Fitzwilliam Notebook Isaac Newton c. 4,948 words Newton Project London 2006 Newton Project, Imperial College

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 1662-69, in English, c. 5,045 words, 34 pp. on 118 ff.

Miscellaneous notebook containing Newton's accounts for 1665-9, a series of increasingly complicated mathematical problems, and a highly revealing personal confession. At Whitsun 1662, Newton compiled a list of all the 47 sins he could remember having committed in his life, from stealing cherries to "threatning my [step]father and mother ... to burne them and the house over them". The accounts section charts the beginning of his study of alchemy in 1669, with purchases of books, materials and a furnace to equip the makeshift laboratory he set up in the grounds of Trinity College.

Described and partly published in Brewster (1855), 1: 31-3. The shorthand section deciphered and discussed in Westfall, 'Short-Writing and the State of Newton's Conscience, 1662'.

Contains expense lists, a confession of Newton's sins, and miscellaneous problems in mathematics and physics

Flyleaf inscribed 'Isaac Newton/ pret. 8d'. This is followed by a sequence of letters (the key to a cipher?), reading:

'Nabed Efyhik

Wfnzo Cpmfke'.

The book proper begins with shorthand notes on 3 pp., dated 1662, and detailing Newton's sins before and after Whitsunday of that year. Then follows a list of expenses, 7 pp., dated from 23 May 1665 to April 1669 (about 140 entries), including assorted chemicals, two furnaces and a copy of the Theatrum chemicum [ed. Lazarus Zetzner, 1659-61: H1608] bought in April 1669. On f. 10v another hand has listed the names of four German noblemen.

The other end of the book begins with 'Nova Cubi Hebræi Tabella' on 1 p., followed by various problems in geometry and the conic sections (ellipsis, parabola, hyperbole, etc.), with diagrams, 24 pp. On the back flyleaf in Thomas Pellet's hand: 'Sep 25 1727/ Not fit to be printed/ T Pellet'.

 34 pp. on 118 ff.

in English

 Fitzwilliam Museum, Cambridge, UK UKCambridgeFitzwilliam Museum SL210

Bought at the Sotheby sale by Maggs Brothers for £180 on 14 July 1936. By the end of the month it had been presented to the museum by the Friends of the Fitzwilliam Museum with financial assistance from Sir Thomas Barlow.

32

 1662-69 England English Latin Holograph Thomas Pellet Unknown Hand (1) Unknown Hand (2) Newton's Notebooks Catalogue information compiled by Rob Iliffe, Peter Spargo & John Young Tagged transcription by Linda Cross Checked against original by John Young Coding audited and updated to Newton v2.0 DTD by Michael Hawkins Coding reviewed and upgraded to Newton v2.1 DTD by John Young Updated to Newton V3.0 (TEI P5 Schema) by Michael Hawkins Catalogue exported to teiHeader by Michael Hawkins

pret 8d

NabedEfyhik,WfnzoCpmkfe
This and the following two pages are written in Thomas Shelton's shorthand notation and were deciphered by R.S. Westfall in 'Short-Writing and the State of Newton's Conscience, 1662', Notes and Records of the Royal Society 18 (1963), 10-16. Before Whitsunday 1662. Vsing the word (God) openly1Eating an apple at Thy house2Making a feather while oneon Thy day3Denying that I made it.4Making a mousetrap on Thy day5Contriving of the chimes on Thy day6Squirting water on Thy day7Making pies on Sunday night8Swimming in a kimnel on Thy day9Putting a pin in Iohn Keys hat on Thy day to pick him.10Carelessly hearing and committing many sermons11Refusing to go to the close at my mothers command.12Threatning my father and mother Smith to burne them and the house over them13Wishing death and hoping it to some14Striking many15Having uncleane thoughts words and actions and dreamese.16Stealing cherry cobs from Eduard Storer17Denying that I did so18Denying a crossbow to my mother and grandmother though I knew of it19Setting my heart on money learning pleasure more than Thee20A relapse21A relapse22A breaking again of my covenant renued in the Lords Supper.23Punching my sister24Robbing my mothers box of plums and sugar25Calling DerothyDorothy Rose a jade26Glutiny in my sickness.27Peevishness with my mother.28With my sister.29 Falling out with the servants30Divers commissions of alle my duties31Idle discourse on Thy day and at other times32Not turning nearer to Thee for my affections33Not living according to my belief34Not loving Thee for Thy self.35Not loving Thee for Thy goodness to us36Not desiring Thy ordinances38Not long longing for Thee in 39
40Fearing man above Thee41Vsing unlawful means to bring us out of distresses42Caring for worldly things more than God43Not craving a blessing from God on our honest endeavors.44Missing chapel.45Beating Arthur Storer.46Peevishness at Master Clarks for a piece of bread and butter.47Striving to cheat with a brass halfe crowne.48Twisting a cord on Sunday morning49Reading the history of the CnChristian champions on Sunday
Since Whitsunday 1662 1.Glutony2.Glutony3.Vsing Wilfords towel to spare my own4Negligence at the chapel.5Sermons at Saint Marys (4)6Lying about a louse7Denying my chamberfellow of the knowledge of him that took him for a sot.8Neglecting to pray 39Helping Pettit to make his water watch at 12 of the clock on Saturday night
5 1665 RdReceived 10li May 23d whereof I gave my Tutor 5li————5 . 0 . 0Remaining in my hands since the last Quarter ————3 . 8 . 4In all —8 . 8 . 4.PdPaid Iohn yethe Taylor————2 . 0. 0.PdPaid MrMaster Bychiner————0 . 3 . 6.To Caverly————0 . 1 . 0.To my Laundresse————0 . 0 . 6.To my Bedmaker————0 . 5 . 0.A paire of Gloves————0 . 2 . 0A paire of Stockings————0 . 5 . 4A hatband————0 . 2 . 0.PdPaid Goodwife Powell for my Laundresse————0 . 5 . 0.Given more to my Tutor————5 . 0 . 0My Iourney to Cambridge Mar 20.0 . 6 . 6.In all8 .10 .10.Lent MrMaster Newton————0 .18 . 0
March 25 1666. Lent Wilford——X——0 . 1 . 0.To yethe Poore on yethe fast————0 . 1 . 0.To MrMaster Babintꝰtons: Wom, 6d. Porter 6d————0 . 1 . 0.Spent wthwith Rubbins 4d.————0 . 0 . 4Lent to SrSir Herring————1 . 6 . 0.Lent to SrSir Drake————1 . 0 . 0.Payd my Laundresse————0 . 5 . 6.ffor a paire of shoos————0 . 4 . 0.Caverly————0 . 0 . 4.
6 Payd Iohn Falkoner————0 .11 . 6.A paire of shooestrings————0 . 0 . 8.Payd my Bedmaker0 . 5 . 0.Dew from Iohn Euans————0 . 1 .10.EuansThe summe of my expences1 .10 . 4.+8 .10 . 10In all10 . 1 . 2Dew to mee————3 . 5 .10More from MrMaster Guy————0 .10 . 0Lent In all————3 .15 .10.
1666. RdReceived 10li March 20th————10 . 0 . 0Remaining in my hands————8 . 8 . 4.In all18 .8 . 4Expences & wtwhat I lent deducted yethe rest is————4 .11 . 4.1667 Apr 22 Received10 - 0 - 0In my hands besid debts14 -11 - 4dMy Iourney to Cambridg0 - 6 - 6.Two paire of shoos————0 - 8 - 0A Cap——X——0 - Cloths & dying & mending 0 - bordering twice——X——0 - Lynings0 - 6 - 6Lath & Table————0 -15 - 0Iron worke for it0 - 9 - 0
Drills, Gravers, a Hone & Hammer & a Mandrill0 . 5 . 0A Magnet————0 .16 . 0Compasses————0 . 3 . 6Glass bubbles————0 . 4 . 0Chappell Clarke————0 . 2 . 6My Bachelors Act————0 .17 . 6.At yethe Taverne severall other times &c————01 . 10 . 0Spent on yetheMy Couz Ayscough0 .12 . 6..On other Acquaintance————0 -10 : 0Shoos————0 . 4 . 0Cloth 2 yards & buckles for a Vest.2 . 0 . 0ffor Woosted Prunella 78yds 12.1 . 5 . 6ffor yethe lining 4yds————0 . 9 . 4Philosophicall Intelligences0 . 9 . 6.yethe Hystory of yethe Royall Ssoc:0 . 67 . 0.Shoe Strings————0 . 1 . 0To Goodwife Powell————0 . 7 . 6To my Laundresse————0 . 8 . 6To Caverly————0 . 1 . 6To the Glasier————0 . 1 . 0New fire cheeks & pointing yethe chamber & windows————0 . 1 . 6Gunters book & sector &c to DsDominus ffox0 . 5 . 0Letters, wyer, files, boats,————0 . 2 . 6.ffor a ffellows key————0 . 1 . 0A Cap turning————0 . 1 . 4.To the Taylor Octob 29. 1667.————2 .13 . 0To the Taylor. Iune 10. 1667————1 . 3 .10For keeping Christmas————0 . 5 . 0Lost at cards at twice0 .15 . 0
7 At yethe Taverne twice————0 . 3 . 6. 612 sacks of coales, carriage & sedge————0 .11 . 0 Shoos & mending————0 . 4 .10. Two paire of Gloves————0 . 5 . 0 wthwith MrMaster Lusmore, Hautrey, Salter0 . 3 . 6 Received of my Tutor wchwhich I lent Perkins0 .10 . 0 I went into yethe Countrey DecembrDecember 4th 1667. I returned Ian to Cambridg Feb 12. 1667. Received of my Mother————30 . 0 . 0 My Iourney————0 . 7 . 6 ffor my degree to yethe Colledg5 .10 . 0 To yethe Proctor————2 . 0 . 0 ffor 3 Prismes————0 . 3 . 0 4 ounces of Putty————0 . 1 . 4 To yethe Painter————0 . 3 . 0 To yethe Ioyner————1 . 1 . 8 Lent to DsDominus Wickins——X——1 . 7 . 6. To yethe shoe maker————0 . 5 . 0 Bacons Miscelanys————0 . 1 . 6 Expences caused by my Degre0 .15 . 0 Subscribing 6d, Reading Græke.0 . 5 .10. A bible binding————0 . 3 . 0. Humphrey 1668————0 . 1 . 0. 18 yards of Tammy for my MrMaster of Arts Goune1 .13 . 0 Lining —— 3, 6————0 . 23 . 6. Making ytthat & turning my Bachelors Goune————1 . 0 . 6. Received of MrMaster Io: Herring0 .10 . 0 Payd my Laundresse————0 . 5 . 6. Payd to Caverly————0 . 5 . 6. Payd Gododwif Tablbot from Feb 12 to Mar 25 16680 . 2 . 6 Payd to my Laundresse0 . 2 . 6. To yethe Porter————0 . 5 . 6. ffor oranges 1667 for my sister————0 . 4 . 2. Bedmaker & Laundresse0 .10 . 0. Shoemaker————0 . 5 . 8. A Hatt————0 .19 . 0. Taverne0 .10 . 0. Carpets of Neats Leather0 .18 . 0 AMy part of A Couch.0 .14 . 0. 1 Bowling Greene————0 .10 . 0 To MrMaster Ieffreys for a Suit3 . 6 . 0 A Tickin for a ffeatherbed.1 .10 . 0 New ffeathers————0 . 8 . 0 A Hood————1 . 3 . 6. Making &c of my last suit————1 .11 . 9 8 Dew to Iohn Hauxy——X——1 .10 . 0. Spent in my Iourney to Londōon65 .10 . 0 As also 4li 5s myore wchwhich my Mother gave mee in yethe Country4 . 5 . 0 Received for Chamberrent1 .11 . 0. Received from my Mother11 . 0 . 0. I went to London on Wednesday Aug 5t & returned to Cambridge on Munday Sept 28, 1668. Bedmaker & Laundresse0 . 4 . 0 Lent DsDominus Wickins——X——0 .11 . 0 Lent MrMaster Boucheret——X——0 . 5 . 0
Aprill 1669. Lent to MrMaster Wadsley————0 .14 . 0 16 yards of Stuffe for a suit2 . 8 . 0 ffor making &c————1 .13 . 0 For turning a Cloth suit1 . 3 . 03 For shoe strings &c————0 . 2 . 0 For Glasses in Cambridge0 .14 . 0 For Glasses at London————0 .15 . 0 For Aqua ffortis, sublimate, oyle y erbe, fine silver, Antimony, vinegar Spirit of Wine, White lead, Allome Niter, Tartar, Salt of Tartar, ☿2 . 0 . 0. A ffurnace————0 . 8 . 0 A tin ffurnace————0 . 7 . 0 Ioyner————0 . 6 . 0 41 Theatrum Chemicum————1 . 8 . 0 Lent Wardwel 3s & to his wife 2s————0 . 5 . 0 Carrriage of yethe oyle————0. .2 . 0 Payd I Stagg————0 .18 . 6 Payd yethe Chandler————0 . 8 . 0 A Table cloth————0 .10 . 0 Six Napkins————0 . 6 . 0 40 Friedadolff Lewys Langerhanss. Carolus gGottlob de Theler Georgius Bernhardus de Theler Equites ex Superiore Lusatia Iohannes Christophorus Ritter Wurcenâ-Misnicus. The following material is written from the opposite end of the notebook. Sep. 25 1727 Not fit to be printed T Pellet Nova Cubi Hæbræi Tabella There follows a table of Hebrew characters with Latin annotations. De Triangulis rectangulis. Of right angled triangles. h = hypotenusa. b = basis. c = Cathetus. p = perpendicular. hdc = diff: hypot & Cath bdc = diff: basis & cathet: bdh = difference basis & hypot dsh = diff: seg: hypoten: sh = segment: hypoten: bh = greater seg hyp: ch = lesse seg: hypot: I. Any two leggs given to find yethe other 1. bq + cq = hq. 2 r: khq - bq: = c. 3 r: khq - cq = b Ought. Eucl. lib 1. pr: 47. 4 b x ch = II yethe b. c. & h given to find p. 1. b x ch = p EucidEuclid 6 .8. III c. h. p. given to find dsh. 1. H - 2r: bq - pq: = dsh. IIII. b. p. h given to find dsh. 1. 2r: bq - pq: h = dsh. V. b. c. h given to find dsh. 1. H - 2r: cq - Q: b x ch: = dsh. 2 2r: bq - Q: b x ch: + h = dsh. VI b.c or b. h or h. c given to find bp: 1 b x cr: bq + cq: = p 2 b x r: hq - bq:h = p. 3 c x r: hq - cq:h = p VII b. h. or c. h. or b. c given to find dsh. Theorem 1 As yethe difference twixt yethe base & cath (in rectang: triang:) is to yethe greater side:: so is yethe difference of yethe segmsegment of yethe base; to yethe greater segmntsegment of yethe base & perpendicular. Theorem 2. As yethe difference twixt yethe base & cathetus to yethe less side:: so yethe diff of yethe segmtssegments of yethe base to yethe lesse segment of yethe base & perpendicular Theorēem 3d. base – Cathetus: hypotenusa:: :: greater segsegment: base - less seg base : base + Cathetus. Theor. 4. If wthwithin a circle be indescribed an Ellipsis touching yethe Circle in 2 opposite points if yethe Diameter cut it at right angle in any points except yethe touch point yn a line drawn fm either touch point perpendicular to yethe former diameter will bisect it & being produced will cut yethe ☉circle in yethe other touch point & all yethe lines drawne twixt yethe ☉circle & ytthat line parallell to ytthat diameter shall be dideddivided by yethe Ellipsis so as one segment shall bee to yethe other as yethe segments of yethe semidiameter are to one another they being divided by yethe same Ellip: let ab bee equall to 10 pts. eb = 157979 = Periph: & priph - Rad: Rad:: Rad: db. db = 175, 1938394. de = 18,1142067 To describe an ellipsis Let fe & gc be two lines ef make righright angles wthwith gc. let a point be taken in bd as at a & let ytthat point move along yethe line gc. & d yethe one end of yethe line db move on yethe line ef & yethe other end b shall describe yethe Ellipsis gbc. f. Let c & a bbe two fixed points about wchwhich let a loose cord be put haveing both ends tyed together. as is signified by yethe 3 lines cb. ba. ac. Strech it out wthwith another point as b. & keeping it so streched out draw yethe point b about & it shall describe yethe Ellipsis bd. Chartesij Dioptr Let yethe line ae be infinitely extended in it take yethe point o about yethe line oc shall turne at yethe point c in oc let yethe point c in yethe line ab be fastened & yn let a yethe end of yethe line ab move on yethe line ae & oc turning round, each point of yethe line ab betwixt ac will describe an Ellipsis whose transvers axis is equall to oc & parallell to ae but each point on yethe other side c describes Ellipsis whose righright axis is equall to oc & parallell to ae Extend de both ways take yethe lines ca & ab & ab equall to one another fasten together at one end as at a. set yethe other end of ca at yethe point c in db. & let yethe other end of ab slide on db. yn take a point in ab as o & turne ac about & it shall describe yethe ellipsis dgoe Shooten in lib. 2d Cartesij Geometria: Cut yethe cone abc so yethethat yethe diam of yethe section ed produced cute yethe base of yethe triangle ac produced wthwithout yethe cone as at r & makes right angles wthwith gh yethe base of yethe sectiōon If eg be moved twixt yethe lines ed & gd. a point in it as (θ) shall describe an ellipsis whose semi-axis ad is equall to bd & semiaxis dc = eb If dc revolve abute yethe center d. & to yethe other end b be fastend a triangle bca & db = ba = bc & yethe angle a moves on yethe line ad yethe other end c will describe yethe streighstreight line cd & yethe angle cba = 2cda & a point in yethe line (ca) as (e) shall describe an Ellipsis ehg whose diam 2dh =2dg = 2ec & yethe other diameter conjugated to it is od & od = 4db x db - ec x ec -  2 ec x ea  for op = ec. oq = ea. dp = 2db. & if in yethe line bc be taken a point as s, it shall describe an ellipsis yethe one diam: being 2ab + 2bs, yethe other diam = 2cs. If o & a be yethe foci & cp = oa & ca = op = it theire section in s shall describe an ellipsis If ab = bc = ci = ai = if or greater yn (if) & bh = fp & ac bisects yethe angles bai. bci. yn if bh turne round yethe intersections of bh & ac shall describe an Ellipsis. & hi & i are yethe foci. To describe a Parabola Let bc fall perpendicular on ad & let c yethe one end there of move perpendicular uppon ad a given line & if bc x k a given line be equall to ac x cd yn shall b yethe other end of bc describe yethe Parabola afd. Draw ah perpendicular to ap. & ab from ah parallell to ap divide bh into equall ꝑparts as bcdefgh. & divide ap into parts equall to yethe former as iklmnop. draw lines cros to each part of yethe lines ah & ap as cb. kc. ld. me. nf. &c wthwith half of each line describing a circle as brc wthwith 12 cb. from bu in yethe point cut by yethe diameters of yethe circle draw lines perpendicular to yethe diameter untill they reach yethe circle from whose diameter they are drawne as yethe lines pw, qx, ry, sz, t&, u+. Erect those lines perpendicular to yethe line bu as p♉, q♈, r♊, s♋, t♌, u♍. & by yethe end of those lines draw a line & it shall be a parabola . as b♉♈♊♋♌ If abc be a cone: de (yethe diameter of yethe Section fgd) parallell to ac: & fg (yethe base thereof) cutting bc at right angles yn is yethe section dfg a Parab Make db perpendicular to ef on yethe center b let yethe right angled figure pbgh turne. Let gh move perpendicularly on ef ever intersecting ef & bh in one point yn pbgh moveing rownd yethe intersections made twixt pg gh describe yethe parabola qbg. If ab = bd = do = ao is greater then ac & ac = cs yethe corner (a) fastened to yethe focus (a) . & yethe line de fastened to yethe corner d & moveing perpendicularly onr on sd & yethe line boe crossing yethe corners b & o. yn yethe line boe & de at theire intersections shall describe a Parab & yethe line boe always toucheth yethe Parabola in (e) &c If (d) be yethe focus od = oe yethe ruler fc = to yethe thred fad & thred fastened to yethe ruler at f & to yethe focus d & yethe ruler move perpendicdicular to ce & parallell to de. yn yethe parteing of yethe thred from yethe ruler as at (a) shall describe a Parabola To describe an Hyperbole Let fa fall on ag suppose at right angles let one end of yethe line lg move up & downe in yethe line fa & towards yethe other end let it cut yethe line ga in g. let mp keepe parrallel to df haveing one end p moveing in yethe line fa but yet keeping an equall distance frōom l yethe iend of gl. ytthat is let yethe triangle npl be immutable. let yn yethe lines mp & gl thus move to & fro & theire intersections shall describe an parabola Hyperbola. & yethe rectangle mad de x e de x ea = ic x cb = qo x op. Cartes Geom: ffasten a pegg as at a & another as at b upon wchwhich let yethe line de be turned at yethe pin a fasten one end of a cord & yethe other at e yethe end of yethe line de. yn streching yethe cord from a & e wthwith yethe pin c turne de about & yethe pin c will slip towards e & describe 12 yethe Hyper: oce If the rectangle twixt ad & db is equall to yethe rectangle twixt ae & ec so ytthat each point c in yethe Hyperb: bc is found by makeing ec = ad x dbae or ae = ad x dbec. also be x ce = be x da - db x ec Cut yethe cone abc so ytthat yethe diaterdiameter of yethe section er produced cuteth one side of yethe Cone bc produced as at d. yethe base thereof gh cutteth ac yethe base of yethe triang: abc aat right angles. If (of) touch yethe Hyperb: & (as) be its transverse diam: & (gb) keepe parallel to (eo) & (cag) aways pass through (a). yethe vertex of yethe parab Hyperb. & (bc) be always in yethe line (fh) fastend to (gb) & equall to fd = de = fh4. yn yethe lines (agc) & (gb) moveing by theire intersection shall describe an Hyperbola whose asymtotes are oea, fe; eb, eb, & wx is a right line conjugate to yethe transverse diameter (as.) viz: it is yethe right diameter If dk = er be (latus transversum) & de = kr, be latus rectum yn shall is sd = sr = se = sk = sa = sx. at (a) & (x) fatenfasten 2 pins on wchwhich let yethe (acbp, xobq) revolve, & if ac = ox = zi = dk = er, & co = ax yn yethe intersection of yethe lines cabp, & qbox (when they move) shall describe a Hyperb Parabola whose focus is a, & yethe opposite Hyperbola (whose focus is x is described by yethe same lines after qbox, esk & cabp are parallell If de = dc = ex = cx is not lesse yn ix = az & 2 of theire ends loose pind together at (e) & 2 at (c) on wchwhich 2 corners lyes yethe line (coe) two of yethe theire ends are loosely pinnd on yethe focus (x) yethe last two are pind on yethe line (adp) at (d) soe ytthat yethe ruler adp being pinend to yethe focus (a), ad = zi yn yethe intersections of yethe lines (adp, coe) describe yethe Hyperbola oiq. & after they are parallell they shall describe yethe opposite Hyperbola hzk. The Asymptotes aq, an, & (m) point yin yethe Hyperbola draw mq || an. & mn || aq. Then draw en at a venture & make er = mc || er & r shall bee a point in yethe Hyperbola If yethe position of yethe Asymptotes (ad) (ab) bee given & any point as (c) in yethe Hyperbola. then draw cbfu ucbf || ad. ud || ab || fg making bf = bu = 4bc. Then at a venter draw bewh, through yethe point b. & make ak = fh = uw Or dw = bk & from yethe point k draw ke, wchwhich shall touch yethe HyperpolaHyperbola. in n, if kn = ne. The foci (a, d) & (c) a point in one Hyerbola given to describe them. Draw ac, cd, frōom the given point c to the foci, yn upōon the center c wthwith any radius ce describe yethe circle erf. soe ytthat ec = ef. yn wthwith yethe Rad ae & df upon yethe centers a & d describe yethe circles hep fhp their points of intersection p, h, shall bee in yethe hyperbola. The intermediate distance twixt divers points thus found may bee completed wthwith by yethe helpe of tangnttangent lines or circles or a steady hand. The properties of yethe Parabola ab = a. bc = b. ac = c. eb = d. ei = x. fi = y. b : c :: x : (ik)cxb. a : c :: d : (es, or il)cda whence yy = ccdabx. ab : cc :: d : (en)ccdab. ccdab = r. rx = yy. ytthat is ne a given line multipling ei = if square. Or breifly a : c :: d : (es or il)cda. b : c :: cda : (en)ccdab = r b : c :: Ne is called latus rectum of Apollon & Parameter by Mydorgius. gh is its base ed its Diameter. ang pbh = phg. kg parallell to ac tangent no parallell to yethe tangent ac. yn nm = mo. (2). db x bk = kg x kg. kg x kg : nm x nm :: db x bk : db x bm :: bk : bm a = foco. ac = 14 lateris recti. ac = oc. ah = do. sit (sh) Parallela ad. (dr). & (rh) contingat Parab: in h. & (dh) perpend: ad (dr) erit ang : ahr = rhs. If cs = sb & su parallell to ab yn yethe triang cea : cab :: l : 4. & so it may be saide infinitely. If ab & cd, are ordinately applyed yethe Parabola ceadb is to yethe triangle cda as Eight to five six. & rf x rf = = rs x re. or, re : rf :: rf : rs. If rs, is parallell to gx yn are yethe 2 segments of Parabolas gproxa = gcsqxa) equall & po = cq. & if ga = ax then yethe diameters ar as cut yethe line rs in its touch points. The properties of yethe Hyperbola rx + acxxbb = yy. ☞ rx + rq xx = yy. for acqbb = r. & acbb = rq. am = a. mb = b. mc = c. de = q. ei = x. di = q + x fi = y. b : c :: q + x : (il) cq + cxb b : a :: x : (ik)axb. il x ik = yy = cqax + caxxbb bb : ac :: q : (en)acqbb(r). bb : ac :: q : acqbb :: :: x : (qpcron) acqxabbq or acxbb. whenc acqx + acxbb = rx + acxxbb = pi x ie = yy More breifly thus. b : c :: q :: (es)cqb : b : a :: cqb : acqbb (= r) de is called latus transversum & en latus rectum by Appolonius. but Parameter by Mydordgius. mn = pd = bq = q. fg = db = pq = p nu = x. au = y. ha = ck = b. st = r (1) q : r :: qx + xx : yy. & yy = rx + rxxq = yy. (2) 2by + bb :: = 14 pp. (3) q : p :: p ; r. (4) q : r :: qq : pp. (5) yy : qx + xx :: qq : pp. pq = fg = db = axi secundo, & recto & diam rectæ pd = mn = qb = axi primo, transverso & lateri sive diametro transversæ. st = r = Lateri recto. If xt = p. sr = q : r = Paramrameter & iry = a. eno = b. in = y en = z. rn = x. then if pq = r as in iry Then if p = q = r as in (a) : (a) is yethe simplest of all ParaHyperbola's, & yn, yy = xx + qx. & if (q) is yethe same in both (a & b) & (xt = p) is propper to (b) then yy : zz :: qq : pp. & therefore In Hyperbolas are to one another as theire rigt axis are to supposeing theire transverse axes equall. viz iryeon : eron :: in : en :: p : p. therefore if (rs) is parallell to ao, & ae = co. yn (arextc = = csoext.) & if at = te = cx = xo tr & xs (cutting rs in yethe touch points) are ordinately applyed to yethe Diameters & bisect yethe Pa Hyperbolas. If (o & a) are yethe foci & (u) a point in one of yethe Hyperb: s. then au + ei = ou & if as = ei = or. & rs yn us = uo. & rs = oa & (iu) bisecting yethe angle (ria.) it shall touch yethe Hyperb in u. The Properties of yethe Ellipsis rx - acxxbb = yy. that is rx - rqxx = yy for acqbb = r & acbb = rq am = a. bm = b. cm = c. ed = q. ei = x. id = q - x fi = y. en = r b : c :: q - x : (ib)cq - cxb. b : a :: x : (ik)axb ki x il = cqax - acxxbb = fi x fi = yy bb : ac :: q : en = aqcbb = r. bb : ac :: x : on = acxbb wherefore rx - (onx) = acxxbb. = yy. Af = q = axi transo: sive primo : ch = p. fg = r = lateri recto. ad = x df = q - x. dh = y. (1) q : p :: p : r. (2) yy : xq - xx :: r : q. therefoerefore 3 q : qr : q 3 yy = rx - rxxq as before. af is yethe first & transverse axis or side ch is yethe seacond & right axis fg is yethe Parameter or right side sit p = qn. nc = no. erit segmentum oeth ad segmgmentum cbd, ut cbd ad gbhcd :: fh : ab :: :: (afbhcd) elipsis : (ahbg) circulum. If yethe lines (pq, rs) are parallell & co yethe common axis of both yethe Ellipses yn are yethe 2 Ellipses equall to one another, for ax = = be. yethe conjugated diam: cut yethe touch points of pq, rs & parallells to these are also conjugated. If ab tougch an Ellipsis & (o) & (x) be yethe foci yn yethe angle aco = bcx. & if (ocx) be bisected by (cr) yn acr = bcr = right angle If xu = ot = ys. & uo bisected in a then uac = oac = to a right angle. If also ut = ox & ut & xo be produced till they meete in h. yethe angle uho shall be bisected by yethe line acb.