Hyperbola. its asymptote & . & . to draw a tangent to yethetangent line , I consider ytthat, increasing of increasing of increase of increase of motion of yethe point towards motion of towards , if . Therefore I make . & yethe diagonall or shall touch yethe line . Or if . . . . ynthen doth touch yethe line at .Tangents to mechanichanicall lines may sometimes bee found by finding such a point wchwhich is immoveable in respect of yethe line described & also doth varyin distance from yethe describing point. Then in yethe for yetheSicunfthrough ytthat point. Thus in yethe Trochoides when yethe point toucheth yethe plaine tis immoveable, & tis ever equidistant from yethe describing point
( both of ymthem fin points in yethe Globe). Therefore yethe line drawne fro=fromyethe describeing point to yethe touch point of yethe Globe & plaine is perpendicular to yethe trochoides. But in yethe spirall though yethe point is from ytthat. Instead in yethe third example . ThereforeAdd 4004 (c)(51) Therefore absolute & whole motion of towards (or ) whole motion of towards (or ). Soe ytthat makeing . & . The point will be moved to yethe line & in same times wchwhich cannot bee unlesse it move to (their common intersection). The point therefore move in yethe line wchwhich doth therefore touch yetheQuadratrix at : (The same is done by makeing . & drawing yethe tangent through yethe common intersection of & .)
To resolve these and such like Problems these following propositions may bee very usefull.May 14. 1666. Prop 1. If yethe body being in a circumference of yethe circle or sphære doth move towards its center its accelerationmotion or velocity towards each point , , of yethe circumference is ynthen as yethe cordes , , , drawn from ytthat body to those points are. This may be Demonstrated by Teheorem R pag 57. Prop 2d. If is a parallelogram although they bee not in yethesame plane & three bodys move uniformely from . yethe first from to yethe 2dfrom to , yethe 3dfrom to their motions beingeach to otheras yethe directing lines , , , are in yethe same time, & is a parallelogram then is yethe motion of yethe third body compounded of yethe other two. Demonstration. For makeing f & adba makeing yethe motion of yethe first body towards is to its motion towards as is to (prop 1); & yethemotion of yethe second body towards is to its motion towards as is to (prop 1). Therefore But . Therefore &cProp 3d. If a moveing line keepe parallel to it selfe all its ptspoints have equall motion.Prop 4th. If a line move in plano, so ytthat all its points keepe equidistant from some common pointcenter the motions of those points are as their distances from ytthat center.Prop 5t. If yethe motion of a line in plano bee mixed of parallell & circular motion, yethe motion of all its points are compound (see prop 2) of that motion which they would have, had yethe line onely its centrall parallel motion, & of ytthatwchwhich they would have, had yethe line onely its circular motion.Schol:Scholium All motion in plano is reducible to one of these three cases, & in yethe 3d case any point in ytthat plaine may bee taken for a center to yethe circular motion. Prop 6t. If yethestreight line doth rest & doth move: soe ytthatyethe point fixed in yethe line moveth towards : Then from yethe moveing line drawing , & yethe same way wchwhichyethepoint moveth; These motions, viz of yethefixed point towards , of yethe intersection point in yethe line towards , & of yethe intersection point towar in yethe line towards , shall bee one to another, as their correspondent lines , , & are. Prop 7th. If yethestreight lines , , move, soe ytthatyethe point fixed in yethe line moveth towards , & yethe point fixed in yethe line moveth in yethe towards : Then from yethe line each line to yethe other draw two lines , parallell to the mo the line , draw ∼ & yethe same way: & from yethe line draw , & yethe contrary way, to make up yetheTrapezium . And if any two of these foure lines , , , , bee to any ∼ correspondent two of these foure motions, viz: of yethe point (fixed in yethe line ) towards , of yethe point (fixed in yethe line ) towards , of yethe intersection point moveing in yethe line according to yethe order of yethe letters , & of yethe intersection point in yethe line according to yethe order of yethe letters , : Also all yethe foure lines shall be one to another as those foure motions are.Note ytthatin yethe two last propositionsifyethe moveing lines may bee crooked so ytthat, , bee tangents to them in yethe point .Note also ytthat by yethe place of a body is meant its center of gravity.To resolve Problems by motion yethe 6 following prop:propositions are necessary & suffcient.May 16. 1666. If yethe body in yethe perimeter of yethe circle or sphære moveth towards its center . its velocity to each point of ytthat circumference is as yethe cordes , , , drawne from ytthat body to those points are.Prop 2. If yethe△striangles, are alike though in diverse planes; & 3 bodys move from yethe point uniformely & in equall times, yethe first to , yethe 2d to , yethe 3d to : ynthen is yethe3d's3d's motion compounded of yethe motion of yethe 1st & 2d.Note ytthat by a body is meant its center of gravity.Prop. 3. All yethe points of a body keeping parallel to it selfe are in equall motion.Prop. 4. If a body onely move circularly about some axis, yethe motion of its points are as their distances from that axis. Call these 2 simple motionsProp. 5. If yethe motion of a body is considered as comp mixed of simple motions: yethe motions of all its points are compounded of their simple motions, so as yethe motion towards (in prop 2d) is compounded of yethe motion towards & .Note ytthat all motion is reducible to one of these 3 cases: & in yethe 3d case any line may bee taken for theaxis (or if a line or superficies movein plano any point of ytthat plaine may bee taken for yethe center) of motion. Prop. 6. If yethe lines being moved doe continually intersect; I describe yethe Trapezium its diagonall : & say ytthatyethe proportion & position of these five lines , , , , being determined by requisite data they shall designe yethe proportion & position of these 5 motions: of yethe point fixed in yethe moveing towards ; of yethe point fixed in yethe line moveing towards ; of yethe intersection point moveing in yethe plaine towards (for those 5 lines are in yethe same plaine though & may only touch yethe plaine in their intersection point): of yethe intersection point moveing in yethe line parallely to & according to yethe order of yethe letters , : & of yetheintersection point moveing in yethe line parallelly to & according to yethe order of those Note ytthat a streight line is said to designe yethe position of curved motion in any point if toucheth yethe line described by yethe motion in ytthat point, or when tis (as , , ), or tis parallell to such a (as , ). Note also ytthat one line resting (as in Fig 3 & 4) yethe points & are coincident & yethe point shall bee in yethe line if bee streight (fig 3), otherwise in its tangent (fig 4) . Haveing an equation expressing yethe relation of two lines & described by two bodys & whose motions ; Translate yethe termes to one side & multiply ymthem, being ordered according to progression &c: & being ordered by yethe dimensions of multiply those by &c. yethesumma of those products equation expressing yethe relation motions & .To draw a tangent to yethe Ellipsis Suppose yethe Ellipsis to be described by yethe thred , & ytthat is its tangent. Since yethe thred is diminished with yethe same proportion velocity ytthat be increasedth, ytthat is, ytthatyethe point hath yethe same motion towards & , yethe angles , , must bee equall, by prop 1. I And, so of yetheotheotherconicks.To draw a Tangent to yethe Concha. Suppose ytthat, , are yethe rulers by wchwhichyethe concha is usually described, & ytthat, & . And (since equality is more simple ynthan proportionality) suppose ytthat is yethe velocity of yethe point towards , or of towards . Then is yethemot circular motion of yethe point about (prop 1); & yethe circular motion of yethe point fixed in yethe ruler , ( prop 4). And is yethe motion of yethe intersection point (ytthat is, yethe velocity of yethe point ) moveing in yethe line from (prop 6). Now since a two fold velocity of yethe point is known nemely toward & towards , make ; & yethe motion of yethe point shall bee in yethe line yethe diameter of yethe circle passing through yethe points (prop 1) & therefore tangnttangentto yethe Concha.To find yethe point wchwhich distinguisheth twixt yethe concave & convex portion of yethe Concha. Those things in yethe former prop:proposition being supposed, make △triangle like or : & ∼ , & draw . Now had yethe line onely parallel motion directed by or , (since ) yethe motion of all its points would bee , (prop 3): & if it had onely circular motion about , yethe motion of yethe point fixed in ytthat line would bee (prop 4): But yethe motion of yethe point is compounded of those two simple motions, & is therefore (prop 5 & 2); & yethe motion of yethe intersection point made by yethe lines , & moveing in , shall bee , (prop 6). Now if yethe line touch yethe concha in yethe required point, tis easily conceived ytthatyethe motion of ofyethe intersection point is infinitely little; & therefore ytthatyethe points & are coincident, & being one streight line, & yethetriãglestriangles, being alike.Which may bee thus calculated. Make . . . ynthen is . . . & . Therefore . Or .In stead of yethe ordinary method de Maximis et minimis, it will be as convenient (& perhaps more naturall) to use ∼ This; Namely To find yethe motion of ytthat line or quantity ∼ ∼ ∼ & suppose it equall to nothing, or infinitely small. But ynthenyethe motion to wchwhich tis compared must bee finite. That is, yethe unknowne quantitys ought not to bee at their greates or least, both at once. Example, In yethe triangle , yethe side being given & fixed. yethe side being given & circulateing about yethe center , I would know when is yethe shortest it may bee. I call . . . .ynthen is yethe. & . & . or . & . (prop 7). And makeing , tis . Or . (For signifieing yethe motion of towards may bee finite though, , its motion towards doth perish). Wherefore . or . . . . are yethe greatest & yethe least valors of yethe line .Should I have taken , instead of . The effect would not have followed because both yethe motions & would have vanished at once in yethe point . But I might have taken yethe tangent for , or any other line wchwhich wouldcoincidere wthwith at its being greatest or least. Example 2d. If is yethe Conchoid (. . . .) fo parallell to it. Then is .Then is . & . & . etponatur esse motus puncti & esse motus puncti versus . Erit (prop 6). supposeing (For when is yethe least ytthat bee yethe point is ytthatwchwhich distinguisheth twixt yethe concave & convex portion of yethe Conchoid, & ynthenyethe motion vanisheth.) it will bee . . & . Or .(52)Concerning Equations when their rootesrelation twixtratio of their rootes is considered.May. 1665. If totwo of yethe rootes of an Equation are in proportion yethe one to yethe other as to Then multiplying yethe termes of the Equation by this progression :&c. &c. (And that rootor by yethe same progressionaugmented or diminished by any quantity, as if it bee augmented by it will bee . &c. Or were it augmented by it would be ). Then shall yethe roote wchwhich is correspondent to be a roote of yethe resulting equation: but inverting yethe order of yetheprogresionprogression, ytthat roote wchwhich is correspondent to shall bee a roote of yethe equation resulting from such multiplication.As for example did I know ytthat two of yethe rootes of yethe Equation were in proportion as 1 to 2 & would I have yethe lesser roote (viz ytthatwchwhich is correspondent to 1) I make . . And soe the progression will bee . &c Or . &c by adding . Or by adding one more it will bee . &c. By any of wchwhich progressions yethe Equation may bee multiplyed, as by yethe 1st, . WchWhich produceth . Or by yethe 3d. WchWhich produceth . Or by the first Otherwise by destroying yethe 1st terme. . WchWhich produceth . &c the rootes of wchwhich products are, viz: of yethe first , & . Of yethe last , & . There I conclude to be yethe lesse, & consequently the greater of those rootes wchwhich arein proportion as one to of yethe Equation . wchwhich are in double proportion But was yethe greater of those rootes desired ynthen inverting yethe progression it would bee . Or . The first producing whose rootes are , . The 2d produceth whose rootes are , . And consequently is the greater & the lesse of yethe rootes in duplicate proportion.Would I draw If in yethe circle is yethe diameter, a perpendicular to yethe end of it from wchwhich I would draw , wchwhich should intersect perpendicular circle in yethe points & soe ytthat bee triple to , ytthat is quadruple to . Then calling . . . The equation expresingexpressingyethe relation twixt & is . yethe rootes of wchwhichequation must be quadruple yethe one to yethe other: Therefore would I find yethe lesse roote I make . . And yethe progression will bee, . &c. by wchwhichyethe Equation being multiplyed yethe product is . Or . Therefore drawing , Or . from wchwhichthe point , or raise yethe perpendiculars, or . & soe draw .Would I have to be a square ytthat is . Then to find I call it & . Soe ytthatyethe lesse roote is to yethe greater as to . Making therefore , , The progression will be . Or, . by wchwhichyethe Equation being multiplyed produceth , Or , Or . And consequently . Or . Or . By wchwhichyethe Equation , must be multiplyed. & it produceth . Or Or . & congsequently . ytthat is . And . Or . And consequently . Or it might have beene done thus. substracted from yethe precedent progression it will be, . &c by wchwhichyethe Equation being multiplyed produceth . Or . And by extracting the roote, . And therefore Or . Or . That is . And . And therefore .Reductions of Equations may bee very often & readilyperhaps performed by this method As in that problem recited by D:Des Cartes pag 83, viz: The square & yethe right line being given , to produce to , soe ytthat drawn towards yethe point may bee equall to yethegiven line . Putting for yethe unknowne quantity. , & . The Equation will bee . wchwhich having 4 rootes the Equation must have 4 divers resolutions; that is yethe lines , , produced both ways indefinitely, there may bee 4 divers lines drawne through the point , whose parts intercepted twixt yethe crosse lines , , are equall to the given line : And they are , . And therefore the rootes of this equation are (two affirmative) , , (& two negative) , . Because , , Therefore , , , . Soe ytthat. That is one roote of this equation is to another as to . Therefore I may multiply this Equation by this progression . And there resulteth . That is . Or, . Which result is yethe same wthwithyethe first Equation the reason of wchwhich is, ytthat if I make then is . Or if , ynthen is . Or if , ynthen is . Or if then is . Soe ytthatyethe relation twixt allyetherotesrootes beingthe same & reciprocally the same & not distinguishing one roote from another, tis noe wonder if they bee all indifferently expressed in yethe resulting Equation. Otherwise yethe reduction must have succeeded.Suppose 3 rootes of an Equation are in proportion to each other as , , . Then if that roote wchwhich is correspondent to be required, multiply yethetermes of yethe Equation by this progression any of these Progressions 1. 2. 3. &c.As if 3 ofyethe rootes of this Equation .were to one another as . And would find yethe roote wchwhich is correspondent to . Then I make , , , & soe I may have have by yethe first progression this. By yethe 2d; . &c. By yethe first of wchwhichyethe Equation being multiplyed produceth . That is . Or . & . Or were it multiplyed by yethe 2d progression thus . It would produce . Or . Soe ytthat being yethe least roote, yethe other two rootes must be & .If it be desired to know the length of & when the rootes of this equation arein p, when yethe rootes of it are in proportion aas I multiply it by one of yethe precedent progressions & it is . Or . wchwhich valor of inserted into its places in yethe Equation there results . Or . Or thus If it be desired to know yethe length of & in this equation when yethe rootes are in proportion as I multiply it by yethe precedent progression & yethe results are . Or . And . Or . . Or . And consequently .Likewise were the proportion of 4 or 5 or more rogotes given I might set down progressions to find them but it will bee better to set downe yethe method of finding thosetheseprogressions, And it is this. Suppose two of yethe rootes of an Equation That Equation will bee of this forme , or of some forme of it; And if corresponds to yethedesired roote of yethe Equation desired this equation will bee of this forme . Then assuming two termes () third progression by wchwhich I multiply this equation by which multiplyed produceth Or . And termes of yethe(56)Soe ytthat I have thus much of yethe progression . And by yethe same proceding might continue it or get termes on yethe other side of yethe cipher. As if I multiply yethe Equation by this progression there is produced . Or . And . Againe multiplying yethe Equation by . It is . Or , And . Soe that I have thus much of yethe progression viz: .The proceeding is same when yethe proportion of 3 rootes to one another are given, but there may bee some difficultydifferencewnwhenyethe ciphers are far distant, as theyre bee three termes betwixt them, then yethe opreration may be done thus. Let yethe quantitys, wchwhich beare such proportion to one another as yethe rootes doe bee, , , . let correspond to yethe roote wchwhichwchwhich must be knowne And ynthenytthat Equation will bee of this forme, . or else compounded of it. Then assuming some quantity (as ) for one of yethe termes of yetheprogression & placing it conveniently, (as it no equidistant from yethe ciphers) feigne two other quantitys as , , for yethe deficient termes and there will r progression will bee . By wchwhich I multiply the Equation . Or .Soe have I yethe progression by wchwhich I againe multiply yethe Equation & there is produced correspond to yethe roote wchwhichwchwhich must be knowne And ynthenytthat Equation will bee of this forme, Or . Soe that I have the progression, by wchwhich I againe multiply yetheEquation & there results . Or . wchwhich valor of substituted into its place in yethe valor There will bee thus much of yethe progression.The same done otherwise.Did I know ytthat 2 of yethe rootes of this Equation , were in proportion as , . Then I suppose one roote to be , yethe other That is . . By one ofwchwhich I divide yethe Equation as first by And yethe operation is; . Againe I divide yethe Quotient by yetheother roote . Thus . By yethe last division I have this equation . Or . And . Therefore yethe rootes of yethe Equation are, . .If I would have & of such a length ytthatyethe rootes of this equation . be in proportion as , , . I suppose , . . And soe first divide yethe Equation by . Againe I divide this product by . Lastly were it necessary I should have again divided this quote by yethe 3d supposed roote of yethe Equation (viz ). By yethe 2doperation. Or . And by yethe first . Or Soe ytthat If I make the rootes of this Equation should bee (57)Published by p. 20NovembrNovemberyethe 13thTo find yethe velocitys of bodys by the lines they describe..R. An Equation being given, expressing yethe Relation of two or more lines , , , & described in yethe same line by two or more moveing bodys , , &c to find yethe relation of their velocitys , , &c:Resolution.Sett all yethe termes on one side of yethe Equation ytthat they become equall to nothing. And first Multiply each terme by soe many times as hath dimensions in ytthat terme. Seacondly multiply each terme by soe many times as hath dimensions in --> it. Thirdly multiply each terme by soe many times as hath dimensions in it &c. The summme of all these products shall be equall to nothing. Which Equation gives yethe relation of , , &c. Or more generally thus. Order yethe Equation acordingaccording to yethe dimensions of , & (putting & for any two numbers whither rationall or not) multiply yethe termes of it by any pteparte of this progression viz : : Also order yethe Equation according to & multiply yethetermes of it by this progression: . Also order it according to yethedimentionsdimensions of & multiply its termes by this progression viz . The summme of all these products shall bee equall to nothing. Which Equation gives yethe relation of , , &c.Example 1st. If yethe propounded Equation bee . By yetheprecedent rule yethe first operation will produce . The seacond produceth . Which two added together make . (Now suppose a yarde to bee an unit & ytthataA hath moved 3 yardes, ynthen (by yethe 1st equation) hath moved two; i,e,i.e., . And at that time by yethe last Equation . Or velocity of velocity of . Onely if increaseth ynthen decreaseth, ytthat is, & move contrary ways because & are affected wthwith divers signes).Example yethe 2d. If yethe Equation bee . The first operation will produce . Or . The second produceth . The third . The summe of wchwhich is . (Note ytthat in this Example there being three unknowne quantitys , , , There must be two of them & theirtwo velocitys supposed thereby to find yethe 3d quantity & yethe third velocity. Or else there must be some other equation expressing yethe relation of yethe th two of these , , . (as in yethe first example) whereby one quantity & one velocity being supposed yethe other quantity & velocity may be found & ynthen by this 2d Example yethe 3d quantity & yethe 3d velocity may bee found)Example 3d, Of yethe more generall rule. If yethe Equation bee . yethe first operation gives Or . the 2d gives . The summe ofwchwhich two products is equall to nothing. &c.Demonstration. Lemma. If two bodys move uniformely yethe from to in yethe same line ynthen are yethe lines & & &c as their velocitys .And though they move not uniformely yet are yethe infinitely little spac lines wchwhich each moment they describe as their velocitys are wchwhich they have while they describe them. As if yethe body wthwithyethe velocity describe yethe infinitely little line in one moment. ThenIn yethe momentyethe body wthwithyethe velocity will describe yethe line . For . Soe ytthat if yethe described lines be & in one moment, they will bee & , in yethe next. [or better . &c]Now if yethe Equation expressing yethe relation of yethe lines & be . I may substitute & into yethe place of & because (by yethe lemma) they as well as & doe signifie yethe lines described by yethe bodys & . By doeing so there results . But by supposition: there remaines therefore . On divideing it by tis . Also those termes in wchwhich is are infinitely lesse ynthen those in wchwhich is not therefore I blotting ymthem out there rests. Or .Hence may bee observed: First, ytthat those termes ever vanish in wchwhich is not because they are yethe propounded Equation. Secondly yethe remaining Equation being divided by those termes also vanish in wchwhich still remaines because they are infinitely little. Thirdly ytthatyethestill remaining termes consist of y will ever have ytthat forme wchwhich by yethefirstrootrule they should have. [partly appeare by Oughtreds Analyticall table].The rule may bee demonstrated after yethe same manner if there 3 or more unknowne quantitys , , &c.By helpe of yethe preceding probleme divers others may bee readily resolved.Of tangents to Geometricall lines. 1. To draw tangents to crooked lines (however they bee related to sreightstreight ones).Resolution Find (by yethe preceding rule) in wtwhat proportion those two lines to wchwhichyethe crooked line cheiflychiefly related doe increase or decrease: produce ymthem in ytthat proportion from yethe given point in yethe crooked line at those ends draw perpendiculars to ymthemlines in which those ends are enclosed to move through whose intersection yethe tangent shall passe.Example 1st. If . . & . Then is . (by yetheformer rule) Therefore I draw . The point is inclined to move in a parallel to & yethe point in a parallel to (for & (by supposition) moves parallel to ymthem selves yethe upon ) Therefore I draw & . & through yethe intersection I draw touching yethe crooked line at . Soe ytthat.Hence may bee pronounced those theorems in pag 47 Fol 47 Example yethe 2d. If . . (wchwhich move about yethe centers & as in yethe Hyperbola or Ellipsis by a thred) And yethe equation bee . ynthen is . or . therefore I make . (note ytthat I draw & yethe one forward yethe other backwdbackwardbecause & have contrary signes) yethe points & are inclined to move yethe one in a perpendicular to yethe other to (for yethe circle they move in circles whose centers are & ) therefore I draw & & the tangent through yethe point .Of tangents to Mechanicall lines. 2. Hitherto may bee reduced yethe manner of drawing tangents in mechanicall lines. see Fol 50.Of yethe crookednesse of Geometricall lines. 3. To find yethe quantity of crookednes in Geometricall lines. Resolution Find ytthat point of yethe perpendicular to yethe crooked linewchwhich is in least motion, let ytthat bee yethe center of a circle wchwhich passing through yethe given point shall bee of equall crookednesse wthwithyethe line at ytthat point. This point of least motion may bee found divers ways, as First. From any two points in yethe perpendicular to yethe crooked line draw 2 parallel lines in such proportion as yethe perpendicular moves over ymthem: through their ends draw another line wchwhich shall intersect yethe perpendicular in yethe point required.Example. Suppose . . . . . motion of from motion of from . & motion of from . . . As if . . yethe tangent of yethe crooked line. . & as yethemotion of from to yethe motion of from so to . Then, drawing through yethepoints & , in yethe radius of a circle as crooked as line at .Example. Suppose . ynthen is . . And . motionvelocity of from , velocity of 's increase velocity of 's increase. therefore , or (since ) tis . & Lastly (or . if ) ytthat is . . & .Hence may bee pronounced those theorems in Fol 49.(58)Addition connects affirmative qua numbers into an affirmation sume, & negative ones into a negative one. as Substractions takes yethegreaterlesse number from yethelessegreater, the difference having yethe same signe prefixed wchwhichyethe greater quantitynumberhats as . .Multiplicacation adds one factor soe often to it selfe as there are units in yethe other, & if yethesignes of yethe factors bee yethe same yethe product is affirmative, if divers tis negative. As to multiply by doe thus Or thus . Or thus . Thus to multiply by the operation willbee Alsoe multiplyed by will produceDivision takes yethe number wchwhich signifies how often yethe divisor yethedivisor, yethe sign ofwchwhich number or Quote is affirmative if yethe dividend & divisor have not divers signes, but negative if they have. For if . then , Or . Suppose to be divided by . First get a Table of yethe Divisor drawn into yethe 9 first units as . cut at yethe bottome close to the figures. Then looke wchwhich of those 9 quantitys are most like yethe dividend. As in this case yethe 7th is therefore substract it from yethe dividend , & there will remaine , & then set downe its caracteristick in yethe quote. I make a prick twixt those figures wchwhich have or might have beene altered & those wchwhich could not bee altered by the subtraction, & the places of yethe pricks will will skew the places of yethe figures in yethe quotient. Againe I substract from &c: & set in yethe quote &c.If was to be divided by (63)The resolution of yethe affected Equation . Or First having found two or 3 of yethe first figures of yethe desired roote viz (wchwhich may bee done either by rationall of Logarithmicall tryalls as Me Oughtred hath thought, or Geometrically by Geometricall descriptions of lines, or by an instrument consisting of 4 or 5 or more lines of numbers made to slide by one another wchwhich may be oblong but better circular.) this knowne pteparte of yethe root I call , yethe other unknowne pteparte I call then is . Then prosecute yetheReductionResolution after this manner (making . . &c.) . . . by supposing . Againge supposing Then . . . . That is yethe. & yethe difference twixt this & yethe former valor of is . Therefore make . Then is &c. the first figure of wchwhich being added to yethe last valor of makes . Then wthwith this valor of presecuting yethe operacotion as before tis . . wchwhich valor of substracted from yetheprecedent valor of yethediffdifferenceis . Also yethediffdifferencetwixt tis & yethe precedent valor of is . Therefore I make . That is (65)65May 30th 1665Of the construction of Problems.The resolution of plaine problems by yethe Circle. If yethe equation to be resolved bee . Or yy−ay+bb=0 in wchwhichyethe roote of yethe last terme (viz b) is knowne, they may bee resolved conveniently resolved by D. Cartes his rules. Otherwise yethe rootes of ytthat terme must bee first extracted as in this yy−py+q=0. Where I take . & soe describing yethe circle smf erect lm⊥ln & from m yethe point of intersection draw mr∥ln. yethe rootes of yethe Equation shall bee mq & mr. ln being yethe radius & n the center of the circleOrOr it may bee done thus. Let the Equation bee . Then in the indefinite line af take . erect yethe perpendicular . & wchwhichyethe Radius cd erect the perpendicular db=c. And from yethe point & towards b draw wthwithwchwhich radius describe yethe circle edf & ae, af shall bee yethe rootes of yethe Equation. When note that any quantity may be taken for c, soe ytthat if it may bee yethe valor of dc may bee noe fraction, & that db & dc bee as equall little differing as may bee. Soe ytthatyethe operaconcionmay thereby be made convenient, & to ytthat purpose the difference twixt db & dc must bee as little as may bee,(that is twixt & 4cc.) soe ytthatyethe circle intersect not (ef) over obliquely nor yethe circle be over greate. As if I had this Equation yy+6y−9=0 Or yy=−6y+9. Then must I make . Then if I make c=6 it will bee . Therefore I take . . . And soe describing yethe circle efc, I have one affirmative roote af, another negative ae. Or had I taken any other convenient valor for c as 1, or 3. or 4 the line ae & af would still have bene yethe same.Had I this equation yy−8=0. or yy=8. Then is . soe ytthat Or makeing c=2; tis: dc=3. Soe ytthat since p is wanting yethe p I take ab=0. ad=c=2 dc=3. & describing a circle yethe rootes will bee ea, af.Note ytthat if dc is negative or not greater then yethe circle cannot intersect yetheline eaf & therefore yethe rootes of yethe equation are imm̄mmaginarie. Or they may bee construed by drawing streight lines onely. thus. Let yethe Equation be . or First I makethe ab equall divide aa+b into square numbers (as forof ymthem as may bee) (It may ever bee divided (though not) into (yethe fewest) squares by taking the greatest squareout of aa+b & yethe greatest out of yethe remainder &c) as if in numbers yethe Equation were yy=2y+4 Or . I take then square 4 out of 5 & there rests 1 wchwhich is also a square. Then I draw ab . & . & make ab⊥bc. soe is . to wchwhich I add ad=1. & soe is .Were yethe Equation yy=−4y+34. Or . Then is 38−36=2. 2−1=1. & 38=36+1+1 wchwhich are square numbers. Therefore I make . . & draw . & draw . from wchwhich take ab=2, & there rests .Were yethe Equation . Or . Find & before, &c:(68)68The construction of sollid — & Linear Problems If yethe Probleme be sollid it may bee readily resolved by the intersection of yethe Parabola & circle.as D: Cartes hath shewed If it bee of 5 or 6 dimensions it may bee resolved by yethe intersection of yethe line b y3−byy−cdy+bcd+dxy=0. Or y3−byy+bcd+dxy=0 & yethecircle when pp= 4q. & q & v affirmative. as D: C: hath explained. Or it might beee done by yethe intersection of a circle & one of these lines, viz y3+byy−hx=0 when yethe equation is reduced to such a forme ytthatpp=4q. Or this y3+byy+gy−hx=0. Or this y3+gy−hx=0, s being affirmative & p=0. Or this y3+d−fyx=0 when p=0, & q & v affirmative. &c.But since all Equations But all Equations in Generall may bee resolved by yethe line a2x=y3, after this manner. First (making a=1.) describe yethe line x=y3 uppon a plate. (as cadce. Then in wchwhich ab=x. bc=y). Then suppose yethe Equation to bee resolved bee y9*+my7+ny6+py5+qy4+ry3+syy+tq+v=0. (in wchwhichyethe letters m, n, p &c: signifie yetheA Generall rule wherby any Probleme may bee resolved.termes knowne quantitys of each terme affected wthwith its signe + or −). I describe another line cdce, whose nature (making ab=x, bc=y) is the exprest & letting & let fall perpendicular frome every point where these two lines intersect as, df eg, they shall bee yethe rootes of yethe propounded eqation.In like manner was yethe Equation to bee resolved y10*+my8+my7+py6+qy5+ry4+sy3+ty2+ry+w=0the nature of yethe line cdce woluld bee . Or else it might bee . Or had I this Equation y10+ly9+my8+ny7+py6+qy5+ry34+sy3+ty2+vy+w=0. The nature of yethe line cdce would bee, . Or, . Or it might bee, . If yethe resolved Equation have fewer dimensions ytthat is if some of yetheultimate termes as, w, v, t &c: (or intermediate termes as m, n &c be blotted out: Or if yethe Equation have more ynthan 10 dimensionsthe nature of yethe lines cdce to bee described may be known by yethe same manner observing yethe order of yethe progressionTis evident alsoe ytthat there are 3 divers lines any of w in by wchwhichany Probl: may bee resolved unless some of them chanch to be yethe same, the easiest whereof is to bee chosen. It appeares also how Equations of 2 & 3 dimensions may be resolved by drawing streight lines; of 4, 5, & 6 by describing some conick section; of 7, 8, 9, by describing a line of 3 dimensions; of 10, 11, 12, by a line of 4 dimensions, &c: but yet y is never above 2 dimensions & consequently all these lines may bee described by yethe rule & compasses.Some Those Equations of more then 9 dimensions may bee (though seldome soeHad I this line y4=x. described on a plate & this Equation to bee resolved viz: y13+ly12*+ny10+py9+qy8+ry7+sy6+ty5+vy4+wy3+ayy+by+c=0. It might bee resolved by describing yethe line whose nature is . A line of yethe2d sort. Whereas by yethe preceding rule was required ytthat a line of yethe 3d sort should have By what lines a Problem may bee resolved. beene described. And here observe ytthat taking yethe square number wchwhich is next greater ynthanyethe number of yetheresolvend equation termesdimensions of the resolvend equation,. That Equation may bee resolved by lines, yethe number of whose dimensions is not greater than the roote of yethe square number. And the rectangles of those numbers wchwhich signifie yethedimensions how many dimensions the lines have, must even either equall or greater may always bee greater or equall but never lesse ynthanyethe number of dimensions of yethe resolvend Equations. For yethe number of points in wchwhich two lines may intersect can never bee In how many points two lines may intersect. greater ynthanyethe rectangle of yethe numbers of theire dimensions. And they always intersect in soe many points, excepting those wchwhich are im̄mmaginarie onely. Soe that all Equations guished into sorts.) St will often bee very intricate to resolve Equations of many dimensions by the simplest line by wchwhichthey may be resolved & also for yethe most ꝑpart will regaine a description of two lines for every probleme. And then if maybe often end to use two lines whereas compound yetheother more simple & As perhaps an Equation of 16 dimension may bee more speedily resolved by two lines of 6 dimensions then by two lines 4 dimensions.How by the Parabola plaine problems are resolved. But it will not bee amisse to shew more particularly how these resolutions may bee performed. And that firs by yethe parabolaT Suppose therefore I had yethe parabola x=yy exactly described & would resolve plaine problemethe Equation yy+ky+l=0. I take ag=l. gf=k. fh=1=lateri recto Parab: & so draw yethe line gh & from yethe intersection points d, e, draw db, ec perpendicular to yethe axis gc. wchwhich shall bee yethe rootes of yethe Equation wchwhich are affirmative when they fall on yethe contrary side to fh, but negative if onyethe same, as in this case.How sollid ones. But were I to resolve a sollid problem the Equation being of 4 dimensions, I take away yethe 2d terme, makeing it of this forme y4*+lyy+my+n=0. Then take . . Then perpendicular to ap draw af=aq. Also draw fk∥ap, & from yethe point of intersection k draw kh=n. lastly draw kr⊥ap, & wthwith the radius wr upon the center q describe yethe circle tsm. (or, wchwhich is yethe same, take . & soe erecting yetheperpendicular rw, wthwithyethe Radius rw describe yethe circle tsm) & from yethe points where it intersects yethe Parabola let fall perpendiculars to yethe axis, (tv, nm) they shall bee the rootes of yetheEquation yethe affirmative ones falling on yethe contrary side to pq. when m is affirmative.If n=0, that is if y3*+ly+m=0, then must the circle bee described wthwithyethe Radius aq; for then is wr=fa=aq. If I would resolve yethe cubick Equation y3+ky2+ly+m=0 (wchwhich multiplyed by y−k=0 produceth ) I make . . . fk∥ae⊥af=aq. kd=km. And wthwithyethe radius cg upon yethe center q describe yethe circle wf. Or else doe thus (since k is one of yethe rootes of yethe Equation ) make k=ab+ar & draw bw∥ae (or make ar=kk, & wr⊥ap) & describe a circle wthwithyethe radius wq. Then letting fall perpendiculars from yethe intersection points, they (being yethe rootes of yethe Equation ) shall all, except wr=k, bee yethe rootes of yethe Equation y3+kyy+ly+m.This operation will bee much shortened when yethe 2d terme is wanting for ytthatsince k=0. it will bee . . & aq yethe radius of yethe circle. And if yethe last terme vanish that is if I would resolve this equation yy+ky+l=0. by yethe intersection of a circle & parabola. I must take . . . & soe wthwithyethe radius aq upon yethe center q describe a circle, & yethe perpendiculars from yetheintersection points to yethe axis (a, tv) are yethe rootes excepting one wchwhich is equall to k.(69)69Constructions performed by a Parabola of yethe 2d kind. x=y3 If I had yethe crooked line described fig 1st whose nature is x=y3, & would resolve yethe Equation y3*+lyy+m=0. (calling ad=x, dg=y; Or a=-x. ce=-y) I take ab=m. bd=l. df=1. & df⊥bd & draw bf infinitely both ways. From yethe intersection points (as e) letting fall perpendiculars, they shall bee yethe rootes of yethe Equation y3*+lyy+m. as ce wchwhich in this case is negative because on that side on wchwhich y is negative.Would I resolve this equation yy+ky+l=0. (wchwhich multiplyed by y-k produceth ) I take ab=kl, (fig 2d) bδ=l, δd=kk. df=1, & soe through yethe points b & f draw the streight line bfλ(Or wchwhich is yethe same take ab=kl. k=ah⊥ab. & draw hλ∥ab untill it intersect yethe crooked line in λ (i. e. untill hλ=k3 & soe through the points λ & b draw λbfe). Then from yethe intersection points to yethe axis letting fall perpendiculars they (being yethe rootes of yethe Equation .) shall all, except βλ=k, be yetherootes of yethe Equation yy+ky+l=0. Would I resolve the Equation y4+ z4+az3+bzz+cz+d=0. It may bee done by a circle thus. Multiply it by this Equation zz−az+aa−b=0, & it will produce , Of this forme z6**+mz3+nzz+pz+q=0. In wchwhich (n) ought to be affirmative, & if it bee not, ynthen augment or diminish yethe rootes of yethe Equation z4+az3+bzz+cz+d=0. & then repeate yethe operaconcion again untill there bee an Equation of this forme z6**+mz3+nzz+pz+q=0 in wchwhich n is affirmmative. Then (dividing this equation by it is therefore) take . . & wthwithyethe radius , describe yethe circle dk & yethe perpendiculars (as dh ck) multiplyed by shall bee yethe rootes of yetheEquation.(70)70Theoremata Optica.Si radius divergens a puncto dato A vel convergens ad punctum idem A incidit in Sphæram CVD ad punctum D, et refractus ejus convergit ad puncto B vel divergit ab eodem, sitsphæræ centrum C in recta AB situm, secetAB sphæram in v tr sit sphæræ centrum C, & secet AC producta sphæram in V et radium refractum in DR in B: a punctis D et C ad AB, AD, BD demitte normales DH, CI, CR; sit sinus incidentiæ ad sinum refractionis seu Ci ad CR ut I ad R; et facto R,AC.I,AV∷CF.VF. erit F focus, seu locus imaginis puncti A radios quaquaversum emittentis.2do A puncto v versus A cape VG ad VA ut est R ad I et error radij refracti DR a loco imaginis seu in axe AV, seu distantia punctorum B et F erit AC, BC, BG, VD AC, FC, sive quamproxime. Sed regula prior in f radios est.3. Ubi punctum infinite distat ita ut radius incidens parallelus sit axi, pro AC scrito AV, et pro FG scripto (nam hæ jam sunt æquipollentia) error BF fistet, vel , vel .4. Si radius non refringitur sed reflectitur a superficie sphærica VD, eadem regula obtinet si modo ponatur S.R∷ 1.-1. et perinde capiatur VG ad contrarias partes VA fipsi VA æqualis. Erit enimadhuc error . vel AVq×CV. ACF×FG∷VH.FG.Hujus autem Theorematis inventio totis est.. . Et extractis radicibus, &c &c fluxio ipsius &c Defluxio ipsius &c in VH, fl VH &c Qusi nihil esset radij omnes accuratè refrigerentur ad focuF. Tunc enim AD et DF fluerent in data ratione, juxea quæ Cartesins in Optica probavit: A' Sed quia nihil nonest, errorfluxioobliquitatis superficiei erit VD ut illud . Et ut error ille sive defluxio a legitima figuraobliquitate ita error angular radij refracti. Jam vero est Ergo fluxioerror angularis radij refracti est ut in , fl VH. Vel etiam ut in , fl VH seu ut , in fl VH seu , fl posito fl VH=1 & . Datur autem ratio ad ergo substituto posteriore fiet error ille ut . Duc in VFq et error in axe FB erit ut quado circuli radius determinatur. Divide per radium circuli et fiet ut error BF in omni casu. Dato igitur errore illo in uno casu datur in omni. At in eo casu ubi est radius incidens axi pallrallelus datur error eodem cum quantitate ergo semper idem est cum hac quantitate.(Between f. 71 & f. 72) (MS. Add. 4004)me.Ce∷d.y. mO.CO∷e.y. . . DO.DR∷Dq.Dp d, Cq, Cp−Cpm=e, Cp, Cq-Cm. d, Ce, CO−Cm=e, CO, mC+Ced, Cp. d−e=f. .. Se CD = 0 sit CO.CR∷Cq.Cp.=Cm. Be.BCBF∷Bn.BQ.BR. Be.BF∷Bn.BRt2ft−2ffar−tfz±tfar−ttff Written upside down at the bottom of the page.(72)72Probl.Habita Lente convex plano-convexa, invenire tum convexitatem, tum refractionem vitri.Sit Lens RS, ejus superficies plana RTS, convexa RVS axis KF. Lentis superficie plana solem respiciente, observentur imaginum solantumubia radijs tumtrajectis tum reflexis congr convergentibus in charta obversa distinctissimè pictærum loci duo F et FG; F locus imaginis trajectæ G locus reflexæ: et mensurentur quam accuratissime distantiæ VF, TG, ut et crassities vitri TV. Dein fac [VF+2GT.GT∷TF2TV..KT, et erit 2KV semidiameter circuli RVS. Et sinus incidentiæ ex aere in vitrum erit ad sinum refracionis ut KT ad GT, vel ut +KF ad VF. vel] ut TVF+2TV ad VF−2GT, itasinus incidentiæ ex aere in vitrum ad sinum refractionis, itaKT ad GT et erit 2KV radius circuli RVSProbl.Habita Lente quavis convexo-convexa, invenire veetiam convexa-concava cujus concavitas convexitmajormulto minor, invenire tum refractionem vitri, tum convexitLentis.Sit Lens RS, superficies magis convexa RVS, minus convexa vel concava RTS, axis KF, vertices V ac T. Lentis hujus superficie minus convexa vel concava RTS solem directe respiciente, observentur quam accuratissimesolem imaginis in charta obversa distinctissimè pictæ tatamtrajectæ locus F quam reflexæ locus G, et mensuredistantiæ VF, TG, et crassities vitri TV. Dein alterLentis superficie RVS solem respiciente observetur quælocus imaginis reflexæ H et mensuretur distantia VHquæ est imaginis illius a vitro. Biseca TV in X. Et fac . Et . Et . Et Lplano-convexa ex consimili vitro confectæ cujus vertices siunt T, V, et convexitas versus F sita æqusummæ convexitatum RTS, RVS et versus F sita estin fig. 1 vel differentiæ convexitatis et concavitatis in fig. 2, projectissolis imaginem refractam ad locum priorem F, reflexam vero ad locum quamproximè. Unde †† juxta proxima præceus, erit VF−2gT ad VF+2TV ut sinus refrationis ad sinum incidentiæ. si fiat (juxta Problema prius) VF−2gTgT∷VF+2TV.KT erit sinus incidentiæ ex aere in vitrum ad sinum refractionis ut KT ad gT, vel ut KF ad VF. Secista ratio. Sec ad R, et erit radius circuli RVSsit ista C−−D−pH57.72IEPet poni debet .Exempli gratia. In Telescopij cujusdam vitro objectivo observabam VF=13ped.11digit.TH=. Et . Ergo Seu VF= VF=167dig.TG= VH=1,8125dig. TG=28,8125dig. TV=0,2222 &c dig.. Adeo . . . Unde prodit A= 0,0002384 dig. B=0,0348127dig. gx=28,72dig56dig.KT=34,532. dT=28,536 T VF+2TV=167,4444&c VF−2gT= 109, 109,60771. Ergo 167,444. 109,76071∷I.R. vel in minoribus numeris 29.19∷I.R aut magis accuratè ∷ I.R. =17161dig=1430ped. CirUnnde alterius RVS semidiameter erat quasi 7ped 4dig. At hæc ita se habebant in vitro objectivo Telescopij DrisDoctorisBabington.In altero Telescopio quad erat in archivis Academiæ, measuravedistantiam imaginis trajectæ a vitro objectivo VF=14ped3gis.(73)73Telescopij novi delineatioVitrum objectivum CD parallelos radios refringat versus O. Imago O per refractionem concavæ superficiei GEH transferatur ad P, et inde per reflexionem superficiei specularis ad Q, et inde per refractionem secundam superficiei GEH ad R ubi a speculo obliquo T detorquetur per vitrum oculare perexiguum V ad oculum.Sit imaginis translatio angularis ab O ad P et a P ad S tanta quanta debendiscorrigendis vitri objectivi refractionibus erroneis ab inæquali refrangibilitate ortis sufficit et erit refractangularis translatio imaginis a Q ad R tanta quanta est a P ad S, et punctum S invenietur faciendo ut sit BE.EO∷EO.ES.Sit X centrum circuli specularis JFK et Y centrum circuli refrigentis concavi GEH. Et quoniam imaginis angulares translationes PX, XQ æquales sunt, ut et PS, QR; erunt etiam translationes SX, RX æquales: adeo si fiat ES.SX∷EQR.RQRX, vel ES+SX.SX∷EX.QRX, ex dato puncto X habebitur ultimæ imaginis locus R, e cujus regione consistet oculus.Sit insuper Y centrum superficiei concavæ GEH, et quoniam est QP EP.EQ∷PX.QX, et I×PI×OE.R×OY∷EP.YP. et I×ER.R×YR∷QE.QY: inde derivabitur hæc conclusio. Hac ER.EX∷OR.P, et et habebitur circuli GEH centrus Y. Ubi nota quod usurpo pro ratione sinus incidentiæ ex aere in vitrum ad sinum refractionis. Et suppono in super vitri crassitiem EF ad instar nihili esse. ffFac . et . Et habebitur circuli GEH centrum Y. Ubi nota quod usurpo pro ratione sinus incidentiæ ex aere in vitrum ad sinum refractionis: et suppono insuper vitri crassitiem EF ad instar nihili esse.(76)76Ghetaldus in his Promotus Archimededs computes yethe wheights of yethe following equall bodys to bee in yethe proportions following.If Sphæres bee made of yethe following metalls each of whose diameters are each one foote their weights will bee as followeth. Note ytthat 1lilibra=12′. 1′=24″. 1″=24‴OrA Sphære of tinn whose diameter is six inches weighs Thirty and Eight pounds. The following line being 6 of Ghetaldus his inches, wchwhichis half yethe Roman foot by Villalpandus account from yethe Farnesian urn. Soe ytthatyethe weight of a circumscribed cilinder is 57 lib& of such a cilinder of water lb lb. And of a circumscribed cube (viz whose side is 6 inches)isliblibOr more exactlyli. Or more exactly -.that is . Or more exactly And such a cube of water lblibra or more exactly 9,80752lib.& a foot cube of water lblibra or more exactly 78,46016lblibra.The Pes Regius Gallorum, the Rhinlandick foot, the old Roman foot, & the English foot are as GaBut by yethe Farnesian Urn printed by Villalpand the French royal foot is to old foot as 12 to 11or perhaps 11 to 10. The urn conteined a longius of water weighing 10 Roman pounds of 12 Gaffendus by weighing found it contein 7 french pounds of 16 ounces. of water Eight such vessels make a Roman foot cube called a Quadrantal or Amphora Romana weighing 56 French pounds. A cubic French Foot of water by Mersennus trial weighed 74lblibra, by the vulgar estimator 72, suppose 72 & two meane proportionals between 72 & 56will be as to or 12 to 11. Suppose 74lblibra two meane will be as 11 to 10 or 12 to & this is yethe proportion of yethe french foot to yethe Roman. Royal French foot 12. . 11. . (The Roman to yethehorary foot as 8 to 9. 9 .Some Problems of Gravity & levity &cProb 1. To find yethe proportion of yethe weights of two equall bodys yethe one being sollid yethe other liquid. Resp. If yethe Sollid body A bee heavier ynthanyetheliquid B weigh it in yethe aire & in yethe liquid Body B; & yethe difference of those two weights is yethe weight of soe much water as is equall to ytthat sollid body & so much wyer as deppend in yethewater. But if it bee lighter ynthanyethe liquid body B, hang a heavier body C at to it, ytthat will sinke it; & weigh ymthem first yethe body A being in yethe air & C in yethewater; 2dlyliquid body, 2dly both A & C being in yethe liquid body; & yethe difference of theses weights is yethe weight of so much water as is equall to A (& also to yethe soe much thred wyer or hayre as was weighed both in yethe water & air.)Note ytthatyethe weight of soe much water as is equall to ytthatꝑparte of yethe wyer (to wchwhich A & C are fastened) wchwhich was weighed both in yethe air & water, bee subducted from yethe whole weight of yethe water &yethe remaining weight shall bee yethe weight of yethe water required.__________________________________________________________________________________________________________________________Or Thus, Make yethe scoale B as light as may bee, yet soe ytthat it will sinke it selfe in yethe liquid body CC, & sinkeyethe body A also, if A chance to bee lighter ynthan CC. Suppose ytthat f yethe weight of scoale B in yethe water, & g in yethe aire; & ytthat e is yetheweight of B+A in yethe water, & d in yethe aire: ynthen is d−g+f−e yethe weight of soe much water as is equall to A. Note ytthatyethe water must reach 'at both weighings to yethe same point of yethe wyer & ytthatyethe wyer bee as small as may bee.P Cor: Hence yethe proportion of all weights of all bodys sollid or liquid or both may bee gathered. & wee may hereby make hence deduce tables of yethe weights of equall bodys, & of yethe quantitys of bodys equally heavy.Prob 2. Two bodys D & E given to find yethe proportion of their quantity. Weigh ymthem in yethe scale B, let their weight & yethe weight of yethe scale in yethe air bee h & k, in yethe water, m & n ynthen is D E∷h−g+f−m:k−g+f−n. For their weight in air is h−g, & k−g; in the water m−f & n−f. &c.Prob 3. A compound body cd being given to find yethe weights & proportions, of its two compounding parts c & d. Answer. a & c, b & d, are of yethe same matter. That yethe weights of yethe 5 bodys a, b, cd, c, d, in yethe aire are ef e, f, n, m, n−m; & in yethewater g, h, q, p, q−p. Then is . & . Therefore . Or yethe weight of C in yethe aire. Also . & . & . & hfeq+−hfgq−fhen+fhgn:−gfhn+ghfn+fheq−hfeq∷c:d. (Prob 1). feq−hen+hgn−fgq:gfn−efq+ehq−ghn∷c:d. And, gfn+ehq−ehn−gfq:efq+ghn−ehn−gfq∷c+d:c.Prob: 4. A body f compounded of 3 severall sorts of matter d, e, f−d−e, begin given;:wthwithyethe proportion of yethe weight of yethe two bodys d & e as 1 to r. To find yethe weight of yethe body d. Resp. Suppose ytthatyethe bodys a & d, b & e, c & f−d−e are of yethe same matter; & & ytthatyethe weight of yethe bodys a, b, c, f, d, e, f−d−e, are in yethe aire are g, h, k, p, x, rx, p−x−rx; & in yethe water l, m, n, v, s, t, v−s−t. Then is g:l∷x:s. & h:m∷rx:t. & k:n∷p−x−rx:v−s−t. Therefore . And ghkv−ghnp=hklx+gkmrx−ghnx−ghnrx. Or .If yethe weights g=h=k, (as may bee either by experience or calculaconcion (see coroll: Prob 1) Then is . Now because gold is usually allayed by mixing wthwith it brasse & silver of each an equall weight; suppose ytthat a & d are brasse, b & e silver, c & f−d−e gold, & ytthat x=rx, or r=1. Then is yethe weight of yethebrasse or siversilver in yethe masse f, & yethe weight of yethe gold in it.(78)78Descriptio cujusdam generis curvarum tertysecundi ordinis.Concipe lineas PED datum angulum PED continentes ita moveri ut una earum EP perpetuo transeat per polum P positione datum, et altera ED datæ longitudinis existens perpetuò tangat rectam AB positione datam . Age PA constituentem angulum PAD æqualem angulo PED sit CD æqualis AP et quodvis punctum C in recta ED. datum describet curvam secundi ordinis.Age CB constituentem angulum CDBD æqualem angulo PED, et ad AD demitte normalem CF. et dictis . DC=c. CE=b. AB=x BC=y. et posito 1. E∷BC.BF, erit Et nota quod ubi angulus PED rectus est, et recta ED bisecatur in C, curva erit cissoides veterum.(8)(80)80A Method Whereby to Square lines Mechanichally.Lemma:Prop 1. Supposing ab=x⊥bc=y. If yethe valor of y (in yethe Equation expressing yethe relation twixt x & y) consist of simple termes, Multiply eachterme by x, & divide it by yethe number of yethe dimensions of x in that terme, & yethe quote shall signify yethe area acb.Example. If ax=yy. Or . ynthen is . Soe if . ynthen is . Soe if . ynthen is Soe if . ynthen is .Lemma 2dProp 2. If any terme in yethe valor of y bee a compound terReduce it to simple ones by Division or Extraction of Rootes or by Vies Method of Resolving Affected Equations, as you would doe in Decimall Numbers; & ynthen find yethe Area by Prop 2d 1st.Example. If . bee divided as in decimall fractions it produce &c. & by yethe21dst Proposition . yethe Hyperbolas Area.As if a=1=b=ab=bc. & x=0,1=be The Calculation is as followeth,The summe of these two summes is equall to yethe area dbfc, supposing ad=0,9. And their difference is equall toyethe area bche, supposing ae=1,1. & ab=1=bc∥df∥he. that isIn the manner If a=b=1=ab=bc. & x=0,2=be. The calculatio is as followethThe summe of these two summes is Equall to yetheArea dbfc, supposing ad=0.8. And their Difference is equall to yethe area bche, supposing ae=1,2. & 1=ab=bc∥df∥he. that is such respect to yethe superficies bcfd, bche yethe numbers(viz as yethe lines ad(8)(81)81Soe ytthat since 10×100=1000 &c. 10×1,1=11 &c. The Superficies answereing to these lines, 2. 3. 9. 11. & their products (of one of ymthem multiplying another) may bee found. Viz. if ab=1=bc⊥abIf yethe line ak is Then, yethe superficies bcgh is______________________________________________________________________________________________________________________________________________________Having already found yethe areas correspondent to yethe lines 1,1. 0,9. 1,2. 0,8. tis easy by yethe help of those operations to ding yethe areas correspondent to yethe lines 1,01. 1,001. 1,0001. &c: 0,99. 0,999. 0,9999. &c: 1,02. 1,002 &c: 0,98. 0,998. 0,9998. &c. And since . &c. Therefore yethe areas correspondentto yethe lines 7. 13. 17. &c: are easily found, as followeth. Viz: if x=0.02. ThenThe sume & difference of wchwhich two summes give yethe areas bcfd, bche as before. That isAnd since . & . ThereforeIf yethe line ck is The superficies bcgk is ___________________________________________________________________________________________________________________________________________________if x=0,001. The operation is as followeth.Where. & . Thereforeak is Which Area may bee otherwise thus found supposing x=db=-0,0016. Viz. As was found before excepting ytthat their differenc is e in yethe two last figures is 28. Which agreement could scarce thus happen in more ynthan 50 figures, were not yethe reas, corresponding to yethe lines 2. 3. 5. 7. 13. &c, calculated arightin so many figures.Octob. 1676.Memorandum. The letters baccdæ13eff7i3lgn4049 rr458t12vx in my second epistle to M. Leibnitz contein this sentenceData æquatione quotcun fluentes quantitates involvente, fluxiones invenire: et vice versâ.The other letters in yethe same Epistle, viz: 5accdæ10effhui4l3m 9nboqqr8s11t9v3x: 11ab3cdd10eæg10ill4m7n603p3q6r5511t8vx, 3acæ 4egh5i4l4m5n80q4r3s6t4v aaddæ5eüjmmnnooprrr5sttuv, express this sentence. Una Methodus consistit in extractione fluentis quantitatis ex æquatione simul involvente fluxionem ejus. Altera tantum in assumptionæ seriei pro quantitate qualibet incognita ex qua cætera commodè derivari posunt, et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei.(82)82The use of these differences is for composing rules to find the differences of yetheterms of a table wchwhich to be interpoled by yethe contiuall addition of those differences & also following a geo Prob. Recta aliqua AA9 in æquales quotcun partes AA2 A2A3, A3A4, A4A5, A5A6, A6A7, A7A8, A8A9 &c divisa et ad puncta divisorum erectis parallelis AB, A2B2, A3B3 &c: invenire curvam geometricam quæ per omnium erectarum extremitates B, B2 B, B2, B3 &c transibit.Erectarum AB, A2B2. A3B3 &c quære differentias primas b, b2, b3, &c secundas c, c2, c3 &c tertias d, d2, d3 &c & sic deinceps us dum veneris ad ultimam differentiam i. Tunc incipiendo ab ultima differentia excerpe medias differentias in alternais differentiarum et arithmetica media inter duas medias inte reliquorum ordinum.♀♀pergendo ad us seriem primorum terminorum A, A2, A3, &c. Sint hæc k, l, - - - s & Nempe k ultimæ diff. l medium arithmeticum inter duas penultimas, m antepe media trium antepenultimarum differentiarum &cCas 1Igitur Si numerus prignorūum. terminorum A, A2, A3, &c sit impar ultim medius servimus eorum erit ultimus terminus series ejus k, l, m, &c. Et tunc sic pergendum erit. Sit numerus primerum terminorum 9 et erit k=i, &c[sint lak, l, m, n, o, p, q, r, s, , nempe k ultimæ=i differentiæ ultimæ, l= , m=g2, , o=e3, , q=c4, s=A5; nam series terminorum A, A2, A3, hic est instar serieri differentiarum, adeo ut medius ejus terminus A5 ultimo termino seriei k, l m, &c existente vel medio termino seriei hujus A, A2, A3 si constat imparei numero terminorum ut in hoc casi, vel arithmetico mediæErige sgordinatim applicatam PQ et bisetpa AA9 in A5 die A5P=x, PQ =y pergendo us ad seriem primorum terminorum AB, A2B2, A3B3, &c. Sint hæc k, l, m, n, o, p, q, r, s &c quorum ultimum significet ultimam differentiam, penultimum medium arithmeticum inter duas penultimas differentias, antepenultimum mediam trium antepenultimarum differentarum, & sic deinceps us ad primum quod erit vel medius terminorum A, A2, A3, vel arithmeticum medium interduos medios. Prius accidit ubi numerus terminorum A, A2, A3 & est impar, posterius ubi par.Cas. 1.In casu priori sit A5B5 iste medius terminus, hoc est A5B5=k, , c4=m, , e3=0, , g2=q, i=s. Et erecta ordinatim applicata PQ die A5P=x, due terminus hujus progressionis &c in se continuò & orientur coefficientes termini &c &c. in se continuo Et orientur termini 1. x. &c per quos si termini seriei k, l, m, n, o, p &c resspectivè multiplicentur, prodibunt aggregatum factorum &c erit longitudo ordinatim applicatǽ PQCas 2.In casu posteriori sint A, A5A4B4, A5B5 duo medij termini, hoc est =k, b4=l, , d3=on, e2+e3=o, f2=p, & h=r,. Et erecta ordinatim applicata PQ, biseca A45A5 in O et dOP=x due terminos hujus progressionis &c in se continuo. Et orientur termini &c per quos si termini seriei k, l, m, n, o, p &c respectivè multiplicentur, aggregatum factorum &c erit longitudo ordinatim, applicatPQ.Sed hic notandum est j. Quod intervalla A,A2, A2A3, A3A4, &c hic supponuntur esse unitates, Et quod differentiæ colligi debent ferendo inferiores quantitates de superioribus A2B2de AB, A3B3de A2B2, b2 de b &c, faciendo AB−A2B2=b, A2B2−A3B3=b2, b−b2=c &c adeo quando differentia illa hoc modo negativæ sigma-earum ubi mutandemp sunt.(783)83For taking of unknowne quantitys out of intricat Equations it may be convenient to have severall formes. Now suppose x, was to be taken out of yethe Equations ax3+bxx+cx+d=0 & fx3+gxx+hx+k=0.I feighe yethe 3 valors of x in yethe first Equation to bee -r, -s, & -t. [ is [. . . . & that is] yethe summe of yethe rootes is ; of their squares is of their cubes is ; of their rectangles is &c] that is,] supposing a=1, every r is b every rr=bb−2c. every r3=b3−3bc+3d, rs=c, rrs=bc-3d, r3s=bbc−2cc−bd. rrss=cc−2bd. r3ss=bcc−2bbd−cd. r3s3=c3−3bcd+3dd. rst=d. rrst=bd. r3st=bbd−2cd. rrsst=cd. r3sst=bcd−3dd. rrssttr3s3t=ccd−2bdd. rrsstt=dd. r3sstt=bdd.r3s3tt=cdd. r3s3t3=d3— — — — — — — — — — — — — — — — —Or thus, every r4=b. rs4=c. rr4=bb−2c. rst4=d. rrs4=bc−3d. r34=b3−3bc+3d. rrst4=bd. rrss=cc-2bd. r3s4=bbc-2cc-bd. r44=b4−4bbc+4bd+2cc. rrsst4=cd. r3st4=bbd−2cd. rrrss4=bcc−dc−2bbd. rrsstt4=dd. r3sst4=bcd−3dd. *r3sstt4=bdd. r3s3t4=ccd−2bdd. r3s3tt4=cdd. r3s3t34=d3. *r3s34=c3−3bcd+3dd.Now supposing k (or any other quantity of yethe second Equation) to bee ian unknownequantity, it must have 3 severall valors by reason of yethe 3 valors of x in yethefirst Equation, & therefore x being taken away, h will bee of three dimensions inyethe resulting equation.The 3 valors of h are −grr+hr +fr3−grr+hr=k. & fs3−gss+hs=k & fts ft3−gtt+ht=k. Which I multiply into one another that they may produce an equation expressing yethe 3 fold valor of k: out of wchwhich equation I take out r, s, t by writing b for their summe c for yethe sum̄mme of their rectangles rs+rt+st. bc−3d for yethe summe of all theire rectangles of this forme rrs (viz: for rrs+rrt+rss+rtt+rtt+sst+stt) &c as in yethe Table. WchWhich substitution may bee most briefly done in yethe said multiplication, thus; writing a to make up six dimensions For solving this Problem generally Datis quotcun punctis Curvam describere quǽ per omnia transibit: Note these differencesProb Curvam Geometricam describere quæ per data quotcun puncta transibit. Sint ista puncta B, B2, B3, B4, B5, B6 B7 &c. Et ad rectam quamvis demitis AA7 demitte perpendicula BA, B2A2, B3A3 &c Et fac , , , , ,. . Deinde . . &c. Tunc . . &c. Tunc . . &c. Differe Sic pergendum est ad ultimam differentiam. Differentijs sic collectis & divisis per intervalla ordinati applicatorum: in altererisearum columnis sive seriem vel ordinibus excerpe medias incipiendo ab ultima et in reliquis columnis excerpedua media arithmetica inter duas medias, pergendo us ad seriem primorum terminorum AB, A2B2, &c. Sunto hæc k, l, m, n, o, p, q, r &c non quorum ultimus terminus significet ultimam diff. penultimus mediam arithmeticum inter duas penultimas, antepenultimus mediam trium antepenultimarum &c. Et primus erit medias duarum ordinatim applicata si numerus p datorum punctorum est impar, vel medium aritthmeticum inter duas medias si numerus earum est par.Cas 1In casu priori sit A4B4 ista media ordinatim applicata, hoc est A4B4=k, . c3=m, , e2=0. f+f2=p. g=q. Et erecta ordinatim applicata PQ et in basi AA5 sumpto quovis puncto O dic OP=x, et duc in segradatim terminos hujus progessionisprogressionis &c et ortam progressionem asserva. Vel quod perinde est duc terminos hujus progressionis &c in se gradatim et terminos ex ortos duc respective in terminos hujus progressionis . &c et orientur termini intermedij: tota progressione existente &c Vel dic OA=α, OA2=β, OA3=γ, OA4=δ, OA5=e, OA6=ζ, OA7=η, . . Et ex progressione collige teriminos e quibus multiplicatis per 1×. x−θ.× x−μ. x−λ&c collige alios terminos intermedios tota serie prodeants 1. x−δ. xx−δ−θx+δθ. x3−−δ−2θxx++γε+δθx−γ.Per cujus terminos multiplica terminos seriei k. l. m, n. o &c et aggregatum productorum erit k+x−δ×l+xx−δ−θ×m &c erit longitudo ordinatim applicata PQ.Cas. 2.In casu posteriori sint A4B4 et A5B5 dua media ordinatim applicata hoc est . b4=l. . d3=n. . f2=pEt k, m, , o, & coefficientes orientur en multiplicatione terminorum hujus progressiones &c et ubiquorum coefficientes en multiplicatione cujus progressionis . (784)84Of the nature of EquationsEvery Equation as x8+px7+qx6+rx5+sx4+tx3+vxx+yx+z=0. hath so many roots as dimensions, of wchwhichyethe summ is −p, the summ of the rectangles if each two +q, of each three −r, of each foure +s: &c: & of all together ♉z.Also the summe of theire squares, cubes, &c is as followeth.Also yethe summe of their square cubes &c is as followeth Or thus If their sume is −p=−a. Then is yethe sume of their squares ap−2q=b of their cubes −pb+qa−3r=c. square squares pc−qb+ra−4s=d. sq: cubes −pd+qc−rb+sa−5t =−e. cube cubes +pe+−qd+rc+−stb+ta−6v &c.Non of these rootes some are true other some false & some imaginary. &c Hoc est eritk+x−OA4+ &c=Ordinatim applicatæ . Sive dic . =ρ. . . =ν. . . Et erit k+πl+ρm+σn+τo+νp+φq χr+ψs=PQ.Probl(785)85Of Equations,Every equation hath soe many roots as dimensions of wchwhich some may be true some false & some imaginary or impossible.If there bee some imaginary then the true & false rootes may be knowne by yethe signes of yethe Equations termes: Namely there are soe many true rootes as variations of signes & soe many false ones as successions of yethe same signes. When any termes are wanting supply theire voyd places wthwith ±0.But if any because imaginary roots are properly neither true nor false roots bee imaginary, this roote soe far admitts of exception. Thus the signes of this Eq: x3−pxx+3ppx−p3=0. show it to have three true roots, wherefore if it bee multiplyed by x+2a=0 the resulting equation have threetrue x4+px3+ppxx+6p3−q3x−2pq3=0 should have thre true rootes & a false one, gbut the signes shew it to have three false & one true,. I conclude therefore that the two roots wchwhich in yethe one case appeare true, & in ayenother false are neither, but imaginary; & that of yethe other two roots one is true yethe other false.Hence it appeares ytthat to know yethe particular constitution of any Equation it is chifely necessary to understandwtwhat imaginary roots it hath. And this in some of the simplest Equations is easily discovered, thus in xx±ax+bb=0, both roots are imaginary if 4bb<aa, otherwis both reall. And thus in px3−apx−q=0 two roots are imaginary if 4p3<27qq, otherwise all reall. But to give accurate rules for determining the number of these roots in all sorts of Equations would bee a thing not onely very difficult, but uselesse:, because in Equations of many dimensions yethe rules would bee more intricate & laborious than yethe to put in practise then to solve the Equations either by lines or numbers. Soe ytthatyethe accurate determinations of those roots is for the most part esilyest acquired by solving the Equations.But yet because the discovery of these roots is very usefull I shall lay downe rules whereby they may bee many times discovered at first sight, & almost always wthwithout much labour.First then if you see any terme betwixtFirst ynthen if in any three termes together yethe two extreame termes having yethesame signes bee neither of them as little or lessegreater (tis as morelittle or lesse remote from nothing) then yethe terme betwixt them, conclude there are two imaginary roots. Thus in+aaax−b3 x3−23xx+2x−4=0 has two roots imaginary because neither 3 nor 4 are lesse then 2. And yethe like of x3−3xx−2x-4=0. And soe of x3+2x−4=0, or x3+0xx+2x−4=0, because neither 1 nor 2 is lesse then 0.Secondly if uppon sight you discover three such termes together that the two extreames having the same signes their rectangle bee as greate or greater then yethesquare of the meane terme, conclude there are two imaginary roots. Thus in x3+−pxx+3ppx−q3=0 are two imaginary roots because 1×3pp<−px−p. And soe of x3−pxx+ppx−2p3=0 because −px 1×pp=-px-p. or −px−2p3= pp×pp.Thirdly ifFirst then the reall roots of an Equation are not more then the number of its termes.First ynthenyethe number of impossible roots is always eaven. If one bee impossible there must bee two, if three there must bee foure &c. And hence Equations of odd dimensions must have one roote reall at least.Secondly the number of reall roots of any Equation are not more then the number of its termes. If Thus x4−2x+3=0 have all foure roots reall & therefore must have two imaginary. Thus x5−3x4+4=0 can have but thre reall roots & then other two beemust bee imaginary. Hence are to bee excepted equations wchwhichwantall their odd termes as x6−2x4+3xx−2=0. And sin like cases write y for xx. And so many terme roots as yeproduct y3−2yy+3y−2=0 hath times soe many reall roots, halfe true halfe false, the other shal & four times soe imaginary ones yethe other shall x6−2x4+3xx−2=0 shall have.Thirdly, if under the termes of any Equation you set a progression of fractionseach havingyethe dimension of the terme above it for its & number denominatingytthat terme first second third &c for its denominator & if any together so ytthat the rectangle of the first have multiplyed viz the fraction first square of the terme multipl by yethe fraction conclude there are two imaginary roots if one in all throughout yethe equation conclude also there are two imaginary roots at least. If equall in all cases throughout the Equation, conclude that all the roots of the equation are equall. If it be greater or equal to it in two plaIf the said factus be greater or equall to yethe said squces of yethe equation & not in all places betwixt conclude there are foure imaginary roots at least. If it bee greater or equall to it in three places of yethe equation & not in all places betwixt, conclude there are six imaginary roots at least. And soe of the rest.Thus ifThus if the Equat be x3−3xx+4x−2=0.Thenyethe progress is . & because , I conclude there are two roots imaginary at least.there are two imaginary roots.all the 3 roots are equall.conclude two rootes are imaginary.imaginary roots at least, also by the three last termes (1) therefore there are two more imaginary rootsall 4 roots are imaginary unlesse the like happen in the three middle termes. I try therefore & find & soe can conclude but two rootes imaginary. Therefore I conclude all four roots imaginary.are two imgaginary roots. Also the 3d 4th & 5t terme give therefore since by the 2d 3d & 4th terme tis I conclude ther are 4 roots imaginary. Also by yethe 4th 5t & 6t termes I find but thence nothing can be concluded because those three termes are of the same condition wthwithyethe 3d 4th & 5t termes wchwhich immediately precede them. Lastly I find by the three last termes ; And by the termes preceding them . Therefore I conclude there are two more imaginary roots; imaginaryytthat is in all 6 & but one reall.Had it beenThus in litterall Equations, if , because 1×3ppxtherefore what ever numbers are taken for p and q two roots shall bee imaginary. And soe of the rest.You may set theThis rule may be otherwise thus exprest. Over yethe termes of yethe equation set a series of fractions each having yethe dimensions of the terme under it for its numerator, & the number denominating yethe terme first, second third&c for its denominator. Then in every three termes observe whither the square of the middle terme multiplyed by the fraction above be greater equall or lesse ynthanyethe factus of the termebefore & after it multiplyed by yethe fraction over yethe terme before it. If greater write yethe signe + underneath; if equal or lesse write the signe − underneath yethe middle terme & lastly set + under yethe first terme of yetheequation. Then observe how maymany changes there are from + to − & conclude that there are soe many paires of imaginary roots.unlesse allyethe roots bee equallThus hath 2: & hath 4 many roots(786)86If you would bee more exact set downe after their signes the differences of yethe said squares & rectangles multiplyed. And then if you see three differences together wthwith the same signe soe ytthatyethe square of the meane diff bee less then yethe rectangle of the other two change the signe of the said meane differenceIf you would bee yet more exact, augment yethe roots of the Equatio the more the better, & at least soe much as to make them all true. then set yethe afforesaid differences wthwith their signes underneath as before. And under them theprogression of fractions squares. Then if you see three differences together wthwithyethe same signe soe ytthatyethe square of the middle difference multiplyed by the fraction under it bee not greater ynthanyethe rectangle of the other two differences multiplyed by yethe fraction under the first: change yethe signe of yethe middle difference.Any Equation being propounded, set down a series of so many fractions as yethe Equation hath dimensions, whose numerators are a prog & denominators are a progression of units backward & forward. Divide each fraction by ytthatpepreceding it & set the quotes in order overal yethe middle termes of the Equation. Then observe whither of every thee term such of every middle terme whither it square multiplyed by yethe fraction over it bee greater equall or lesse ynthanyethe rectangle of yethe two termes on either hand. If greater write + underneath, if equall or lesse write −. Lastly set + under yethe first & last terme & soe many changes as there are from + to & there shall bee soe many impossible roots as there are changes of signes. Unlesse it happen ytthat all yethe roots are equall, for &c:Thus if x3−63xx+46x−24=0. The series fractions will bee & dividing by , & by their quotes will be , , to be set over yethe middle termes of the equation thus . Then I observe in yethe 2dterme that is lesse then 41×6 & therefore I write − under it. so in yethe3d terme I find , therfore I write − under it. Lastly seting + under yethe first & last terme I find two changes of sines & so conclude there areThus if x5−5x4+5x3−xx− x5−24x4+x3−2xx−45x−64=0. The series of fractions will bee And dividing by & by &c there results to bee set over yethe middle termes of the equation thsthe third : therefore I write −. In the 4th(-20) therfore I write +. In yethe 5t therefore I write +. lastly under yethe first & last terme I write +. And soe finding two changes of termes I conclude two roots to bee impossible.Sometimes there may bee impossible not by this meanes discovered, wchwhich if you suspect, augment or diminish yethe roots of the Equation a little, not soe much as to make them all affirmative or all negative, or at most not much more. & try the rule againe. And if there bee any impossible roots twill rarely happen ytthat they shall not bee discovered at two or three such tryalls. Nor can there bee an Equation whose impossible roots may not bee thus discovered.Thus if x3−3ppx−3p3=0, because ifin wchwhich noe impossible appeare I put x−p x=y−p & the result is y3−3pyy−p3=0 in wchwhich two appeare, Or if I put x=y−2p the result is y3−6pyy+9ppy−5p3=0 in wchwhich also two appeare.Thus if., I set yethe signes + & − under it as before and find two imaginary roots & to try if it have any more I suppose x=y+1 & yethe results is x45+6x4+10x3+9xx+5x+0Now by this rule false roots may bee often discovered at first sight; as if you see a terme wanting twixt two others of the same signes, or if it bee lessegreater there either of those two or its square not greater then their rectangle; conclude there a paire of impossible roots, may soe many paire as these caseses happen. Thus x7+3x5−5x4+4x3set the signe − under that termeconclude conclude there is a paire of impossible roots at least & set the signe − under ytthatterme also set yethe signe + on either side the term wanting. As in this . In wchwhichit appears there are 4 if not 6 impossible rootsIf there bee two or more termes wanting set signes under them successively tothe best advantage indicating impos: rootsbegining wthwith a negative only end wthwith an affirmativeif the terms on either hand have contrary signes. As in so in The first shows 4, yethe last two roots imaginary: Soe. in x9 * x8 *wchwhich hath 8 roots imaginary(788)88To reduce severall equations of fourby divisors of three dimensions.Get yethe divisors of 6 or 7 or 8 such numbers as were described before. Add & sustract them from 29. 8. 1. 0. -1. -8. -27. Make some number in the Take any three numbers out of the three middle ranks, r, s. t. Make -s=c. . Then see if you can find 4a+2b+c in yethe rank pecedingpreceding these & 9a+3b+c in yethe rank preceding that also 4a−2b+c in the rank following these & 9a−3b+c in the rank after that; if you can try the division by x3−axx+bx−c.Or takebetter multiply all yethe numbers in yethe middle rank by 45 &910. Let s, ov signify yethe products, out of yethe two ranks on either side take any two eaven or two odd numbers out of yethe two ranks on either hand. Let those be r & t. Then observe if you can findmakeing of. observe if you can find m−any s 4m+any s 4m+2n+any s in yethe rank preceding those or 4m−2n+any s in yethe rank following those or&ynthen 9m+3n+any v in yethe 2d rank preceding those & 9m−3n+any v in yethe 2d rank following them. If so try yethe division by .Or yet better. Do not add & subduct yethe divisors from 9, 4, 1, 0, 1, 4, 9 but try if of those in yethe first & last rank yethe difference of any two eaven or two odd ones be divisible by 6. Call that difference G & yethe summ of yethe same terms H. Then try in yethemiddle collumn there be any termswchwhich subducted from H, or added to it produces a number divisible by 9. Call that term + c if it be subducted or − c if added. [Then try if in that column next before or after yethe middlemost wchwhich has leastfewest divisors there be any termwchwhich added to or subducted from produces a number divisible by 9] Then making & yethe summ or difference K, & putting , & G+−=b. Try if you can find 8−4a+2b−c onyethe2d rank next above yethe middle one or 84+4a+2b+c in yethe2d rank 2d nextra below it. If so try yethe division by x3−ax2+bx−c. Or thus against those divisors added & subducted from 207. 8. 1. 0. 1. 28. 27. set 9a+ b+c. 4a+2b+c. a+b+c. c. a−b+c. 4a−2b+c. 9a−3b+c. then choose yethe three ranks of the fewest terms & in them those numbers one in eachbby wchwhich get yethe valor of a, b, & c. & those gotten will for yethe give you numbers to be sought in the other ranks, wchwhich if you find there try the division by x3−axx−bx−c. otherwise close there other numbers out of the firstsame 3 raknks, & doe so till you have gone through all variety.Note that if yethe last term of yethe æquation divided by cbe p yethe last but one q, Then if c & must ust have a common divisor wchwhich divides not q, that c is to be rejected. Also if is greater then +×yethe grea or be greater then yethe greatest term of yethe æquation ytthatb or a is to be rejected.Note also ytthat if yethe æquation to be reduced be of six diimensions it is not necessary both to ad & substract yethe divisors fro 207. 48. 1. 0. 1. 8. -27Or thus best. LIf the Let yethe numbers wchwhich arose by substituting 2. 1. 0. -1. -2 for x be G. H. I. K. L. Seek If I end not in 5 or 0 substitute 10 & -10 for x & let yethe numbers arising be F. & M but if I end in 5 or 0 increase yethe root of& H or K do not, increase or decreaseyethe root of yethe æquation or littleby an unit. Do so also if H be an eaven number and H or K an odd one wthwith fewer divisors & then substitute 10 & -10 for x.How numeral æquations are to be reduced by divisors of 3 or 4 or more dimensionsSubstitute 5, 6, 3, 2, 1, 0, -1, -2, -3, -, -5 & also 4 & -4 if need be, for x, & find allyethe divisors ofsupposeyethe resulting terms, wchwhichset to be F, G, H, I, K, L, M, N, O, P, Q. Find all their divisors set those of F & Q together by pairs whose last figures are equal . Let yethe sumTake any one of these pairsor differ by 5. Gather the summs & differences of these pairs. Let Lettheirt summ of any two be R, the tenth parts of their difference S if their last figures, But for finding this differences you must subduct yethe divisor ofQ from yethe divisor of F not ytthat of F from ytthat of Q so ytthat S will be negative if yethe divisor of Q be yethe greater ✝✝ And if yethe æquatiōonbe not of more ynthan5 dimensions so ytthatit must be divisible (if at all) by a divisor of 2 dimensions set down xx∓Sx to be tried for such a divisor. Where R & S must have yethe same signes if s yethedivisor of F was greater then yethe divisor of K otherwise contrary signes. Quadruple yethe divisors of L. L F Take one& if the two last figures of any one be the same wthwithyethe two last figures of 2R, take it from 2R. Let yetheresidue be T divided by 100, be T. Or if yethe two last figures of any one added to yethe two last figures of 2R make 100 add it to 2R & let yethe summ divided by 100 be T & let yethe number whose quadruple is added to or subducted from 2R be a And [if yethe æequation be not of more then 5 dimensions so that it must be divisible (if at all) by a divisor of but two dimensions, set down xx+25+Sx. a to be tried for a divisor. But] if yethe æquation be of 6 or 7 dimensions & no more set down x3+Txx+S-25−+a to be tried Where note that S & T must be negative if they were found so above & a must be negative if it was added to 2R to make T, or els ffirmative if it was subducted. & the same is to be observed of the signes in yethe following operations.But if yethe æquation be of more then 7 dimensions then look in yethe columns among yethe divisors of K for a number wchwhich added to or subducted from S+T+a gives a number divisible by 24. (all This number divided by 24, call v & the divisor wchwhich gave it call βAnd set down to be tried for a divisor if the æquation be not of more then 9 dimensions. Where v must be negative if yethe number β was greater then S+T+a & subducted from it. so S+T+a in sed or if it was so above.But if yethe æquation be of more then 9 dimension then Cook in yethe column among the divisors of M for a number wchwhich added to or subducted from −S+T+a gives a number divisible by 24. This number divided by 24 call W & yethe divisor wchwhich gave it γ. And set down to be tried for a divisor if yethe æquation be not of more then 11 dimensions or supposed divisible by − Divisor of not more ynthan5 dimensions At ita in infinitum pergiturNow the trial of these divisors is this suppose yethe divisor be a+bx+cxx+dx3+ex4+fx5 &c And observe if be among yethe divisors of L, if this Divisor ascend but to twodimensions; & a+b+c++1+c+1 among yethe divisors of K if ascend but to two or 3 dimensions & a−b+c−d−1 divisors of if it ascend but to 2, 3 or 4 (789)89dimensions & a+2b+4c+d+e+1 among yethe divisors of I if it ascend not to more then 5 dimensions & a−2b+4c−d+e−f+1 among yethe divisors of N if it ascend to no more then six dimensions & a+3b+9c+27d+81e+243f+&c among yethe divisors of H if it ascend not to more then 7 dimensions, & so in infinitum In all wchwhich put c=1 & d, e, f = 0 if yethe divisor be but of 2 dimensions or d=1 & e, f, = 0 if but of 3 dimensions, & so on. equall, otherwise they differ by 5 let the difference of yethe numbers be R & their sum tenth ꝑpartpart of their Summ S. And when you have tried all yethe divisors wchwhich may be found by this rale, If& rejected those wchwhich will not hold this trial: if there remain more or if yethe æquation be not reducible divisible by any of those wchwhich remaine, you may conclude yetheæquation irreducible by any rational divisor.Veterum Loca solida componere restituta.Cartesius de hujus Problematis confectione se jactitat quasi aliquid præstitistet a Veteribus tantopere quæsitum̄mm, cujus gratia putat Apollonium tractatumlibros suos de Conicis sectionibus scripsisse. Sed cum tanti viri pace rem Veteres: neutiquam latuisse crediderius.Tradit enim Docet enim Pappus modum ducendi Ellipsim per quin data puncta et eadem est ratio in cæteris Con. sect. Et si Veteres norint ducere Conicmissectionem per quin data puncta, quis nori videt eos cognovisse compositionem loci solidi. Imo vero eorum methodus longe elegantior est Cartisiana. Ille rem peregit per calculum Algebraicum qui in verba (pro more Veterum scriptorum) resolutus Adeo prolixum et perplexum evaderet ut nauscum crearet neposset intelligi. At illi rem peregerunt per simplices quasdam Analogias, nihil judicantes lectu dignum quod aliter scribereter, & proinde celantes Analysin per quam Constructiones invenerunt. Ut pateat hanc rem eos non latuisse, conabor inventum restituere insistendo vestigijs Problematis Pappiani. In cum finem proponohæc Problemata. 1 Conicam sectionem per quintria data puncta ABC describere quæ datum centrum O habebit. AO dat A duobus punctis AB ad centrum O age rectas AO BO et eos produc AO ad AOQOP ut sit AOP=AO., et OQ BO, et puncta P et Q erunt ad curvam. A tertio puncto C age CS parallelam AO et occurrentem OB in S et cape ST ad AO, et erit etiam punctum T ad curvam. Biseca TC in V, et ipsi OV parallelam age CR occurrentem AO in R erit CR ordinatim applicata ad diametrum AP. Et latus rectumerit ad AP ut RCq ad AR×RAP,. Figura existente Ellipsi si punctum R cadit intra A et P, aliter Hyperbola.2 Per data quinque puncta A, B, C, D, E Conicam Sectionem describere ge Junge quatuor puncta A, B, C D duo puncta AC et alia duo BE sit jungentium intersectio K. Ipsis AC BE age parallelas Dg DG occurrentis AC, BE, AC in punctis H et F, et facto , puncta G et I erunt ad curvam. Biseca ergo ID et AC in m et n ut et BE ac GD in p et q et actarum mn, pq intersectio O erit centrum Conicæ sectionis.nisi ubi Habito autem centro, describe figuram per puncta A, B, C ut supra, Quæ quidem ellipsis erit si punctum R cadit inter A et P. Aliter Hyperbola Quod si mn et pq parallelæ sint Figura erit Parabola, et rectæ AOCujus determineutiquam difficilior. Quo casu produc PQ ad V ut sit BPq.GQq∷PV.QV. et erit PV diameter V vertex et latus rectum figuræ . His præmissis nihil aliud restat agendum in solutio compositio loci solidi quam ut quin puncta quæramus per quæ figuratransibit. Id quod in Problemate Veterum facillimum est. SuAP, AT, ST, AG, SG quatuor positione datæ rectæ et ad hac ducendæ sint sum in datis angulis a puncto aliquo communi C aliæ quatuorCD, CF, CB, CH ea lege ut rectangulum duarum primarum CD ×CF datam habeat rationem ad rectangulum ereliquarum CB×CH curva in qua punctum C perpetim reperitur transibit per intersectiones datarum A, G, S, T, nam ubi FC nulla est, rectangulum FC×CD nullum erit adeo et rectangulum CD×CH, et rectarum CD, CH. Si CD; punctum C incidet adT, si CH; ad S. At ita ubi CD nulla est punctum C incidet vel in A vel in G. Dantur ita quatror puncta A, G, S, T et sola restat quinta investiganda. Id quod facillimum est. Nam per punctum A agatur recta quævis AC et in ea quaratur punctum C quod Problemati satisfaciet. Et ergo Jam adeo ratio DC ad BC, et proinde etiam ratio CH ad FC siquidem ratio DC×CH ad BC×FC detur. Age ergo rectam SC ea lege ut sit CH ad FC in ratione ista data, et hæc secabit rectam AC in puncto quæsito C. Eadem lege unum era puncto invenire possunt sed uno aliquo invento habebimus C quin puncta quæ ProbConicam Sectionem jaxta præcedentia determinando Problemati satisfaciunt.Et hæc videtur methodus naturalissime solvend problematam quod non factum quod addum simplsit pars Problemat ab ipso Cartesio proponitur) est invenire punctum aliquod (datam habens conditionem et secunda parsdeinde cum infinita sint ejusmodi puncta, determinare omnia locum in quo ejusmodi punctaista omnia reperiuntur. Quid autem magis naturale quam reducere difficultates hujus posteri oris partis ad difficultates prima prioris determinando locum ex paucis punctis inventis. Proinde cum veteribus constiterit rati ducendi conicam sectionem per quin data puncta, nullus dubitaverim eos hoc medio loca Solida composuisse.(90)90ProblemaData quavis refringente superficie CD quæ radios a puncto A divergentes quomodocun refringat: invenire aliam superficiem EF quæ refractos omnes DF convergere faciet ad aliud punctum B.Junge AB. Eam secent refringentes superficies in C et E. Et posito d ad e ut sinus incidentiæ ex aere in medio ACD in medium EDC ad sinum refractionis, cape B produc AB ad G ut sit BG.CE∷d−e.e, & AD ad H ut sit AH=AG, et DF ad K ut sit DK.DH∷e.d. Junge KB et centro D radio DH describe circulum occurrentem KB productæ in L. Ipsi DL parallelam age FBF et erit F punctum superficiei EF quæ radios omnes ADF convergere faciet ad punctum B. Nam de fluxio DF est ad fluxionem AD+BF ut e ad d: adeo CE−DF.AD+BF−AC−BE ∷e.d.Haud secus Problema resolvitur ubi tres sunt vel plures refringentes superficies.(91)91De resolutione Quæstionum circa numeros.Primo numeri quæsili redigendi sunt ad æquationem secundum conditiones quæstionis, Deinde exponendi sunt per basem et ordinatam curvæ lineæ quam æquatio illa designabit. Sit curva ista DC et numeri AB, BC, curva existente tali ut numerus BC ordinatim applicatus ad basem A numerum AB in dato angulo ABC semper terminetur ad eam. Deinde inquirenda erunt puncta curvæ quæ efficient numeros AB, BC rationales. Casus autem in quibus hoc fieri deprehendo sunt sequentes.1. Si numeri in æquatione non ultra gradum qquadraticūumascendant ita ut curva sit Conica sactio: et detur aliquod punctum FF in curvæ quo efficitur ut numeri AH. HF. sint rationales ex hoc unico exemplo regula generalis sic elicietur. In AH cape HE cujusvis rationalis longitudinis, age EF, secet hæc curvam in G et demissa GH pararallelaparallela CB numeri AK, KG erunt rationales. Si Si punctum F reperitur in linea AH, FH existente nullâ, tunc cape HN cujusvis rationalis longitudinis, Erige NE parallelam KBC, & cujusvis etiam rationalis longitudinis. Age HE occurrentem curvæ in G et erunt AK KG rationales.Quoniam ita hic unicum saltem exemplum requiritur, primo inquarendum est ejusmodi exemplum, dein regula generalis inde elicietur ut supra. Exempli autem inveniendinotand esthec erit methodus.ni melior occurrat. Sit æquatio axx+bxy+cyy+dx+ey+y=0 ubi x et y designant numeros. Et si terminus f desit, puncttum istud erit A. Si x in æquatione axx+bdx+f=0 sit rationale punctam erit duo erunt in AB. Si cyy+ey+f sit rationale puncta duo erunt in recta quæ ducitur ab A parallela BC. Si bb−4ac sit quadratus numerus affirmativus tunc curva erit Hyperbola Et recta ducta a puncto A p vel B parallela Asymptoto secans curvam in G exhibebit numeros AK GK rationales. Sunt alij innumeri casus quibus enumerandis non immoror2 Si æquatio ascendit ad tres dimensiones, et tria habentur exemplo rationalia non in arithmetica progressione possunt inde innumera alia reperiri sint enim P, Q, R puncta Curva ad ista exempla Junge PR, RQ, PQ et punctum S, T, Y ubi PR, RQ, PQ secant curvam dabunt alios duastres numeros. Dein junge QS et punctum x dabit alui quo QS secat curvam dabit aluim numerum. et Sic in infinitum. Punctum B circumferentiam contingit a quo si duæ agantur rectæ una ad datum punctum A altera in dato angulo c ad rectam positione datum & ad datum punctūum terminatam DC, quadratum prioris ABq æquales sit rectangulo sub posteriore abscissa ac data DC×E Vel sic. Si ABq=FC×E punctum B contingit circ.(92)92Loca plana.1. Si datur linea AB, et angulus ACB, punctūumest in circulo per A et B transeunte.2. Si dantur puncta A, B et ratio AC ad BC punctum C est in circulo Et si C est in circulo recta BC vergit ad locum B.3 Si a dato puncto A ad rectam positione datam BD ducatur recta quævis AB et in ea atur punctum C si ea lege ut detur rectangulum BAC punctum C est in circulo, transeunte per A4 Si detur punctum A et rectangulum BAC vel BA×DCet punctum B ac D est ad circulus, etiam punctum C erit ad circulum.5 Si detur punctum A et proportio AB ad AC vel AD ad DC.et punctum B est ad circulum etiam C erit ad circulum. Sin punctus B est ad rectam erit C a rectam6 Si dentur puncta duo A, B et differentia quadratorum ACq−BCq punctum C est ad rectāam7 Si dentur puncta duo A, B et summa quadratorum ACq+ BCqet punct C erit ad Circulum 8 Si dentur puncta plura A, B, D et quadratorum ex lineis AC, DC, BC vel quadratorum quæ ad ipsa sunt in datis, rationibus summa vel quod subducendo aliqua ab alijs restat vel proportio aggregati unius ad agrgregatus alterum aliud, punctus C erit in circulo.10 Si dantur puncta B, D, E et productam BD secat quævis EC in A, et sit rectangulum EAC æquale rectangulo BAD erit punctūum C ad circulum.11. Si dantur puncta E, D et concurrant rectæ CE BD ad rectas positione datam AF extdetur rectangulorus. EAC, DAB differentia vel nulla est vel æqualis rectangulo sub AF et recta data: sit autem punctum B ad circulus erit punctus C ad circulum.12 Si detur quadrilaterum ADE quod cujus anguli oppo in circulo inscribi potest, et a puncto C ad latera ejus ducantur in datis angulis rectæ quatuor CF, CFG, CH CI ita ut rectangulus sub duabus CG×CI datam habeat rationem ad rectangulus sub alijs duabus CF×CH, et in uno aliquo casu punctus C est in circumferentia circuli transeuntis per ABDE, semper erit in circulo illoAffinia sunt 1, 2, 6, 7, 8. Item 3, 4, 5. Item 10. 1113 Si dentur puncta A, B. Et centro circulous centro A radio dato AD descriptus utcun secetur an D et E a ducta AC ac demittatur normalis CF, csit cautem rectangulum DCE=2ABFpunctum C erit in circulo.Si trapezij ABDE anguli A ac D recti sint et ad AB et AE. demittantur perpendicula CF CI a puncto quovis C secantia BD ac DE in H et G erit rectan et fuerit rectangulum FCH=GCI erit punctum C in circulo transeunte per puncta ABDE. Et viceversa. Idem eveniet si ad latera singula a puncto C demittantur perpendicula Si in circulo quovis ABCD inscribatur trapezium A B C, D, et a circumferentiæ puncto quovis E ad latera trapezij ducantur lineæ EF, EG, EH, EI constituentes cum lateribus conterminis AB, BC parallelogrammum EFBG et cum alijs duobus lateribus conterminis AD, DC parallelogrammum EHDC. quod sub ductis ad opposita duo latera continetur rectangulum GE×EH æquale est rectangulo EF ×EC sub ductis ad reliqua duo latera contento.NB Idem eveniet si puncto E ad latera trapezij demittantur pependicula. Ut et si duct ad duo latera contermina EH, EC ad AD, CD æquales angulos cum EHD, ECD vel EHA, ECD cum ipsis constituant conficiant et ductæ ad altera duo latera EF EG æquales angulos cum ipsis.Et hinc si ductæ quosvis angulos conficiant cum lateribus trapezij, et rectangula GEH, FECI sunt in data ratione facile est cognoscere utrum punctum E sit in circus ferentia circuli. Nam ad latera duo opposita AB, CD, duc EK, EL in angulis EKB ELD æqualibus illis quos ducta ad altera latera opposita faciunt cu in quibus ad altera duo latera ducta sunt lineæ æqualibus.Verbi gratiaEK in angulo AKE=ang CGE et EL in angulo DELE =ang DHE. Et se ratio rectanguli FEI ad rectangulum GEH componitur ex ratione FE ad KE et EI ad LEL ita ut rectangula KEL GEH æqualia sint, erit punctum E in circulo secus non erit in circulo. Si ABE sit circulus et detur A & rectangulum BAC erit punctum C in recta. Et si punctus C in recta sit converget recta CA ad datus punctus A. Et in fig 2 si datur rectang BAC erit C in circulo. Si punctarum A linearum AD, BD, CE poli ABC in linea recta sunt, et puncta intersectionum duo DE lineaas rectas describunt tertia intersectio F lineam rectam describet. Idem eveveniet si linea DE parallela est lineæ BC. Ut et si siAB puncta ABC non sint in directum si modo loca punctorum D, E se secant in recta BC. Si dati anguli DBCA, DCA circa polos B, C volvantur et angulis ABCD, ACD æquales capiantur CBF BCF sit punctum D in recta transeunte per punctum F vel etiam in conica sectione transeunte per puncta tria BCF erit punctum A in recta. Et si D in recta sit non transeunte per punctus F aut in con. sectione transeunte per duo E tribus punctis B, C, F, non autem per omnia tria, punctus A erit in conica sectione. Si per unicum tantum E tribus punctuis BFC transit conica sectio, punctum A erit in curva primi gradus tertij generisSi per nullum erit primi generis quarti gradus. Si circ dua E AD se secuerint in A et per A agatur recta ACDB et CD ad DB in circulo transeunte intersecti priorum.Si recta CD datasdata et secetur cujus data est ratio Si secetur punctum AD, BD, CD vel AB, BC agei B circulum (93)93 Datis positione lineis AE BE et punctis AB: Si recta quævis CD secat alteras in C, D ea lege et rectangulum AC, BD æquætur dato rectangulo AE×BE, comple parallologrammum AEBP et locus ad quem recta CP vergit erit punctum P. Si AB datur positione et longitudine & AD BC longitudine sint CE DE æquales cape AF.BF∷AD.BC et CD verget ad datum pun locum puncti F. Sei a datis punctis A, B ductæ AC., BC datam habeant summam vel differentiam N: dDuc CD parallelam AB et in ratione ad AC qualm habet N ad AB et punctum D erit in recta quæ perpendicularis est ad AB. Debet vero CD ad plagam versus A duci ubi datur summæ AC+BC, ad plagam versus B ubi datur differentia. AC−BC vel BC−AC. Si datur circulus ABD et rectangulum ACABdatampunctus C erit in circulo idem habens centrum cum circulo ABCD Si per data puncta AB transit circulus secans in E rectam ipsi AB perpendicularem et arcui BE æqualis sit arcus B EF erit F in circulo cujus centrum est A et si F in tali circulo sit, et biseecetur BF in E erit E in recta.De Loco rectilíneo.Si locus crura duo infinita opposita habet, et non plura, aut rectus est, aut tertij, quinti, septimi vel imparis alicujus generis curva linea.Si rectæ locum tangentis plaga determinat est rectus est locus.Si recta nulla ad plagam infiniti infiniti cruris tendens potest locum secare rectus est locus.Si per datum loci punctum recta transiens non potest locus alibi secare rectus est locus.Si a loci puncto quovis ad rectas duas positione datas in datis angulis demittantur aliæ duæ rectæ, et progrediendo per additionem subductionem et rationes datas, alterutra demissarum ex altera assumpta vel utra ex assumpta tertia determinari potest rectus est locus.Si in recta quavis ad datam non infiniti cruris plagam tendente determinabile est loci punctus per simplicem Geometrirectus est locus.Si rectæ per punctum datus extra locum transeuntis et loci intersectio determinabilis est per simplicem Geometriam, rectus est locus.Si rectæ cujusviscujusvis assignatæ et loci intersectio determinabilis est per simplicem Geometriam rectus est locus. 1 A datis punctis A, B ductæ conveniant AC BC in C et si dentur ipsorum A, B, summa vel differentia (loc. C solid.) proportio (loc. circ) differentia quadratorum (rect)summa quadratorum vel adliud quodvis compositum ex quadratis (circ) rectangulum (lineare) Area ABC (rect) angulus ACB (circ) differentia angulorum AB−BA(solid) vel summa 2AB+C (solidHyperb &rectdiff AB−C vel summa A2B+BAvel 2C+A (vel lineare ellipsis) 2A=B vel et. 3A+C (Hyperb.) 2A=C (Lin ) 2C=A (Lin ) A =B (rect) DB=C (circ) 2 Detur Ang AD positione et ang. DAC et punctus B. Et si dentur etiam ipsorum AC BC summa, differentia (Parab erit C in) Proportio (loc. sol.) summa quadratis differentia quadratorum (Parab) summa quadratorum vel aliud quodvis compositum ex quadratis (Loc solid) rectangulum (Lineare)(94)94Quæstionum 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 QPRD∥ADB et erit PD.DC=AD−QR∷ ∷PR.QR. Ergo Q in conica sectione est. Pone QR infinitūumet erit AD−PQR.PQQR∷PD.QPQR. seu PR =−PD. Pone PR infinitus et erit PD+PR.PR∷AD.QR ergo AD=QR. et AB Asymptotos. alteras Capere ergo PS=PD et per S parallelam AD age alteram Asymtoton & his Asymptotis per punctus P describe Hyperbolam 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.PQ∷PE.PC. PR.PE∷PQ(BC).PC∷RQ.EC.Et PQ.(BC).FC∷DC.PC Et PR×PQ Ergo PQ×PC=FC×DC =ECq−EFqPR.PQ Et Seu BQC,PR.PE,PRRQ−PR,EF∷ Si PR infinitumQ Ergo Q locatur in Conica sectione cujus diameter PR, ordinata RPPR. Sit RQ=0, erit PR=0 et In EP producta cape ergo 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 centro P radio PQ circulum & per eorum intersectionem Q age rectam PC.Corol. si ang. PEC rectus est Problema planus erit. Nam circuli centrum incidit in axem 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 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 . Pone HB infinitus ergo et erit EH.GE∷HB.−GP−2HB∷HB.−2HB Ergo . Quare biseca PE in S et per S age Asymtotos parallelas EH et HB et per punctum E vel P describe Hyperbolam secantem circulum in BD. Et per B age PC.Corol. Hinc si ang PEG semirectus erit PE axis Hyperbolaadeo Problema in eo casu planum.(95)95Quæstionum solutio Geometrica1 Datis trianguli cujuvis angulo latere et summa vel differentia reliquorum laterum datur triangulum, Detur latus AB reliquorum laterum AC+BC summa vel differentia AD. Si detur angulus datus dato lateri conterminus est, sit iste A. Et dabitur triangulum DAB. Angulorum vero CDB ADBD differentia in priori casu summa in posteriori est ang ABC.Sit angulus datus dato lateri oponitur, sit iste C datum ang DCB dabuntur ang CDB CBDest dabitur trdabuntur 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 casuEx ratione utravis & uno angulorum 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 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 comprehendentur. Si istorum summa vel diff. datur ada quadratuso summæ aufer duplum rectangulus vel ad quadratum differentiæ laterus adde quduplus rectangululumgulum et habebitur priori casu quadratus ,differentiæ posteriori quamdratus differentiæ s summæ laterum: Ex datis autem summa ac differentia laterum dantur latera. Si angulus datus basi conterminus est problema erit solidum.5 Datis angulo A latera AC vel BC et differentia segmentorum basis 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 circuloerit radio CB centro C descripto. [Et et AB.ergo D determinatur per Geometriam planam simplicicem. Sed et ubi punctum D incidit in B positio rectæ ADB determinatur. Ergo locus puncti D conica est sectio]Ut et in Chonchoide Polo Basymptoto AC intervallo AD descripta.Vel sic. Dato angulo ACDB 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 etuna cum latere alteratro vel summa differentia ratione laterum aut summa vel differentia □lolaterum aut area, perpendiculo vel segmentato basis aut summa vel differentia lateris alterutrius et perpendicula vel segmenti basis. &c Datur triang. Nam data basi et angulorum ad Basem differentia, C erit ad Hyperbolam; et ex dato tertio, punctum C erit ad recta aut circulus aut conican aliquāam sectionem.8 Ubi datar angulus verticalis et differentia segmentorum basis et tertium aliquod, habebitur aliud triangulum ADC ubi datur differentia angulorum ad basem, et tertium aliquod.9 Dat basi ratione laterum et tertio quovis ut ⊥ segmentbasis, angulo aliquo, ratione ⊥ ad lateripeh ad segmentum dDtis. Nam data lat. Datur circulos (96)9696Quæstionum solutio Geometrica. Prob 1Circulum ABE per data duo puncta A, B describere quæ rectam FG positione datam continget.Junge A,B. D biseca eam in D. Erigenormalem DF occurrentem FG in F. Produc AB donec occurrat FG inSit E punctum contactus. Produc AB donec occurrat FG in G et erit EG medium proprortionale inter datas AG,BG.Prob. 2Circulum ABE per datum punctum A describere qui recta duas FE, FH continget. Center F, radis FA describeRecta FD biseca 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. 3Circulum ABE per data duo puncta A, B describere qui alium circulum positione datum EKL continget.Puta factumSit 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−BMB. 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 pro MK×ML scribe MGq+FGq−KF erit AM×BM− AG−MG×BG−G(AM×BM)=MGq+FGq−KFqest AG×BG−FGq+KFq=AG+BG×MG. Seu 2AG.BL∷BK:MB. Undecum sit BL.BO∷MBN.BK, erit 2AG.BN∷BN.BM. Quæ solutio versalis est.Prob 4Circulum BDE per datum punctum B describere qui datum circulus & rectam lineam AD postione datam continget.AB est 2CD−AH. NF est 2CQ−NS posito CQ=CF=CS. Ergo HF est 2CD− AB−NF est 2CD−AH−2CQ+NS seu. AB−NF+DQ estAdde 2DQ, erit AB+DQ−HF=NS−AH=. Dividendoa est ita data AH in D ita ut datoæquale sit, nempo dato AB(Hb)+DQ−HF, seu bk. ADqDHq×AB AD =AB, HK, bL. DH=AD AH−AD.DHq=AHq−2DAH+ADq.AD×Kq−2AAH−2DAH+ADq−. [AB−HK(bK).AH∷AB.AV.AHq−2DAH]. Fact AB−HK.AH∷BA.AV. & AH. . ADq−DAV+D AVq=AV×PVPAD−AV PV=DV.AD AVPV−AD+AV=AD×PV =DV. Age ergo BK occurrentem AH in VHK ad HA versus A si HKversus b aliter A ad bk Nota etiam quod Problematis quatuor sunt sunt 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 ang GED agatur GD datam per A transiens posito AE quadrato, quære summam radicum Fd, FD Ad AD erige ⊥ DK age erit CK F ErgoAK summa illa, et CDq+CKq(DKq)+GDq=GKqAufer BGq seu CKq et restabit CDq+GDq=BKqDatur ergo summa AK. Quare cum ang ADK rectus; super diametro AK describe circulus secantem FE in D, dSuper datis rectis tribus AB, CD, EF, tria constituere triangula quorum vertices erunt ad idem punctum G et anguli ad vertices AGB, CGD, EGF æquales.SuperJunge AD, BC. Biseca eas in r, s. Produ AB 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 segm. ad circulos . Per intersectionem circulorum AB, CD age rectam, ut et aliam rectam per intersectionem circulorum CD, FE: nam hæ rectæ se secabunt in puncto GThe Problem in Schooten de tribus baculis may be solved more easily by supposing yethe Ellipsis to be a circle first & then reducing it to yethe desired circle.In triangulo DEFdato ABC aliud triangulum DEF dato def simile inscribere cujus latus EF transibit per datum punctum G. Nemper vierticis trianguli DEF locus 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.(97)97 Ex observationibus proprijs Cometæ anni 1680.A stella major et orientalior duarum in orientaliaustrali pede Persei, B stella minor earundem. AB stellarum distantia 1gr.46′6″. α, β, γ &c loca Cometæ(98)98Observationes Cometæ habitæ ab Academia Physicomathematica Romana anno 1680 et 1681, a Ponthæo æditæ.Observationes ejusdem cometæ habitæ a R. P Ango in FlecheNovem 28 hor. 5 matitin. in medio erat inter stellas duas exiguas quarum una est minima 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 post occasum solis duodecima Cometa erat in ≏ 23grcum lat Aust. 1gr 30′Decem 1, Erat in hor 5 matutina erat in 27. 51 ≏.Decem 2 erat in ♏ 2. 33Decem 4 erat in ♎ 12. 52Credidit M. Montenari latitudinem ad us finem harum observationem augæri.Observationes Hevelij destituti instrumentisAnno 1680 Decem. 2 Cometa erat in ≏ 25 cum lat. Austr. 5grDecem 3 Arcte ortum ☉is hora sexta erat in ♏ 4 cum lat. austr 4gr.Decem 4 mane hor 6 20′ erat in ♏ 10 cum lat austr. 3gr.pH. 8.4 I.EPObservationes Cometæ Mense Novembri anni 1680Canterburiæ per Artificem quendam nomine Hill, instrumento cujus radius erat 4 pedum Die Veneris Novemb 11 tempore matutino, Cometa inventus est in ♍ 12gr cum lat. bBoreali 2gr. Locus ☉is ♏ 29gr. 53′Romæ per Marcum Antonium Cellium observationes hæ factæ sunt.Romeæ per Galletium hæCantabrigiæ 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. Jan: 6. 5hor. 34′Ejusdem posterioris Cometæ Observationes Grenovici habitæ(99)99Observationes de Cauda Cometæ priorisNovemb 19 Cometa juxta spicam virginis existens caudam projiciebat ad us caudam Leonis, spectante juvenes quodam.Postea caudam versu per merideiem versus occidentiem projici longam satis & ad horizontem obliquam capite vel sub horizonte vel pone ædificia delitescente vidit Humf. Bab. S. T. D.De cauda Cometæ posteriorisDecemb 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 citante Flamstedio. Unde Cometa inquit Flamstedius tunc borealem habebat latitudinem & cum solenondum conjunctus fuerat. Apparebat autem cauda latæ divergens et ex corpore egressa aer prius quam ♀ Decemb 10. duabus horis post occasum Solis, bat cauda per medium distantiæ inter caudam serpentis Ophicha 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 stellamslucida secundæ magnitudinis in eductione colli Aquilæ, et estellam tertiæ quæ penultima est in cauda ejus, ac stellæ illi penultimæ [duplo quidem] propior existebat quàm alteri in eductione colli. Flamstedius in Epistolis ad nos datis. Desinebat igitur cauda in ♑ cum lat. bor. 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 mægnitudinis (Bayero α, β,) in capite seu glyphidæ Sagittæ. (Flamst. 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. (Flamstib) Linquebat igitur stellas 5tæ et 6tæ magnitudinis, δ et ζ in tribulo sagittæ, quasi 40′ ad occidentem, et ultra per 3grad circiter vel fortæ 4 extendens desinebat in ♑♒ 4 cum lat bor circiter vel 424334 43. Desinebat uti e regione superiorisduarum informium 4tæ magnitudinis quæ supra sagittam sunt non et ultra extendebat. Nam cauda ensiformis nobis visa sagittam paulo longius superare quam Flamstedio, in viam lacteam nihil extendens & termino acuto paulatim languescens. Caeter in Astrolabio Flamstedij, cauda hac nocte desinit accurat ad stellas duas exiguas prædictas in tribulo sagittæ.☿ Decemb 15 hor lucida Aquilæ erat in medio caudæ fere Pergebat vero cauda Ancon item austrinus Aquilæ erat la parte caudæ prope terminum ejusmedio caudæ fere prope terminūum ejus ad latus australe vergens. (Ipse ego & Bainbro et ellis partim ex observationepartim ex circusstansijs) Erat autem cauda 50 grad. longa (steed epist. 1) nec tutrentenuem extremitatem ejus propter Lunæ novel splendorem oliquam apparuise probabila est.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 tangebat, aut quasi; ut et lucidam in ancone austrino cygni tangebat eodem latere 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 P.M. Transibat cauda per Delphini caput dein latere suo boreali stellam penultimam in austrina ala Cygni stringebat, tendans inde versus lucidam in Cassiopeiæ cathedra et quasi 60 gradus longa existens ( 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 .Decem 17 cauda inferiùs duos gradus lata , superius non-nihil latior, ad caput Cepher extendebat. Decem 22 cauda grad longa ad Cassiopeiam us extendit: (minor tamem& minorquam intra ob D splendorem apparuit. Decem 23 cauda tenuis et per Cassiosu extendit, 72gr Conica existens circiter. Decem 28 orta fortior et clarior apparuit sed 56gr. inter Alaet lum ge in femure Andromedæ ad us Persei caput extendit? (Observator quidam Hamburguesis, qui præ cæteris caudam longam ad ultimam descripsisse videtur.♄ Decem 18 Cauda linquebat stellas Delphini ad dextram. Penultima in Ala austrina Cygni. (quæ tertiæ magnitudinis est et in Tabulis Bayeri ζ diatur) llunat per caudam quarta parte latitudinis caudaa latere australi ejus distans. erminus ejus habebut multitudinem seu distantiam ab horizonte cum stellas quibus extrema cauda cygni, Bayero dictis ♊. Deigitur♓lat bor 52gr. 20′.♂ Dec 21 In cauda stella nulla apparuit sed cauda incuvata versus ad onminò in loco qui seu pectore Cassiopeiæ et alia tertiæ magnitudinis stella in summa fere cathedra prope brachium dextrum (Bayero β dicta) triangulum æquilaterum constituit, tantum ab utra distans quantum utra ab invicem, (Flamsted. epist. subs.) adeo in long. ♓ 23gr.54′ lat 47gr 24′ desinebat. Cauda jam 70 Cadas fere longa duas Cata tendebat versus intervallum inter schedir et lucidam cathedræ( Epist. 2). In Astrolabio vero Flamstedij, Aris caudæ productus secabat ab intervallo inter caput & pectus Cassiopeiæ tertiam ejus partem versus pectus, desinebat autem e regione schedir. Transibat axis ille per caput Delphini, dimidio gradu a stellis duabusorientalibus in capite equiculum versus distans. Dein a distantia ultimarum duarum in ala austrina cygni auferebat quin nonas partes distantiæ illius versus stellam ultimam in ala. Postea a spatio inter terminum catenæ Andromadæ & stellam proximam in capite Cephei auferebat tertiam partem distantiæ illius versus terminum catenæ♀ Decem 24 Cauda transibat per medium intervallum stellarum duarum borealium in manu superiori Andromedæ et vix ultra Schedir extendebat (Flamst. Epist. post.) Desinebat igitur in long. ♉ 4 vel 5gr, lat. gr. In Astrolabio Flamstedij cauda desinebat e regione pectoris Andromadæ. Transibat autem (sed on rectæ) per medium punctum inter genu dextrum seu australe Pegasi et stellam illam informem ad pedem dextrum quartæ magnitudinis cujus Long ♒ 29 55 lat bor 36. 11. Dein per prædictas stellas duas in manu superiori Andromedæ. Dein per Juxta Astrolabium erat stellæ duæ γ, δ invictu equiculi, et Cometa triangulum rectangulum constituebant Angulus rectus erat ad stellam occidentaliorem γ. Cometa boream versus distabat ab hac stella tertia parte distantiæ stellarum.☉ Dec 26 Genu sinistrum Pegasi (quæ stella tertiæ magnitudinis est et Bayero dicitur η erat inmedio caudæ Flamst. Epist. ult.). Sed hac nocte et præcædentiebus caudæ terminus ob Lunæ splendorem haud satis definiri potuit (Flamst. epist. 2.) Unde die 24 gradus unus forte et alter ad caudæ longitudinem addi debet. Cauda vero hactenus semper curva apparuit, sed non valde curva. Convera sui parte austrum respiciebat: qua etiam parte lucidior et distinctiùs terminata apparuit quam altera.♂ Decem 28 Cauda 56grad longa distantiam inter Alamac et lucidam in femure Cassiopeiæ bisecans ultra pergebat ad us Persei Caput (Observat Hamburgens.)☿ Decem 29 Cauda tangebat Scheat sitam ad sinistam & intervallum stellarisin pede boreali Andromadæ accurate complebat (Flamsteed epist ultstella φ in femure boreali Andromade erat in medio caudæ, etDecem 30, hora Situs erat humerus Pegasi seu scheat in latere australi caudæ ita ut per caudam laceret, a termino caudæ quinta circiter vel sextaquartavel quinta sexta parte latitudinus caudæ distans. Implebat autem cauda quasi vel intervalli inter Scheat & genu sinistrum Pegasi (Bayero η) Stella φ in femure boreali Andromedæ erat in medio caudæ.Bbisecabat axis caudæ intervallum stellarum in pede Borelali Andromadæ, & cauda intervallum illud plusquam implebat. Desinebat vero in medio loco inter stellam τquintæ magnitudinis in capite Persei & extemam in borealis pede Andromadæ, sive inter stellam γ tertiæ magnitudinis in humero boreali Persei & punctum qua distantia duarum in pede boreali Andromadæ bisecantur. Un(Ego.) Unde caudæ longitudo tota erat grDeflectio caudæ ab oppositione ☉is, seu angulus quem linea jungens caput et extremitatem caudæ effecit cum Cabe linea jungente solem & cometam, 5gr. Latitudo caudæ juxta duas λ, μ in pectore Pegasi (hoc est 5gr a capite Cometæ) erat dimidium distantiahumeri illius duarum illarum stellarum una cum triente distantiæ (nempe 1gr30′ circiter). Ejusdem juxta humerum Pegasi (seu gr a capite) latitudo erat dimidium distantiæ humen illius et Pegasi genu sinistri orientalis, Pegasi adeo 2gr 30′ circitem. Ejusdem inter put Andromadæ et annulum qui est in termino catenæ, (hoc est gra capite) latitudo caudæ erat quinta par distantiæ stellarum illarum feré adeo caudæ latitudo adhuc (100)100 aliquantulum us ad extremitatem fere, ita ut tandem evaderet 5gr vel paullo major (Ego).♃ Decem 30 hor 8& hor Scheat sita erat e latere caudæ ad dextram, et australior duarum in boreali pede Andromedæ erat in medio caudæ. ad dextram(Flamsteed. Decem 1st Epist. 2)& EgoIdem et ego observabam hora 9.) Ultra vero hanc stellam australiorem cauda quasi ad 7gr vel gr extendebat circiter.(Ego)☾ Jan 3 Hor Cauda transibat per medium intervalli inter Alamac et australiorem in pede boreali Andromadæ & partes distantiæ stellarum (id est grad) ibi (hoc est 30gr a capite) lata erat. Tendebat verdus lucidam in latere Persei sed magis accuratè vedrsus stellam ι quartæ magnitudinis in dorso Persei lucidæ proximam quæ tamen sex vel decem minutis circiter distabat ab axe austrum versus. Desinebat verò quasicauda e regioneina medij lucie inter stellam illam quarta magnitudinis, et aliam ejusdem magnitudinis in humero dextro seu clypeo Persei quæ Bayero θ dicitur. Desinebat igitur in ♉ 22gr. 27′ & lat. bor. 30gr 50′. Si borealiorum duarum μν in angulo Andromadæ distantia dividatur in tres partes æquales & una pars sumatur versus mediam trium trium in angulo μ, ibi erat medium caudæ (hoc est in ♈25g. 4′ lat 30gr 52′) et ibi hoc est 18gr a capite) latitudo ejus æquabat distantiam stellarum illarum, vel paullo superabat adeo erat✝✝ 9? 2gr 6′ circiter. Ex his colligitur caudam curvam fuisse & convexo sui latere austrum respexisse concavo boream. Cauda jam haud multò lucidior erat quam partes lucidiores viæ hacteæ, si partes capiti proximas excipias, et quidem per ultimos duodecim vel quindecim gradus non erat illis u lucidior. Caput jam multo magis conspicuum erat quam cauda at Decemb 15 cauda maximè conspicua erat caput vero instar stellæ adeò exiguæ apparuit (a crepusculo; scilicet et luce lunari obscuratum) ut nudis oculis ne quidem videre possem quamvis adstantes digitum ad eam intenderent. Longitudo caudæ41gr. Distantia circul termini caudæ a circulo solem et cometas jungente 4gr 30′ ′. Delinatio caudæ ab oppositione ☉is 7gr.♂ Jan 4 Hor 9 Cauda juxta caput Cometæ tendebat versus lucidam in eductione cruris sinistri Persei, sed postea vergebat ad lucidam in latere Persei et ubi aer admodum defæcatus erat, et meo et sinetiūum judicioextendebat ad us stellam ι in dorso Persei. Axis caudæ non transibat per stellam ι, sed paucis minutis australior existens, dirigebatur accurate versus. Media rium in cingulolucidam in eductione cruris Persei Algeb. dictam. vel potius versus punctum 5′ aut 6′ australius.Andromadæ erat in mediao caudæ. Latitudo caudæ e regione capitis Andromadæ erat partes distantiæ medij caudæ a capite Andromadæ: Inter Alamac et lucidiorem etin altero pede Andromadæ æquabat seu vel potius partes distantiæ stellarum illarum: Juxta cingulum Andromadæ æquabat distantiam duarum obscuriorem in cingulo. Caudæ limes australis lucidior erat et distinctius terminata, item convexior quam limes borealis. Limes borealis ferè recta erat vel potius nonnihil concava. Caput in centro lucidius, inde ad circumferentiam paulatim languescens, apparebat per tubum duodecim pedum sine stella aliqua vel ubi globo lucido in centro, simillimum vero stellæ alicui vel planetæ per nubem densum lucente ita crassam ut stella distinctè cerni nequeat. Totius lucis in capite diameter erat 12′ vel 14′ circiter. Caput nudis oculis instar stellæ quartæ magnitudinis apparebat. (Ego) Hinc Nox hæc superiori clarior erat & Cometa longius distabat ab horizonte. Unde omnia melius definiebam. (Ego) Hinc distantia termini caudæ a circulo jungente solem et cometam 4gr 45′ Angulus quem cauda juxta caput. Cometæ efficiebat cum circulo illo gr, juxta terminum caudæ 10 vel 11gr, quem chorda caudæ efficiebat cum eodem circulo 8gr. Longitudo caudæ 42gr. Latitudo e capitis Andromadæ (hoc est 43gr a capite cometæ) 1gr 15′ circiter juxta cingulum Andromedæ (hoc est gr a capite) 2grInter lucid stellas in pedibus Andromedæ hoc est (28gr a capite) grad.☿Jan 5. Stella π in pectore Andromadæ caudam ad ad dextram Flamsteed epist. ult.♃ Jan 6 hor cauda transibat per medium prima et secundæ in cingulo Andromadæ, sed ob aeris crassiliem ultra lineāamjungentem Alamach & lucidiorem in altero pede Andromadæ cerni non pouit, quamvis aer non ita crassus esset quin stellæ quartæ magnitudinis apparerent. Caput cometæ cum tota luce sua vix æquabat stellam quartæ magnitudinis.♄ Jan 8 hor 8 Cauda, quæ ex australi latere lucidior distinctior & nonnihil convexa erat, a capite incipiens primùm tendebat versus Mirach (seu primam in cingulo Andromadæ) quæ sita erat in medio ejus nisi quod sex vel octo minutis circiter distabat ab ipso medio versus austrum: Postea flectebatur versus Alamach qua sita in ipso medio ejus. Ultra Alamach ad tres vel quatuor gradus luce languescente extendebatur: nec ultra facilè cernebatur quamvis aer adeo clarus esset ut stellæ sextæ magnitudinis apparerent. Aliquando tamen ubi aer solito clarior erat subobscura caudæ vestigia cernebantur us ad lineam jungentem stellas ι, χ in tergo et latere dextro Persei & nonnunquam us ad medium locum inter hanc lineam et stellas duas exiguas σ, ψ in cibasi Persei, & semel quidem ul paulo ultra ita ut stellarum illarum exiguarum citeriorem σ videretur attingere. Nam versus stellas illas duas σψ accuratè dtendebat. (Ego) Hinc longitudo caudæ minima erat 24gr circiter, media, maxima 35 & semel 376 vel 37′. Distantia termini caudæ a circulo solem et cometam jungente 5gr. Inclinatio caudæ ad hunc circulum juxta caput cometæ 7gr 30′ juxta extremitatem alteram 10gr40′ Inclinatio chordæ caudæ ad eundem circulum 8gr 48′ 9gr 10′. Caput cum tota sua luce stellis quartæ magnitudinis cassit eas quintæ paullo superavit. Diameter totius lucis circa caput 12′ Lux caudæ semper argentei erat coloris sed jam per totam caudam obscura valde.☉ Jan 9 hora Caudæ longitudo constans erat 15gr vel 16grextendebatur enim paullo ultra pedem sinistrum Andromedæ Alamach seu pedem australem Andromadæ. Aliquando tamen ubi aer erat solito clarior luce tenui superare visa est dimidiam distantiam inter Alamach & præfatas duas stellas σψ in cibasi Persei ad quaru citeriorem σ nocte superiori semel extendebat, ita ut longitudo ejus tunc esset 24gr circiter (Ego) Caudæad latus boreale tetigit Mirach, desis verò ad υτ in femore genu Andromadæ. Flamsteed Epist ult.☾ Jan 10 hora 6, 8, 10, cauda desinebat ad Alamach,.aAliquando tamen ubi aer erat solito clarior, luce suboscura se extendebat ad stellam χ in australi lateri Persei, vel potius ad puctum duodecim vel quindecim minutis borealiorem quàm stella illa. Seribit Flamstedius caudam hac nocte desysse sub Alamech, directam vero fuisse versus stellam illam illam χ in latere Persei, id est si recta producas; at ob curvaturas cauda ubi eo us visibilis extitit deflectebat a χ ad punctum 12′ vel 15′ borealiorem.♂ Jan 11 hora 8, 9, 10 cauda satis distincta erat ad us Alamech, et paulo ultra, subobscura ad us stellam præfactam exignam χ in latere Persei, per quam axis ubi terminabatur axe caudæ per stellam transeunte. Distantia termini caudæ a ♃ Jan linea sole circulo solem et Cometas jungente 3 erat igitur 3gr. 50′. Inclinatio chodæ caudæ ad circulum illum gr. At distantia illa et inclinatio paullo majores extitissent si modo cauda æque longe in sig Persei visibiliter extendisset ac aute Caput jam cum tota sua luce stellas quinta magnitudinis æquabat.♃ Jan 13 Cauda luce perobscura desinebat e regione stellæ præfatæ χ in latere Persei, luce satis sensibili inter Alamach t Algol. terminabatur. Jan 23 & 24cometam rursusbeneficio sensibili impetus vidi sed Cauda ejus ob Lineasplendorem netiquam apparuit caput ejus inter nubecula cunda apparuit reliquo cao haud lucidioris ut sentiri ægre rit. Jan 25 Luna sub horizonte cauda cometæ denuò sensibilis potuit ad longitudinem gradusvel septem. sequente ad longitudinem graduūum(101)10112 aut paullo ultra sed luce obscurissima et agerrimà sensibili: Tali uti luce extendebatur ad lineam jungentem Algol & Pleiadas. Dirigebatur vero axis ejus ad capillamlucidam in humero orientali Auriga accuratà. Unda deviatio caudæ ab oppositione solis boream versus 10gr.Caput Cometæ cum ommi sua luce stellam septimæ magnitudinis æquare videbatur, aut quillonon superare. Jan 30 Caudæ non nisi vestigia quædam obscurissima restabant quæ tamen tam is oculis quam armatis sentin potueteextrudebantur hæcce caudæ vestigia magis luce para magis sensibili ad lineam jungentem Algol et stellas informes in nube arietis, luce minus sensibili ad us lineam jungentem Algol & Pleiadas Quinimò nonnunquam sentire visus sum vestigia quædam lucis rarissimæ ad us lineam jungentem Algol et stellam 3 tertia magnitudinis in australi pede Persei. PTendebat vero axis reliquiarum caudæ vdus punctum inter lapellam at lucidam in humero orientali Aurigrai punctum paullinter genu lucidum Persei & lucidam in humero orientali Aurigæ, quam proximènempe versus punctum trientegradus australius quam lucida illa in humero circiter, adeo ab oppositione solis deflexit 10gr 4circiter. Caput Cometæ cum omni sua luce stellis septima magnitudinis cessit. Ex hoc tempore caudam nudis oculis observare destiti. Telescopio vero septupedali caudam vida us ad Feb 10quo tempore duos circiter gradus longa videbatur, & versus puncgrada uno et altero australius quam lucida in humero orientali Aurigaæ dirigi, magis et magis ab oppositione solis deflectens. Posthaæ cometam a Feb. 25 ad Mart. 9 demò vidi sed sine cauda. Nam et caput ipsum jam adeo tenue evaserat ut ope Tubi septupedalis cum apertura duarum unciarum cerni vix posset.Interim ubi me Cometam nudis oculis observasse affirmo nolim credas Myopem vitro concavo cænisse quo visio redderet distincta. Tali vitro, sed optimo, semper usus sum.Cæterum cauda quoad directionem, has observabat leges. Ad singulas observationes in globo per caput cometæ et extremitat caudæ in globo due circulos maximos se secantis in A B C D E &c Divide AB, segmenta AB, BC, CD &c in duplicate ratione tempointer observationes utrobi factas intercedentium. Per puncta divisionusduc un superficie globi lineam uniformem quæ segmenta illa AB,BC, CD &c in punctis divisionum contengat, et in omni casu circulus per caput co& extremitatem caudæ ductus tanget linem illam, uniformem quamproximè. Unde cauda, dato tempore, quoad positionem duci potest. vero quod segmenta AB, BC, CD, DCE &c divisa per sumtemporum duorum observationes utrobi factas intercedentium (ABsummam temporis preimi et sedi, BC per summam temporis secundamtertij &c seu AB per tempus inter observationem primam ac tertiam BC per tempus inter secundam at quartam CD per tempus tertiam et quintam &c) debent esse in progressione seu geometrica seu arithmetica alia aut alia aliqua quavis regulan. Et hinc collatis inter se observationibus cognosci potest an situs caudæ fuerit rectaè observatus.Si inter capellam et polum eclipticæ sumatur punctum tribus gradibus distans a capella, cauda Cometæ, a Jan Decem 15 ad Jan 8 versus punctum illud satis accuratè, dirigebatur pratim circa JDec 18, 25, Jan 4.Si in globo ducatur circulus maximus qui sel ecliplipticam 10gr 20gr in angulo 54gr transiens per stellam αala septentrionali sagillæ, dem per stellam θ quarttæ magnitudinis orientali brachio cassiopeiæ, deni per stellam in tergo Persei aut per punctum gradus circiter australius: Cauda Comedtæ ab imdad us Jan : 4, imd ad Jan 8 salis accuratè terminabatur. Excipe tantum a Dec 15 ad Jan Dec 26 ubi lux tenuior in extremitate caudæ ob Lan splendorem videri suon potuit. Si tamcaudæ longitudines Flam stedianæ juxta observationes Hamburg ensnonnibil augeantur. upon circullo b altaeri queproximè proxime. A longidttudine cauda aufer dimidiam latitudinem, et habebitur longitudo correcta. circuluscirculus termino hujus longitudinis descriptus secat eclipticam in grcirciter in angulo 54gr circiter. Via cometæ secat eclipticam in 21 in angulo 30gr circiter. Ubi hæc via præfatum circulus correctum secat, hoc est in 20gr4′ & latitudine boreali 40′ circiter, ibi erit punctum per quod planum, in quo cometæ movit, transire debet. Secuit igitur planum illud eclipticam in 20 vel 20 circiter.Cauda Feb Decem 10, 11, 12, &c angustior apparuit, Decem 15 paullo latior, Decem 29 & 30 ultòlatior, ut et Jan 3 & 4. Unsepropiorremotior a nobis fuit extremitas caudæ Decem 10, 11, 12 quam Decem 29, 30, & Jan 3, 4, adeo cæteris paribus minforem parallaxim habuit imo imò capila. remotior quam caput, a capite Cometæcauda in regiones ulti & nobis app aversas pergense.Halleius mihi narravit se iter Parisias instituentem Dec 8 stylo veteri tep caudam cometæ vidisse perpendiculariter ex horizonte orisurgentem ad instar trabis ignece ad longitudinem decem vel maiore quimdecim graduum paulo ante ortum solis. Quod cauda hæc nonexcorpore solis non prius disperetet quam sol oriens inciperet supra horizontem conspici: ad solis autem fulgorem mox evanesceret. Et quod Cauda e corpore solis exire videretur, ita ut caput cometæ esset soli proximum. Deni quod ipse quid esset ipse quid esset hoc Phænomenon nesciret pquam donec Cometa e radijs solis egressus se omnibus conspicuum exhiberet.MonsrMonsieurRicher sent by yethe French King to make observations in yethe Island of Cayenna (north Lat 5gr) having before he went thither set his clock exactly at Paris, found then (viz at Cayenna) that it went too slow so as every day to loose two minutes & an half for many days together & after his clock had stood & went again it lost 2 minutes every day as before. Whence MrMister Halley concluded that yethe Pendulum was to be shortned in yethe proportion of to to make yethe clock go true at Cayenna. In Goree yetheObservation was less exact. They there found(102)102Decem 12 Caput per Telescopium Flamstedio apparuit Iovè minus nec rotundum quidem sed inæquale ad instar quadorati cujus anguli fortuitò & irregulariter diffracti fuissent. Lumen capitis jam fuscum admodum & lumine saturnio multis gradibus deterius.Decem 21 Caput per Telescopium apparuit ut nubeslocus nubilosus in cælo nudis ofculis apparere solet: excepto quod per faceiem ejus puncta quædam lucida sed exigna valdè irregulariter spargebantur. Capitis diameter erat plusquam minuti unius sed non bene terminata nec lucida sed nebulosaDecem 26 Caput nudis oculis minus apparuit quam Os Pegasi & pallidius, sed per Telescopium ut ante, nisi quod puncta lucida mindistincta erant. Exinde caput minus & tenuius perpetuò evasit. Hæc Flamstedius epist. ult.(103)103103Ex Hookij Cometa edito ann 1678.Apr 21 1677 Cometa ab Hookio vesus est inter basem trianguli et stellas informes in nube Aristis, in recta linea jungentemCor cCassiopeiæ & Alamak. Distubat ab Alamak austr versus distantiæ cinguli & pedum Andromadæ. Cauda æquabat distantia ejus ab Alamak, & dirigebatur accurate verus stellam in nasu cCassiopeiæquartæ magnitudinis. Unde caput dirigebatur non versus solem qui erat in ♉ 11gr sed versus ♉ 14gr. Caput æquabat stellam primæmagnitudinis & lumine magis fisco. Stella in medio capitis æque(per Telescopium qumidecim pedum conspecta) æque lucida apparebatac ♄ubi prope horizontem versatur. Rotunda erat, sed non distindefinita. Diameter ejus erat 25″. Comæ verò lotius caput ampbientis latitudo seu diameter 4′ 10″. id est decuplo major quam diameter capitis. Angustior erat coma et melius terminata solem versus.Apr 23 Cometa erat in medio puncto inter Algol et lidem informium in nube Aristis, nempe in ♉ 14 lat. bor. 17gr. Unde orientem versus movebater sed in linea nonnihil ad austrum deflectente. Caudrecta erat et versus stellam tertiæ magnitudinis in femore Cassiopeiæ dirigebatur quasi 7 vel 8 gr longa existens. Caput it versus ♉ 17 dirigebatur, sole tamen existente in ♉ 13 .Capitis lumen densum erat et compactum & saturno feræquale, caput tamen limbo æquabili ut saturnus non definitum. Et capitis partes aliquæ lucidiores erant aliæ mius lucidæ. Hæ non prorsus permanentes sed notabiliter mutabiles sese ostentabantis.Imo Hevelius in schemate quidem viam cometæ infra rostroscorvi desibit at in observationibus non item. Dicit erum, Decemb , 5h mat. cometam properostrum corvi a se detectum esse a rostro illo favonium versus vix gradu distantur. Item suo calculo facitVidit calculum HeveliusVidit item Hevelius Cometam Dec per unam tampunctorum Cometam. Et ocul Leporis & seCometam paullcisante posteriorem observationem supra oculus Leporis ad distantiam circiter transitisse, unde infra humerum Leporis transivit & fere tegit. Hevelius præterea cometam parsim omnia quamdiu cometa magnam habuit latitudinem australēemaustraliorem pointquam Auroutius. Fortè quod Auroutius refractiones neglexit vel prooribus habuit.Cometæ anni 1664 observationes optimæ in lucem edita sunt Hevelij, Ægidij Francisci de Gottignies in urbe Roma Professoris & MPetiti Pariciensis qui Observationes Aurantij edidit. Hevelius tamen Gottignies viam Cometæ illius infram stellam in rostro corvi descriPetitus autem (quocum consentit Hugenius in observationibus quibusab Hookio visis) viam ejus supra stellam illam seu ad boream status Hevelius præterea ubi fere australiorem facit viam cometæ quam Petitus et Gottignies, & verbi gratia, cùm illi viam supram stultertiæ magnitudinis in humero dextro Leporis describunt, nie pomit infra. Gottignies in prima sua tabula statuit Cometam in ♊ 4 lat austr 33 in secunda in ♊ 4 lat austr 34 utrumeodem tempore nempe decem anno 1664.Comam Cometæ anni 1677 juxta nostrum seu stellam in capite, lucid e latere nuclei quod soli oppocebiturlucidior errca reliqos partes nuclei: quæ quidem partlucidior llium euda Constituebunt nes capitis esil apparuit in cuada ne regio soli opposita obseumorsuit quam regio soli obversa ut opporteret si caput cometæ corpus opacum esset & lucis expers. Nucleus vero cum coma comeanni 1664 collatus minorem rationem ad comam obtinebat sub finem ubi Cometa longies a Sole recesserat.Via cometæ anni 1664 & 1665 juxta delineationem Hookij.Nota. distantia Cometæ stella B Feb 7 juxta Hookium fuit pars distantiæ primæ et Sedæ Arietis id, est 5′ 12″ seu . Situs autem erat cometa in medio inter stellas B et C.Longitudines & latitudines subse quentium stellarum, ex catalogo Tychomico, ad annum completum 1664 suppost collectæ Distat inquit Auroutius stella A a secundæ ♈tis dextram versus 45′ vel 46′ a prima vero 1gr20′ Angulo recto existente qui a lunes ad stellam illiam a prima et secunda ♈tis ductis contintetur. Ait et Hevelius setellam distare 46′ a secunda ♈tis & 1gr 15 vel 20′ a prima. A prima ♈tis niquit Auroutius Cometa Feb ‡‡ Hookius facit distntiam duabus minutis tanto spatio distitit quanto ab eadem stellula A removetur hocest 1gr 20′. Unde concludit Auroutius Cometæ Longitudinem trinc fuisse 27gr circ. & Lat. boreal. 7gr 4′ vel 5′.Feb ait Auroutius Cometa 12 vel 13 movebatur a priori loco et 9 pringles propior factus est prima ♈tis. Feb & aut circiter cometa a primæ♈tis in minima fuit distantia, quæ distantia erat ad summum 50′, inquit idem Auroutius, Porro cometa in Mart 7stylo novo, aidt Auroutius, cometa non ultra 7′ vel 8′ uno die movebatur.Juxta Observationem R. P Gottignies,& Vll Aureoutij Cometa Matrtjam modo prætergressus fuerat Cornu sinistrum ♈tis quasi spatio quivel quintæ partis itineris uno die confecti id est 1′ 30″ vel 2′circiter: quocum satis consentiunt Hookius et Auroutius. Ad Distantia primæ et sedæ ♈tis, quæ est 1gr 33′, Hookius in delineatione quiponit distantiam cometæ a seda ♈tis eab tem Gottignies in delineationesuapinil distantiam cometæ a seda♈is esse ut 4 ad 4, Hook ut 4 ad 45, Petitus ut 2 ad 17 sed Petitus in delineationibus suis hand satis assecutus est mentes Aurouutij, facil cometam propius ueiessisse ad stellam A quam ad primam ♈tis contra qui facit Hookios & Gottignies. Sit ergo distantia illa distantiæ primæ et secunda ♈tis hoc est 8′ 16″ circiter & cometæ longitudos ea temporibusset Mart 1 Rora 8′ esp erqit 1′ circiter major quam longitudo primasecunda ♈tis adeo in♈, 29gr 18′ 30″ Latitudo verò 18′ 15″ mquam latitudo ejusdem stellæ adeo 37′ 15″.(104)104Cometæ anni 1661 loca ex HevelioMaxima Cometæ latitudo Australis 49gr 33′ vel 49gr et locus maximæ latitudini in ♈♋ 27 gr. Ut ex circulo maximo per loca duo cometæ transeunte ex pius cum ex pluribus observatiionibus colligitur. Securit autem Cometa eclipticam in ♋♈ 28gr 58′ id Jan 15 hora 9 P. M circiter in angulo. CometæDecemb 28 pro ratione calurarum observationem latitudinem 23′ vel 4′ justominorem habere videtur. Cætera quoad latitudinem inter se bene consentiment. Et hinc latitudo maxima cometæ forte 49gr 35′ vel 36′gr melius statuitur, quam 49 33′.Decem Cometa detecta prope rastrus Corvi Favonius versus vix gr ab eo distans.Decem hor 2 vel 3 mat paucis cometa transibat 20′ supra oculum Leporis. hor 8 vesp cometa infra stellas in eridano (dictam tertiam a primo flera) a semigradum ferè liberotum versus incedebat, sic ut hora 10 commain suam decurtaseper stellam istam in ea tunc clare emicantem projiceret. hora 4 vesp. Cometa jam modo mandibulam fere occultarat, non tamen occultabat ommino ub aliqui volunt, nam hora 9 Mandibula in ipsa cauda apparuit. Hora 9 Cometa inter Mandibula et stellam in ore Cati in linea fere recta apparuit. hor 7 vel 8 vesp circiter Cometa cus Mandibula et illa in core triangulus æquilaterum ferè constituit Et postea Cometam inter mandibulum et Caput Andromedæ existente, Cæpit Hevelius distantiam ab utra stella ut cognosceret an summa distantiarum æquabat distantiam stellarum Inde motum Cometæ quoad progressum in orbita sua exhinc accuratèdeterminavit, præsentim sub initio mensis February. Cometa distabat una, sui diametro ab inferiore duarum stillularum , quæ eaudem quo distantiam ferè habebunt ab vivicè nempe 2′ circiter vel 3′ d. Binne illa stellæ limbo orientali comet ad hærebant. Inferior et lucidior binarum vix tolade limbo exiet superior maxima sui parte suo tenet maxima adeita latebat ut ea proptus etisem minors et obser videretur. Inferior igitur lem hum a capite cometico plane tectafuerat superior lem hum strinscerat. hor. 7 vesp Cometæ limbus a hinis illis stellæ una cometæ diametro distabat JadeHave Hevelius. Unde colligit cometæm vix ultra 6 jam per diem movisse. Movebatur autem, inquit, sursum ita ut longitudo ejus vix quicquam mutaretur.Dixi latitudinem Cometæ Decemb 18 insto minorem esse per 3′vel 7′, imò non consentit cum cæteris observationibus nisi 8′minusaugeatur. Pro 49gr24′ lat. Aust. scribe igitur 49gr32′ lat austr. et maximam Cometa latitudinem 49gr 40′ in ♋ 27gr45′. (105)105 A, B, C, D, E, F, G are yethe stars in yethe greater evain. L, m, n, s, t stars in yethe Bears bright hinder leg. H, J, K stars in his head & neck. Anno 1682 SeptOn Satturday at 1h 20′ after midnight I saw yethe comet in V in a right line wthwithyethestars F & s, distant from yethe star s twice as far as that star was from yethe star t. The tayle pointed directly towards the star K in yethe eye or cheek, & was about six degrees long reaching of yetheway to that star.Sunday at 9h 20′ before midnight the comet was in X. Xs & sn were equal & a little greater then Xn. ms, mX & 1Xs were equal. Xs was equal to 3st The taile ended over against the middle of st & produced cut of or of qr towards q.Munday at 8h 40′ at night vYo were in a right line Yo=1st. The taile ended over against mn or alittle beyond those stars suppose about a degree beyond. & pointed towards a little star p not noted (I think) in yethe globe.Tuesday at 9h. 0′ The comet was in Z. oZ was a little greater then DE almost as great as CD. The comet passed about 8′ or 10′ above yethe star o wchwhich is a little scarce noted in yethe globes. The tayle was crooked, the convex side southward was sensibly brighter, then yethe concave side. The head in this & yethe former observations scarce so luminous as a star of yethe first magnitude but more luminous then one of yethe 2d. The taile went exactly in yethe middle between the stars m & L or a very little nearer to m & pointed almost at yethe Pole star, vizt as much below it as yethemiddle star in yethe little bBeares taile was above it & reached up within a degree or two to over against it or very nearely. The tayle produced would have wiped the star A wthwith its north concave side.(106)106Problemamm solutiones juxta sequentes RegulasReg. 1. Circumspicera quid ex datis consequatur ut ex pluribus datis facilius assequamur quod propositum est. Item crcircumspicere quomodo schemata constmantur ut en datis aliquid colligamus. In hune cognoscendæ sunt proportionalium legès et transmutationes, eo quod Geometria, proportionales ob simplicitatem magis quam per æquationes amat progredi. Cognoscendæ sunt etiam Figurarum proprietates quæ in elementis sunt & determinationes simpliciores: Et quando triangula vel quadrangula dantur specie, quando specie et magnitudine Determinatæ item sectiones veterem quæ sunt æquationes recentiorus in promptu essae debent. Ut et Locorum determinationes. Nam Geometria tota nihil aloud est quam inventio punctorum per intersectiones Locorum. Sectio determinatæ dici potest simplex duplex triplex &c proudin uno, duobus, tribus punctis &c fit, vel ut recentis loqucantur prout æquatio unius duarum trius dimensionum est.Si secanda sit recta data AB in x ita sit ut Ae.Ax &c ∷Bx.De. vel rectangulum AxB sequetur dato rectangulo AeD: sit angulus DBAD rectus. Biseca BD in C Radio C. Centro C radio Ce describe circulo secantem AB in x. At hoc modo construi potest omnis æquatio quadratica. Sed rem longius prosecutus est Apollonius.Igitur is in recta aliqua A dantur trio puncta A, B, F et secanda sit recta in x ita ut sit Ae.Ax∷Bx. Fx componendo vel dividendo erit Ae.Fex∷Bx.BF unde solvetur Problema uti prius.Si din recta dentur quatror puncta A, B, F, G et seconda sit recta in x ita ut rectangulūum AxB, sit ad rectangulus FxG in data ratione AH ad HG, erige perpendiculum HQ quod sit medium proportionale inter AH et HG. Iunge AQ, GQ. Super BF constitue trianIn angulo Aeg AFP æquali AGQ et GBR æquali GAQ age rectas FP, BR occurrentes AQ et GQ in P et Rsuper BF constitue triangulum BSF simile triangulo, AQG et ad easdem partes rectæ AG si punctum x quæritur vel inter A et B vel inter F et QG, Aliter ad partes contrarias. Super diametro PR describe circulum secantem rectam AG in xAddo si recta QG secanda sit in x ita ut rectangulus AxB sit ad cessum quodiferentiam inter rectangulam FxG separat& rectangulum datum mXn: seca rectam illum in T et V ita ut rectangulaTG FQGsecundaæquater dato rectangulo M Dein rursus seca in x ita ut rectangulum AxB sit ad rctectangulum TxV in ratio ratione illa data.Si recta secanda est in X ita ut rectangulum AxB sit ad summam rectanguli FxG et rectanguli dati mXn. Erit dividendo AxB÷mXn ad FxG in ratione data et inverse FxG ad AxB÷mXn in radtione data. ergQui casus est superioris propositionis.Ad hoc casus faciele est cæteros reducere.(109)109"Signo verò ± additionem et subductionem abiguaè denoto.(128)128Geometria.Lib. 1.I Gradus Problematum: Problemata pro numero solutionum quas admittunt distingunntur in gradus. Quæ unitc tantum admittunt solutionem sunt primi gradus, quæ secundi, quæ tres tertij, & sic deincepedeincepo in reliquis. Ut si recta data recta AB producenda est ad D ita ut punctum D daohabentintervallodistantiamdistet a puncto aliquæ C quod in sublimi datur: Solvetur problema si centro C et intervallo isto dato describatur circulus datam illam rectam secans. Et ista solutione prodibit ostendit Problema secundi gradus esse.Et duplici intersectione in D et dprodibit duplex responejus solutio, Producendblematis. Ad utrum vis enim punctum D vel d produci potest recta AB. Quod ostendit Problema secundi gradus esse.II Quantitates positivæ et subductitiæ cum earūum notis Quantitates autem quibus quæstioni respondetur aliquando directæ & positivæ sunt aliquando retrorsæ vel subductitiæ quas et negativas vocant. Ut si datum illud intervallum BCdeturmajus sit quam distantia BC ita ut circulus rectam illam sectet in Δ et δ, respondebitur quæstioni vel directe producendo AB ad δvelet contrario modo ducendo BΔ retrorsum. Directas quantitates signumnotamus præfigendo notamsignum + retrorsas præfigendo signum −: eut in his, + ABBδ & BΔ. Et utbi neutrum signum proæfigitur quantitas directa estest. Quantitates per quas æquarHis signis etiam ad additionem et subductionem significamus. Unt in ADB + Bδ & AB − BA ubi Bδ addi, BΔ subduci intelligitur. IISigno III Quantitates impossibiles. Hœ quantitates quibus sestijoni respondemus aliquando etiam impossibile evadunt; Ut in hoc casu quantitates BD et Bd ubi intervallum CD minus assignatur quam ut circulus rectam AB secare possit. Impossbilium verè numer semper est par Et quando duæ vel forte quatuor vel aut plureetiam servant plures impossibiles sunt (nam numerus impossibilium semper est par) gradus Problematis non æstimabitur ex numero solarum realiumprossibilium sed ex omnium numero omnium, id est omnium qui in quocun casu Problematis generaliter propositi possibilesreabes evaderetpossunt. Problema verò generaliter proponi dico in quo quantitates nullæ ita. limitantur datum possun limitantur quin possint additis vel subductis datisin datis retienibus datis differentijsadditis vel subductis datis majores vel minores sumi in uno termino quam in alia. Ut si inter A et B inveniendæ sint duæ mediæ proportionales x et y, unica tantum est Problematis hujus solutio realis, nec tamēenideo primi erit gradus Problema. Nam si omnes ejus termini.exprimantur & datis quibusvisdeminantur singulæ in datis rationibus, datis differentijs datis quibusvis CDquotquot aliquo modo limitantur, ut C, D, E, F, G, Hindefinite&c augeantur vel diminuantur, Problema generaliter enuniciabitur hoc modo Invenire quas quantitates x et y ita ut sitnAC + A ad B + x utA −+C ad x −+C & + x −+ C ad et y ± D ad B in eadem ratione. Hujus generalis serpe problematis casus est ubi C, D, E, F, G et H nulla sunt id est ubi sæpro tres suntpossunt essesolutiones reales, adeo casus ejus ubi C, D, E, F, G et Hnulla sunt id est ubi x et medosunt proportionales inter A et B, problema grades quamvis duæ ex solutionibus hic evaserint impossi in omtres ejusdem sunt gradus cum generaforte per conditiones quesdein de quibus ad gradum aliquaor inferiorem a(129)129Ubi Si linea A ducitur in lineam B rectangulum genitum signamus scribendo A×B vel AB et si id rursus ducatur in lineāamC parallelipipedum genitum signatrus scribendo A×B×C vel ABC Latus vero quod orit oritur applicando rectangulum vel parallelipipedum, illudillud ad lineam quamvis D sic notamus . Et sic in reliquis. Sed et exposita linea aliqua ad quam tanquāammensuram universalem referantur aliæ omnes lineæ referanturscribimus A×B vel AB atit AB designandum quartam proportionales ab hac linea ubi duæ mediæ propor sunt A et B et A×'B×'C ad designandam etiam quartam ab eadem linea ubi A×' et C sunt duæ medaiæ et sic in infinitum. Et si ab eod linea illa sit prima Et et continuè proportionalium et alia quævis A secunda tertiam desi designamus AA vel A91vel A2 quartam quintam & sextam et sequentis sic A3, A4, A5, &c. sic Ac vel A3 quintam sic A99 vel A4per ha notas A9, Ac, A99, A9c, Acc, &cubi Ac, A3, A4A2 diei possunt A duarum, trisun, quatuor quimfrationum et sic A est[ intelligendo AA dimidiæ rationis seu media proportion inter mensuram universalem et A quadratum, cubum, quadrato-quadratus, quadrato-cubum, cubocubum de A et sic in superioribus infinitum. Nam et quadratum et cubum super latere A constitutum designamus ijsdem notis . Unde et ac tertium quartum, proportionalem,. Unde et reliquis proportionalibus per analogiam nomina derivanturdantur quæ. Quæ et analogicè etiam dicun dicuntur dimensiones ac potestates lineæ A. Sitc A4 dicitur A quo quatuor dimentionus vel A potestatis quadrato-quadraticæ quamvis revera nihil utra trinam dimensionem et potestatem cubicam in gGeometria reperiatur. eEt simili analogia dicimus proportionales A9, Ac, A99 generari ducendo A in se et A9 in A et Ac in A. Et Bquartam proportionalem AB generari ducendo A in B. Et vicissi A99 applicatum ad A producere Ac et AB applicatum ad B producere A, applicatum vero ad C producere .] Deni ad designandum tum latus quadrati æqualis retrigo areæ A'B tum medium proportionale inter mensuram universalem et A'B scribo vel et ad designandus tum latus cubicum solidi A,'B,'C−B,'C9 ct perimum e duo bus medijs proportionalibus inter mensuram universalem et A'B'C−B'C9 scriicaso vel & etad hujus quadratum designandum scribo. Eadem notarum ratio in magis compositis tenenda est.Terminis Arithmeticis multiplicandi dividendi et extrahendi radices non utor quod hos non opus estimpropriéæ sint et minimè necessariéæ et scientias diversi generis confundere nolui. Quantitas per aliam quantitatem multiplicare absurdè dicitur. Solus numerus est per quiem potest multip possumus multiplicare. Tràs homines p multiplicari possunt per 4 non atutem per uatuor homines et linea trium pedum per quatuor non autem per lineam quatuor pedum. Si linea exponantur et Multiplicatio non fit per lineas nisi equatimus hæ per numeros exponi hac deciman antur il est pede propria natura in quantitates meticas convertantur. Est autem Arithmetici exponere et speculare quantates omnis generis per numero Geo lineas superficies et solida QuæQuæ de causæCerte Veteras ut Geometriam servarent incontaminatas a terminis illis exoticis maxima abstinueruntTrite Inter has scientias tritammaximam esse affinitatem animadverterunt., ita ut ex analogia termino; geometricos quadrati cubi et similium in Arithmeticam introducerent et Euclides scripta Geometrica libros Arithmeticos miscereaGeometricis; sed tamen Geometriam tamen quæ scientiarum Mathematicarūumregina est terminis exocticis contaminare noluerunt. Inventa est uti Geometria ut ejus succinctis operationibus in terris metiendis effugeremus tædium computi Arithmetici Proinde ut est sed computis quantum fieri potest vacare debet, sic etiam a computi nominibus ne horum usu etsad rem significatam plus nomio invitenurad caleputa et sic sei entiam nobilissimam contra institutcid ejus cum Arithmetica tandem confundemamus. Hae igitur in re si veteres sequar reprehendi non debiam si vVeteres sequar.(130)130IV Quibus lineijs problemata solvuntur. Geometria per intersectiones linearum solvit omnia Problemata, singulas ejusdem problematis solutiones per totidem intersectiones una vice exhibens Nam solutionum omnium eadem est lex et natura ita ut una exhi Geometricè exhiberi non possit quain reliquæ eadem constructione simul exhibi prodeant. Unde fit ut ad constructionem cujus Problematis lineæ duæ adhiberi debent quæ se mutuò in tot punctis secare possunt quot Problema admittit solutiones. Ad constructiones omnium problematum primi gradus sufficiunt lineæ rectæ ad eas secundi requiruntur recta et circulus vel duo circuli ad eas tertij requiritur linea magis complexa quæ rectam aut circulum in tribus punctis ad minimum secare possit et sic in infinitum.V GeneraOrdines Linearum. Et hinc pro numero punctorum in quibus linaea quævis secari potest a linea recta secari potest, oritur distinctio linearum in gradus. Primi gradus vel generisordinis est linea quam recta in unico tantum puncto secare potest vel cujus intersectionēemcum imperata quavis recta determinare Problema est primi gradus. et hujusmodi sutnt solæ rectæ leineæ. Secundi gr gradus linea est cujus intersaectionem cum datarectaquavis recta determinare Problema est secundi gradus et sic in Ter hujusmodi sunt circulus et reliquæ lineæ illæ omnes quas Comicas Sectiones appellant. Tertij verò generisgradus linea est cujus intersectionem cum recta determinare problema est tertij gradus est lissoida Veterumest et sic in infinitum. Sic etQuas quidem lineas omnes sic licebit exprimere.Lineæ vero quas recta in punctis infinitis secare potest (qualls sunt Spiralis Quadratrix Trochoides & similes) meritò dicēenturordinis ultimi gradus ordinisDatis positione quotcun rectis AB, mF,, HP IQ, KR concipe rectam BC super datorecta AB in dato angulo AC incedere et interea productam secare reliquas positione datas in FP, Q, R, &&c. termino suo cC lineam cCc describere ea lege ut sit ssemper sint AB ad BC, PC ad BD, QD ad BE, RE ad BF in eadem ratione.Modus exprimendi lineas Concipe rectam BC parallelo motu ad latus ferri et interea secare datosrectaspquotcun positione datas AB in B, mF in F HP in P, IQ in Q, KR in R netc non curvas nec in C, dd in D ee in Eet punctis in ea mobilibussuis C, D, E, F alias lineas cc, dd, ee, ff describere. Sit Determinari autem concipiantur longitudines BC, BD, BE, BF hac semper lege ut sint AB ad BC et PC ad BD et QD ad BE et RE ad BF in eadem ratione. Et si linea dD ad quam ratio secunda desinit recta est, tunc linea cC ad quam prima desinit erit secundi generis et aliquando primi nes ulla est linea secundi generis quæ non potest hoc modo exhiberi. Sin linea eE ad quam ratio tertia desinit recta assumitur; linea vel ad quam prima desinit erit tertij generis et aliquando secundi vel primi ne ulla est linea tertij generis quæ neu potest hoc designarisic. Quod si linea fF ad quam quarta ratio desinit recta statuaturtunc linea cC ad quam prima desinit erit aut quarti aut inferioris alicujus generis. Et sic novas in infinitum lineas designare licet, et numerus rationum gradum altissimum lineæ cC semper æquabit. Tot enim punctis &(131)131non pluribus possibile est Curvam illam cC a recta BC secari quot sunt rationes. Nam si verbi gratia tres sint rationes et linea BC detur positione dabuntur puncta AB BE, BP, BQ et BC invenienda erit ea lege ut sit AB.BC ∷BC+BP.BD∷BD+BQ.BE quod Problema triplicem admittere solutionem ex superioribus constat adeo linea BC triplex est. Secu Tria sunt igitur igitur et non plura possunt esse puncta C in quibus recta BC occurritt Curvæ cC, proinde Curva illa tertij est generis.Facilius autem imaginamur has curvas ubi per motus locales linearum inter se cohærentium tanquam per organa quædam delineari describi concipimus. Ut si regulæ PC PD datum angulum CPD continentes volvantur circa datur datum punctum P quod in anguli illius vertice est et similitur regulæ QC QD circa datum angulum Q continentes circa punctūum Q ea semperlege ut regulæ PQ, QD se mutuo semper secent ad rectam aliquam lineam positione datam AD et interea reliquarum regularum PC, QC intersectio C motu suo lineam cC describatdesignet: Erit hæc linea cC linea secundi gradus et aliquando primi. Et hac ratione possunt omnes lineæ secundi generigradusdescribidesignari. Deinde si loco rectæ AD substituatur curvalinea aliqua secundi greneris per neutrum punctorum PQ transiens et inter regularum intersectio D in hac movere cocipiatur, altera intersectio C, describet lineamdesignabit lineam quarsi gradus aut etiam tertij. gradus. Qua ratione et omnes hujustertij gradus lineæs quarum commoda aliqua descriptio organica hactemus reperta fuit describeredesignare liceat. At ita ad lineas superiorum generum pergitur, licet omnes non possunt hoc modo describio Quod idem fiet si regularum .duæPC, PD non volvantur circa polum P sed parallelo motu ferantur ita ut concursus earum P moventpergat in recta aliqua data positione data. Sed Et ad majorem describerignandi copiamvice rectarum regularum adhiberi possunt curvæ.Locus linearis puncti vagi. Lineam vero ut cC in qua punctum aliquod indeterminatūumut C perpetuò reperitur veteres dixerunt puncti illius locus et quoniam problematum solutionesconstructiones pendebant a descriptioneab inventiam duorum locorum puncti quæsiti pendebant quasi s in quorum intersectionessitum essetinveniretur, ideo. Veteres ad hujusmodi locorum inventiorum ac determinnationem quam compositiones eocompositionem ut loquebantur id est ad eorum inventiorum ac determinationem summis viribus intebuntur. Duo autem hic requiruntur. Primum ut sciamus sciamus sadatis loci conditionibus, sciamusæqualis sit et quomodo describendus deinde ut in quocun d in quolibet problemata locasimplicissima inveniamus quæ simplicissima sunt et facillisà determinari ac describi possint sed quam hec de his agamus proprietates curvarum cognoscendæ sunt. Insigniores autem sunt hæ.Curvarum proprietates generales. Si paraquotcun quavis AG, DF, GS agantur curvam in tot punctis A, B, C ac D, E, F, G, H, quot curva a recti secari potest Dein tertia agatur recta secantisprioribus ita secans in K et ofut(132)132 utrius parts vel summa partium ad curvam extensarum ex uno latere æqualis sit parti vel summæ partium ad curvam extensarum ex altero latera, vizvizt KA+KB=KC et LD=LE+LF: tunc rectapartes etiam reliquarum parallelarum hinc inde æquales erunt MG=MH +MI. Linea vero quæ parallelas ita secat in partesDiametrum Curvæ appellamus et partes parallelarum quæipbas ordinatim applicatas ad Diametrum, utpuncta itemut R, S, T ubi Diameter secat curvam ut R, S, TVerticesejus, et conjugatam Diametrum quæ ordenatum applicatis parallelaest siqua tatis situ conjugabam Diametrum quæ ordinatim applicatas parallela est siqua tatis sit. Nempe Porro ubi Diameter curvamverticum duarum quarumvis intervallum latus rectum transversus et partem Diametri inter vert verticem quamvis et ordinatis applicatam tum Diamtri. Quibus nominibus analogiam inter Conicas Sectiones et superioresinsinuare solui. Porrò ubi diameter curvam in tot punctis R S T secat quot ipsam recta secare potest et ab aliqua ordinatim applicatarum ita quæita divibditur in ut pars vel summa partium ex una latera ad vesegmentum vel summa segmenrtorum ex uno latere ad puncta illa seu vertices extensam conjunctim æqualis patsit segmento velvel summæ latium partiumsegmentorum ex altero latere, illa ordinatim applicata dicetur Diameter conjugata, Diametrorum verò intersectio centrumvel unum ex centris & summapartium primæ diametri ex utra latere secundæ jacentium summaLatus transverum Figuræ et linea illillæquævislatrectam quæ ita sunt ad latera transversa ut contentussub onimnibus ordinatim applicatis AK, BK, CK, ad contentus sub omnibus segmentis diametri RK, SK, TK verbi gratia ut contentum sub MgG, MH M, mg MMG, MH, Mg ad contentus sub MKR, K MS, MT. K. Nam in omnibus figuris contentum sub ordinatim applicatis est ad contentum sub segmentis diametri in data rationesi modòtot suntsegmenta totapplicata &segmentiorūum quot ejus generis numerus pro genere figuræ ple est. Quinetiam si datis positione rectis vx, xy utrun ducanturparallela duæ IG, RT utrun ducantur secantes se mutuo in M curvam vero in tot punctis quot rectæ curvam ejus generis secare potest punctain G, H, I et R, S, T Rectangul contentum sub omnibus partibus unius rectæ inter curvam et alteram rectam sitis MG, MH, MI erit ad contentum sub omnibus ejusmodi partibus alterius rectæ inMR, MS, MT in data ratione. Et hinc recta duci potestTangentes ad Curvas descriptas ducaenture. quæ curvam quamvis descriptam in puncto imperato tanget.secetve in dat anguloSit illud punctum PR. Per illudquod age duas quasvis rectas PRP, PRT, se secantes insecantes curvam in pleno numero punctiorūumquæ, I, M et uni earum PRI parallela GIG secantem altera RT in M quæ omnes etiam secent Curvam in pleno numero punctorum IR, G;M, E, T; P, R, Q; R, S, T; IG, H, I. Sit In IG cape MN ita ut sit rectangulum s contentum sub PR, QR, MN ad contentum sub RS, RT, RM ut contentum Sub GM, HM, IM ad rectangulus sub ut contentum sub RS, RT ad contentum sub MS, MT et acta RN tanget curvam in R, si modo RN capiatur in in eo angulo PRM quiem curva secat. Nam concipe gm,ξ parallelam ess PQ et ad eam accedere interea d secat curvam in g, h, i et evanescentium, RM, MhN erit ea quæ est RM ad MN ubi RN curvam. Est autem rectangulum , contentum sub gMhM ,iN ad contentum sub GM, HM, IM ut contentum sub MR, MS MT ad contentum sub MR, MS, MT. Hic pro ratione RM ad MR substituaturinter rationem NM ad MR et coalescentibus lineas PQ et ig sub scribo R pro M, Q pro g et P pro i et hatincidesin proportione quar tangentem RN determinavimus.(133)133De cruribus infinitis et Asymptotis curvarum. Lineae Cæterum hæ lineæ utplurimum crura habent in infinitus serpentia quæ aut Hyperbolici sunt generis aut Parabolici. Concipe punctum B secundum lineæmcurvæm crus AB delatum abire in infinitum et interea curvam a linea mobili BC rectas GD secante insemper tangi. Sicet autem tangens illa BC rectam positione datam GD Incidat autem semper a puncto atiquo G in tangentem illam perpendiculum GC, et ubi punctum B in infinitus abit si GCC fit infinite longum tangente BC prorsus evanescente crus illud AB HyperParabolicum est, sed si GC non fit infinitè longum, crus estHyperbolicūm est et tangens in ultima positione seu recta illa uti DE qu(uti DE) quacum tangens ultimò convenit, cruris illius Asymptotos Aappelatur. Crus vero infinitum semper habet socium suum qui nunc ad eandem nunc ad oppositum plagam tendit. Et paris Hyperbolici semper eadem est Asymptotos. Est hæc Assymptotorum insignis proprietas, quod si curva cujusvis generis plenè Hyperbolica secetur a recta in pleno numero punctorūumet recta G, H, I et recta illa secet etiam omnes Asymptotos puta AB, AC, BC in D, E, et Fsegmentumpars vel summa segmentorūumpartiumrectæ Asymptoto vel ab Asymptotis ad curvamcrura totidem versus unam plagam tendentes extensarūum æqualis est parti vel summæ partium similium a reliquis Asymptotis adversus alteram plagam ad reliqua crura tendentium DG=EH+FG, vel EG+FH=DI. FiguramCurvāamverò plenè Hyperbolicam voco qua alia ejus generis cruva non potest habere plura paria crurum Hyperbolicorum. Habet autem tria paria si sit tertij generis quatuor si quarti et sic deinceps. Et par crurum Parabolicoram æquipollet duabus paribus crurum Hyperbolicorum.Quomodo curare in species distinguendæEx crurum infinitorum numero et et diversitate endet distinctio curvarum in gradus Qu species principales. Sunt autem alia crura conspirantia seu ad eandem plagam tendentiæ, alia oppositadivergentia seu pvergentia in oppositas plagas et utra rursus velad easdem partes convexa vel ad contrarias. Par crurum Parabolicorūum Hyperbolicorum æquipolle conspirantium æquipollet duobus paribus Hyperbolicorum divergentium, et Par Parabolicorum divergentium contraries partes convex æquipollet tribus: saltem in curvis tertij generis. Unde sciri potest quot crura cujusvis generis curva quævis habere potest Ut si curva terti generis habebt par Parabolicum conspirans sive Parabolicum sive Hyperbolicum conspirans: non habebit misi aliud par Hyperbolicum divergens. Sed et Ellipses conjugatæ considerandæ sunt quæ et aliquando in puncta conjugata contrapuncturquæ puncta conjuga cujusmodi et polus Conchordis, aliquando prorsus evanescente: Asymptotorum item situs an parallelæ sunt vel inclinatus et aliæ quædam differentia notæ est. De De curvis supelineis autem enim superiorum generum fase disserere non est instituti. nec in demonstrandis quæ dicta sunt tempus non tantum quæ de lineis secundi ab Apollonio demonstram habentur (134)134ita commemorare ut line eadem lineis etiam superiorum generūumcompetere insinuarem. CDe curvarūum tangentibus Cæterum quoniam cognitio determinatio aclinearumcurvarum maximè pendet a crubribus infinitis, hæc autem cum eorum Asymptotis noscuntur ex tangentibus; tum etiam quia tangentium inventio posthecalijs inservient usibus: sub methodum jam subjungam ducendi rectas quæ curvas quasvis nondum descriptas postquam describuntur tangent. Sed notæ operationum quibus in hujusmodi operationibus tam Analyticis quam syntheticis quamGeometricis utimusr sunt prius expicandæ.Notarum quarundam explicatio. Ubi linea aliqua AB ducitur in aliam lineam BCD rectangulum genitum significamus scribendo AB'BCD AB et si id rursus ducatur, in quamvistertiam lineam CEF ad experimendum parallelipipedum genitum scribimusmusAB'C vel ABC AB'CD'EF. Latus verò quod oritur applicando quadraturrectangulum illud ad lineam quamvis DGH sic notamus ,.aAt ita in alijs. Sed et exposita linea linea aliqua ad quam tanquam mensuram universalem aliæ omnes (ut fit in decimo Elementorum) referantur scribimus AB'CD ad designandam quartam proportionalem ab hac linea ubiubi per mediaæssduaæssuntsunt AB et CD et AB'CD'EF ad designandam etiam quartam ab eadem linea per ubiubi mediæ duæsunt AB'CD et EF et sic in infinitum. Et si linea linea mensuralinea illa sit prima continuè proportionalium et alia quævis AB secunda, tertiam sic designamus AB2, quartam sic AB3, quintam sic AB4 at ita deinceps. Et inter lineam illam et aliam quamvis AB notamus mediam proportionalem sic AB, primam e duabus medijs proportionalibus sic AB secundam sic AB. Inter lineam illam et AB'CD similiter est media propor denotat tum latus quadrati æqualis rectangulo AB'CD tum mediam proportionale inter mensuram illam universalem et AB'CD vel quod perinde est intermedio proportionalem inter AB et CD. Sed AB'CDest quarta proportionalis inter a mensura illa ubipemubi mediæsuntsuntAB et CD. Et has quantitates nominibus usitatis significamus præterquam quod a vocabiulis Arithmeticis certas ob rationes cum veteribus abstinendum esse duximus. Porro quantitates compositæ eodem legemodo signantur. Si cA+B'C+DA+B2 denotat quadraA+B'C−D denotat rectangulum sub A+B et C−D et AR±BS2quadratum ipsius A±B. Quæ quantitates juxta pam propositionemetiam partibus juxta secundáāam Elementorum in se ductis possunt et sic scribintur A'C+B'C−A'D−B'D et AR2±2AR'BS+S2. Ubi notes quod pars positiva ducta in subductitiam vel subductitia in positivam producit subductitiam duæ vero subductitiæ in se ductæ producunt possitivam. Sit ABcujumvisseulineo seumenara linearum omnium ad quam ubiræ omnes referuntur secentur AB AD parallelis BD, EG ita ut sit AB ad AD ut AE ad AG et possita AB mensura illa ad quam lineæ omnes referuntur AG erit AD'AE. Diminuatur AE donec evanescet et postea evadat retrorsa Ae et A G simul cadiminuetur evanescet & convertitur in retrorsam AG. Diminuatur etiam(135)135etiam AD donec evenescat et postea retrorsa evadat Ad et retrorsa Ag simul diminuetur evanescet et evadet directa Aα et convertetur in directam Aα. Duæ igitur retrorsæ Ae, Ad by retrorsam id est directam Aα efficiunt.Fluxiones quantitatum.F Præter F Notis intellectisproscognitis præmittenda est etiam methodus determinandi fluxiones linearum f et fluxionum plagas. Per Fluxionem intelligo celeritatem incrementi vel decrementio quan lineæ cujusvis indeterminatæ ubi lineæ aliquæ super alias in descriptione curvarum moveri concipiantur aut quomodocunet inter movendum augeri vel diminui aut motu punctorum describi. Unde et quantitates illas fluenteindeterminatias illas quantitatesfluentes nominare licebit. Proponantur datiæ aliquot quantitates a, b, c, d, et fluentes x, y, z et sit ea fluentiūumrelatio inter se ut datur semperquæ in hac æquationehoc aggregatumcomplexio exprimitur+yx= +zz−yz. Fluatnt x y et z donec et in ea fluxione fiat sit sit ro augmentum ipsius x, so ipsius y & ipsius z ita ut x y et z jam fiant x+ro, , et perinde has pro x y et z scribrespectivè scribendo æquatio superior fiet −yz−yto−zso−stoo. Quantitates se per æqual vi primæ æquationes æquales dele et restab restabunt . Applica omnia ad o et fiet −sto.Proponantur datæ aliquot quantitates A, B, C, D, et fluentes V X, Y, Z quarum fluxiones respectivè designentur minusculæ v, x, y, z. et requiratur fluxiones linearumalicujus quæ ex his generaturfit ut linea X'Y. Maneat primum X et Y fluendo fluat Y donec ipsa fiat Z et XY fiat XZ et quia Y et XY fluendo non mutant rationem fluxiones earum erunt ut ipsæ hoc est ut 1 ad X. Unde cum fluxio Y sit y fluxio XY erit Xy. fluat jam Maneat jam Z et fluat X donec ipsa fiat V et XZ fiat VZ et fluxio XZ erit Zx ut in casu superiorem priore. Fluant jam X et XY simul donecmotuceleritate priore donec XY fiat ut XV una vice fiat VZ et quia fluxio Xy sufficit ad mutandum XY in XZ et fluxio Zx ad mutandum XZ in VZ Fluxio tota qua XY mutatur in VZ erit Xy+Zx. Pone V æqualem X et Z æqualem Y ut fit ipso fluendi initio et fluxio initialis ipsius XY erit Xy+Yx.Proponatur jam factum XYZ et ponendo V= XY=V erit XYZ=VZ. Ergo Cujus fluxio jutxta casūum priores est Vz+Zv.Sed et ob XY=V est Xy+YX=v. Pro V et v scribesublimus æquipollentia et Vz+Zv hoc est fluxio ipsius XYZ fiet XYz+XZy+YZx. Et progressionis modum observando colligitur universalitur quod facti cujuscun , utVXYZ, fluxio semper invenietur substituendo sigillatim in facto illo pro unoquofactore fluxionem ejus et sumendo resublantium terminorum aggregatum. Quot regulax'y'z', x2, y3 et similius Fluant X et Y donec evadent x+ro et in ea fluxione sit ro augmentum x+ ipsius X et so augmentum ipsius Y ita X et Y jam evaserint X+ro & Y+so et X'Y fiet X'Y+Xso+Yro+srosroo adeó augmentum ipsius xX'Y erit Xso+Yro+sroo. Jam cum fluxiones Defluant jam quantitates X+ro et Y+so donec evase iterum ad X et Y redierint et erit ultimæ rationis partiusm evanescentium ro, so, Xso+ Yro+sroo quæ est eritquæ esteæ erunt quæ sunt fluxiorum x, y et xyx, y, xyquantitatum fluentium X, Y et XY Sunt igitur fluxiones illæ ut r, s, Xs+Yr+sro Sunt igitur x, y, xy ut id ut Rationes ille sunt ipsorum r, s, Xs+Yr+sro Applica partes illas ad communem factorem o, et iaod pars ro jam nulla estevanuit dele sro et fluxiones erunt ut r, s, Xs+Yr, hoc est si r denotet fluxionem ipsasiss X et s eam ipsius Y, Xs+Yr denotabit eam ipsius XY. Pro r et s substitue æquipollentes x et y et Xy+Yx denotabit fluxionem ipsius XY.Simili argumentatione fluxio ipsius XYZ invenietur XYz+XyZy+YZx et fluxio ipsius VXYZ invenietur VXYz +VXZy+VYZx+YXVYZv et sic in infinitum fluxio fasti semper invenietur substituendo sigillatim pro unoquo factore fluxionem ejus et sumendo resultantiun terminorum aggregatam. Quæ regula etiāam(136)136etiam obtincet ubi aliqui factores æquales sunt. Ut si X et y æquales iut XY valeat X2erit ejus fluxio XY+ Xy+ Yx id estfiet 2Xx. Et similiter fluxio3X2x erit ipsius X3erit 3X2x etfluxio eristt 3X2x et ipsius X2xX2Z fluxio X2z+2XzZx. At ita in compositis, fluxio ipsius AX−3X2 erist Ax−6Xx Nam fluxiones partium simul sumptæ sunt fluxio totius.In lateribus applicatorum ad fluentia methodus hæc est Proponatur latus. Pone ipsum æquale V et erit XZX=YV adeo 2Xx=Yv+Vy nam æqualium fluentium semper æqualium fluxiones æquales sunt. Aufer utrobi Vy et reliquum divisum per Y nemper erit v. hac estEst autem v ipsius V id est ipsius fluxio quam invenire oportuit.Similis est methodus in medijs proportionalibus. Proponatur XY. Pone ipsum æquale VZY et erit XY=X2V2. Adeo Xy+Yx=2Vv et .Similis est methodus in med proportionalibus. Proponaturmed proportionalibus lateribus æquilaterorumlateribus quadraticis, cubicin alijs.medium P medium AX-X2|lateribus æquilaterorum. Proponatur quadrati abicujus latus AX-X2|. Pone ipsum æquale V, et erit AX-X2=V2adeo Ax−2Xx=2Vv, et .Ad AC demitte normalem BG et quia BG2+GA2=ABA2et BG datur erit 2GA'ga=2ABA'ba seu GA'ga=BA'ba. Sed fluxiones ga et ca æquatæ ædem sunt ergo 2GA'ca =2ABA'ba et .Fluat jam CA donec evaserit Ca. Dein maneat Ca fluat etiamCB donec evaserit Cb et demisso ad AH normali CB linea AB fluxione sua converfit linea convertitur in lineam aB et similiter linea aB deinceps fluxione sua convertitur in ab Ergo utra fluxione conjunctim convertitur AB in ab. Id est si CA et CB simul fluant erit fluxio ipsius AB, it quia Ab et aB initio æquales sunt, erit eadem.In figuris hæc est methodus. In triangulo ABC dentur angulus C et latus CB, fluant cætera. Et ubi CA fluendo evasit Ca simul BA evadat Ba. In Ba cape BD æqualem BA, et age A partes genitæ erunt Aa et Da. Defluant jam lineæ Ca Ba donec ad priorem magnitudinem et positionem CA et BA redierint et partium si ad CA demittatur perpendiculum BG, ultima ratio partium evanescentium Aa, BDa erit ea quæ est linearum AB AG eo quod in eo casu AD perpendiculare fit ad Ba adeo triangulūum aAD simile sit triangulo aBG id est triangulo ABG eavadit simile. Quare cum fluxiones sint in ultima ratione partium perevanescentium erit AB BG ut AB ad BAG ita fluxio ca ad fluxionem ba seu .Fluant jam ruirsus CA et BA donec evaserint Ca et Ba et dein fluat etam CB donec ipsa evaserit CB et Ba evaserit ba et hac recta secund demisso ad CB normali AH fluxio ipsius aB hac secunda vice ex jam inventis erit . AB fluxionesua priori convertitur in aBa et illud aBa fluxione posteriori convertitur in aba Ergo utrà fluxiosle conjuxtim BA convertetur AB in ab.: id est si(137)137Si CA et CB simul fluant utita ut AB fiat it evadall ab fluxio ipsius AB erit AH seu quia initio fluxionis AB et aB æquales sunt, erit ea .Ubi CA et CB fluendo evaserint Ca et Cb fluat insuper angulus ACB et fluendo fiat ACB. In aB cape aF=aβet junge bF, bβ. Et ipso fluxionis initio bF perpendiculare erit ad ab vel AB adeo perallelum KI et bβ perpendiculare erit ad Cb adeo perpendiculare parallelus AK. Unde triangula βFb & AIK æquales habentia angulos ad b et K et rectos ad F et I similia erunt et bβ erit ad Fβid est augmentum lineæ ab ipso fluxionis initio ad augmenttum arcus subtendentis angulum ACB radio Cb descripti vel quod perinde est fluxio lineæ ab ad fluxionem arcus illius ut AI ad AK, hoc est ut CI ad CB.Cæterum ad fluxionem arcus angulum ad ascirculi dati significandam pono angulum literis minisculis nominando illud ejus latus ejus ultimoloco quod radius est. Aut si neutrum anguliejus latus radius est ubi angulum literis minusculis posui subjungo radium literis majusculis cum litera r ijsdem prefixasuffixa. Est it fluxiolineæ ab fluxio ad arcus præfati fluxionem acab ut CI ad BC. Adeo sunt vel quod perinde est fluxio est ipmlineæ ab. Hanc fluxionem quia ab convertitur in aβ adde fluxione qua AB conversa fuit in ab et habebis totam fluxionem nempe AG'c qua AB per fluxionēem laterūumCA, CB et anguli C convertitur in aβ. Est ita AG'ca +BH'cb+CI'cABr=AB'ab. relation inter fluxiorelation inter fluxiones trium laterum et anguli trianguli cujuscun, cujus beneficio si tres ex his fluxionibus cognos cantur possumus quartam invenire.In figuris pro significanda fluxione lineæ alicujus pono lineam ileam literis minusculis.: ut ab p bc prosignificanda fluxione lineæ BC. Angulorum verò fluxiones expono per fluxiones arcuum quibus subtenduntur ad datam distantiam. A Et distantiam illam quæcun tandem assumatur designe per literam R; fluxionem arcus per angulum literis minusculis serppentum, lineam tam quæ oppositam ordinatatim applicatam in in hoc circuloid est s sinum cujus arcus per literam s angulo præfixam distantiam ordinatæ centroseu id est sinum complemenrationis per literam s angulo præfixam & horum senuum fluxio s per literas easdem s it 'sangulo minusculis literais noto præteras. Sit ABC angulus quilibet flucus DC arcus suo ad datamdistantis Bc subtenditc& DE [ordenatim applicata est sinus ad latus angulis BC sinus ejus et vel significabit datum illamsustantiam BC illam dataim BC vel BD, I vel abc fluxione arcus CD, sB lineam DE, s'B lineam BE&sb s'b fluxiorum earum fluxiones (138)138Illud etiam præmittenduni est, fluxionem arcus esse ad fluxionem sinus ejus ut Radius ad sinum complementi et ad defluxionem com sinus complementi ut radius ad sinum. ad sb ut R ad s'B et b ad−s'b ut R ad sB. Fluant enim omnes aliquantulum donec HI fiat HM, IK fiat MN et CD fiat CF, DE fiat FG et BCE fiat BCG et ipso fluendi initio fluxiones erunt ut augmentea incipientia IM, ML, FD, FH, HD id est ut BD, BE, DE.His præmissis proponatur triangulum aliquod ABC et demissis ad latera singula perpendiculis AE, BF, CG erit exerit (ut e demonstratis trigonometricis notum est) sA.sB ut sA ad sB utut sA ad sB ita BF, ad AE & ita BC ad AC adeo sA'AC=sB'BC. Ergo sA'ac+AC'sa =sB'bc+BC'sb. Sed sa.a∷s'A.R∷AG.Ac Ergo AC'sa=AG'a. Et eodem modo est BC'sb= BG'b. Quare sA'ac+AG'a=sB'bc+BG'b. Quo theoremate conferre possumus fluxiones angulorum duorum et laterum oppositorum trianguli cujuscun et ex tribus cognitis invenire quartam.** Et cum summa trium angulorum detur adeo aggregatum fluxiorum omnium nullum sit, vel quod perinde est duorum fluxio æqualis sit defluxioni tertij si pro +BG'b scribas +−BG'a+c, & idem utrobicons AG'addæ BG'BG'auferas fiet sBsA'ac+AB'a+BG'c=sB'bc, Theorema ad comparandas fluxiones duorum angulorum totidem laterum ubi quorum unum angulis illis in terijcitur.Rursus * est R.s'A∷AC.AG ergo seu s'A'AC=R'AG. Ergo s'A'ac+AC's'a=R'ag. Est et (per Præmissa) s'A'a ac∷sA Ra.−s'a vel −a.s'a∷R.sA∷AC.CG. Ergo pro AC's'a scribendo CG'−CG'a, fit s'A'ac−CG'a=R'ag. Eodem modo est s'B'bc−CG'b=R'bg. Et æqualibus æqualia addendo fit s'A'ac+s'B'bc−CG'a+b=R'ab. Ob datam summa trium angulorum pro a+b scribe −c et fit s'A'ac+s'B'bc+CG'c=R'ab. Theorema ad comparandas fluxiones trium laterum et anguli cujusvis.Simili argumentatione possunt alia Theoremata colligi ubi perpendicula triangulorum et segmenta basium aliæve lineæ considerantur. Sic AC'ac−BC'bc+BG'ab=AB'ag Theorema est ubi latera tria et segmentum basis considerantur, et, posito X commmunni trium perpendiculorum intersectione, est BX'ac +AX'bc=GX'ab+AB'gc Theorema ubi agitur de lateribus ut perpendiculo. Sed hæc et similia necessaria sunt et a Geometris ubi usus eorum inciderit, ex inventis haud difficulter colligentur. Sed hæc non prosequor. Satis est investigandi methodum aperuisse. Horum verò Theorematum beneficio possumus in propositus quibus figuris fluxiones flinearum et angulorum haud secus ac in computo trigonometricos lias ab alijs colligere donec ad quæsitam pervenimus. Ut si dentur positione lineæ AB, AD, DE et BC data longitudinis moveatur perpetuo subtendæts angulum A et producta secans rectam ED in E, et ex cognita vel desideretur fluxio lineæ EC: primum in trianguli ABC per secundum i Theorematum invenietur sA'ac+BG'c=0 evanescunt enim fluxioterminiSimili argumentatione possunt varia Theremata colligi ad datur ubi sega perendicula triangulorum et segmenta baisium condiderantur. Ut si perpendiculorum communis intersectio sit x, erit Sic AC'ac−BC'bc+−BG'ab=AB'ag Theorema est ubi latera et segmentum basis consideratur et si trium perpendiculorum communis intersectio sit x, erit BX'ac+AX'bc =GX'ab+AB'gc Theorema ubi latera et perpendiculum in quæstione sunt.Porro in triangulis ACG, ACE ubi fluxio anguoirumrectoirum E nullæ est per Theorema in tribus novissimis primum sA'ac+AG'a=R'gc mutatis mutandir sit sC'ac+CE'c= R'ae. Est et in triangulo ABC, per Theorematim secundum, sA'ac+AB=sB'bc−AB'c−BG'c. Quod hac epsi sA'ac æquale est substitue in prioribus et orietur sB'bc+CE−BG'c−AB'a =R'ae, Theorema ubi Perpendiculum, angulus vertticales, basis et angulus alteruter ad bm in quæstione sunt Ubi si pro −a sub(139)139stituas æquipollentem b+c, fit sB'bc AGc+AG+BG'c+CE+AG'c +BAB'b=R'ae. Quæ deiuo Theoremata casus omnes determinant ubi perpendiculum basis et anguli duo quilibet in quæstione sunt. Sic et ubis Perpendiculum basis et latera duo in quæstione sunt, estposito. X communi intersectione trium perpendiculorum, etinvenietur hoc theorema BX'ac+AX'bc=GX'ab+AB'GC. Alia ejusmodi Theoremata ubi bas perpendicula et segmenta basium aliæve lineæ considerantur Geometra, pre rectaquoties usus eorum inciderit, insistendo vestigijs methodi hic patefactæ, facilè inveniet; adeò ut rem plenius presequi superacaneum ducam.Cæterum Theorematum quæ frequentius usui fuerint et ex quibus cætera, siquando opus erit, licebit derivare licebit,seriem sub convenit subjungere ut promptius citori possint.De Proportionalibus Th. 1. Si quotcun continue proportionalium unum S, T, A, V, X, Y, unum A datur, fluxiones eorum erunt ut ipsa multiplicata per numerum locorum quibus distant a dato termino s.t.v.x.y∷−2S.−T.vV.2X.3Y.Th. 2. Positis tribus continuè proportionalibus V.X.Y, si summa extremorum datur, erit ut excessus unius extremi supra alterum ad duplum medij ita fluxio medij ad fluxionem extremi illius alterius vel ad defluxionem prioris. V−Y. 2X∷x.y∷x.−v. Et si differentia extremorum datur, erit ut summa extremorūumad duplum medij ita fluxio medij ad fluxionem alterutrius extremi. V+Y. 2X∷x.y∷x.v.Th. 3. Iijsdem positis si summa medij et extremi datur, a duplo medio aufer alterum extremum & residuum erit ad extremum prius ut fluxio alterius extremi ad fluxionem medij vel ad defluxionem extremi prioris Detur V+X, erit 2X−Y.V∷y.x∷y.−v. Et si differentia medij et extremi datur, ad duplum medium adde alterum extremum, et summa erit ad extremum prious ut fluxio alterius extremi ad fluxionem tam medij quam extremi prioris. Detur V−X vel X−V, erit 2X+Y.V∷y.x∷y.v.Th. 42. Positis quatuor proportionalibus XV.X∷Y.Z fluxiones extreorum mutuò ductæ in extremas æquantur fluxionibus mediorum mutuò ductis in medias. Vz+Zv=Xy+Yx.De PTriangulis ContentisTh. 53. Contenti fluxio ut quæ fitet quæ fit ducendo sigillatumilateris cujus fluxionem in contentum illud applicatum ad latus illud et summam productorum capiendo. Et fluxio aggregatio ex contentis componitur ex fluxionibus partium Sic fluxio XY est Xy+Yx et fluxio AY est Ay. Unde conjunctim fuxio AY+XY est Ay+Xy+Yx.(140)140De TriangulisTh. 6. In triangulo quovis acutangulo, perpendiculo ad basem demisso, differentia fluxionum laterum ductarūumin sinus conterminorum angulorum ad basem, æqualis est differentiæ fluxionum arcuum subtendentium angulos ad basem ductarum in contermina segmentaDe Triangulis.Th. 64 In triangulo quovis ad basem acutangulo, si perpendiculis ad basem demittatur, et fluxio lateris utrius ducta in sinum anguli sibi contermini ad basem, seorsim addatur fluxioni arcus subtendentis angulum illum ductæ in conterminum segmentum basis; æquales erunt summæ. In triangulo ABC, demisso ad basem AB perpendiculo CG, est sB'bc+BG'b=sA'ac+AG'a. Ad quod Theorema recurrendum est quoties anguli duo et latera duo opposita in quæstione sunt.Th. 75. Fluxio In triangulo quovis ad basem acutangulo, si perpendiculum demittatur ad basem, flauxio lateris alterutrius ducta in sinum contermini anguli ad basem æqualis est summæ fluxionis lateris alterius ductæ in sinum contermini anguli ad basem & fluxionis arcus subtendentis angulum illum ductæ in basem et fluxionis arcus subtendentis angulum ad verticem ductæ in segmentum basis lateri primo conterminum. In triangulo ABC demisso perpendiculo CG est sB'bc =sA'AGac+AB'a+BG'c. Ad quod Theorema recurrendum est ubi anguli duo et latera totidem interjectum et oppositum in quætione sunt.Th. 86 In triangulo quovis ad basem acutangulo, fluxio basis ducta in radium æqualis est summæ fluxionum laterum seorsim ductarum in sinus complementorum conterminorum angulorum ad basem & fluxionis arcus angulum verticalem subtendentis ductæ in perpendiculum ab angulo isto ad basem demissum. In triangulo ABC demisso perpendiculo CG est R'ab=s'A'ac +s'B'bc+CG'c. Adeo ad Theorema et ubi angulus et latera tria in queatione sunt.Scholium casus triangulorum habent aangulorsusad basem, Theoremata extenduir ad omnes casus. Nempe si angulus alterate ad basem obtusus sit, sinus complementi ejus et conterminam segmentum basis pro retrorsis haberi debent(141)141debent et perinde signa eorum de + in − mutari, terminispositivis Theorematum terminis in quibus positive ponunturrepericintur transformatis in subductitios. Et quamvis hæc Theoremata de quatuor fluentibus proponuntur, continent tamen omnes casus triangulorūumubi tria vel duo tantum fluentia sunt. Numerus enim qua ternarius ex datis terminis trianguli implen semper implendus et in Theoremata delendæ fluxiones datorum. Ut si in triangulo ABC duratur angulus A et latBC oppositus et ex fluxiones alterius anguli B invenire vellem fluxionem lateris huic oppositi AC: quatuor termini quæestionem ingredientes erunt tam anguli duo datus A et fluens B et latera duo opposita datum BC et fluens AC. Vide Consulo qigitur Theorema sextumquartum quod hunc casum includit, et ibi deleta fluxione Anguli A dato datorum terminorum A et BC prod invenio BG'b=sA'ac; id est fluxionem lateris AC esse ad fluxionem arcus subtendentis angulum B ut BG ad sinum anguli A. Et eodem modo ubi in quovis triangulialicujus. dantur duo termini quilibet, Theoremata se semper resolēunt in proportiones. Unde et proportiones illas nomine Theorematum tunc licebit citare perinde ac si eædem in Theorematis expressæ fuissent. In triangulis autem rectangulis angulus rectus pro dato termino semper habendus est, et numerum quatuor terminorum implet. Hoc modo semper incides in quatuor terminos;.qQuibus cognitis in promptu Theoremata consulereestestper annotata in calce cujus Theorematum legitimum Theorema consulere. Nec minori promptitudine recurretur ad Triangulorum easus sequentes.De motuum plaga Cas. . De erpendiculo bas et lateribus.De motuum plagis et fluxione Curvarum.Th. 7 Si recta mobilis BD rectæ positione datæ BDet ad datum punctum A terminatæ AB tanquam basi insistat et habeantur fluxiones basis illius AB rectæ insistentis BD et arcus subtendentis angulum quem hæ rectæ comprehenurit exponantur fluxiones illæ tres per totidem rectus S, T, V respectivè.De mottum plaga et celeritateTh. 7. Ubi puncti alicujus motus pendet a diversis linearum et angulorum fluxionibus colligenda sunt sigillatim loca in quæ fluxiones singulæ seorsim vel etiam plures earum conjunctim temporibus æqualibus punctum illud transferrtet si modo plagas et celeritates servarent inmutatas servarent inmutatas quas habent ipso fkuendi initio. Et locos ultimus locorum ultimus vel ultimorum intersectio ipse erit locus ad quem omnes fluxiones conjunctim eodem temporis spatio punctum illud rectà transferent: adeo si a loco primo ad hunc locum recta linear ducatur, hæc et Curvam motu puncti descriptam tanget et puncti describentis celeritatem exponet longitudine suaexhibebitseu fluxionem curvæ exponet.Exempli gratia si linea mobilis inmobili AB feratur et cognoscantur fluxiones lineæ illius mobilis AB, arcus angulum ABC ad datam distantiam subtendentis et lineæ mobilis BC: exponantur fluxiones illæ tres per totidem lineas S, T, V: In BC producta cape CD æqualem S, erige normalem DE quæ sit ad T ut BC ad rRadium, et ipsi A age CF parallelam AB et æqualem V. Et eodem tempore quo fluxio lineæ BC si sola esset faceret punctum C transferri ad D, fluxio anguli B si sola esset ipsum transferri faceret ad E, et fluxio lineæ AB si sola esset, ad F. Duco igitur DG parallelam et æqualem CE et GH parallelam et æqualem CF, et concipiendo quod æqualibus temporibus punctum C sola fluxione prima transferretur de C in D dein sola secunda de D in G, postea sola tertia de G in H concludo quod omnibus conjunctis rectà perget eodem temporeis spatio de C ad H. Comple parallelogramma ECDG & CGFH si motus transferre valet punctum C de linea CE ad lineam DG et alius eodem tempore de linea CD ad lineam EG uter conjunctim transferent in diagonali CG ab utrius linea concursuus G. Et ex motu de C ad G et de C ad F rursus componetur motus in diagonali CD. Quare linea CD curvam puncto C descriptam tanget & puncti illius celeritatem seu fluxionem curvæ exponet. Duo Lotus CD, CE componunt motum CG in diagonali parallelogrami ECDG, et motus ille cum tertio motu CF componit motum CH in diagonali parallelogrammi FCGH. Erit igitur hæc linea CH tam tangens curvæ motu punctis C quàm expone fluxionis ejus.Rursus si a rectis decabus positione datis AB, AI ad idem curvæ alicujus punctum C conveniant lineæ duæ mobiles BC, IC, et ex cognitis fluxionibus anguli B et linearumBC(143)143BC, AI, IC determinandus esset motus puncti C motum quem punctum C haberet si linea BC solum modo flueret exporo quoad plagam et celeritatem per lineam CD;.dDein motum quem idem punctum haberet si linea angulus B solummodo flueret expono similiter per lineam etiam quoad plagam et celeritatem per lineam DG: Demi quia fluxio lineæ AB ignoratur, motum puncti C ab ea oriundum expono quoad plagam tantum per lineam indefinitam GH ductam parallelam AB. Et considero lineam GH ut locum ultimum indefinitum puncti C. Tum pergens ad alteram lineam mobilem IC a motum puncti C a fluxione lineæ IC expo oriundum expono per lineam CK et motum ejus a fluxione lineæ AI oriundum per lineam KL, motumet ultimò motum ejus ab ignota fluxione anguli I ex pono quoad plagam per indefinitam lineam LM. Cum igitur hi otus faciant ut punctum C ultimo locetur alicubi in linea LM, et priores motus ut ultimo locetur alicubi in linea GH, necesse est ut ultimo locetur in harum linearum communi intersectione H, adeorecta CH motum ejus exponet et Curvam ab eo descriptum tanget.Ne res difficilior est si detur fluxionum duarum summa fifferentia vel proportio. Detur fluxio anguli B et summa fluxionum linearum AB, BC. Motum puncti C a fluxione anguli illius oriundum expone per lineam CG, dein positionem rectæ OQ ea lege quære ut si summa fluxionum linearum AB et BC dividatur utcun in duas partes et motus duo puncti C qui ab his fluxioni partibus seorsim orirentur exponantur per quodplagam et quartitatem per per GP et PQ, punctum Q semper incidat in hanc rectam. Et erit hæc recta locus ille ultimus indefinitus puncti C cujus intersectione cum alio ejusdem ultimo loco ex alijs quibusvis datis inveniendo determinabitur tangens. Determinabitur autem positio rectæ OQ producendo GP ad C ut sit PO æqualis PQ et jungendo OQ.Qum etiam si plurium linearum fluxiones sigillatim ducerentur in cognitas quantitates et productorum aliquorum summa poneretur æqualis sum reliquorun, problema nihil minùs solvi posset. A mobili puncto C ducantur tres lineæ CA ad datum punctum A, CB et CD in datis angulis ad rectas positione datas BI ac DK, et fluxionibus earum respectivèductis(144)144ductis in tres datas lineas L, M, N æquentur producta duo priora producto tertio, nempe L'ac+M'bc=N'dc.. Et Curvæ a puncto C descriptæ tangens ita ducetur. Ad AC erigo normalem CE quæ plagam motus puncti C circa A gyrantis exhibeat, et rectis BI DK parallelas CF duco CF CG quæ plagas motuum puncti C secundum rectas BI DK exhibeat. Dein quæro punctum aliquod H ita est si ad CE ducatur HE parallela AC, et ad CF aut autem HF parallela BC et ad CG HG paralella CD, fiat L'HE+M'HCF=N'GH et fluxiones linearum AC, BC, DC motus relativi puncti C inde oriundi exponentur per lineas HE, HF, HG et motus absolutus puncti C per actam lineam CH. Hæc igitur erit tangens Cqurvæasita. Quomodo vero inveniri potest punctūum aliquod H ita a quo si ad rectas quotcum positione datas CE, CF, CG totidem aliæ in datis angulis ducgantur quarumagantur earum aliquæ ductæin datas lineas ductæ æquentur alijs in datas etiam ductis. patebit e sequentibus.Sunt et alij casus difficiliores, ad quæ pergere liceretsed ex his credo sensus et viset vis Theorematis satis constabit Quapropter pergo jam exemplis aliquot methodum hic propositam illustrareProponatur Hyperbola Ellipsis vel Hyperbola ADB cujus centrum sit G, vertices A et B,diameter AB ordinatim applicata ad diametrum CD, et latus rectum N et ex natura figuræ erit ut ACB ad CD utut AB ad N ita ACB ad CD2 et (ab datam hanc rationem) ita prioris fluxio prioris AC'cb+CB'ac, id est AC'cb−CB'cb seu 2GC'cb, ad posterioris fluxionem 2CD'cd. Exponantur (juxta Theor 7) fluxiones cb et cd per FC et CD ita ut FD tangens fiat Ellipseos ad Desconverietur 2GC'cbGC'FC et 2CD'cd in inGC'2FCG' et 2CD2: quorum ita dimidia sunt ut ACB ad CD2, adeo GC'CG et ACB æqualia sunt. Cape igitur FC ad AC ut CB ad CG ut acta FD figuram tangent in D.Propo(145)145In figuris hæc est methodus. Puncti mobilis considero semper motus diversos juxta diversas plagas quarum principalis sit via puncti. Et hos motus expono vel saltem exponi imaginor describendo per punctum illud circulum quemvis cujus centrum sit in via illa et in singulis plagis ducendo rectas us ad hunc circulum. Ut si punctum A moveaur in linea BA, per illud A decribo circulum quemvis cujus centrum sit in BA et cui illa BA aliæq lineæ CA quævis CA, DA, EA occurrant in F, G, H, I, et linearum partes intra circulus AF, AG, AH, AI erunt inter se ut motus puncti A in illarum plagis. Adeo ut si motus puncti A a B exponitur per AF, motus ejus a C exponatur per AG et sic in reliquis: Aut quod perinde est si fluxio lineæ ABA ex parte termini A exponitur per AF, aliarum linearum CA, DA EA ad idem mobile punctum A semper desinentium fluxiones ex parte termini illius A exponantur per AG AH, et lineæ EA defluxio per AI. Unde ex cogniis motibus duorum punctorum ad quæ linea quævis utrin terminatur, cognoscetur et exponi potest ejus fluxio absoluta: quippe quæ summa est fluxiorum ejus ad utrum terminum, vel excessus fluxionis ad unum terminum supra defluxionem ad alterum. PorroPorro motus punctorum circa polos quosvis ijdem sunt et easdem habent exponentes cum ac motus in plagis perpendicularibus ad radios. Sic motus puncti A cirum quemvis in linea CA situm exponens est normalis AK circulo occurrens in K. Expositis vereo duorum punctorum rectæ cujusvis motibus circumpolaribus, recta alia per terminos exponentium acta secabit rectam illam in Polo suo. Et per harum exponentium rationem ad radios id est ad distantias suas a Polo, exponere licebit motum angularem lineæhujus rectæ seu fluxionem angulorum quos hæc lineaea cum rectis positione datis continet.Et ut ex motibus punctorum invenire et exponi possunt fluxiones linearum et angulorum sec vice versa ex horum fluxionibus colligere licet motus punctorum. Nimirum considerando lineam AF in qua punctum quodvis A movetur et ut exponentem motus ejus, et exponentis illus terminum ulteriorem F ut metam ad quam punctum illud A tendit, et lineas omnes FA, FG, FH, FI per metam transientis ut loca rectæ ex inventione duorum ejusmodi locorum, meta quas utrasque inter intersectione est determinabitur. Loca verò si invenientur. Quando mobile punctum ex assumptione duarum quarum vis vel plurium determinatum et stabile redditur, inveniendus est motus ill quem punctum illud haberet si una quantitatum assumeretur et alterius tantum vel reliquarum fluxio maneret et motûs illius quoad plagam et quantitatem tam designasexponens ducenda est. Detero Cognoscenda est etiam plaga motus quem punctum idem haberet si vice versa illa una quantitas flueret et altera vel reliquæ assumerentur. Et in plaga illa per terminum exponentis acta recta loc erit metæ unus locis metæ.Quando motus puncti ex diversasdarbus vel pluribus quantitatum fluxionibus certas ac determinatus redditur, quarum una ignota est vel ut ignota spectatur, ex altera vel reliquis invenendus est motus ejus qualis foret si fluxio illaignota(146)146ignota esset nulla. Et si motus inventi exponens ducatur & per terminum ejus in plaga qua punctum vi solius fluxionis ignotæ pergeret, recta agatur, erit hæc unus e locis metæ. Metâ vero ex duobus locis inventa, simul habetur exponens motus quæsit. Hoc modo a motibus punctorum ad fluxiones quantitatus et vicissim ab fluxi harum fluxionibus ad illorum motus pergelicebit donec quoadus libuerit perventum sit. Et ubi exponens motus puncti curvam propositam describentis inventa est, hæc et cuevam in puncto illo tanget et exponens erit fluxionis ejus. Sed res exemplis clarior fiet.Exempla primaA mobili puncto A qua linea curva describit ad rectam KL positione datam in dato angulo ALK.Super recta KL positione data et ad datum punctum A terminata incedit recta AL in dato angulo ALK et termino suo A curvam lineam KA describit. Datur relatio linearus KL et AL ad invicem et recta ducenda est quæ cuevam hanc tangat in A.Quoniam punctum A ex assumptione fluentium KL et LA determinatur, postquam earum fluxiones exposin per LM et AN motum quem punctum A haberet si ALA assumeretur et fluxio solius KL maneret expono quoad plagam et quantitatem ducendo AD parallelam et æqualem LM, eo quod punctum illud A moveret per hanc AO eodem tempore quo KL evaderet KM. Deinde quoniam plaga motus puncti A foret AN si vice versa flueret ALA et altera KL assumeretur in hac plaga per exponentis terminum O duco indefinitam lineam OF et concludo hanc OF esse unum locum metæ. Et simili argumento quoniam AN exponens est tam motus puncti A ubi KL assumitur quam fluxionis lineæ KLA per terminum ejus N in plaga motus quem punctum A ex sola fluxione ipsitus KL haberet duco lineam indefinitam NF pro altero loco metæ. Et ad locorum concursus F meta erit ad quam tangens quæsita AF ducenda est. Quam conclusionem ut concinniorem reddas produc AF donec occurrat LK etiam productæ in Q, et ob similitudinem triangulorum QLA, AOF erit QL ad LA ut AO ad OF velseu LM ad AN adeo vice exponentium AG et AN ad LM et AN adhibe si possunt QL et LA, qu rationeeo utdeterminainvenietur punctum Q. Comple parallelogrammum KLAH cujus latus HK tangentem secet in I & ob proportionales AH, HI et LM AN , vice exponentium LM et AN adhibere licet AH, HI eo ut invenietur punctum I. Utrumvis punctum Q vel I proet commodum videbitur quære.(147)147A mobili puncto A qua curva quævis EA describitur ad rectas duas positione datas DB, DC in datis angulis ducuntur rectæ duæ AB AC et ductarum relatio ad invicem habetur. Ducenda est recta quæ curvam hanc tangat in A.Ut hot fiat exponentur ductarum fluxiones per AG et AH. Jam quia punctum A, assumptione fluentium DB, BA determinatur, et ubi earum una DB assumitur et altera BA solummodo fluit, linea AG exponens est tam motus puncti quam fluxionis lineæ BA, ubi vero vice versa altera BA assumitur el prior DB fluit punctum A movetur in plaga lineæ DB, recta GF quæ per exponentis terminum G in plaga lineæ DB parallela ducitur erit unus locus Metæ. Et simili argumento recta HF quæ per exponentis AH terminum H in plaga lineæ DC ducitur erit alias locus Metæ. Et locorum intersectio F metam dabit ad quam tangens quæsita AF ducenda est. Quam conclusionem sic concinnare licebit. Lecet tangens rectam DB in M et ipsi DC parallela agatur MN occurrens AC in N et AB, AM, AN erunt inter se ut AG, AF, AH, adeo vice exponentium AG, AH adhiberi possunt AB, AN: qua ratione longitudo AN at adeo punctum M ad quod tangens duci debet invenietur.Ut is relatio inter AB et AC sit quod rectangulum sub AC et data quavis recta R æqualessit quadrato AB2, æquales erunt etiam horum fluxiones R' ac & 2AB'ab.Hic Pro fluxionibus ac et ab substitue earum exponentes AG AH, AHvelpotius harum vice lineas AN, AB, et fiet R'AN=2AB2. Unde R'AN et 2R'AC æquales sunt utpote eidem 2AB2æquales; adeo AN=2AC. Cape ergo CN=AC&Per N ageipsi CD parallelam age NM occurrentem DB in M et raecta AM curvam propositam tanget in A.Hand secus si ad definendam relationem inter AB et AC ponatur R'AC−AC2 esse ad AB2 in data ratione, colligentur horum fluxiones R'ac−2AC'ac & 2AC'ab & 2AB'ab, et inde R'AN−2AC'AN in eadem ratione. Unde R'AC−AC2 & æqualia erunt, utpote eandem rationem ad AB2 habentia. Caeiatur ergo AN ad AC ut R-AC ad et, actâ MN parallelâ CD, habebitur tangens AM. QuodPorrò Curvæ EA hec sit propietas ut si inter datos duos circulos EL a dato circulo FK per data puncta P, Q ducantur rectæ duæ LI, LK, concurrentes ad idem curva ad dat circulum EL, ponatur AB æqualis LG et AC æqualis LK: ducantur circulorum tangentes IM KN, LR et fluxio arcus ER exponatur per LR cujusvis. longitudinis. Super diametro LR describatur circulus secans PL productam in S et QL in T, et erit LS exponens fluxionis rectæ PL et LT exponens fluxionis retrogradæ rectæ QL. Erigantur normales LV ad LP et LX ad LQ occurrentes(148)148occurrentes circulo LTR in vV et X et erunt hæ exponentes motuum puncti L circa polos P et Q. Erigantur etiam normales YI ad PI et ZK ad QK ita ut sit YI ad IP ut VL ad LP et ZK ad KQ ut XL ad LQ et erunt hæ exponentes motuum punctorum I et K circa polos eosdem P et Q. R punct AI IY et KZ erige normales YM, & ZN occurrentes IM et KN in M et N et erit YM exponens fluxionis ipsius PI eo quod æqualis sit segmento quoad parti lineæ PI productæ quæ intra circulum per puncta I et Y descriptam et cujus centrum sit in IM circa centrum constitutum in IM descriptum caderet. Et simili argumento NZ exponens est fluxionis retrogradæ ipsius QK Conciper per puncta I et Y circulum describi cujus centrum sit in tangente IM et pariter per puncta K et Z alium circulus cujus centrum sit in tangente KN, et horum circulorum diametri IM KN exponentes erunt motuum punctorum I et K in circumferentia circuli JK: item YM exponensæqualis erit exponenti fluxionis lunæ PI et N ZN æqualis exponenti fluxionis totius IL et TL+ZN exponens fluxionis retrogradæ totius KL. Cape ergo AG=SL+YM et AH=LF+ZN, sed obf fluxionem retrogradam ipsius LK vel AC cape AH ad partes ipsius A versus C, et HF acta parallela DC secabit GF actam parallelam DB in Meta F ad quam tangens quæsita AF duci debet.Quod si vice rectarum LI, LK adhibeantur circulorum arcus EL, FK ponendo AB æqualem arcui EL et AC æqualmem arcui FK, tunc AG sumenda erit æqualis arcui LR et AH æqualis KN, eo quod LR et exponens sit fluxionis arcus EL et KN exponens defluxionis arcus FK, et actæ GF, HF ut prius tangentem determinabunt. Ne problema difficilius esseteritsi vice circulorum EL FK adhibeantur aliæ quævis curvæ lineæ L quarum tangentes LR, KN ductæ habentur. Sed et alijs modis innumerris relatio inter AB et AC exprimi potest, imò et vice rectarum DB, DC curvæ quvis adhiberi ad quas quam tangentessent DB et DCAB, AC ducantur in datis plagis et quarum tangentes DB et ad puncta B et C sint DB et DC. et quad quas AB, AC ducantur i datis plagis.Ducatur verò jam linea DB, DC non in datis plagis sed ad data puncta B et C, et earum fluxiones exponantur per AG et AH. Et quoniam assumptione anguli ABC et longitudinis BA determinatur punctum A, et ubi angulus ille solummodo assumitur exponens motus puncti A est linea AG, ubi vero e tra angulus ille fluit et longitudo BA assmitur plaga motus puncti A perpendicularis est ad BA, recta GF in plaga illa per exponentis terminum G ducta erit unus locus metæ. Et simili argumento recta HF per terminum exponentis AH in plaga perpendiculari ad CA ducta erit alius locus Metæ(149)149Metæ. B Et meta erit in locorum intersectione FMetæ. Et meta in utro loco consistens erit in eorum inter sectione F, adeo AF ad intersectionem illam ducta curvam motu puncti A descriptam tanget in A.Ut si ea sit natura curva hujus ut summa vel differentia linea fluentis AB et datæ cujusvis R sit ad fluentem AC in data ratione (qui casus est quatuor Ovalium Cartesij) fluxiones illarum AB et AC erunt in eadem data ratione, adeo si in plagis fluxionum illarum capiāantur AG ad AH vel quod perinde est AN ad AC in illasi in plagis contrarijs capiantur An et AC in illa ratione et ad terminos captarum erigantur perpendicula concurrentia in F vel M acta AF vel AM curvam propositam tanget in A. Unde si ratio illa est æqualitatis (qui casus est Hyperbolæ et Ellipsis) tangens bisecabit angulum CAN.Ponamus jam super plano immobili in quo puncta P et K et recta infinita KD positione data habentur, planum mobile BCA motu parallela ita ferri ut puncta duo B, C in sBCA curva aliqua CA terminatum, ita ferri, ut puncta duo recta BC in eo data semper coincidat cum linea in recta KD, et interea secum trahere regulam PB per punctum suum B perpetuo transeuntem et circa polum P rotantem cujus ,& ejus intersectione cum termino suo curvilineo CA describere curvam lineam PAL in plano immobili, et requiratur hujus curvæ tangens ad punctum quodvis A. Quoniam assumptione recta KD et DA rectæ KC et curvæ CA deturminatur punctum A assumatur solummodo curva AC et sit CQ exponens fluxionis punct lineæ KC et huic parællela et æqualis AG exponens erit motus puncti A, et GF ductaoparallela tangenti curvæ AC ad linea in plaga motus quem punctum A haberet si vice versa KC assumeretur et curva CA solummodo flueret id est ducta parallela rectæ AD quæ curvam AC tangit in A, erit unus locus metæ. Rursus quoniam punctum A assumptione longitudinis KB et proportionis PA ad PB determinatur, assumamus solummodo proportionem illam et punctum movebit in linea AG erit motus ejus ad motum puncti B ut PA ad PB. Exponatur ergo motus ejus per AH quæ sit ad alterius exponentem id est ad CQ vel AG ut PA ad PB, et per punctum H in plaga motus (150)150quem haberet punctum A si vice versa KB assumeretur et ratio PA ad PB flueret, id est parallela PB acta recta HF erit alter locus metæ. Et Habetis autem duobus metæ locis habetur Meta in eorum intersectione F una cum tangente AF quæ ad metam duci debet. Quæm conclusionem si concinare animus est, produc tangentem donec secet BK in N, et ob similes figuras AFGH, NADB erit BN ad BD ut AH ad HG hoc est ut AP ad ABUt si Curva CA Parabola sit qu cujus vertex C diameter CK ordinatim applicata AI, (quo casu AL Parabola erit Cartesij)imprimis ducenda erit AD quæ Parabolam CA tangat in A quod fiet si capiatur CD æqualis CI, dein capiendà est BN ad BD ut AP ad AB et acta AN tanget curvam AL in A.Quod si AC circulus sit centro AB descriptus, quo casu AL Conchoides erit Veterum, erigenda est ad AP normalis AD occurrens BN in D, hàc enim circulum illum tanget. Dein capienda est BN ad BD ut AP ad AB. Vel brevius capiendaest BM=AP et erigenda normalis MN occurrens BD in N et acta AN figuram AL tanget in A.pag 130 post verb pag 130' post verba [ - Meritò dicentur ordinis ultimi] adde This edit was later made to the text on f. 130r by Newton.Genera Linearum ejusdem Ordinis Si linea aliqua oculo extra planum ejus sito spectetur per planum translucidum, et in plano illo locus ejus apparens vel (ut voce mathematica utamur) projectio notetur, erit linea projecta ejusdm ordinis cum projiciente. Si projiciens est recta projectio erit recta, si curva est quæ rectam secare potest in duobus vel pluribus punctis, projectios ejus projectionem rectæ in totidem punctis secare potest. Et hinc habita linea aliqua cujusvis ordinis possunt aliæ plures ejusdem ordinis inde derivari. Sic Veteres ex circulo derivarunt omnes secundi ordinis figuras et inde Conicas sectiones nominarunt, considerantes spatium illud solidium quod radijs per circuli spretati perimetrum transeuntibus terminatur ut conum quem planum figuræ projectæ secat. Stic et figuræ superiorum ordinum possunt omnes a simplicioribus quibusdam ejusdem ordinis figuris per successivas projectiones derivari, et inde distingui in genera coordinata pro numero figurarum quæ ad omnium projectionem requiritursufficiuntpositis illis ejusdem esse generis quæ ab eadem figura derivantur. Nam hæ omnes & solæ in se mutuò per projectiones transeunt et ea ratione cognotæ sunt, a cæteris verò in quas non transeunt alienæ. Hac lege unicum tantum est genus f linearum secundi ordinis, eo quod omnes derivantur a circulo: at ordinis tertij genera sunt quin.Exemplum in lineis terijtij ordis. In recta infinita EAB dentur puncta duo A, E et ad tertium quodvis ejus ppunctum B in datos angin dato angulo erigatur ordinata.BC ejus longitudinis cujus quadratum, si præterea dentur rectæ duæ M et N, æquale fuerit rectangulo sub N et AB una cum cubo ex AB applicato ad M . Et curva liniea ad quam hujusmodi recta omnis BC terminatur erit Parabola ata Parabolæ casus sunt quin principales; primus et simplicissimus ubi linea N nulla est: Secundus ac tertius ubi N negativè ponitur et præterea AE est , et secundus quidem ubi AE capitur ab A versus D seu versus alas figuræ, tertius verò ubi AE capitur ad contrarias partes ipsius A: Quartus et quintus sunt ubi AB est atrius, cujusvis longitudinis ubi Parabola illa secas lineam utrin puncto quintus verò tribus. Primo casu habetur Parabola Neiliana cujus uti longitudinem ubi Neilius noster primus invenit: secundo haetur, Parabola longitudinis parabola campaniformis ellipsin conjugatam ab punctum quam punctum conjugatumumhabenshabens conjugaum:puncto parabola cum jam formissolitaria quinto Parabola sum Ellipsihabens conjugataquæ si in punctum contragiturcons illud conjugatum in casu tertio. Et hæ qum figuræ cum profi cons qum fenera curvarum tertij ordinis quinta nulla ipsius generis prosicit aliquam alterius omnes verò quæ generis per successivas projecti in se mutuò curvæ omnes tertij ordinis comprehenduntur et eadem ratione curvæ superiorum ordinum subdistinguuntur in genera.(154)154Species Linearum ejusdem Generis. Quinetiam per casus Projectionum distinguuntur genera curvalinearum in species. Nominenuis planum illud Horizontem quod per oculum transit et plano figurælineæ projectæ parallelum est, et lineam illam Horizontalem in qua Horizon secat planum figurælineæprojicientis. Et linea omnis projiciens dubit tot projectionum species quot sunt casuspositionis lineæ Horizontalis casus respectu lineæ projicientisspecies positionum lineæ Horizontalis.ad respecta projicientis Si linea Horizontalis alicubi secat projicientem intersectio illa generabit in projectione cruræ duo Hyperbolicai generis cirva eandem Asymptoton ad oppositas plagas in infinitum tendentia, et linea quæ projicientem in puncto intersectionis tangit projiciet Asymptotonid ex eodem Asymptoti latere si intersectio sit in puncto flexus contrarij, aliter ex latere diverso, et Asymptotos erit projectio rectæ quæ curvam projicientem tangit in puncto intersectionis, tot ejusmodi crurum paria quot sunt in projectione quot sunt intersectiones lineæ Horizontalis cum projiciente. Unde curvalinea secundi ordinis non nisi duo paria crurum Hyperbolicorum habere potest, curva terlLinea tertij ordinis non trisi tria paria Linea quarti quatuor &c; et eiarum Asymptoti tris vel plures se secabunt in uno puncto si tangentes se secant in uno, bus vel pluribus; & si Projicitur senel vel secat in puncto deccrissatioris, duæ parallelæ erunt.Si Linea Horizontalis generunt crura duo Parabol generis ad eadem pla in infitum et concavis partibus picientia, insi ubi cont est puncto casa crura Parab ad mod celerum oppo et ex eodem latere concave erunt vertice centocurva alicujus quum obliquissimè tangit Projicientem seu ut projirièangulo contactus crura Parabolica ad plagas oppositas ut si di latere concavæ erunt, at si tangit ipsam angulo qui rectilineo æqualis sit(1155)155 sit contactas ille generabit crura duo Hyperbolica ex eodem latere ejusdem Asumptoti ad eandem plagam in infinitum tendentia.D Si linea Horizontalis et Asymptotos Projicientis crura Hyperbolica quæ circa Asymptoton illam sunt, convertentur in Parabolica: Et vice versa si linea Horizontalis tendit ad plagam crurum Parabolicorum crura illa convertentur in Hyperbolica. Omnia vero crura infinita quæ non tendunt ad plagam lineæ horizontalis in omni casu evanescunt.Si deni Linea Horizontalis linea transit per punctum conjugatum, generabitur curva linea cujus punctum conjugatum in infinitum abijt. Et ne punctum conjugatum infinite distans absurdum videatur scias projectiones hujus curvæ projectione curvæ haberæ puncta conjugata finitè distantia quæ sunt projectiones puncti illius infinitè distantis projectiones.At hæ sunt mutationes linearumli curvarum linearum quæ projectione fiunt: quarum casus omnes et eorum complexiones siquis ad curvam aliquam projicientem enumeraverit, is simul enumerabit figurarumlinearum species omnes quæ sunt ejusdem generis cum projiciente: saltiem si in lineis altiorum ordinum Projicius satis latè sumitur.Exemplum in lineis secundi ordinis ac tertij ordinis. Sic ubi Projiciens est circulus, Linea Horizontalis hunc circulum aut secabit in duobus punctis aut tanget in uno aut tota cadet extra circulum, et perinde Projectio aut quatuor habebit crura infinita crura Hyperbolica aut duo Parabolica aut nullum. Unde hujus ordinis tres erunt species Hypeborbla Parabola et Ellipsis præter Circulum. At in generibus linearum tertij ordinis casus sunt pluresIn primo GenereExemplum 1. Si oculus infinitè distat, vel si planum projectionis plano projicientis parallelum est, projectio erit Parabola ejusdem speciei cum projiciente id est Parabola cuspidata quam Neilianam quam etnominavimus. cuspidatam nominave licet.2. Si Linea Horizontalis Projicientem in vertice cuspidata secattransit per verticem cuspidatum Projicientis id in angulo contactus, Projectio erit Parabola corcumflexaWallisiana, habens crura duo Parabolica ad oppositas plagas in infinitum tendentia et ex diverso latere concava. et centrum in puncto flexus contrarij.3. Si linea illa Horizontalis transit per verticem cuspidatum et tendit ad plagam infinitorum crurum Projicientis Projectio erit Crux Hyperbolica cuspidaparbijnga ad Diametrum librata, habens duas Hyperbolas ex eodem habere unius Asymptoti ex diverso alterius. B veroParemvco lineam cujus area comprehensaitaæquales partes a recta aliqua ut ordinata ad ex utro distantis : non potest ita secariBrijngamLibratam vero vaco curvam quæ diametrum rectieineāamhabet ordinatas h ad ordinatas inde æquales terminantem: non quæ 4. Si linea illa transibis cuspidatum tendit ad aliam quamvis plagam: Projectio hyperbolica nonbrijngalineata habens Hyperbolas duas duarum duo crura ex diverso latere unius ad eandem plagam altera duo ex diverso latere deteriusproti ad plagas oppositas tendunt. 5. Si tendit ad plagam crurum infinitorum et Projicientem nec secat nec tangit Projectio erit Cissois librata, et uno casu Cissois Veterum(156)156Veterum6. Si tendit ad plagam crurum infinitorum et secat Projicientem in duobus punctis Projectio erit Hyperbola triplex librata cuspidadtæ. Hyperbolarum una quæ cuspidata erit jacebit extra angulum Asymptotorum alteræ duæ jaceb non cuspidatæ jacebunt intra.7 Si secat Projicientem in unico tantum puncto et non transit per cuspidem ejus Projectio erit Cissois circa Asympotom torta.8 Si tangit Projicientem extra cuspidem, at adeo in alio etiam puncto secat Projectio erit Crux Parabolica cuspidata. Ejus crura duo Parabolica tendunt ad eandem plagam et concavitate se mutuò respiciunt, in vertice verò non junguntur sed postquam convergendo unum eorum processit in cuspidem, divergunt denuò et ad plagas oppositas cruribus Hyperbolicis ex diverso latere ejusdem Asymptoti cruribus Hyperbolicis in infinitum tendunt.9 Si secat Projicientem in tribus punctis Projectio erit Hyperbola triplex cuspidata non librata. Hyperbolarum illa quæ cuspidata est jacebit extra angulum Asymptotorum suarum, altera jacebit intra, tertia uno crure jacebit intra altero extra.In secundo Genere.1. Si oculus infinite distat vel si plana Projectionis et Projicientis parallela sint, Projectio erit ejusdem species cum Projiciente id est Parabola nodosa.2. Si linea Horizontalis tendit ad plagam crurum infinitorum Projicientis et præterea secat projicientemnullibi secat nec tangit Projicientem nec secat nec tangit Projectio erit Cissois nodosa if librata.3. Sin Projicientem tangit in vertice Projectio erit Crux Parabolica nodosa librata4. Si secat eam inter vertiem et nodum projicietur Hyperbola triplex librata cum nodo in pari Hyperbolarum.5. Si secat alta nodum versus in ipso nodo, Projectio erit Hyperbola triplex librata duas ex tribus asymptotis parallelas habens.6 Si secat ultra nodum versus crura infinita Projectio erit Hyperbola triplex librata cum nodo in impari Hyperbola.7. Quod si linea Horizontalis non tendit ad plagam crurum infinitorum et secet occurrit Projicienti in unico tantum puncto, Projectio erit Cissois nodosa circa Asymptoton torta8. Si præterea tangit Projicientem inter verticem et nodum projicientur Crux Parabolica nodosa , non librata, clausa in vertice.911. Si secat eam bis ad partes nodi versus verticem et semel alicubi projicietur Hyperbolæ triplex non librati cum nodo in pari Hyperbolarum102. Si secat ad versus verticem et bis in nodo projicietur Hyperbola triplex non Asymptotos parallelas habens, Hyperbolam concavo habens et præterea in contrariomodosi modo linea Horizon secat Projicientem in ipso vertice.1113. Si secat eam bis in nodo et semel cum sum versus crura infinita Projectio erit Hyperbola triplex non librata duas ex Asumptotis parallelas habens et inter eas Hyperbolam ad easdem partes ommino concavam.(157)157Si secat unam Projicientis lineam in nodo et ibidem tangit alteram, Projectis erit Tridens Parabolica sive Parabola illa cujus proprietates Cartesisin Geometria, explimit.Si Secas nodumProjicientem in tribus punctis extra nodūumversus crura infinita, Projectio erit Hyperbola triplex non bifida nodum habens inHyperbolaSi secat ParabolasSi tangit eodem in crure infinito9 Si tangit projicientemeam bis nodo Projectio erit Parabola Carte.10. Si tangit eam ultra projectio erit Crux Parabolica nodosa, non librata sepvertice(158)158In tertio genere.1. Si oculus infinite distat vel si plana projectionis et projicientis parallela sint, projectio est, ejusdem speciei cum projiciente id est Parabola campaniformis cum puncto conjugato.2. Si Linea Horizontalis vel tendit ad plagam crurum infinitorum vel transit per flexum contrarium Projicientis et præterea transit ultra punctum conjugatum Projectio erit Concha librata punctum habens conjugatum ad convexitatum verticis.3 Sin transit per punctum conjugatum, orietur Concha librata cum puncto conjugato ad infinitam distantiam.4 Si transit inter punctum conjugatum et Projicientem Projectio est Concha librata punctum conjugatum habens ad concavitatem verticis5 Si tangit Parabolam Projicientem fit Crux Parabolica punctotis librata cum vertice aperto et puncto conjugato ultra verticem.6 Si secat ParabolamProjicientem inter verticem et puncta flexus contrarij fit Hyperbola triplex librata cum flexibus contrarijs in pari Hyperbolerum et flexu con puncto conjugato inter tres Asymptotos.7 Si secat Projicientem in pun utro flexu contrario fit Hyperbola triplex, trifariam librata, sine flexu contrario, cum puncto conjugato in centro trianguli Asymptotis inclusi, quod centrum est figuræ projectionis.8 Si secat ParabolamProjicientem ad alteras partes alterutrius vel utrius flexus contrarijrer fit Hyperbola triplex librata cum flexibus contrarijs in impari Hyperbola et puncto conjugato intertres Asymptotos.9 Quod si linea Horizontalis nec tendit ad plagam crurum infinitorum nec transit per punc flexum contrariūum;transit verò ultraper punctum congjugatum, fit Concha flexu contrario circa Asymptoton torta cujus punctum conjugatem in infinitum abit quæ insuper centrum habebit in flexu contrario ubisi modò linea Horizontalis secat Projicientem in Vertice transit per verticem Projicientis.10 Sin transit ultra vel citra punctum conjugatum et Projicientem secat in unico tantum puncto extra flexus contrariosfleProjectio erit Concha flexu contrario circa Asymptoton torta cum puncto conjugato ad finitam distantiam.11 Quod si transiens ultra vel citra punctum conjugatum tangit Projicientem projicientemEt habebiturautem verticem ex fluxi ultrariorūumprojicitur Crux Parabolica non librata aperta in versus Asymptoto conjugado12 Si deni secat Projicientem in tribus punctis projicitur Hyperbola triplex non libratacum puncto conjugato inter tres Asymptotos. Et una Hyperbolarum ultra Asymptotos sua altera secet uno ultraIn quartgenereSpecies 1. 2. 3. 4. 5. 6. 7. 8. 9 dem seritdescribuntur ac in Genere tertio species 1. 2. 5. 6. 7. 8. 10. 11. 12 respectivé, nisi quod projectiones hic non (habent punctum conjugatum. Et 4. 5. 6. 9 casus sunt implicissimi ubi tres Asymptoti in unico puncto concurrunt.(159)159In quinto genere.Species 1. 2. 6. 7. 8. 9. 10. 11. 14. 15 ijsdem verbis describuntur ac in Genere tertio species 1. 2. 4. 5. 6. 7. 8. 10. 11. 12 nisi quod loco puncti conjugati Ellipsis conjugata ponenda est.3. Si linea Horizontalis vel ad plagam infinitorum crurum tendeus vel per punctum flexus contrarij transiens tangit Ellipsin ad partesm exteriorem Projectio erit Parabola librata cum Concha quæ convexitate sua Parabolam respicit.4 Sin secat Ellipsin Projectio erit Hyperbola duplexduplex cum triplex librata, quarum Hyperbolarum una est Conchordatis inter alias duas sitacum concha interjecta: cujus casus est simplicissimus ubi tresse concurrunt in eodem puncto.5 Quod si tangit Ellipsin ad partes interiores seu versus Parabolam campaniformem Projectio erit Parabola librata cum concha quæ Parabolam concavitate sua Parabolam respicit.12. Si tangit Ellipsin et non tendit ad plagam infinitorum crurūumnec transit per flexum contrarium Projectio erit Parabola non librata cum concha flexu contrario circa Asymptoton torta.13. Si secat Ellipsin in duopbus punctis et alibi extra in tertio extra flexum contrarium; Projectio erit Hyperbola duplex non librata cum cConcha flexu contrario circa Asymptoton torta: et præterea centrum habebit in flexu illa contrario si linea horizontalis per tres figuræProjicientisvertices transit; quo casu tres etiam asymptoti per centrum illud transibunt.At hæ sunt species linearum tertij ordinis quarum formas et particulares proprietates conditiones fusius describere non operæ pretium duxi quoniam has ubi opus est Geometræ speculando formam situm et conditiones Lineæ Projicientis haud difficulter colligent. Malui propositionibus quibusdam poristeispaucis inventionemCaeterum qua ratione generaliores proprietates linearum inventis particularibus eruantur non pigebit paucis insinuaresgeneraliorum proprietatum linearum aperire. UConsidero igitur quod quæ conveniunt duabus linearum speciebus convenire solent generi et quæ conveniunt duobus generibus convenire solent ordini et quæ conveniunt duobus ordinibus observato progressionis tenore convenire solent ordinibus universis: demide quod combinatio duarum simpliciorum linearum quarūumordines conjuncti efficiunt ascendunt ad ordinem teneæ minus simplicis,superiorem vicem obire potest lineæ illius ordinis superiorissuperiorvicem obira erdie vicem obira obit generisgeneris linearum illius ordinis superioris. Ut Combinatio Duarūum rectarūumcombinat vicem obitbis generis linearum secundi ordinisU combinatio duarum linearum primi ordinis vicem lineæ secundi, combinatio trium, quatuor vel pluris vicem generislinearum Vertij quarti ut superioris ordinis Et recta & lineacombinatio unius linea primi superiores secundi ordinis vicem generistertijs ordinis et sic in lineas. Nam linea superioris ordinis sæpe transit combinationem simpliciorum et combinatio cujusvis ordinis recta linea tenævis ejusdem ordinis igitur proprietates combinatio incipienatus combinationibus rectarum simplicioribus, et in serie rectarumincipiendo a simplicioribus considero proprietates rectarum combinatorium in infinitorum deindein in proprietates circuli vel alterus cujusvis non curvæ : cum rectis in infinitum. Nam quæ duabus generibus combination combinationum consolent curvis per ordinis universosinveni, fieri vix potest conveniant lineis et linearum combinationibus universis.(160)160Porismata15 Si punctis a duobus datis punctis A, B, C ad rectam Dz positione datam inflectantur duæ rectæ Bz, Cz secantes rectam Ay ipsi positione datam sit Ayet parallelam Dz, habebunt Ax, Ay, xy datas rationes ad invicem.Est enim Ax.Dz∷AB.DB∷dat.dat et Dz.Ay∷DC.AC∷ dat.dat. Ergo aa 8. Dat. Ax.Ay∷dat.dat et bb 5 Dat Ay.xy∷dat.dat. Q.E.D.26 Si a duobus datis punctis A, B, C ad rectam Dz positione datam in punctum tertium z concurrant duæ recta Bz, Cz datus secantes rectam Ay positione datam in x et y et habeat Ax ad Ay datam rationem parallela erunt Ay Dz tanget punctum z rectam positione datam.Nam quia Ax est adAgatur enim per punctum Dz ipsi Ay parallela. Et quia Ax estDz∷AB.DB et Dz.Ay∷DC.AC et conjunctis rationibus Ax.Ay∷ AB×DC DB×ACAB.DB+DC.AC datur ratio AB×DC.DB×AC. datur ratio AB×DC ad DB×AC sed datur etiam ratio AB ad AC ergo datur ratio DC ad DB et divisim ratio DC ad datam BC at adeo datur punctum D. Datur autem puncto illa et angulo etiam angulus D et proinde rectasDz quam punctum z tangit datur positione. Q. E. D.37 Si a duobus datis punctis A B,C ad rectam positione datam Dz inflectantur rectæ duæ Bz, Cz secantes rectam Ay ipsi Dz parallelam in punctis x et y et detur ratio Ax ad Ay datur Ay positione.Nam ob parallelas Ay, Dz sit Ax.Ay∷AB×DC.DB×AC ut supra. Sed Ergo datur ratio AB×DC∷DB×CAC sed datur etiam ratio DC ad DB etrgo et ratio AB ad AC, ut et dat AB ad BC . Et inde of datam BC datur AB. dat Dato autem tum puncto A tum angulo BAy datur positione Ay. Q. E. D.478 Si a dato puncto B agatur recta Bzz secare parallela duas positione datas in x et z capiatur autem Ay et Ax in data ratione et jungatur zy converget zy ad datum punctum C.EsEst enim Ax ad Dz ut AB ad DB hoc est in data ratione et Ax ad Ay in data ratione adeoaa 8 Dat Dz ad Ay in data ratione ergosed estDC ad AC in data ratione est D ad Ayin eadem ratione ergodatdivisimratio DC ad AD dat datur et punctum ergoet indepunctum58 9. sdem positios dantur zy yC.Nam zy.yC∷DA.AC. 610. Easdem positios dantur ratione yxz, AxB, DzB, DAxz,DAyz, BzC, ByC, ACy seu Dz in datum.DPDPorism. 1 Si a datis duobus punctis B, C ad rectam DAz positione datam concurrentes rectæ secent in punctis x, y rectam Ay a dato puncto A ipsi BC parallelam ductam, erit Ax ad Ay in data rationeN Nam si Az producta occurrat BC in D erit Ax.xy∷DB.BC ∷dat.dat Q. E. D.Porism 2 Et si a datis duobus punctis B, C ductæ rectæ Bz, Cz secent Ay in data ratione, erit punctum z tanget rectam positione datam.Nam pro Age rectam zAD etoccurrrentem BC in D et erit Ax.xy∷DB.BC ergo ergo datur ratio DB ad BC. Ergo datur punctus D.Porism 3 Et si a dato puncto B agatur Bxz occurrens rectis positione datis Ax, Az in x et z detur autem ratio Ax ad xy. inclinabit zy ad datum punctum C. Est Per B Ipsi Azy parallela, agatur DBC occurrens rectis zA, zy in D et C. Ergo aa 30 Dat datur punctumlinea DB. Est BC.DB.BC∷Ax.xy .dat Ergo datur BC. Ergo datur punctum C. Q. E. D. Porism 4 Si a datis punctis B, C concurrentes rectæ Bz Cz secentin v et y rectas a datis punctis In rectas positioine datas et ipsi BC parallelas ductas Iv, ny Sit Iv ad ny in data ratione tanget punctum z rectam positione datam.Age BI occurrentem ny in L et erit Iv ad Lx ut IB ad LB hoc est in data ratione Ergo Lx est ad ny in data ratione. In eadem ratione capiatur KL ad Kn et erit erit Kx ad Ky in eadem data ratione. Ergo (per Porism 2) punctum z tangit rectam zK positione datam. Q. E. D. Porism. 11. Si a datis punctis A, B, C conveniuntconcurrant rectæ duæ Bz Cz secantes rectas positione datas Ax Ay in data ratione jaceant autem puncta ABC in directum, punctum z tangit rectam positione datam.CuCas. 1. Junge xy et triangulum Axy dabitur . Jam si xy parallela Se sit ipsi BC, junge xy secundum Az in E, etproduc xy ad E ut sit Ex ad xy et AB ad BC et concurrentesrectæ A puncto z Atqui datam Ax ad xy Ex ad datur ratio A Ex Ax. Ergo Ax et recta AE positione. Ergopositione datas. Q. E. D. Cas. 2. Sin perfectio. due vy ipsi BC paret ob rationem x ad xy vx ad xy dt ratios Aspeciefiguram Avx ratio Azv ad Avx In istarationes capeDC ad EB ad AB et debitur punctum ad DC ad AE facutsit AC ad EC nec non in rationemAEC ad Avut sit DC ad DB et dabitur punctum D. Ipsis Ax Ay age parallelas DC Dn occurrentesBz Cz in et n. Convertem rationem nomissimam et fiet EC DC∷AB.DB∷Ergo EC ad Ax.DO Ergo sit Av.Ax∷AC.EC et Ax.DO.∷.DC erit denægno Av.DO∷AC.DC. Sed in eadem ratione est Ay.Dn ergo Avn DOn . Ergo On parallela est BC. Ergo et ratio(1785)185 ratio DO ad Dn datur. Ergo (per cas 1) punctum z tangit rectam positione datam. Q. E. D.Cas 3. Si datur A ratio Ax ad Tay age AyT ipsi Tay parallelam et dabitur ratio AyT ad Tay: quippe quæ eadem sit rationi AC ad aC. Ergo datur ratio Ax ad AT. Ergo per cas 2 punctum z tangit rectam positione data. Q. E. D.Porisma 12DefinitioQ Magnitudo P magnitudine major est sui parte quam in ratione quando ablata sui parte, reliqua ad eandem habet rationem datamPorismaIysdem positis est si Ax est ad Aay ut datum data parte ipsius Ax auctum vel diminutum Aad datum, tangit punctum z rectam positione datam.(1786)186The pricked circle is the Moon according to the parallex of M.C. 46′.20″. And so the digits by the type are 11.43′. which were observed 11.22. The luminous part alwayes seemeth broader than it is.In this Type the Sun standeth as in the former, for the time is the same. Now because then was the greatest Observation, it is manifest that the ☾ was then at s, k not at q where the Tables place it. the Tables gô give the ☾ 9' too must in Longitude, as you may measure with your compasses in this Type.The following paragraph is written upsidedown at the bottom of the pageThe following paragraph is written upsidedown at the bottom of the pageEclipse of the Sun observed at Ecton A.D. 1652. marrs 29.h.10.32′. mins tempore apparente; sed tempore æqualis 10.26′. Digits eclipsed 11.22′.This type agreeth with the Tables of M. lunitia & the Rudolphim. the other type repesenteth the observationFor the Altitude of the SunUt rad. ab sim. anguli orientis: ita sinus distantiæ ☉ab ang a Decendente ad sim alt. ☉ 41.50′.In the same manermanner I find the altitude of the next superiour degree in the Ecliptic to be 41.54′. & the altitude of the next inferiour degree to be 41.45′. The one being 4′ more than the Suns altitude, & the other 5′ lesse; I take the meane &c. for the distance of the Almicenters, sc. of the Almic. of the Sun, & the Almic. that cutts the Ecliptic either one degree before the Sun, or one degree behind him. & this number I keepNow I trace a line (AB) for the moones Orbit. & because the Eclipse hapneth in the 9th degree from ♌I prick that degree behoren 8 & 9 from a scale of one degree, or from my Sector set for the purpose, by where I can measure with my compasses to the 6 part of a minute. The Lat. of 8°. is 43′.58″. I take it into my compasses from my Scale or Sector & setting one foot in 8 of the Orbit with the other I draw the arch about h. & the Lat. of 9° being 49′.20″ I take likewise & setting one foot in 9 of the Orbit I draw a second arch below the Orbit. about & by the outsides of these arches I draw the Ecliptic in his true situation.Then from 8 in the Orbit I let fall a perpendicular (8h) upon the Ecliptic. which perpendicular falleth short of 8 in the Ecliptic by the quantity of the Reduction, which here is h 8 being 2′.03″. set 8 therefore in the Ecliptic so much to the left hand (s.s.s.) from the perpendic. & 60 minutes furtherest 9.Then I prick the center of the Sun upon the Ecliptic 30′.34″. from 8 toward 9. I & I prick the ☾in her Orbit 41′.06″ from 8 toward 9.From the Sun measure one degree in the Ecliptic bd. Take in your compasses the . before ned for this purpose, & setting one foot in d, with the other draw the arch at e. & laying a ruler to the Sun & to the outside of this arch draw a strait line which shall be the Almicenter of the Sun. Then from the Sun raise a perpendicular at right angles with the said Almicenter, & it shall be the Azimuth of the Sun. & draw a parallell thereto through the Moones in her Orbit, & that shall be the Azimuth of the Moones.I measure from the Ecliptic downward in the Suns Azimuth so much as his parallax of altit.comes to (which here is 45″) & there set the apparent center of the sun (as at c) & there upon with his semidiameter 15′.12″ I draw his circle. Also from the Moones place in her Orbit herAzimuth I measure her parallax of alt. and (m n being 46′.20″. according to my Tables & where the parallax ends prick the moones apparent center (at n. ) & therewith her semidiameter (16′.10″.) describe her circle. you a ꝑperfect type in which you may measure with your compasses what you will and if you would know the posture of the Luminaryes an hours or half or quarter or before prick the points of 8 & 9 into another paper. & by these points draw then rectifie the places of the Sun & moone by adding or subtracting the here you must a orient angle, & altitude of the Sun the distance of the labour of 1191199ed=x=distantiæ solis a planeta Rad=gb=a. abg=medio motai. b=ghb sini velcosinui medii motus ab aplelio. af=q= diametro màximo ellipseos=be+ed. bc bd=c=distantiæ focorum. eb=q−x. qq−2qx+xx−zz−=xx−zz+2cz−cc. .. & therfor2bcq−2bcx=aqq+acc−2aqx. . . . . That is. . Therefore soe is cd to de. & is yethe secant of the angle edc.In yethe Ellipsis of yethe Earths motion, ad:df∷ diameter of ☉ at f: Diamet of him at a. &Or ad:de∷Diam of ☉ at e:Diam of ☉ at a &c. & by this meanes yethe foci of yethe Ellipsis may be found.Haveing yethe Aphelion viz ∠akl, yethe middle motion of yethe☉ viz ∠eba+∠akl & yethe☉s apparent place viz: ∠edp, taking any quantity for af to find yethe distances of yethe foci bd Nayethe given quantitys bg=rd=Rad=a. gh=b. rs=c. bd=x. eb=y. af=q. bh=e. ds=d. Then, . . . . . .Or if yethe angle (edb) bee right, & af=q. bg=a. bd=x. eb=y. gh=c. ynthen. . . . . . As for example if yethegreatest difference twixt yethe middle & apparent place of ☉ytthat is yethewhen he is at e ytthat is yethe ∠bhg bee 2degr..2′.54″.. The signe of it 357425, yethe cosine 9993609=gh, & yethe rad=gb=af=100000000. Then is & ∠edb=90 degrees.. ytthat is, af:bd∷10000000:178770. And this is yetheexactest way to find yethe Ellipsis of ☉. For in March & September when ☉ is about 90d2′ 54″ of his meane motion from his Apogæ heeplace may perhaps be observed to bee 90d from his Apogæ of his apparent motion. That is yethe ∠bgh=2d.2′.54″. when ∠edb=90d.Having yethe middle motion of a planet in its orbe viz , to find abe+fka. ba+ad=af=q=Radio. bh=b. . bg=a. be=x Then . . . . . Or . &, to yethe distance of a planet from ☉. also , & . Making af=bg=q=radio for brevitys sake, ynthan to yethesecant of yethe angle eda. Or thus, to yethecosine of yethe angle eda. Note ytthat after yethe first operacon yethe calculacōion will bee very short. for haveing once found 2cqq & cc+qq I call, cqq=m. & . Soe ytthat in all other operacons yethe wherein m & n vary not as in yethe same planetyethe equation is , soe ytthatyethe middle motion & consequente(b) being given sd the cosine of eda is readily found. By this meanes yethe☉s place in yethe Ecliptick may always bee found This equation may be ordered so ytthat n, or e be a decimall