Fitzwilliam Notebook

Author: Isaac Newton

Source: Fitzwilliam Museum, Cambridge, UK

Published online: October 2003

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pret 8^d

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

R^d 10^li May 23^d whereof I gave my Tutor 5^li	————	5 . 0 . 0
Remaining in my hands since the last Quarter	————	3 . 8 . 4
	In all —	8 . 8 . 4.
P^d Iohn y^e Taylor	————	2 . 0. 0.
P^d M^r 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.
P^d 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 M^r Newton	————	0 .18 . 0

March 25 1666.

Lent Wilford	——X——	0 . 1 . 0.
To y^e Poore on y^e fast	————	0 . 1 . 0.
To M^r Babintꝰ: Wom, 6^d. Porter 6^d	————	0 . 1 . 0.
Spent w^th Rubbins 4^d.	————	0 . 0 . 4
Lent to S^r Herring	————	{illeg}\|1\| . 6 . 0.
Lent to S^r 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 M^r Guy	————	0 .10 . 0
Lent In all	————	3 .15 .10.

1666.

R^d 10^li March 20^th	————	10 . 0 . 0
Remaining in my hands	————	8 . 8 . 4.
	In all	18 .8 . 4
Expences & w^t I lent deducted y^e rest is	————	4 .11 . 4.
1667 Apr 22 Received		10 - 0 - 0
In my hands besid debts		14 -11 - 4^d
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 y^e Taverne severall other times &c	————	0\|1\| . 1\|0\| . 0
Spent on y^e\|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 y^e lining 4^yds	————	0 . 9 . 4
Philosophicall Intelligences		0 . 9 . 6.
y^e Hystory of y^e 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 y^e chamber & windows	————	0 . 1 . 6
Gunters book & sector \&c/ to D^s 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 y^e 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
\|w^th\| M^r Lusmore, Hautrey, Salter		0 . 3 . 6
Received of my Tutor w^ch I lent Perkins		0 .10 . 0

I went into y^e Countrey Decemb^r 4^th 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 y^e Colledg		5 .10 . 0
To y^e Proctor	————	2 . 0 . 0
ffor 3 Prismes	————	0 . 3 . 0
4 ounces of Putty	————	0 . 1 . 4
To y^e Painter	————	0 . 3 . 0
To y^e Ioyner	————	1 . 1 . 8
{illeg}\|Lent\| to D^s Wickins	——X——	1 . 7 . 6.
To y^e shoe maker	————	0 . 5 . 0
Bacons Miscelanys	————	0 . 1 . 6
Expences caused by my Degre		0 .15 . 0

<7v>

Subscribing 6^d, Reading Græke.		0 . 5 .10.
A bible binding	————	0 . 3 . 0.
Humphrey \¹⁶⁶⁸/	————	0 . 1 . 0.
18 yards of Tammy for my M^r of Arts Goune		1 .13 . 0
Lining —— 3, 6	————	0 . 2\|3\| . 6.
Making y^t & turning my Bachelors Goune	————	1 . 0 . 6.
Received of M^r 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 y^e 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 M^r 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 4^li {illeg} 5^s my\|o\|re w^ch my Mother gave mee in y^e Country		4 . 5 . 0
Received for Chamberrent		1 .11 . 0.
Received from my Mother		11 . 0 . 0.

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

Bedmaker & Laundresse		0 . 4 . 0
Lent D^s Wickins	——X——	0 .11 . 0
Lent M^r Boucheret	——X——	0 . 5 . 0

Aprill 1669.

Lent to M^r 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 3^s & to his wife {illeg}\|2\|^s	————	0 . 5 . 0
Carrriage of y^e oyle	————	0. .2 . 0
Payd I Stagg	————	0 .18 . 6
Payd y^e Chandler	————	0 . {illeg}\|8\| . 0
A Table cloth	————	0 .10 . 0
Six Napkins	————	0 . 6 . 0

<9v>

Friedadolff Lewys Langerhanss.

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>

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 & 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 y^e 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 y^e 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 y^e difference twixt y^e base & cath (in rectang: triang:) is to y^e greater side:: so is y^e difference of y^e segm of y^e base; to y^e greater segm^nt of y^e base & perpendicular.

Theorem 2.

As y^e difference twixt y^e base & cathetus to y^e less side:: so y^e diff of y^e segm^ts of y^e base to y^e lesse segment of y^e base & perpendicular

Theorē 3^d.

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

Theor. 4.

If w^thin a circle be in|de|scribed an Ellipsis touching y^e Circle in 2 opposite points if y^e Diameter cut \it/ at right angle in any points except y^e touch point yⁿ a line drawn fm either touch point perpendicular to y^e former diameter will bisect it & being produced will cut y^e ☉ in y^e other touch point & all y^e lines drawne twixt y^e ☉ & y^t line <4v> parallell to y^t diameter shall be dided {sic} by y^e Ellipsis so as one segment {illeg}|sh|all bee to y^e other as y^e segments of y^e semidiameter are to one another they being divided by y^e 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 w^th gc. let a point be taken in bd as at a & let y^t point move along y^e line gc. & d y^e one end of y^e line db move on y^e line ef & y^e other end b shall describe y^e Ellipsi{illeg}|s| gbc. f.

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

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

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

Cut y^e cone abc so y^e {sic} y^e diam of y^e section ed produced cute y^e base of y^e triangle ac produced w^thout y^e cone as at r & makes right angles w^th gh y^e base of y^e sectiō

If eg be moved twixt y^e 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 y^e center d. & to y^e other end b be fastend a triangle bca & db = ba = bc & y^e angle a moves on y^e line ad y^e other end c will describe y^e streigh {sic} line cd & y^e angle cba = 2cd{illeg}|a| & a point in y^e line (ca) as (e) shall describe an Ellipsis ehg whose diam 2dh =²dg = ²ec & y^e other diameter conjugated to it is od & od = $\sqrt{4db x db - ec x ec -^{2} ec x ea}$ for op = ec. oq = ea. dp = 2db.

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

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

<7r>

To describe a Parabola

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

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

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

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

<8r>

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

If (d) be y^e focus od = oe y^e ruler fc = to y^e thred fad & thred fastened to y^e ruler at f & to y^e focus d & y^e ruler move perpendic to ce & parallell to de. yⁿ y^e parteing of y^e thred from y^e 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 y^e line lg move up {illeg}|&| downe in y^e line fa & towards y^e other end let it cut y^e line ga in g. let mp keepe parrallel to df haveing one end p moveing in y^e line fa but yet keeping an equall distance frō l y^e i|e|nd of gl. y^t is let y^e triangle npl be immutable. let yⁿ y^e lines mp & gl thus move to & fro & theire intersections shall describe an parabola Hyperbola. & y^e 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 w^ch let y^e line de be turned at y^e pin a fasten one end of a cord & y^e other at e y^e end of y^e line de. yⁿ streching y^e cord from a & e w^th y^e pin c turne de about & y^e pin c will slip towards e & describe $\frac{1}{2}$ y^e Hyper: oce

<9v>

{If} the rectangle twixt ad & db is equall to y^e rectangle twixt ae & ec {illeg}|{so}| y^t each point c in y^e 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 y^e cone abc so y^t y^e diater {sic} of y^e section er produced cuteth one side of y^e Cone bc produced as at d. y^e base thereof gh cutteth ac y^e base of y^e triang: abc a {sic} right angles.

If (of) touch y^e Hyperb: & (as) be it{illeg}|s| transverse diam: & (gb) keepe parallel to (eo) & (cag) aways pass through (a). y^e vertex of y^e parab Hyperb. & (bc) be always in y^e line (fh) fastend to (gb) & equall to fd = de = $\frac{fh}{4}$ . yⁿ y^e 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 y^e transverse diameter (as.) viz: it is y^e 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 w^ch let y^e (acbp, xobq) revolve, & if ac = ox = zi = dk = er, & co = ax yⁿ y^e intersection of y^e lines cabp, & qbox (when they move) shall describe a |Hyperb| Parabola whose focus is a, & y^e opposite Hyperbola (whose focus is x is described {illeg}|by| y^e 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 w^ch 2 corners lyes y^e line (coe) two of y^e theire ends are loosely pinnd on y^e focus (x) y^e last two are pind on y^e line (adp) at (d) soe y^t y^e ruler adp being pine|n|d to y^e focus (a), ad = zi yⁿ y^e intersections of y^e lines (adp, coe) describe y^e Hyperbola oiq. & after they are parallell they shall describe y^e opposite Hyperbola hzk.

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

<10v>

If y^e position of y^e Asymptotes (ad) (ab) bee given & any point as (c) in y^e Hyperbola. then draw {cbfu}{illeg} ucbf || ad. ud || ab || fg making bf = bu = 4bc. Then at a venter draw bewh, through y^e point b. & make ak = fh = uw Or dw = bk & from y^e point k draw ke, w^ch shall touch y^e 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 w^th any radius ce describe y^e circle erf. soe y^t ec = ef. yⁿ w^th y^e Rad ae & df upon y^e centers a & d describe y^e circles hep fhp their points of intersection p, h, shall bee in y^e hyperbola. The intermediate distance twixt divers points thus found may bee completed w^th by y^e helpe of tang^nt lines or circles or a steady hand.

<11r>

The properties of y^e 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. y^t 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 y^e 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 y^e triang cea : cab :: l : 4. & so it may be saide infinitely.

If ab & cd, are ordinately applyed y^e Parabola ceadb is to y^e {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 y^e 2 segments of Parabolas gproxa {illeg} = gcsqxa) equall & po = cq. & if ga = ax then y^e diameters ar as cut y^e line rs in its tou{illeg}ch points.

<13r>

The properties of y^e 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 y^e simplest of all Para|Hyper|bola's, & yⁿ, yy = xx + qx. & if (q) is y^e 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 y^e touch points) are ordinately {illeg} applyed to y^e Diameters & bisect y^e Pa Hyperbolas.

<14v>

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

<15r>

The Properties of y^e 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 y^e first & transverse axis or side

ch is y^e seacond & right axis

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

<16v>

If ab toug|c|h an Ellipsis & (o) & (x) be y^e foci yⁿ y^e 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. y^e angle uho shall be bisected by y^e 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.