Newton's Waste Book (Part 2)Isaac Newtonc.16,766 wordsThe Newton ProjectFalmer2013Newton Project, University of Sussex
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c. 1664 - c. 1685, c. 20,543 words, 38 pp.38 pp.Newton's Waste Book (Part 3) [MS Add. 4004, ff. 50v-198v]MS Add. 4004Newton's Waste Book (Part 1) [MS Add. 4004, ff. {cover}-15r]MS Add. 4004, ff. 15v-50r, Cambridge University Library, Cambridge, UKUKCambridgeCambridge University LibraryPortsmouth CollectionMS Add. 4004, ff. 15v-50rc. 1664 - c. 1685EnglandEnglishLatinHolographBarnabas SmithUnknown CataloguerNewton's NotebooksScienceMathematicsCatalogue information compiled from CUL Janus Catalogue by Michael HawkinsTranscription and encoding of 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 find yethe axes vertices Diamiters, Centers, or Asymptotes of any Crooked Line supposeing it have them.
Definitions. If is a crooked line, & touch it at yethe point . & if yethelinelines & all other lines wchwhich being parallell to one another & extended from one side of yethe croked line to yethe other and also bisected by yethe streight line . Then is a Diameter, & one of those lines wchwhich are ordinately applyed to it, & if yethe angles , , &c are right ones Then is the axis, & one of those lines wchwhich are ordinately applyed to yethe axis.
3 The Vertex of a crooked line is ytthat point where yethe crooked line intersect the diameter or axis as at 4 AnTheThe Asymptote of crooked lines are such lines wchwhich being produced both ways infinitely have noe least distance twixt ymthem & yethe crooked line & yet doenoe where intersect it. or touch it as , . 5 Those lines wchwhich are limited on all sides & have axes but as are Ellipses if yethe of yethefirst, 2d, 3d, 4th kind &c6 Those wchwhich are not ellipses & have axes but noe Asypymptotes are Parabolas of yethefirst, 2d, 3d, 4th kind &c. as .7 Those wchwhich have Asymptotes, are Hyperbolas of yethe 1st, 2d, 3d 4th kind, &c as (upon) whose asymptotes are , .8 There are some lines of a middle nature twixt aParab:Parabola & hyperb:hyperbolahave an asymtote haveing an Asymptote for one of its sides wchwhich are but none for yethe other as , one side haveing yetheasymptoeasymptote, yethe other side haveing none.10 If an Ellipsis have 2 axes (as & ) yethe longer is yethe transverse axis (as ) yethe shorter is yethe right axis (as ).If all those lines wchwhich are parallell to one of yethe diameters of yethe crooked line & are terminated by yethe crooked line be bisected9 If two diameters of yethe same Ellipsis be ordinately applyed yethe one to yethe other yethe shortest of them is called yethe right diameter, yethe longest yethe transverse one. () (as & ).If all yethe parallells wchwhich are terminated by yethe same or by 2 divers figures, are bisected by yethe same streight line, ynthen is ytthat line called a right diameter. But if That ytthat diameter wchwhich is ordinately applied to it (i.e. wchwhich is parallell to these parallells) is a transverse diameter.1 If all yethe parallell lines wchwhich are terminated by the same or by 2 divers figures, bee bisected by a streight line; ytthat bisecting line is a diameter, & those parallellines, are lines ordinately applied to ytthat diameter.2 If those parallell lines intersect yethe diameter at right angles ytethe diameter is an axisA line is said to bee of yethe first kind whIn an Ellipsis yethe point where 2 diameters intersect, is yethe center.The center of anfigureEllipsis is ytthat point where two of its diameters intersect. The center of two diers figures of toppositeHyperbolas is ytthat point where two of their diameters intersect one another or else where theretheir Asymptotes intersect.(16)
Propositions. The lines ordinately applied to yethe axis of a crooked line are parallell to yethe tangent of yethe crooked line at its vertex. Demonstr. Suppose a Parab & (being ordinately applied to yethe axis ) not parallell to yethe tangent but to some other line like . If bee understood to move towards continually decreasing untill it vanish into nothing at yethe conjuction of yethe points & . but sinc & since mustalways be equall to at yethe conjuntion of yethepointpoints & . it followeth ytthat cannot decrease so as to vanish into nothing at yethe same time wchwhich doth & therefore cannot allways be =equal to . Otherwise. if is not parallell to yethe tangent but to some other line as . Then doth not bisect all yethe parallell lines (as ) wchwhich are terminated by yethe crooked line . & therefore cannot bee its diam:diameter2dly. If is yethe axis of a crooked line & , is ordinately applied to . ytthat is if . Then must be found noe where of odd dimensions in yethe Equation expressing yethe nature of yethe line . For (supposeing to be yethe unknowne quantity) hath 2 valors & equall to one another excepting ytthatyethe one is affirmative, yethe other is negative. wchwhich two valors canotcannotbee exprest by an equation in wchwhich is of ododd dimensions for suppose . ynthen isis, or , since . & , since . & therefore is , or . soe if . ynthen is , & or but if . ynthen. but not . soe if. ynthenbut not . The same reason is cogent in compound equations. as if . Then, . where though yethe root is affirmative & yethe roote may bee negative yet they can never be equall in length, & though yethe 2 roots of an equation wchwhich differ in signes should bee equally long yet ytthat is when there is but one quantity considereyethe Equation is fully determined.Prop:Proposition 4th. If is of more dimensions in a quantity not multiplied by ynthen in one multiplied by it (as in ) ynthen is not parallel to one of yethe lines Asymptotes. & e contra. Otherwise & are parallel to yethe Asymptotes of yethe line. et e contra.PropProposition 3d. If is yethe Asymptote of yethe crooked line , & is coincident wthwithit, then & . then in yethe Equation (expressing yethe relation twixt & ,) must be of more dimensions in some one quantity onely where it is multiplyed by or : &c in any other quantity in yethe Equation & of as many. & if is an Asymptote to yethesameline , & bee parallell to it then must noe where be of soe many dimensions as in ifone quantity onely in wchwhichtis drawne into , , or &c.
Demonstracocion. If this be false ynthen suppose yethe equation expressing yethenature relation twixt & to be . first if is coincident wthwithyetheAsymptote ynthen,if (ytthat is if being infinite in length) it is, if (ytthatis vanisheth into nothing if be infinite. ) therefore tansposeingtransposeing into yethe place of , I find . Or that is , or . Now since vanisheth not it followeth ytthat is not coincident wthwithyethe Asymptote. But if I blot out one of yethe terme as So: if yethe equation be . ynthen by substituteing into yethe place of I have , or soe ytthat not vanishing but being infinite cannot be asymptote. But if yethe equation was . ynthen by writeing for , there results, . or . & therefore in this case is coincident yethe asymptote . Soe if , by making . it is . Or by extracting yethe roote . Soe ytthat hath 2 roots yethe one infinitly grategreatwchwhich is . yethe other is infinitely little wchwhich is .bee multiplied by wherever it is of its greatest dimensions. & if is an asymptote to yethe line , & be parallel to it, yethe & terminated by it at yethe point , then must be multiplied by wherever it is of its greatest dimensions. Example: Suppose . because in these 2 termes is of itits greatest dimensions; but in one of ymthem (viz: ) it is not multiplyed by therefore is not coincident wthwithyethe asymptote. If . then since is of its greatest dimens:dimensions in & onely & is drawne into in both of ymthemtherefore is coincident wthwithyethe Asymptote;I Also since is of its greatest dimensions in onely, d (wchwhichbl multiplied by ) therefore is parallel to terminated by an Asymptote. &c.Demonstr:Demonstration If is coincident wthwithyethe Asymptote ynthen when . i:e: is infinite when wisheth. Now suppose . ynthen if is . i:e: is but if . ynthen if it is . soe ytthat is finite & thereforecoincident wthwithyethe de of is likeOctober 1664 Haveing an the nature of a crooked line expresed in algebraicall termes to find its axies if iIishave Draw a line infinitely both ways fix upon some point (as ) for yethe begining of one of yethe unknowne quantitys (wchwhich I call . Then reduce yethe Equation to such an order (if it bee not already so) ytthat may be always found in yethe line . wthwith one end fixed at , & having moving making right angles wthwith it at yethe other end: ytthat end of wchwhich is remote from , describing yethe crooked line. wchwhich may bee always done wthwithout any great difficulty. As may be perceived by these examples. Suppose yethe given Equation was . & soe ytthat. . being perpendicular to & yethe angle being given, yethe proportion twixt & is given, wchwhich I supose as to . ynthen is c . . . & . . or . & . Or . Therefore I write for , & for in yethe equation , & soe I have this equation . wchwhich expresseth yetherelation twixt & , ytthat is twixt & , writeing therefore for , & for : I have this equation o Soe if turned about yethe pole & about yethe pole j describing yethe crooked line by yethe conjunction at yethe extremitys. & yethe equation expresing yethe relation wchwhich they beare to one another is . yethe distance of yethe poles is given wchwhich I call .perpendic:perpendicular to I draw & make . Then, is soe ytthat for I write . Or . wchwhich expresseth yethe relation wchwhich beareth to , & by makeing , , it is, .ExampExample 3d If be always in yethe line . & turning about yethe pole & passing by yetheend of wthwith its other end describes yethe crooked line , soe ytthat calling . ynthen drawing & perpendicular to . , & are given. & I make .. ynthen is . . . or by extracting yethe roote . againe . Or . & by transposeing toyethe other side & so squareing both ptsparts. wchwhich equation expresseth yethe relation twixt , & . & so by calling , & , it is, .Example 4th. if turnes about yethe pole , & (a given line ) slides upon wthwith one end & intersecting at right angles at yethe other end describes yethe crooked line by its intersection wthwith. then makeing , . & therefore or . & so by writing for & for , I have yethe relation twixt & exprest in this equation .Or if yethe relation twixt & was exprest in this equation (making . ) . then as before therefore . first therefore I take away by making . or by ordering it . Then I take away by substituteing itits valor into its roome & it will be . & by □ingsquareing both ptsparts. . & by writeing for & for yethe equation will be .The like may as easily be performed in any other case.After yethe equation is brought to this order observe ytthat if is noe where of odd dimensions ynthenyethe line in (wchwhich is coincident wthwith) is ꝑparte of an axis of y ethe crooked line, as in yethe 2d Example. And if is noe where of odd dimensions (as in this, )Then I draw from yethe point at yethe begineing of . I draw perpendicular to towchwhich is coincident wthwithyethe axis of yethe crooked line. And if neither nor bee of unequall dimensions in any terme of yethe equation then both & may bee taken for axes of yethe crooked line or lines whose nature are expressed by yethe equation. As in yethe 4th Example.But if is of odd dimensions in yethe Equation then ordering yethe Equation according to see if is of eaven dimensions in yethe first terme not found in yethe 2nd. if so take away yethe2nd terme of yethe equation & if there result an Equation in wchwhich is noe where of odd dimensions. Then I draw perpendicular to & equall to ytthat quantity wchwhich added or substracted from ytthat might take away yethe2d terme; through yethe point Pfbee .(17) As in this Example, . Then to take away yethe2d terme I make . & soe I have, . in wchwhich is not of odd dimensions. Then drawing for , for , & for : or wchwhich is yethe same (since ) I make that is I draw & on 2 contrary sides of yethe line . & through yethe point I draw parallell to & make it yethe axis of yethe line Example yethe 2d. . ytthatyethe first terme may beeof eaven dimensions I multiply yethe Then by making I take away the 2d terme. & yethe Equation . in wchwhich is onely of eaven dimensions. Then I draw for . for for, or wchwhich is yethe same (sincsince) I make , ytthat is I draw & on yethe same side of then through yethepoint parallell to I draw for yethe axis of yethe lines . In like manner, if is of odd dimensions in some terme of yethe Equation, yethe Axis perpendicular to may bee found. As for Example. . by makeing, I take away yethe 2d terme and soe have this equation . therefore I draw from yethe fixed point , & , or wchwhich is yethe same (since ) I draw , ytthat is I draw & on two contrary sides of yethe line ynthen throughyethe point , parallell to I draw yethe axis of yethe line Example 2d. . by makeing I have this EquatEquation. In wchwhich is noe where of odd dimensions. therefore assumeing for yethe begining of & making , & , or wchwhich is yethe same if I
I make , since ; that is if is affirmative I take & on yethe same side of yethe line . otherwise I describe ymthem on contrary sides of it. then through yethe point parallell to I draw an axis of yethe lines & . Againe I order yethe Equation according to & it is . & soe since is not in yethe 2d terme makeing . I take away yethe 2d terme, & it is . Therefore I draw , , & . or wchwhich differs not (since ) I make & through yethe point parallell to I draw for another axis of yethelines , & . But if but ifyethe unknowne quantity ( or ) is of odd dimensions in yethe first terme or if both yethe unknowne quantitys are in yethe 2d terme, or if by thisthese meanes yethe equation is irreducible to such a forme ytthat, or, , or both of ymthem bee of odd dimensions noewhere in yetheEquatEquation: Then try to find yethe axes by yethe following method. Observing by yethe way ytthatIf begins at yethe point & extends towards in yethe line ynthen is taken yethecontrary way towards , & all yetheaffirmative lines parallell to are drawne yethe same way wchwhich is but yethe negative lines parallell to are drawn yethe same way wthwithas if from yethe point I must draw a line , I draw it towards but if from yethesmesame point I must draw a line I draw it towards . soe if from yethe point I must draw , ynthen I draw it towards , but if then I draw it towards . Againe if is drawne toward from yethe line , ynthen is drawne from yethe same line yethe contrary way towards , & those lines wchwhich are affected wthwith an affirmative signe & are paralellparallell to they are drawne yethe same way wchwhich is but those lines wchwhich are negative are drawn yethe contrary way. as if then I draw a towards but if ynthen I draw it towards . soe if ynthen I draw it from towards , if , I draw it towards .A generall rule to find yethe axes of any line. Suppose . . & to be yethe axis. ynthen parallel to from yethe point to yethe axis draw . from yethe end of , dd perpendicular to draw . & make & suppose . ynthen is . & . . . therefore . Againe , that is . or for writeing its valor, . Now assumeing any quantity for , ytthat I may find yethe valors of & . I substitute these valors of & into theire roome in yethe Equation. as if yethe equation be . by making . yethe valor of is & yethe valor of is . wchwhich 2 valors substituting into their roome in yethe equation, theirthere results Now ytthat I may have an equation in wchwhich is of 2 eaven dimensions onely I suppose yethe 2d terme & soe have this equation & ytthatyethe terermes in this feigned equation may destiny one another I order it according to & soe suppose each terme . & so I have these equations & . by yethe first I find , or . by yethe 2d. & by substituteing yethe valor of into its roome I find . or therefore from perpendic:perpendicular to I draw . through yethe point paralellparallell to I draw & since therefore I draw . & lastly yethe points & I draw yetheaxis of yethe crooked line .But since there is noe use of those termes in wchwhichϩ is of eaven dimensions yethe Calculation will bee much abreviated by this following table.
. . . . . . &c . . . . . &c.For yethe first equation of yethe first sort. . . . . . &c . . . . . &c.For yethe 2d. . . . . . . . . . . &c. . . . . . &c.For yethe 3d. . . . . . . . . &c . . . . . . . .For yethe 4th. . . : . . . . &c. . . . . . . . . . . . .For yethe 5t. . . . &c . . . . . . . . . . . . . . . . . . . .For yethe 6t &c.. : : : . . . . . . . . . . . . . . . . . . . . . &c.For yethe first Equation of yethe seacond Sort. . . . . . . . . . . . . . . . . .For yethe seacond. . . . . . . . . . . . . . . . . . .For yethe 3d.. . . . . . . . . . . . . . . . . . (18.)The use of yethe precedent table in finding yethe Axes of crooked Lines, declared by Examples. Suppose I had this Equation given, . That I may find yetheaxis of yethe line signified by it, first I observe of how many dimensions one of yetheunknowne quantiesquantitiesor yetherectangrectangle of ymthem both is found at most in yethe Equation, (as in this Example they have noe more ynthan 2) then I take every quantiyquantity in wchwhich one of yethe unknowne quantitys is of or yethe rectangle of ymthem both is of soe many dimensions (wchwhich in this case are .) Then lookeing in yethe Table, (either amongst yethe rules of yethe first or 2d sort &c) for such a rule in wchwhichyethefirst quantity is of soe many dimensions I substitute yethe valors of yethe unknowne quantitys, found by ytthat rule, into their place in yetheEquationselected quantitys & supposeing yethe product , I find yethe proportion of to thereby, that is I find yethe angle wchwhichyethe axis makes wthwithyethe unknowne quantity called . As in this case I take yethe 2d Rule of yethe first sort, & by it I find yethe valors of. . . wchwhich valors substituting into yethe roome of yethe unknown quantitys in these selected termes . I have this equation. . or, . & so ytthat by assuming any quantity for as I have yethe valor of , for . therefore &c. In yethe next place ytthat I may find yethe length of yethe line . I take another rule whose first quantity is not of soe many nor of fewer dimensions ynthan one of yethe unknowne quantitys or yethe rectangle ymthem both is anysome where in yethe Equation. Then select every quantity out of yethe Equation, yethe valor of whose unknowne quantity may be found by this rule, & substituting their valors, found thereby, into yrtheir places in thisthese selected termes make yethe product . & find yethe valor of thereby. As in this example I must take yethe first rule of yethe 1st sort. By wchwhich I find , , : but yethe valor of ciannot be found by it. therefore I onely take yethe termes , & by substituting yethe valors of yetheunknown quantitys into their roomes I have . Then by substituting yetheabout found valors of & into yrtheir places, it is . Or . or . Soe ytthat if I make yethe beginning of , & to tend towards in yethe line , & towards perpendicularly to . then must I draw from yethe point perpendicular to ; & , & parallell to ; ynthen, & parallell to . Lastly through yethe points & draw yetheaxis of yethe line sought. Otherwise suppose :
it may be done thus . therefore I take , & through yethe points & . I draw yethe axis sought. Example yethe 2d. If yethe Equation bee . yethe Rule whose first quantity is of as many dimensions as either of yethe unknowne quantitys in thisEquation, is yethe 3d of yethe first sort or yethe first of yethe 2d sort. Selecting therefore onelyout of yethe Equation (since in neither of theserulesyethe valor of is found) by yethe 3d rule of yethefirst sort I find , . therefore yethe selected termes . & . Or, . In like manner by yethefirst rule of yethe 2d sort tis found . . & therefore . & . Or as before. Soe ytthat. therefore . Now ytthat I may find I take yethe 2d Rule of yethe first sort (whose first quantity is of fewer dimensions ynthan or but not of fewer ,) The quatitysquantitys in yethe Equation whose valors are expressed in this rule are , & for . . Soe ytthat I write instead of . soe ytthat. Or since , it is, . Had I taken yethe first rule of yethe first sort I had found . & . therefore wchwhich is right since . but by this equation hath other valors for or , & . &c. Whence observe ytthatfor yethe most ptpart it will bee most convenient to find yethe by ytthat rule whose 1st quantity hath one dimension lesse ynthanyethe first quantity of ytthat rule by wchwhichyethe proportion twixt & were found.(20)20Example yethe 3d. If yethe Equation be . being of 4 dimensions I take yethe 4th rule of yethe first sort, or yethe 2d rule of yethe 2d sort. By yethe 4th rule of yethe 1st sort I find & since by that rule I can find yethe valor of noe other quantity in yethe Equation I make . Or . Whose rootes are , , & . therefore either , or Which is divisible by , & , & by . therefore either ; or, ; or, . The operation is yethe same if I make use of yethe2d rule of yethe 2d sort. Againe I take yethe 3d rule of yethe 1st sort & by it I find, . . therefore .
& if . then & . therefore . Or . & if , then . or is infinitely long. but if . then . & if , then . Againe I take yethefirst rule of yethe 2d sort & by it I find . . . therefore And Now if . then Or Now if . or if , then yethetermes of this EquatEquation destroy one another soe ytthatyethe valor of may not be found thereby. but if , then I find . Or . Or . Againe I take yethe 2d rule of yethe 1st sort & by it I find . & if then , or If then . or . that is is infinitely long as was found before. If also it may bee found to bee , or but upon this supposition it was not before found & therefore is false, when . If . then I find . or . &c. If . then . . Which valor not being found before I conclude to bee false. Lastly by useing yethe first eq rule of yethe first sort I find, . & by supposeing I have . & . & if , then wchwhich being always found upon yethe supposition . I conclude yethe valor of to be & of to be . & so draw yethe axis parallell to & distant from it yethe length of . But here ofbserve ytthat this might have beene better performed by taking away yethe 2dterme of yethe Equation . Or as was observed before.(21)21November 1664To find yethe Diameter or axis of any crooked line which hath it Suppose yethecrooked line to bee , yethediameter or Axis , yethe undetermined quantitys describing yetheline to be , . to from yethebegining point (yethe begining of ), perpendicular to draw , cutting yethe axis in . paralellparallell to draw . & produce soe ytthat it intersect yethe axis in yethe point . & suppose ytthat is to . as to : or ytthat. & therefore . letbee one be one of those lines wchwhich are ordinately applied to yethediameter . & lastly suppose that is to , as to : ynthen is . & . then .. therefore . & draw perpendicular to yethe Axis & & by ordering yethe Equation it will bee & & by ordering yethe equation lastly suppose ytthat is to as is to : ynthen is ; & then . & . & by ordering yethe Equation it will be, . Againe, . & , or by substituteing the valor of into its ro or by substituteing yethe valor of into its place it is . And ytthat I may abbreviate yethe termes I make ; & ; & so yethe Equation is . Also by supposeing , I lessonlessenyethe termes of yethe Equation , by writeing instead .Now therefore by meanes of t substituting these valors of & into yrtheir stead I take ymthem out of yethe Equation expressing yethereltionrelation twixt ymthem soe ytthatynthen I have an equation expressing yethe relation twixt & . And to that end it will bee convenient to have a table of yethe squares, cube, squaresquares, square=cubs, rectangles &c of yethe valors of & , After yethe manner of ytthatwchwhich follows.. . . . . &c.. . . . . &c. . . &c. . . . &c As for example. If yethe relation twixt & bee exprest in this Equation, . then into yethe place of , , , , I substitute their valors found by this table, & there results . Which Equation expresseth yethe relation twixt & . ytthat is twixt & or . Now ytthat be yethe diameter of yethe line& & be ordinately applied to it, it is required (by Prop 2d) ytthat in this Equation be not of odd dimensions. & that may bee soe yethe quantitys in the 2d terme wchwhich is of must one another wchwhichcanotcannot be unlesse these quantitys destroy one another in wchwhichyethe unknowne quantitys & destroy oneare of yethesamedimensions dimensions. Which things being considered it will appeare ytthat I must divide yethe 2d terme into two ptsparts, makeing, ; &, . & divideing yethe first by & yethe 2d by they will be, . Hitherto useing useingyetheletters , , & for brevitys sake, I must now write their valors in theire stead (ytthatI may findyethe length of , & yethe proportion of to wchwhich determine yethe position of yethe axis, & also yethe proportion of to wchwhich determines yethe position of yethe lines applyed to yethe axis.) & soe insteadof yethe Equation ; there results, wchwhich by squareing both ptsparts is Or . Or . And by squareing both ptsparts it is& ordering yethe product it is. Or .. Which being divided by the product is . Wherefore I conclude ytthat. or ytthat. Againe by inserting yethe valors of & into yethe Equation there results . or by . & by squareing both ptsparts & ordering yethe product it is, . Which is divisible by , for yethe quote will bee . & therefore . Or, . Againe by inserting yethe valorsof & into yethe Equation, , there resulteth, . & by writing instead of & divideing it by there resulteth Or . Thus haveing found yethe proportion of to , & yethe valor of since theirethere remaines noe more equations by wchwhich I may find yethe proportion of to I concluded it to be undetermined, soe ytthat I may assume any proportion betwixt ymthem. As if I make . Then yethe angle is a right one & yethe axis of yethe line, & . & . or . that is ; As in yethe 1st figure. Or if I make . that is . or , then I find ytthat. that is that yethe diameter intersects yethe line at yethe point yethe begining of . & ytthatyethe lines are parallell to as in yethe 2d figure &c. Soe ytthat by assumeing any proportion twixt & , that is, supposeing yethe angles of any bigness, yethe position of yethe diamiter , may be found iafter yethe same manner. As If I would have yethe angle to be an anangle of degrees. ynthen must be double to , & . i.e. . & , therefore . I found before ytthat. or writeing yethe valor of in its roome, tis that is . Or since must be lesse ynthan it must . & since . As in yethe 3d figure. But if I would make yethe angle of degr:degrees then as before , & , & , or since must be greater ynthan tis , as in yethe 4th fig. &c.Example 2d. If yethe Equation expressing yethe nature of yethe line be .(22)22November 1664 To find yethe Axis or Diameter of any crooked Line supposeing it have ymthem.A Suppose ; ; yethe line whose axis or Diameter is sought; its axis or Diameter; its vertex; lines ordinately applied to its Diameter; a perpendicular to drawne from yethe point , i.e. from yethe begining of ; pteparteof yethe line intercepted twixt yethe diameter & ; a line parallell to & drawne from yethe to yethe intersection of & ; & & parallell to ; & a right angled triangle. &, .Then since therefore Then . &, . Againe . &, ; or, .Now therefore by substituteing into yethe place of , & into yethe place of , & theire □ssquares & cubes &c: into yethe place of , , &c. I take & out of yethe Equation expressing yethe relation twixt ymthem & Soe have an Equation expressing yethe relation twixt & . And to ytthat end it will be convenient to have a table of yethe squares, cubes, & rectangles &c: of yethe valors of & , like ytthatwchwhich follows.As for example if yethe relation twixt & bee exprest by, then in stead of , , , , writeing their valors found by this table there resulteth .Which equation espresseth yethe relation twixt & when any valors are assumed for , , , & . And if yethe valors of , , , & bee such that yethe 2dterme in is not of odd dimensions in any termeyethe Equation (that is ytthatyethe 2d terme of this Equation be wa) then (by Prop: yethe 2d) is ord(24)24 is ordinately applyed to yethe Diameter .Now ytthatyethe 2d terme of this Equation vanish it is necessary ytthat those termes destroy one another in wchwhichyethe unknowne quantysquantitysare & are not diverse nor differ in dimensions. Whence it appeares ytthatI must divide the 2d terme into 2 ptsparts making . & . Or by divideing the first of these by , & yethe 2d by . they are, , & . The first being divided by . there results, . Therefore one or both these propositions ; , is trew. by yethe 2d tis found ytthat. Now since by assumeing some quantitys for yethe valors of , , or I cannot find yethe valor of unless by yethe Equation . therefore I conclude . whence it is not necessary ytthat, or yethe proportion of to bee limited soe ytthat by assuming yethe angle of any bigness I may find yethe position of yethe axis . As if I suppose yethe angle to be a right one (i.e. ytthat is yethe axis of yethe line) then are yethe△striangles & alike, & therefore & . & . Or because . therefore . Soe ytthat I draw parallell to & equall& . There Soe ytthat I draw . & parallell to & parallell & through yethe points & I draw yethe axis of yethe line , wchwhich is a ParabParabola.as in figure 1stSo if I would have paralellparallell to i.e. yethe angleg of 45 degrees. then this evident ytthat. & . Threfore through yethe point I draw yetheaxis , so ytthat, as before. &c.& note ytthat since the axis is always paralellparallell to it selfe yethe line is a parabola. Example yethe 2d, . Being first to write yethe valors of & (found by yethe precedent table) into their roome, since I have noe neede of those termes in wchwhich is of eaven dimensions I leave ymthem out, & soe for I write onely . Then sorting these quantitys together in wchwhichyethe unknowne quantitys are yethe same there these 4 Equations (yethe 1st being divided by , yethe 2d by , yethe 3d by , yethe 4th by ) viz: ; ; ; . In yethe first Equation , I extract yethe cube roote & tis . or . In yethe 2d, , or . By the 3d, , or . & so by yethe fourth. Now therefore since . In yethe line from some point as perpendicular to I draw, & , both of ymthem. then from yethe points & through I draw yethetwo lines& bothwchwhich (since they it cuts one anotheryethe lines applied to them at right angles) is are axeis of yethe lines & wchwhich appeares al in ytthat, for therefore , soe ytthat & perpendicular to . Example 3d If yethe nature of yethe given line bee expressed in these termes . Then by supplanting yethevalors of & into theire roome & working as before, there will bee, . & &2dly. & 3dly. & 4tly, . The first of these divided by . is . Or □ingsquareingtis & ordering yethe product tis . Which being . there results . Wherefore I conclude one of these 3 to be yethe valors of viz: . Now ytthat I may know wchwhich of those is yethe right valor of I try yymthem singly, & first suppose ; If so ynthen by yethe 4thEquation , therefore . If , ynthen in yethe 3d Equation all yethetermes vanish except : therefore . & since , all yethe termes in yethe 2dEquatEquation vanish except except , therefore also , wchwhichsince it ought not to bee I conclude ytthat is false. Therefore I passe yethe 2d valor of , or, . & soe divideing yethe 4thEquatEquation by itresults . wchwhich is divisible by & by , Now ytthat I may know wchwhich is yethe right valor of first I suppose : & soe all yethe termes in yetheEquation vanish except, . or, .
& since , by yethe 2dEquation tis or . Which things sincethey agree I conclude ytthat, or ; ; . Since must be parallell to & must bee coincidentwthwith it. then yetheaxis I take some & fro perpendicular yethe Example yethe 4th. If yethe Equation bee . by takeing onely those termes (of yethe valors of & found by yethe precedent table) in wchwhichis of odd dimensions, & sorting those together in wchwhichare multiplied byyethesameunkowneunknowne quantitys are yethe same & of yethe same dimensions as before. there will result these Equations. first . 2dly. & 3dly, . yethe 1st is divisible by , . To know wchwhich of these 2 are yethe valors of first I suppose to be trew, & ynthen all yethetermes in yethe 2d Equation vanish except , or . by yethe 3dEquation vanisheth since & thereforeyethe valor of cannot beefound soe ytthat if I assume some valor for it as now since both & should never bee therefore I conclude ytthat is false & so pass to its other valor . or . & soe by yethe 2d Equation tis . wchwhich is divisible by , . If tis . & soe yethelines ordinat diameter will bee parallel to yethe lines ordinately applied to it wchwhich cannot bee therefore I try yethe other valor of . And if , ynthenyethe 3d Equation vanisheth & soe cannot bee found & is therefore unlimited. Now since I find noe repugnancys in these Equations , & , to be trew I conclude ymthemtrew. & since . I draw perpendicular from yethe begining of , wchwhich shall bee yethe Diameters of yethe lines & . then in ytthat diameter I take some point as or & from ytthat point draw or, i.e. of any length, & paralellparallell to . then from yethe pointe or perpendicular to I draw , or . that is, . & so through yethe points & or & I draw wchwhich shall be ordinate parallel to yethe lines ordinately applied to yethe Diameter .Example yethe 5t. Suppose . Then by selecting those termes out yethe valors of & in wchwhich is of od dimensions , & sorting them together in wchwhichyethe unknowne quantitys differ not, I have, ; ; & 3dly. by yethe first , & therefore yethe 2d vanisheth; & yethe 3ddivided by is, ; or . wchwhich may not bee since .
Now since , & yetheproprortionproportion of to &yethe length of cannot bee found tis evident yethe line hath noe axis or diameter.(25)25November 1664B Observe yethe Axes, Diameters & position of yethe lines ordinately applied to yemthemmay bee for yethe most pteparte easlier obteined by making . . . . . . yethe angles , , , , , right ones. . . . . Then for readiness in these operations make a table of yethe□ssquares, cubes, rectangles, &c of these valors of & . As was done before This line is a streight one yethe equation being divisible by Example If yethe relation twixt & be expressed by . then by inserting those quantysquantitys (of yethe valors of & found by this table) in wchwhich is of odd dimensions, into place of , , , in this Equation, & supposeing those to destroy one another wchwhich are multiplied by yethe same unknowne quantitys there will bee these 2 Equations , & ∼ . The 2d is divisible by & there ∼ results . Now to try wchwhich of these two are true first I suppose Endeavor not to find yethe quantity in these cases, but suppose it givenThere is a line connecting the end of this note to the following one, & soe yethe first Equation will bee . wchwhich is imposibleimpossible unlesse , & ynthenyethe valors of & canotcannot bee found, Therefore is false. And therefore by yethe 2dEquatEquation. & by yethe first . . & . Whence yetheproportion twixt & ytthat is yethe angle is undetermined, & & have double valors viz: when , then . And when , then . wherefore yethe line hath 2 axes.Or else C ☞ For avoyding mistakes
(wchwhich might have happened in yethe 4th Example where I found . & ) it will not be amisse to make . & . & soe it will be . & . Or, . & . And then observe ytthat it can never happen ytthat. or . observe alsoe ytthat if . ynthenyetheline is yethe axis, otherwise yethe diameter of yethe crooked line. when yetheaxis is perpendicular to from yethe point as also if : And ynthen it will be convenient to doe yethe worke over againe changing yethe names of & ytthat is writeing instead of & instead of .(26)26December Haveing yethe Diameter to find yethe Vertex of yethe line. Suppose , or . or . . . ytthat is . soe ytthat into yethe given equation I insert this valor of or of into yethe place of or (wchwhich may more readily bee done). As in yethe first example I found & yethe proportion twixt & to bee unlimited so ytthat if I would to bee yethediameter Axis I make . (vide C) or . & there I found . or sincsince it is . As may bee seene in ytthat example. Now ytthat I may find yethe vertex of yethe line was there exprest in these termes. . I suppose . that is . or . or . & writeing this valor of into its roome in yethe Equation ; there results . or . Therefore from yethe point I draw . & from yethe point I draw yethe perpendicular until it cutcuttyethe axis , ytthat is, soe ytthat. & yethe point shall bee yethe vertex of yetheParab:Parabola. Soe in yethe 2d Example of yethe line , it was found . & . & therefore . or . therefore I write for in yethe Equation . & it is . or . therefore I take & soe draw yethe perpendicular , which shall intersect yethe axis at yethe vertex of yethe crooked line. ☞ & ynthen (calling ) it shall be . Soe ytthat in this case . In yethe 3d Example yethe Equation being , It was found, . . ytthat is . therefore . Therefore by writeing instead of in yethe Equation all yethe termes vanish except ∼ , or . & . soe ytthatyethe vertex of yethe line must bee at yethe point . But in yethe 4th Example, . It was found . or . & was unlimited, I make therefore . & since for I make . Or . , Or , & yethe axis is perpendicular to therefore I insert yethe valor of into yethe equation & there results . or . Wherefore I conclude yethevertex of yethe line to be infinitely distant from towards . ☞ If yethe position of any line (as ) be given yethe point where it intersects yethe given crooked line may be found by yethe same manner; for suppose ∼ or . or . & . . . . angles , , right ones; ynthen, to find yethe point where yethe crooked line is intersected by yethe line , I suppose . ytthat is, . or . & since by yethe nature of yethe line . it follows ytthat. &. & by extracting yethe rootes of ymthem. both, . & . therefore I take . & .By yethe same manner yethe intersection by 2 crooked lines may be found.(27)27F Having yethe nature of any lines expressed in Algebraicall termes, to find its Asymptotes if have any Suppose , & yethe asymptotes of yethe line . & parallel to drawne from yethe Asymptote to yethe line . . . . . . . yethe angles , , , , , to bee right ones. ynthen is, . . . . & . Now for readiness in operation it bee convenient to have in readinesse a table of these valors of & wchwhich will bee yethe same wthwithytthat by wchwhichyethe diameters of crooked lines are determined. viz. This table may be continued when yethe nature of yethe lines are expressed by Equations of 4 or more dimensions. This like yethe former rules will be beste perceived by Examples ynthan precepts. AsExample yethe 1st. To find yethe asymptotes of yethe line whose nature is exprest by . first I write yethe valors of , & (found by this table) into theire places in yetheEquation . & there results . or by ordering it,. Or, . Now by assumeing any valors for ; , ; , . I have, by this equation, yethe relation wchwhich beares to , ytthat is wchwhichbeares to . But ytthatyethe valors of , , , & , may be such ytthat (to wchwhich is applyed) may be one asymptote of yethe line¶lellparallell to yethe other, it is necessary (by Prop:Proposition 3d) ytthat neither nor bee any where of soe many or of more dimensions, ynthen in those termes in wchwhichthey multiply one another. Therefore I consider of how many dimensions is at yethe most in any terme multiplied by , & how many is in any terme multiplied by ; & find ymthem but of one. & therefore conclude ytthat & ought to be found in noe terme in this equation ∼ unlesse where they multiply one another. GMoreover tis manifest ytthatyetheEquation (expresing yethe nature of yethe given line) will ever be of one & but of one dimension more ynthan or in some termes in wchwhich they multiply one another: & therefore this may bee put for a Generall Rule viz. All those termes must destroy one another in wchwhich there is not & wchwhich are of as many, or want but one dimension of being of as many dimensions as yethe Equation is. Now that these termes destroy one another, tis necessary ytthat those be in wchwhichyethe unknowne quantitys & are yethe same. Upon wchwhich considerations it will appeare ytthat in this example I must make, . 2dly, . thirdly. 4thly. Or by dividing ymthem by those quantitys wchwhichneede. they are . 2dly. thirdly, . 4thly, . by yethe 3d, . & since tis not , by yethe first . by yethe 2d. or . by yethe 4th. Therefore from yethe point I draw & . from I draw parallell to . from I draw , soe ytthat. & through yethe points & I draw wchwhich shall be one Asymptote ynthen or which is yethe same I make (since tis not ) I make. . Or, . & soe draw yetheasyptoteasymptote passing through yetheline points & . Againe if from some point in yethe Asymptote , as I draw & from I draw Then from yethe point I draw paralellparallell to & paralellparallell to soe ytthat (assumeing some other proportion twixt & ynthan before if there be any other) . & soe through yethe points & I draw yethe other Asymptote. Or since it is not ; I make . & soe through yethe points & I draw yethe other asymptote, wchwhich shall be parallell to .Example yethe 2d. SuposeSupposeyethe Asymptotes of were to bee determined, Since I have noe use of yethe termes in wchwhich is I onely select those termes out of yethe valors of , & in wchwhich is not & sorting them as was before taught I have these equations, 1st2dly. 3dly, . 4thly. by yethethird. by yethe4th. by the 1st. by the 2dExample yethe 3d, . by working as before I have these Equations . . . . . . Suppose . then by workeing as before I have these Equations, . . . .by yethe 2d. & soe by yethe 4th by yethe 4th. by yethe 1st. or . by yethe 2d(by suposeingsupposeing) tis : & by yethe 4th (by supposeing ) tis . But by yethe 2d (by supposing ) tis . & by yethe 4th (by supposeing ) tis . Whence I conclude ytthatwhen ynthen is , & ; & when ynthen& .(29)29December 1664 To find yethe Quantity of crookednesse in lines. Suppose . . . . & tangentssecants to yethe crooked line intersecting at .yethe angles , , , right ones. & let , be yethe relation twixt & . soe ytthat is a ParabParabola. Then . . . . . That is . Or, SqareingSquareing both sides that is ( by blotting out on both sides, divideing yethe rest by , & then supposeing to vanish) . Or therefore makeing . . . & describing a circle with yetheRadRadius, yethe circle shall have yethe same quantity of crookednesse wchwhichyethe Parabola hath at yethe point .Or thus. If . . . & perpendiculars to yethe crooked line wchwhich intersect at yethe point . . . yethe angles, , , right ones.SuposeSuppose, expresseth yethe relation twixt & . First I find yethe length of (sedefol:folium 8th hujus, or Des=Cartes his Geom:Geometry pag 40) wchwhich is . . .. Or . Out of these termes first I take away by writeing its valor in its romeroomewchwhich in this case is & there remaines results, . Then I take away either or (wchwhich may beeeasliesteasiliest done) by yethehelpe of yethe Equation expressing yethe nature of yethe line wchwhich is now . or . And there results . Now tis Evident ytthatwhen yethe lines & are coincident ytthat is yethe radius of a circle wchwhich hath yethe same quantity of crookednesse wchwhichyethe Parabola hath at yethe point . Wherefore I suppose & 2 ofyethe rootes of yethe Equation , to be equall to one another. & so by Huddenius his method I multiply it . & there results, . againe otherwise , & there results . Soe ytthat if . ynthen. . . then yethe circle described by yethe radius shall bee as crooked as yethe Parabola at yethe point Or better thus. All th Make . . . . . Then . .Or thus. Make . . . . . & . Thus in yethe former Example . . Or if out of yethe EquaTheorema The crookednesse of equall portions of circles are as their diameters, reciprocally.DemonstrDemonstration. The crookednesse of th any whole circle (, ) amounts to 4 right angles, therefore there is as much crookednesse in yethe circle as in . Now supposing yethe perimeter is equall to yethe arch , Then as yethe arch is to yethe circumference , soe is yethecrookednesecrookednesse of yethe arch to yethe crookednesse of yetheperimeter , or of . so is to .(31)To find yethe Quantity of crookednesse in linesDecember 1664. Suppose & perpendiculars to yethe crooked line , wchwhich intersect one another at . . . . . . & yethe angles , , , right ones. Then, . . Or .Haveing therefore yethe relation twixt & (as if it be ) first I find yethe valor of ( see Cartes Geom:Geometry pag 40th. or fol:folium 8th of this) (as in this Example tis ) by wchwhich I take out of yethe Equatioon, (& in this case there results .) then by meanes of yethe Equation expressing yethe relation twixt & I take out either or , wchwhich may easliest bee done (as in this example I take out by writeing in its stead & there results . Or squareing bioth ptespartes Or . & by squareing both ptsparts) Then if I assume any valors for &ytthat is if I determine yethe point , I have an Equation by wchwhich I can find all yethe perpendiculars to yethe crooked line, drawne from yethe point . for if I tooke out of yethe Equation, yethe rootes of yethe Equation will bee (& &c:) all such lines as are drawn from yethe points of intersection , , , , to yethe line (as , , &c) but if I tooke out of yethe Equation ynthenyetheroots of yethe Equation will bee those lines drawne from to yethe perpendiculars (as , , &c. Now by how much yethe nigher yethelines points & are to one another, soe much yethe lesse difference there will bee twixt yethe crookednesse of yethepteparte of yethe line , & a circle described by yethe radius or . And should yethe line be understood to move untill it bee coincident wthwith, taking for yethelast point where they ceased to intersect at theire coincidence, yethe circle described by yethe radius , would have yethe & yethegiven crooked line at yethe point , would bee alike crooked. And when yethe 2 lines & are coincident 2 ofyethe rootes of yethe Equation (viz & , if yethe Equation & ) shall bee equall to one another; WherforWherefore to find yethe crookednesse of yethe line at yethe point I suposesupposeyethe equation to have 2 equall rootes & so ordering it According D: Cartes or Huddenius his Method, yethe valor of any of these 3being given, yethe valor of yethe other 2 may be found. (as in this Example yethe valor of being given I multiply yethe Equation according to Huddenius method & it is Then by divideing both yethe numerators by & yethe denominatorsby , & so multiplying ymthemin crucem & ordering yethe product it is. .Now considering ytthat if , , & bee known, ytthat is, if yethe Ellipsis be determined, & yethe line given, there are onely two points in yethe line (viz: & ) to be considered. And yetheroots of this equationvalors of are (, , ) suchlines as are drawne from yethe line to yethe points where yethe perpendiculars intersect (as ) or to such points as of where two perpendiculars (as & ) ceased to intersect at theire coincidence into one (as & ). Therefore of yethe first roots I get yethe valor of yetheline. as this Equation by ytthat is ; there results . That is dividingit by ; . Which Equation expressethyethe length of yethe lines (, & ) wchwhich are drawne from yethe line to yethe points & at wchwhichyethe coincident perpendiculars last intersected one another before theire coincidence. Now haveing yethe length of or it will not be difficult to find, , or, ; for it was found before ytthatOr . Likewise it will not bee difficult to find or , for (supposeing ; ; ; ; ; ). . or ) it is, , Lastly yethecircle described wthwithyethe radius shall have yethe same quantity of crookedness wchwhichyethe Ellipsis hath at yethe point . Example yethe 2d. Were I to find yethe quantity of crookedness atsome given point of yethe line exprest by ; I might consider ytthatit differs from yethe former Example onely in ytthat there I have hereor , hiere , ytthat is in yethe former was negative in this is affirmative. Soe ytthat this operation will bee yethe same wthwithyethe former yethesigne of being changed soe ytthat it will be found or . &c as before. Example yethe 3d. Had I yethe Par In yethe Parabola, . & . In yethe above mentioned Equation I take out by writeing in its roome & it is . ynthen I take out by writeing in its stead . & by □ingsquareing both sides, . Which is an equation haveing 2 equall roots& therfore multiplied accordindg Huddenius his method soe ytthat be blotted out, & therere resultdivided by it is, . Now tis evident ytthatbeing determinddetermined there are 2 points (viz: & ) from wchwhich perpendiculars being drawne they intersect one another in yethe axis at , wherefore ∼ is one of yethe rootes of yethe Equation & therefore it being divided by , or by there results Or . Then into yethe above found Equation ∼ ∼ , I substitudt this valor of & there results . . Soe ytthat I have . And . & therefore shall be yetheRadRadiusof a circle wchwhich is as crooked as yethe Parabola at yethe point . Or it might have beene done thus, haveing yethe Equation , I might have prtwritwritten in stead of , & soe have had wchwhichmust have 2 equall roots & therefore by yethe Method demax:maximis & min:minimis I bottblott out & there results, . Or, . makeing , . . . . now if bee determined it is manifest ytthat there is but one point of yetheParab:Parabola (viz: ) to bee considered from wchwhichyethe perpendiculars wchwhich are drawne doe noe where intersect one another & therefore this equation hath not superfluous rootes like yethe former. Example yethe 4th. If it bee supposed ytthatyethe nature of yethe line is contained in. & if tis . . . & 2 perpendiculars to yethe crooked line, , & two points where yethe coincident perpendiculars last intersected . . Then is . by wchwhich I take out of yethe above named Equation , & yethe result being divided by , it is, Or (: Then I substitute this valor of into its place in Equation & there results . or by ordering it it will bee Which Equation must have (32)32two equall roots & therefore by ordering it according to Huddenius Method de Maximis& Minimis, I blot out yethe last terme & yethe result is . Or . By what was said before tis evident ytthat the perpendicular drawne from yethe line to yethe point where yethe two perpendiculars intersect, is one of yethe rootes of this Equation. And ytthat I may have a general rule to find yethe line (or had there beene 3 or more perpendiculars, to find all those lines wchwhich are drawne from yethe line to eachevery intersection of yethe perpendiculars) I consider ytthat if be not drawne from yethe line to yethe point of intersection ; ynthen hath two valors as but if they bee drawne to yethe point , ytthat is, if they be ∼ coincident wthwith; ynthen hath ctynthenyethe two roots of are equall to one ∼ another, being yethe same wthwithyethe line . Likewise if be drawne from yethe line to yethe perpendiculars , but not from yethe point where they ∼ intersect; then hath two roots (as , ) wchwhich will also be equall to one ∼ another & coincident wthwithyethe line , when is yethe same wthwith. This being considered; if I would yethe valor of , I must order yethe affore found Equation (in wchwhich was supposed to have 2 equall roots) according to & it will bee . wchwhich must have 2 equall roots & therefore by HudenꝰusHuddeniusMeth:Methodde Max:Maximis & Min:Minimis I take away yethe last terme & Soe I have, ; or, : But if I would have yethe valor of I order yethe Equation according to yethe letter & it is . wchwhich Equation must likewise have two equall roots & therefore takeing away yethe last terme there re by Hud:Huddeniusmeth:methodde Max:Maximis & Min:Minimis there resulteth this, . Or & this is one o ofyethe rootes of yethe Equation , wchwhich was required, therefore I must divide this equation by . ytthat is by , & there will result, . That is . Whence it will not be difficult to find yethe points & & ∼ consequently yethe lines , wchwhich shall be yethe radij of circles wchwhich have yethe same quantity of crookednesse yethe line hath at yethe points & . Makeing .Note ytthat these Equations have not or as one of theire rootes unlesse when yethe axis of yetheaxis line is eitherparalellparallell to ( for ynthenonely a circle whose center is at yethe intersection can touch yethe crooked line in both & together) & then perhaps they may easlyer bee found ynthen by yethe foregoeing rule.May 1665 1. Note ytthatyethe crooked line (described by yethe points & ) is always touched by the (perpendicular) ; & that in such sort as to bee measured by it,they applying themselves the one to the other, point by point; soe ytthat if the shortest of all yethe lines be substracted from there remaines . By this meanes yethe length of as many crooked lines may bee found as is desired2. Also if yethe line is applyed to yethe crooked line point by point, every point of yethe line (as ) shall describe lines to wchwhich (as ) to wchwhich is perpendicular. 3. The line is yethe same (if be a Parab:Parabola) Heura found. & perpendicular to the line , & a tangent & yethe position & point wthwithyethe tangent (as if they were inherent in yethe same body) while yethe tangent glyethe crooked crooked line , soe ytthatyethe point describe dag then from draw perpendiculars to yethe line (or then shall the point , perpendiculars intersect yethe po of yetheleastFeb 1664 The Crookednesse in lines may bee otherwise found as in the following Examples In the Parabola suppose yethe point where yethecrookednessscrookednesse is sought for, & ytthat is the center & yethe Radius of a Circle equally crooked wthwithyethe Parabola at . Then naming yethe quantitys . . . . ByetheBy yethe nature of yethe line . . . , That is, wchwhich Equation must have 2 equall rootes that may be ⊥perpendicular to yetheParab:Parabola& therefore multiplyed according to HuddenHuddenius's Method it produceth . Which Equation hath soe many rootes as there can be drawn perpendiculars to yetheParab:Parabola from the determined point . And two of thersethese rootes must become equall, ytthat may bee the center of yethe required Circle, therefore this equation is to bee multiplyed againe, & it will produce that is . Or : As was found in yethe 3d precedent example. Here observe ytthat in yethe 1st of these 3 Equations hath 4 valors , , & . see fig 2dwhen ; , & are determined. But , , & being determined hath but one valor . And if , , & bee determined ynthen hath 2 valors & . And , , & being determined hath 2 valors & as that first equation denotes byyethe dimensions of yethe quantitys in it. By the 2d of these Equations 2 of yethe valors of are united but not by yethe increasing or diminishing yetheRad:Radiusof yethe ci valor of &c. But 2 rootes one as first suppose yethe circle soe little as noe where to intersect yethe Parabola, it being increased gradually will first touch yetheParab:Parabola at (fig 3d) then ceasing to touch it it intersects it in 2 points & (fig 2d) wchwhich two points growingmore distant at last it untill it touch yetheParab:Parabola in (fig 3d) wchwhich being divided into two intersection points & (fig 2d) the points & draw neeredrnearer untill they conjoyne in yethe touch point & soe yethe circle ceaseth (by still increasing) to touch o is intersectyetheParab:Parabola or intersect it unlesse in & . Whence from one point may be drawne 3 perpendiculars , , , to yethe Parabola . And therefore in this 2d Equation must have 3 valors , , & , when , & , are determined then also hath three valors , , & .By yethe 3d Equation Two of the valors of in yethe 2d Equation are united by incresing or diminishing yethe length of . For begining at yethe point at the point (from wchwhichyethe3 perpendiculars fall upon , & ) if yethe point doth gradually move from , the perpendicular moves from towards Soe ytthatyethe two perpendiculars = & will at last conjoyne into one f, Which shall be yetheRad:Radius of a Circle as crooked as yetheParab:Parabola at .This 3d operation might have beene done by making determined & by = increasing or diminishing . That is by destroying yethe term in stead of in yethe 2d Equation. And so might yethe 2d Operacotion beene done otherwise by determining yethe circle , &Or taking or out of yethe 1st Equation instead of .Another way. There is another way of finding yethe crookednesse in lines & ytthat is not by supposing two perpendiculars ( & , or & ). but 3 intersections of a circle wthwithyethe figure, (fig 2d, , : or , & ). And then shall have 3 equall valors Or . As if (in yethelast example I had this equation . Supposing it to have 3 equall rootes by Huddenius his method tis . (Which equation doth not determine yethe perpendiculars to ) as doth for by . this I can find yethe valor of ( being determined) but by it I neither find yethe valor of nor till one of ymthem is taken out of yethe Equation). That Equation multiplyed yethedimensions of produceth Or . The same may be done thus. If a circle touch a crooked line at one point & intersect it er when two points come together ytthat circle to or As if . . . . . . may ever bee ). Then . Or .Or Or, wchwhich cannot Tby yethe perpendiculars by supposeing yethe circle described by yetheRad:Radius & to (33)33December 1664 Haveing found an Equa (by yethe former rule) an Equation expressingby whichyethe quantity of crookednesse in any line may bee found to find yethe greatest or least crookednes of any that line. In yethe34dth Example I had found . And by a rule there shewed viz : or writeing in stead of It was there found . Now by writeing in stead of & ordering yethe product according to yethe letter it is . Or extracting yethe roote it is Also by yethe nature of yetheline, . Therefore . Also . And Since Therefore ; Supposing The roote of yethe Surde quantity extracted the Equation is . Or . In wchwhich Equation yethe least valor of is to bee found & ytthat should happen when hath 2 equall valors or rootes. But because being determined can have but one valor yethe other 2 rootes being imaginary tis impossible ytthat it should have 2 equall rootes: Therefore I substitute y take away out of yethe Equation &by substituting its valor in its stead & there results . In wchwhich equation or being determined hath 2 valors & yethe other foure being imaginary & when is the longest or shortest that may bee then these two valors become one & then is yethe line more or least crooked. If therefore (ytthat's valors become equall) this Equation is multiplyed according to its dimensions there will result . wchwhich is divisible by , or by (for there results ). And if , ynthen is . Therefore I take & & atyethe point shall bee yethe least crookednesse.Here may bee noted Huddenius his mistake, ytthat if i some quantity in an equation designe a maximum or minimuumytthat Equation hath two equall rootes & wchwhichis false in yetheequation . & in all other equation'sequations which have but one roote.Or becauseAnother way. May 1665 Or because yethe lines & described by yethe points & doe touch one another point'spoints from wchwhich points onely lines drawne perpendicular to yethecrokedcrooked line will bee perpendicular to yethepoint of greatest or least point crookednesse: And also since all those are points of greatest or least crookedness fto wchwhich such perpendiculars are drawne: The difficulty will be to find yethe point . Now suppose ytthat be determined ynthen hath two valors for . And alsoe hath two valors for . Alsoe (when is not parallell to yethe axis of yethe line) hath two (or more) valors . wchwhich valors of , , or become equall if : by wchwhich meanes yethe point may bee found: Excepting onely when , , are parallel to yethe crooked line at ()ytthat is, parallellperpendicular to yethe streightest or most crooked ptespartes of yethe line . But if be determined, then (but if is parallel to yethe axis of yethe line yethe two valors of are equall & soe not usefull). Which valors of , , & become equall if : excepting onely when is perpendicular to yethe most streight or crookedptsparts of yethe line . As for example. In yethe precedent example it was found . But because or is parallell to yethe axis of yethe line, in ytthatEquation hath but one dimension. Therefore substitute either yethe valors of or of into their stead. As if I substitute yethe valor of into its place it will bee or . wchwhich must have 2 equall roots & therefore multiplyed according to 's dimensions tis . Or as before. But if I had substituted 's valor into its stead it would have beene which ving 2 equall roots being rightly ordered is . Or . Or . Or , as before.In yethe first Example of finding yethe quantity of crookednesse in lines yethe found . wchwhichmust have 2 equall rootes & therefore by Huddeniusmethod it is . wchwhich being divided by it is. Or, .That is . or Or ytthat if I take & yethe points yethe greatest or least crookednesse yethe line crookednesse wchwhich foundThese Equations superfluous rootes often as yethe perpendiculars The points of greatest or least crookednesse may bee yet otherwise found by an equation of 4 equall rootes. As in yethe example of yethe 2d way of finding yethe quantity of crookedness in lines it was found . wchwhich being compared wthwith an equation like it . by yethe 2d terme tis , or . In like manner by & . Soe ytthatyetheParabParabola at yethebegining is most crooked (at ).(35)35If yethe bodymove froyethe line & from yethe point two lines , bee drawne yethe motion of yethe body from is to its motion from as is to .Of compound force.Coroll:Corollary 1. The body receiving two divers forces from & & yethe force from is to yethe forcefrom as to , ynthen draw & , yethe body shall bee moved in yethe line . 2d Or if yethe body is suspended by yethethred & is forced from to & from towards , ynthen draw & , & make force from to force from yethe body nd in Equilibrio is .Coroll:Corollary 3d. yethe force of yethe body from is to its force from as to .39vide pag: 15. But here observe ytthat unlesse yethe reflecting line bee drawne from yethe through yethe point the center of motion in yethe whole body yethe determinacotion of yethe motion of will not be yethe same wthwithyethe determinacotion of yethe motion of before reflection (as in yethe first figure) but verge from it (as in yethe2d fig) ytthat is & will not bee parrallellparallell. For since yethecheifechiefe resistanceis in yethe line of yethe body is from its center of motion (prop 32) from towards , & not from towards , the body will find more opposition on ytthat side towards yethe center , ynthen on yethe other side towards & therefore at its reflection it must incline toward from yethe(axaxiom 120) & not returne in yethe line . But if yethebody presse towards ynthen presseth yethe body towards yethe contrary pteparteas from towards (axaxiom 119) & not from towards , if . But if the line pass through yethe point (as in fig: 1st) ynthen38 If yethe superficies (fig 3d) circulate all its points in yethe line move wthwith equall velocity from towards . For make . & & draw than is yethe motion of yethe point from to yethe motion of yethe point from as to . but (for △trianglesimilis△triangle therefore . also similis△triangle therefore . or, & ) therefore yethe motion of from is equall to yethe motion of from .39 If yethe body move reflect on yethe immoveable surface at its corner (fig 4th) its parallell motion (viz from to ) shall not bee hindered by yethe surface , (viz: if yethe center of 's motion were distant from yethe perpendicular an inch at one minute before reflection it shall bee sofarrfar distant from it one minute after reflection). For is noe ways opposed to motion parallell to it, & a body might upon it wthwithout looseing any motion, & if at yethe first moment of contact yethe body should loose its ⊥perpendicular & onely keepe its ∥parallel motion it would (perhaps) continue to slide upon it & not reflect.40 The body reflecting on yethe plaine at its corner all its points in yethe⊥perpendicular line shall move from yethe plaine wthwithyethe same velocity wchwhich before reflection they moved fro to it. For yethepoint (prop 9) moves wthwithytthat velocity bacwardsbackwardswchwhich it before did forwards (viz to ) & all yethe other points (prop 38) move wthwithyethe same velocity from it.(41)41MarchMay 20th 1665 A Method for finding theorems concerning Quæstions de Maximis et minimis. And 1st Concerning yethe invention of Tangents to crooked lines. Suppose . . . . . & . yethe nature of yethe line . Then is . . . Or Or . & since . Therefore . Or . Now ytthat may bee perpendicular to yethe line tis required ytthatyethe points , & conjoyne, wchwhich will hapenhappen when vanisheth into nothing. Therefore in the equation . Or , those termes in wchwhich is must be blotted out, & there remaines . wchwhich determines yethe perpendicular .Theoreme 1st Observation 1st Hence it appeares ytthat in such like operations those termes may be ever blotted out in wchwhich is of more ynthan one dimension.As if yethe nature of yethe line was . Then is since it is . That is . Also . or . Therefore . That is . That is (both ptsparts□edsquared& those terms left out in wchwhich is of more ynthan one dimension) Or . That is . Now if vanisheth ynthen is . And consequently .Observacocion 2d Hence I observe ytthat if in yethe valor of there be divers termes in wchwhich is then in yethe valor of there are those same termes & also those termes each of ymthem multiplyed according to yethe by so many units as hath dimensions in ytthat terme & againe multiplyed by & divided by . As if ∼ ∼ ∼ . Then, . Which operacotion may bee conveniently symbolized by (ordering yethe equation according to yethe dimensions of ) &) making some letter (as . . . . ) to signifie a terme, & yethe same letter wthwith some marke (as , , , , , , &c), to signifie yethe same terme multiplyed according to yethe dimensions of in it as in yethe former example (suposingsupposing. . .) The nature of yethe line is Soe if Then .A And as any particular Equation may be thus symbolized so divers equations may bee represented by yethe same caracters as may represent all equations in wchwhich is of one & two dimensionsNow if a generall Theoreme be required for drawing tangents to such lines it may bee thus found. , , , , , by supposition, . Then by observation yethe 2d, . Or, . Againe that is. . Which valor of put into its stead in yethe termes & in yethe former Equation the result is . And both ptsparts squared it is (by yethe first Observacocion) . WchWhich rightly ordered is . And since yethe points & conjoine to make a perpendicular therefore is & consequently . WchWhich is yethe Theorem sought for. As for example were it required to draw a perpendicular to yethe line whose nature is Then is or .In like manner to draw tangents to those lines in wchwhich is of 1, 2 & 3 dimensions suppose . Then is by 2d observacocion& by writeing yethe valor of ) in its stead in those termes in wchwhich not (viz there results . by it is . Or . That is .By yethe same proceeding of 1, 2, 3 dimensions in it would be found . &cAn universall theorem for tangents to crooked lines, when . Having yethe nature of a crooked line expressed in Algebraicall termes wchwhichare not put one pteparte equall to another but all of ymthem equall to nothing, if each of yethe termes be multiplyed by soe many units as hath dimensions in them. & then multiplyed by & divided by they shall be a numerator: Also if the signes be changed & each terme be multiplyed by soe many units as hath dimensions in ytthat terme& ynthen divided by they shall bee a denominator in yethe valor of .
Example 1st. If . Then . Example 2d. If . Then . Exam:Example 3d. If . Then . And by taking out of yethe valor of ynthen, .See Des Cartes his Geometry. booke 2d, pag 42, 46, 47. Or thus, . And .Note. That haveing given, it will be often more convenient to find by yethe equation expressing yethe nature of yethe line & ynthen having & to find by them both, Then to take out of 's valor & soe to find it by alone.The Perpendiculars to crooked lines & also yethe Theorems ∼ for finding them may otherwis more conveniently be found thus Supposing ; , , , . And if yethe bee supp distance twixt & , bee imagined to bee infinitely little, ytthat is if yethe triangle is supposed to bee infinitely little then . That is . Or .Now suppose yethe nature of yethe line bee . Then is In wchwhich equation instead of & write theire valosvalors & yetheresult is . Or . but these two termes , are infinitely little, ytthat is if compared to finite termes they vanish therefore I blot ymthem out & there rests .Suppose yethe nature of yethe line be Then (by observation yethe 2d) it is . Then writeing yethe valor of in its stead in these termes , There results . Or because yethe difference twixt & is infinitely little it is .An universall theorem for drawing tangents to crooked lines when & intersect at any determined angle And though yethe angle made by intersection of & is not determined whitherwhether it acute obtuse or a right one, yet may yethe line bee found as easily after yethe same manner wchwhich determines yethe position of yethetangnttangent. For suppose . , , , & ytthat. Then (supposing yethe distance of & to be infinitely little) it is, . Now if yethe nature of yethe line is Then is And by putting yethe valor of into its stead in those terms in wchwhichresults . Or Soe ytthatyethe variation of yethe angle makes no variation.Note ytthat the foundacotion of this operacotion of ytthat pag 131) tangents But since Equation is the sastons ytthat it would bee if (50)To draw perpendiculars to crooked lines in all other cases.Although yetheunknoneunknowne quantitys & are not related to one another as in the precedent rules (ytthat that is soe ytthat move upon in a given angle), yet may there be drawne tangents to them by yethe same method.As if is an ElipsisEllipsis described by yethe thred (as is usuall) Then make . . . And let yethe relation twixt & be . . Then is . . And consequently And. And . Againe suppose , , . Then is . That is . Alsoe . And Making . And consequently Or . A= Or . Lastly by yethe nature of yethe line . And . Or . And wchwhich valor substituted into yethe Equation A, the result is .. As ifThe rest of the page is damaged.