<flyleaf>

pret 8d

 Nabed Efyhik, Wfnzo Cpmkfe
<3r> [Editorial Note 1]

Before Whitsunday 1662.

 Vsing the word (God) openly 1 Eating an apple at Thy house 2 Making a feather while one {sic} Thy day 3 Denying that I made it. 4 Making a mousetrap on Thy day 5 Contriving of the chimes on Thy day 6 Squirting water on Thy day 7 Making pies on Sunday night 8 Swimming in a kimnel on Thy day 9 Putting a pin in Iohn Keys hat on Thy day to pick him. 10 Carelessly hearing and committing many sermons 11 Refusing to go to the close at my mothers command. 12 Threatning my father and mother Smith to burne them and the house over them 13 Wishing death and hoping it to some 14 Striking many 15 Having uncleane thoughts words and actions and dreamese. 16 Stealing cherry cobs from Eduard Storer 17 Denying that I did so 18 Denying a crossbow to my mother and grandmother though I knew of it 19 Setting my heart on money learning pleasure more than Thee 20 A relapse 21 A relapse 22 A breaking again of my covenant renued in the Lords Supper. 23 Punching my sister 24 Robbing my mothers box of plums and sugar 25 Calling Derothy {sic} Rose a jade 26 Glutiny in my sickness. 27 Peevishness with my mother. 28 With my sister. 29 Falling out with the servants 30 Divers commissions of alle my duties 31 Idle discourse on Thy day and at other times 32 Not turning nearer to Thee for my affections 33 Not living according to my belief 34 Not loving Thee for Thy self. 35 Not loving Thee for Thy goodness to us 36 Not desiring Thy ordinances 38 Not long {longing} for Thee in {illeg} 39
<3v>
 40 Fearing man above Thee 41 Vsing unlawful means to bring us out of distresses 42 Caring for worldly things more than God 43 Not craving a blessing from God on our honest endeavors. 44 Missing chapel. 45 Beating Arthur Storer. 46 Peevishness at Master Clarks for a piece of bread and butter. 47 Striving to cheat with a brass halfe crowne. 48 Twisting a cord on Sunday morning 49 Reading the history of the Cn champions on Sunday
<4v>

Since Whitsunday 1662

 1 Glutony 2 Glutony 3 Vsing Wilfords towel to spare my own 4 Negligence at the chapel. 5 Sermons at Saint Marys (4) 6 Lying about a louse 7 Denying my chamberfellow of the knowledge of him that took him for a {illeg} sot. 8 Neglecting {illeg} to pray 3 9 Helping Pettit to make his water watch at 12 of the clock on Saturday night
<5r>

1665

 Rd 10li May 23d whereof I gave my Tutor 5li ———— 5 . 0 . 0 Remaining in my hands since the last Quarter ———— 3 . 8 . 4 In all — 8 . 8 . 4. Pd Iohn ye Taylor ———— 2 . 0. 0. Pd Mr Bychiner ———— 0 . 3 . 6. To Ca{illeg}|v|erly ———— 0 . 1 . 0. To my Laundresse ———— 0 . 0 . 6. To my Bedmaker ———— 0 . 5 . 0. A paire of Gloves ———— 0 . 2 . 0 A paire of Stockings ———— 0 . 5 . 4 A hatband ———— 0 . 2 . 0. Pd Goodwife Powell for my Laundresse ———— 0 . 5 . 0. Given more to my Tutor ———— 5 . 0 . 0 My Iourney to Cambridge Mar 20. 0 . 6 . 6. In all 8 .10 .10. Lent Mr Newton ———— 0 .18 . 0

March 25 1666.

 Lent Wilford ——X—— 0 . 1 . 0. To ye Poore on ye fast ———— 0 . 1 . 0. To Mr Babintꝰ: Wom, 6d. Porter 6d ———— 0 . 1 . 0. Spent wth Rubbins 4d. ———— 0 . 0 . 4 Lent to Sr Herring ———— {illeg}|1| . 6 . 0. Lent to Sr Drake ———— {illeg}|1| . 0 . 0. Payd my Laundresse ———— 0 . 5 . 6. ffor a paire of shoos ———— 0 . 4 . 0. Caverly ———— 0 . 0 . 4.
<6r>
 Payd Iohn Falkoner ———— 0 .11 . 6. A paire of shooestrings ———— 0 . 0 . 8. Payd my Bedmaker 0 . 5 . 0. Dew from Iohn Euans ———— 0 . 1 .10. Euans The summe of my expences 1 .10 . 4. + 8 .10 . 10 In all 10 . 1 . 2 Dew to mee ———— 3 . 5 .10 More from Mr Guy ———— 0 .10 . 0 Lent In all ———— 3 .15 .10.

1666.

 Rd 10li March 20th ———— 10 . 0 . 0 Remaining in my hands ———— 8 . 8 . 4. In all 18 .8 . 4 Expences & wt I lent deducted ye rest is ———— 4 .11 . 4. 1667 Apr 22 Received 10 - 0 - 0 In my hands besid debts 14 -11 - 4d My Iourney to Cambridg 0 - 6 - 6. Two paire of shoos ———— 0 - 8 - 0 A Cap ——X—— 0 - {illeg} Cloths & {illeg} \dying &/ mending 0 - {illeg} {illeg} bordering twice ——X—— 0 - {illeg} Lynings 0 - 6 - 6 Lath & Table ———— 0 -15 - 0 Iron worke for it 0 - 9 - 0
<6v>
 Drills, Gravers, a Hone & Hammer & a Mandrill 0 . 5 . 0 |A| Magnet ———— 0 .16 . 0 Compasses ———— 0 . 3 . 6 Glass bubbles ———— 0 . 4 . 0 Chappell Clarke ———— 0 . 2 . 6 My Bachelors Act ———— 0 .17 . 6. At ye Taverne severall other times &c ———— 0|1| . 1|0| . 0 Spent on ye|M|y Couz Ayscough 0 .12 . 6.. On other Acquaintance ———— 0 -10 : 0 Shoos ———— 0 . 4 . 0 Cloth \2 yards/ & buckles for {illeg}|a| Vest. 2 . 0 . 0 ffor Woosted Prunella 7|8|yds $\frac{1}{2}$. 1 . 5 . 6 ffor ye lining 4yds ———— 0 . 9 . 4 Philosophicall Intelligences 0 . 9 . 6. ye Hystory of ye Royall Ssoc: 0 . 6|7| . 0. Shoe Strings ———— 0 . 1 . 0 To Goodwife Powell ———— 0 . 7 . 6 To my Laundresse ———— 0 . 8 . 6 To Caverly ———— 0 . 1 . 6 To the Glasier ———— 0 . 1 . 0 New fire cheeks & pointing ye chamber & windows ———— 0 . 1 . 6 Gunters book & sector \&c/ to Ds ffox 0 . 5 . 0 Letters, wyer, files, boats, ———— 0 . 2 . 6. ffor a ffellows key ———— 0 . {illeg}|1| . 0 A Cap turning ———— 0 . 1 . 4. To the Taylor \Octob 29. 1667./ ———— 2 .13 . 0 To the Taylor. Iune 10. 1667 ———— 1 . 3 .10 For keeping Christmas ———— 0 . 5 . 0 Lost at cards at {sic} {tw{illeg}|ice|} 0 .15 . 0
<7r>
 At ye Taverne twice ———— 0 . 3 . 6. 6$\frac{1}{2}$ sacks of coales, carriage & sedge ———— 0 .11 . 0 Shoos & mending ———— 0 . 4 .10. Two paire of Gloves ———— 0 . 5 . 0 |wth| Mr Lusmore, Hautrey, Salter 0 . 3 . 6 Received of my Tutor wch I lent Perkins 0 .10 . 0

I went into ye Countrey Decembr 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 ye Colledg 5 .10 . 0 To ye Proctor ———— 2 . 0 . 0 ffor 3 Prismes ———— 0 . 3 . 0 4 ounces of Putty ———— 0 . 1 . 4 To ye Painter ———— 0 . 3 . 0 To ye Ioyner ———— 1 . 1 . 8 {illeg}|Lent| to Ds Wickins ——X—— 1 . 7 . 6. To ye shoe maker ———— 0 . 5 . 0 Bacons Miscelanys ———— 0 . 1 . 6 Expences caused by my Degre 0 .15 . 0
<7v>
 Subscribing 6d, Reading Græke. 0 . 5 .10. A bible binding ———— 0 . 3 . 0. Humphrey \1668/ ———— 0 . 1 . 0. 18 yards of Tammy for my Mr of Arts Goune 1 .13 . 0 Lining —— 3, 6 ———— 0 . 2|3| . 6. Making yt & turning my Bachelors Goune ———— 1 . 0 . 6. Received of Mr Io: Herring 0 .10 . 0 Payd my Laundresse ———— 0 . 5 . 6. Payd to Caverly ———— 0 . 5 . 6. Payd God|o|dwif Tab|l|bot from Feb 12 to Mar 25 1668 0 . 2 . 6 Payd to my Laundresse 0 . 2 . 6. To ye Porter ———— 0 . 5 . 6. ffor oranges \1667/ for my sister ———— 0 . 4 . 2. Bedmaker & Laundresse 0 .10 . 0. Shoemaker ———— 0 . 5 . 8. A Hatt ———— 0 .19 . 0. Taverne 0 .10 . 0. Carpets of Neats Leather 0 .18 . 0 A|M|y part of A Couch. 0 .14 . 0. 1 Bowling Greene ———— 0 .10 . 0 To Mr Ieffreys for a Suit 3 . 6 . 0 A Tickin for a ffeatherbed. {illeg}|1| .10 . 0 New ffeathers ———— 0 . 8 . 0 A Hood ———— 1 . 3 . 6. Making &c of my last suit ———— 1 .11 . 9
<8r>
 Dew to Iohn Hauxy ——X—— 1 .10 . 0. Spent in my Iourney to Londō 6|5| .10 . 0 As also 4li {illeg} 5s my|o|re wch my Mother gave mee in ye Country 4 . 5 . 0 Received for Chamberrent 1 .11 . 0. Received from my Mother 11 . 0 . 0.

I went to London on Wednesday Aug 5t & returned to Cambridge on Munday Sept 28, 1668.

 Bedmaker & Laundresse 0 . 4 . 0 Lent Ds Wickins ——X—— 0 .11 . 0 Lent Mr Boucheret ——X—— 0 . 5 . 0

Aprill 1669.

 Lent to Mr Wadsley ———— 0 .14 . 0 16 yards of Stuffe for a suit 2 . 8 . 0 ffor making &c ———— 1 .13 . 0 For turning a Cloth suit 1 . 3 . 0|3| For shoe strings &c ———— 0 . 2 . 0 For Glasses in Cambridge 0 .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, ☿ {illeg}|2| . {illeg}|0| . 0. A ffurnace ———— {illeg}|0| . 8 . 0 A tin ffurnace ———— 0 . 7 . 0 Ioyner ———— 0 . 6 . 0
<8v>
 Theatrum Chemicum ———— 1 . 8 . 0 Lent Wardwel 3s & to his wife {illeg}|2|s ———— 0 . 5 . 0 Carrriage of ye oyle ———— 0. .2 . 0 Payd I Stagg ———— 0 .18 . 6 Payd ye Chandler ———— 0 . {illeg}|8| . 0 A Table cloth ———— 0 .10 . 0 Six Napkins ———— 0 . 6 . 0
<9v>

Carolus g|G|ottlob de Theler

Georgius Bernhardus de Theler Equites ex Superiore Lusatia

Iohannes Christophorus Ritter Wurcenâ-Misnicus.

[Editorial Note 2] <1r>

Sep. 25 1727
Not fit to be printed

T Pellet

<2r>

Nova Cubi Hæbræi Tabella

[Editorial Note 3]
<3r>

Of right angled triangles.

h = hypotenusa.
b = basis.
c = Cathetus.
p = perpendicular.
hdc = diff: hypot & Cath
bdc = diff: basis & cathet:
bdh = difference basis & hyp{ot}
dsh = diff: seg: hypoten:
sh = segment: hypoten:
bh = greater seg {illeg}|h|yp:
ch = lesse seg: hypot:

I. Any two leggs given to find ye other

1. bq + cq = hq.

2 r: k|h|q - {illeg}|b|q: = c.

3 r: k|h|q - cq = {illeg}|b|

Ought. {illeg} Eucl. lib 1. pr: 47.

4 $\frac{b x c}{h}$ = {illeg} {illeg}

II ye b. c. & h given to find p.

1. $\frac{b x c}{h}$ = p Eucid {sic} 6 .8.

III c. h. {illeg}|p|. given to find dsh.

1. {illeg}|H| - 2r: {illeg}|b|q - pq: = dsh.

IIII. b. p. h given to find dsh.

1. 2r: bq - pq: {illeg}h = dsh.

V. b. c. h given to find dsh.

1. H - 2r: cq - Q: $\frac{b x c}{h}$: = dsh.

2 2r: bq - Q: $\frac{b x c}{h}$: + h = dsh.

VI b.c \or b. h or h. c/ given to find b|p|:

1 $\frac{b x c}{r: bq + cq:}$ = p

2 $\frac{b x r: hq - bq:}{h}$ = p.

3 $\frac{c x r: hq - cq:}{h}$ = p

VII b. h. or c. h. or b. c given to find dsh.

<4r>

Theorem 1

As ye difference twixt ye base & cath (in rectang: triang:) is to ye greater side:: so is ye difference of ye segm of ye base; to ye greater segmnt of ye base & perpendicular.

Theorem 2.

As ye difference twixt ye base & cathetus to ye less side:: so ye diff of ye segmts of ye base to ye lesse segment of ye base & perpendicular

Theorē 3d.

base – Cathetus: hypotenusa:: :: greater seg: base - less seg base : base + Cathetus.

Theor. 4.

If wthin a circle be in|de|scribed an Ellipsis touching ye Circle in 2 opposite points if ye Diameter cut \it/ at right angle in any points except ye touch point yn a line drawn fm either touch point perpendicular to ye former diameter will bisect it & being produced will cut ye in ye other touch point & all ye lines drawne twixt ye & yt line <4v> parallell to yt diameter shall be dided {sic} by ye Ellipsis so as one segment {illeg}|sh|all bee to ye other as ye segments of ye semidiameter are to one another they being divided by ye same Ellip: let ab bee equall to 10 pts. eb = 157979 = Periph: & priph - Rad: Rad:: Rad: db. db = 175, 1938394. de = 18,1142067

<5r>

To describe an ellipsis

Let fe & gc be two lines ef make righ {sic} angles wth gc. let a point be taken in bd as at a & let yt point move along ye line gc. & d ye one end of ye line db move on ye line ef & ye other end b shall describe ye Ellipsi{illeg}|s| gbc. f.

Let c & a {sic} two fixed points about wch let a loose cord be put haveing both ends tyed together. as is signified by ye 3 lines cb. ba. ac. Strech it out wth another point as b. & keeping it so streched out draw ye point b about & it shall describe ye Ellipsis bd. Chartesij Dioptr

Let ye line ae be infinitely extended in it take ye point o about ye line oc shall turne at ye point c in oc let ye \point c in ye/ line ab be fastened {illeg}|&| yn let a ye end of ye line ab move on ye line ae & oc turning round, each point of ye line ab betwixt ac will describe an Ellipsis whose transvers axis is equall to oc & parallell to ae but each point on ye other side c describes Ellipsis whose righ {sic} axis <5v> is equall to oc & parallell to ae

Extend de both ways take ye lines ca & ab & {illeg} ab equall to one another fasten together at one end as at a. set ye other end of ca at ye point c in db. & let ye other end of ab slide on db. yn take a point in ab as o & turne ac about & it shall describe ye ellipsis dgoe Shooten in lib. 2d Cartesij Geometria:

Cut ye cone abc so ye {sic} ye diam of ye section ed produced cute ye base of ye triangle ac produced wthout ye cone as at r & makes right angles wth gh ye base of ye sectiō

If eg be moved twixt ye lines ed & gd. a point in it as (θ) shall describe an ellipsis whose semi-axis ad is equall to bd & semiaxis dc = eb

<6r>

If dc revolve abute ye center d. & to ye other end b be fastend a triangle bca & db = ba = bc & ye angle a moves on ye line ad ye other end c will describe ye streigh {sic} line cd & ye angle cba = 2cd{illeg}|a| & a point in ye line (ca) as (e) shall describe an Ellipsis ehg whose diam 2dh =2dg = 2ec & ye other diameter conjugated to it is od & od = for op = ec. oq = ea. dp = 2db.

& if in ye line bc be taken a point as s, it shall describe an ellipsis ye one diam: being 2ab + 2bs, ye other diam = 2cs.

If o & a be ye 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 ye angles bai. bci. yn if bh turne round ye intersections of bh & ac shall describ{e} an Ellipsis. & hi & i are ye foci.

<7r>

To describe a Parabola

Let bc fall perpendicular on ad & let c ye 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 ye other end of bc describe ye Parabola afd.

Draw ah perpendicu{lar} to ap. & ab from ah parallell to ap divid{e} bh into equall ts {illeg}|as| bcdefgh. & divide ap into parts equall to ye former as iklmnop. draw lines cros to each part of ye lines ah & ap as cb. kc. ld. me. nf. &c wth half of each line descri{bing} a circle as brc wth $\frac{1}{2}$ cb. from bu in ye poi{nt} cut by ye diameters of ye {illeg} circle draw lines perpendicular to ye diameter <7v> untill they reach ye circle from whose diameter they are drawne {illeg}|as| ye lines pw, qx, ry, sz, t&, u+. Erect those lines perpendicular to ye line bu as p, q, r, s, t, u. & by ye end of those lines draw a line & it shall be a parabola . a{illeg}|s| b

If abc be a cone: de (ye diameter of ye Section fgd) parallell to ac: & fg (ye base there{illeg}|of|) cutting bc at right angles yn is ye section dfg a Parab

Make db perpendicular to ef on ye center b let ye right angled figure pbgh turne. Let gh move perpendicularly on ef ever intersecting ef & bh in one point yn pbgh moveing rownd ye intersections made twixt pg gh describe ye parabola qbg.

<8r>

If ab = bd = do = ao is greater then ac & ac = cs ye corner (a) fasten{ed} to ye focus (a) . & ye line de fastened to ye corner d & moveing perpendicularly on|{r}| on sd & ye line boe crossing ye corners b & o. yn ye line boe & de at theire intersections shall describe a Parab & ye line boe always toucheth ye Parabola in (e) &c

If (d) be ye focus od = oe ye ruler fc = to ye thred fad & thred fastened to ye ruler at f & to ye focus d & ye ruler move perpendic to ce & parallell to de. yn ye parteing of ye thred from ye ruler as at (a) shall describe a Parabola

<9r>

To describe an Hyperbole

Let fa fall on ag suppose at right angles let one en{illeg}|d| of ye line lg move up {illeg}|&| downe in ye line fa & towards ye other end let it cut ye line ga in g. let mp keepe parrallel to df haveing one end p moveing in ye line fa but yet keeping an equall distance frō l ye i|e|nd of gl. yt is let ye triangle npl be immutable. let yn ye lines mp & gl thus move to & fro & theire intersections shall describe an parabola Hyperbola. & ye 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 wch let ye line de be turned at ye pin a fasten one end of a cord & ye other at e ye end of ye line de. yn streching ye cord from a & e wth ye pin c turne de about & ye pin c will slip towards e & describe $\frac{1}{2}$ ye Hyper: oce

<9v>

{If} the rectangle twixt ad & db is equall to ye rectangle twixt ae & ec {illeg}|{so}| yt each point c in ye Hyperb: bc is found by makeing ec = $\frac{ad x db}{ae}$ or ae = $\frac{ad x db}{ec}$. also be x ce = be x da - db x ec

Cut ye cone abc so yt ye diater {sic} of ye section er produced cuteth one side of ye Cone bc produced as at d. ye base thereof gh cutteth ac ye base of ye triang: abc {sic} right angles.

If (of) touch ye Hyperb: & (as) be it{illeg}|s| transverse diam: & (gb) keepe parallel to (eo) & (cag) aways pass through (a). ye vertex of ye parab Hyperb. & (bc) be always in ye line (fh) fastend to (gb) & equall to fd = de = $\frac{fh}{4}$. yn ye 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 ye transverse diameter (as.) viz: it is ye right diameter

<10r>

If dk = er be (latus transversum) & de = kr, be latus rectum yn shall is sd = sr = se = sk = sa = sx. at (a) & (x) faten {sic} 2 pins on wch let ye (acbp, xobq) revolve, & if ac = ox = zi = dk = er, & co = ax yn ye intersection of ye lines cabp, & qbox (when they move) shall describe a |Hyperb| Parabola whose focus is a, & ye opposite Hyperbola (whose focus is x is described {illeg}|by| ye 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 wch 2 corners lyes ye line (coe) two of ye theire ends are loosely pinnd on ye focus (x) ye last two are pind on ye line (adp) at (d) soe yt ye ruler adp being pine|n|d to ye focus (a), ad = zi yn ye intersections of ye lines (adp, coe) describe ye Hyperbola oiq. & after they are parallell they shall describe ye opposite Hyperbola hzk.

The Asymptotes aq, an, & (m) point y|i|n ye Hyperbola draw mq || an. & mn || aq. Then draw en at a venture & make er = {illeg} \mc/ || er & r shall bee a point in ye Hyperbola

<10v>

If ye position of ye Asymptotes (ad) (ab) bee given & any point as (c) in ye Hyperbola. then draw {cbfu}{illeg} ucbf || ad. ud || ab || fg making bf = bu = 4bc. Then at a venter draw bewh, through ye point b. & make ak = fh = uw Or dw = bk & from ye point k draw ke, wch shall touch ye Hyperpola {sic}. in n, if kn = ne.

The foci \(a, d)/ & (c) a point in one Hyerbo{la} given to describe them.

Draw ac, cd, frō the given point c to the foci, yn upō the center c wth any radius ce describe ye circle erf. soe yt ec = ef. yn wth ye Rad ae & df upon ye centers a & d describe ye circles hep fhp their points of intersection p, h, shall bee in ye hyperbola. The intermediate distance twixt divers points thus found may bee completed wth by ye helpe of tangnt lines or circles or a steady hand.

<11r>

The properties of ye Parabola

ab = a. bc = b. ac = c. eb = d. ei = x. fi = y. b : c :: x : (ik)$\frac{cx}{b}$. a : c :: d : (es, or il)$\frac{cd}{a}$ whence yy = $\frac{ccd}{ab}$x. ab : cc :: d : (en)$\frac{ccd}{ab}$. $\frac{ccd}{ab}$ = r. rx = yy. yt is ne a given line multipling ei = if square. Or breifly a : c :: d : (es or il)$\frac{cd}{a}$. b : c :: $\frac{cd}{a}$ : (en)$\frac{ccd}{ab}$ = r b : c :: Ne is called latus rectum of Apollon & Parameter by Mydorgius. gh is its base ed its Diameter.

<11v>

ang pbh = phg. kg parallell to ac tangent no parallell to ye 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

<12r>

a = foco. ac = $\frac{1}{4}$ 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 ye triang cea : cab :: l : 4. & so it may be saide infinitely.

If ab & cd, are ordinately applyed ye Parabola ceadb is to ye {illeg}|triangle| {illeg}|cda|{illeg} as Eight to five \six/. & rf x rf = {sic} rs x re. or, re : rf :: rf : rs.

If rs, is parallell to gx yn are ye 2 segments of Parabolas gproxa {illeg} = gcsqxa) equall & po = cq. & if ga = ax then ye diameters ar as cut ye line rs in its tou{illeg}ch points.

<13r>

The properties of ye Hyperbola

rx + $\frac{acxx}{bb}$ = yy.

☞ rx + $\frac{r}{q}$ xx = yy. for

$\frac{acq}{bb}$ = r. & $\frac{ac}{bb}$ = $\frac{r}{q}$.

{illeg}|{am}| = a. mb = b. mc = c. de = q. ei = x. di = q + x fi = y. b : c :: q + x : (il) $\frac{cq + cx}{b}$ b : a :: x : (ik)$\frac{ax}{b}$. il x ik = yy = $\frac{cqax + caxx}{bb}$ bb : ac :: q : (en)$\frac{acq}{bb}$(r). bb : ac :: q : $\frac{acq}{bb}$ :: :: x : (qpcron) $\frac{acqx}{abbq}$ or $\frac{acx}{bb}$. whenc $\frac{acqx + acx}{bb}$ = rx + $\frac{acxx}{bb}$ = pi x ie = yy

More breifly thus.

b : c :: q :: (es)$\frac{cq}{b}$ : b : a :: $\frac{cq}{b}$ : $\frac{acq}{bb}$ (= r)

de is called latus tran{illeg}|s|versum & en latus rectum by Appolonius. but Parameter by Mydordgius.

<13v>

mn = {illeg} pd = bq = q. fg = db = pq = p nu = x. au = y. ha = ck = b. st = r (1) q : {illeg} r :: qx + xx : yy. & yy = rx + $\frac{rxx}{q}$ = yy. (2) 2b{illeg}|y| + bb :: = $\frac{1}{4}$ 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.

<14r>

If xt = p. sr = q : r = Param & 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 ye simplest of all Para|Hyper|bola's, & yn, yy = xx + qx. & if (q) is ye 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 = {sic} csoext.) & if at = te = cx = xo tr & xs (cutting rs in ye touch points) are ordinately {illeg} applyed to ye Diameters & bisect ye Pa Hyperbolas.

<14v>

If (o & a) are ye foci & (u) a point in one of ye Hyperb: s. then au + ei = ou & if as = ei = or. & rs yn us = uo. & rs = oa & (iu) bisecting ye angle (ria.) it shall touch ye Hyperb in u.

<15r>

The Properties of ye Ellipsis

rx - $\frac{acxx}{bb}$ = yy. that is

rx - $\frac{r}{q}$xx = yy for

$\frac{acq}{bb}$ = r & $\frac{ac}{bb}$ = $\frac{r}{q}$

am = a. bm = b. cm = c. ed = q. ei = x. id = q - x fi = y. en = r

b : c :: q - x : (ib)$\frac{cq - cx}{b}$. b : a :: x : (ik)$\frac{ax}{b}$ ki x il = $\frac{cqax - acxx}{bb}$ = fi x fi = yy

bb : ac :: q : en = $\frac{aqc}{bb}$ = r. bb : ac :: x : on = $\frac{acx}{bb}$ wherefore rx - (onx) = $\frac{acxx}{bb}$. = yy.

Af = q \= axi transo: sive primo/ : ch = p. fg = r = lateri recto. ad = x {illeg} df = q - x. dh = y.

(1) q : p :: p : r. (2) yy : xq - xx :: r : q. therefo 3 q : q{r} : q{illeg} 3 {illeg} yy = rx - $\frac{rxx}{q}$ as before.

af is ye first & transverse axis or side

ch is ye seacond & right axis

fg is ye Parameter or right side

<16r>

sit p = qn. nc = no. erit segmentum oeth ad segm cbd, ut cbd ad gbhcd :: fh : ab :: :: {sic} (afbhcd) elipsis {illeg} : (ahbg) circulum.

If ye lines (pq, rs) are parallell & co ye common axis of both ye Ellipses yn are ye 2 Ellipses equall to one another, for ax = {sic} be. ye conjugated diam: cut ye touch points of pq, rs & parallells to these are also conjugated.

<16v>

If ab toug|c|h an Ellipsis & (o) & (x) be ye foci yn ye 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. ye angle uho shall be bisected by ye line acb.

[Editorial Note 1] 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.

[Editorial Note 2] The following material is written from the opposite end of the notebook.

[Editorial Note 3] There follows a table of Hebrew characters with Latin annotations.

© 2017 The Newton Project

Professor Rob Iliffe
Director, AHRC Newton Papers Project

Scott Mandelbrote,
Fellow & Perne librarian, Peterhouse, Cambridge

Faculty of History, George Street, Oxford, OX1 2RL - newtonproject@history.ox.ac.uk

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