# Fitzwilliam Notebook

pret 8^{d}

Nabed | Efyhik, |

Wfnzo | Cpmkfe |

^{[Editorial Note 1]}

## Before Whitsunday 1662.

## Since Whitsunday 1662

## 1665

## March 25 1666.

Lent Wilford | ——X—— | 0 . 1 . 0. |

To y^{e} Poore on y^{e} fast | ———— | 0 . 1 . 0. |

To M^{r} Babint^{9}: 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. |

## 1666.

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 |

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.

## 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.

II y^{e} b. c. & h given to find p.

1. = 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: : = dsh.

2 2r: bq - Q: : + h = dsh.

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

1 = p

2 = p.

3 = p

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

## 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

## Theor. 4.

If w^{th}in 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 y^{n} 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 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

## 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}|&| y^{n} 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 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. y^{n} 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^{th}out y^{e} cone as at r & makes right angles w^{th} gh y^{e} base of y^{e} sectio

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

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 =^{2}dg = ^{2}ec & y^{e} other diameter conjugated to it is od & od = 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 ^{2}ab + ^{2}bs, 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 y^{n} (if) & bh = fp & ac bisects y^{e} angles bai. bci. y^{n} if bh turne round y^{e} intersections of bh & ac shall describ{e} an Ellipsis. & hi & i are y^{e} foci.

## 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 y^{n} 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 pts {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} 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 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 y^{n} 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 y^{n} pbgh moveing rownd y^{e} intersections made twixt pg gh describe y^{e} parabola qbg.

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. y^{n} 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. y^{n} y^{e} parteing of y^{e} thred from y^{e} ruler as at (a) shall describe a Parabola

## 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 fro l y^{e} i|e|nd of gl. y^{t} is let y^{e} triangle npl be immutable. let y^{n} 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. y^{n} 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 y^{e} Hyper: oce

{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 = or ae = . 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 = . y^{n} 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

If dk = er be (latus transversum) & de = kr, be latus rectum y^{n} 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 y^{n} 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 y^{n} 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 y^{n} 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

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, fro the given point c to the foci, y^{n} upo the center c w^{th} any radius ce describe y^{e} circle erf. soe y^{t} ec = ef. y^{n} 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.

## The properties of y^{e} Parabola

ab = a. bc = b. ac = c. eb = d. ei = x. fi = y. b : c :: x : (ik). a : c :: d : (es, or il) whence yy = x. ab : cc :: d : (en). = r. rx = yy. y^{t} is ne a given line multipling ei = if square. Or breifly a : c :: d : (es or il). b : c :: : (en) = 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 y^{e} tangent ac. y^{n} 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 = 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 y^{n} 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 y^{n} 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.

## The properties of y^{e} Hyperbola

rx + = yy.

☞ rx + xx = yy. for

= r. & = .

{illeg}|{am}| = a. mb = b. mc = c. de = q. ei = x. di = q + x fi = y. b : c :: q + x : (il) b : a :: x : (ik). il x ik = yy = bb : ac :: q : (en)(r). bb : ac :: q : :: :: x : (qpcron) or . whenc = rx + = pi x ie = yy

More breifly thus.

b : c :: q :: (es) : b : a :: : (= 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 + = yy. (2) 2b{illeg}|y| + bb :: = 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, & y^{n}, 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. |y^{n}| (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.

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 y^{n} us = uo. & rs = oa & (iu) bisecting y^{e} angle (ria.) it shall touch y^{e} Hyperb in u.

## The Properties of y^{e} Ellipsis

rx - = yy. that is

rx - xx = yy for

= r & =

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

b : c :: q - x : (ib). b : a :: x : (ik) ki x il = = fi x fi = yy

bb : ac :: q : en = = r. bb : ac :: x : on = wherefore rx - (onx) = . = 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 - 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

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 y^{n} 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.

If ab toug|c|h an Ellipsis & (o) & (x) be y^{e} foci y^{n} y^{e} angle aco = bcx. & if (ocx) be bisected by (cr) y^{n} 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.