Newton's Waste Book (Part 1)

Author: Isaac Newton

Source: MS Add. 4004, ff. {cover}-15r, Cambridge University Library, Cambridge, UK

Published online: June 2013

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- Revision History
  - 14 February 2011
    - Catalogue information compiled from CUL Janus Catalogue by Michael Hawkins
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    - Transcription and encoding of Isaac Newton's text from ff 64v-198v by Margarita Fernandez-Chas
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Related Texts
- This document is part of MS Add. 4004
  The next part of this document is Newton's Waste Book (Part 2) [MS Add. 4004, ff. 15v-50r]

[Normalized Text] [Manuscript Images][Catalogue Entry]

<ii(r)>

{ $+ 6 = x x$ }{ $+ δ = x x$ }

<ii(v)>

$\begin{matrix} 1 & . & 5 & . & 10 & . & . & 5 & . & 1 & . \\ 1 & . & 6 & . & 15 & . & 20 & . & 15 & . & 6 & . & 1 & . \\ 1 & . & 7 & . & 21 & . & 35 & . & 35 & . & 21 & . & 7 & . & 1 \\ 1 & . & 8 & . & 28 & . & 56 & . & 70 & . & 56 & . & 28 & . & 8 & . & 1 & . \\ 1 & . & 9 & . & 36 & . & 84 & . & 126 & . & 126 & . & 84 & . & 36 & . & 9 & . & 1 \\ 1 & . & 10 \end{matrix}$

$\sqrt{a a} \times$ { $\sqrt{3: a^{3}}$ } = {illeg} $\sqrt{a b} \times \sqrt{3: c d e} = \sqrt{6: a^{3} b^{3}} \times \sqrt{6: c c d d e e} = \sqrt{6: a^{3} b^{3} c c d d e e}$ . to find y^e proportion of two irrationall rootes. to free y^e Numerato{r} or denom from {illeg}surde q{illeg} $\frac{a}{b + \sqrt{c}} = x = \frac{a}{b + \sqrt{c}} \times \frac{b - \sqrt{c}}{b - \sqrt{c}} = \frac{a b - a \sqrt{c}}{b b - c}$ .

<iii(r)>

$\begin{matrix} \begin{matrix} \frac{2 a x}{c} & - & \frac{\sqrt{a^{3}}}{c} \\ 3 a & + & \frac{\sqrt{a b b}}{c} \end{matrix} \\ \begin{matrix} 6 a a x & - & 3 a \sqrt{a^{3}} \\ + & \frac{2 a x}{c} \sqrt{\frac{a b b}{c c}} & - & \frac{\sqrt{a^{4} b b}}{c c} \end{matrix} \\ \begin{matrix} 6 a a x - 3 a \sqrt{a^{3}} + 2 a x \sqrt{a b b} - \frac{a a b}{c} \\ error tantum 1′ in metallo 2″ in radio \end{matrix} \end{matrix}$ $\begin{array}{l} \begin{matrix} \frac{1}{10} & \begin{matrix} \frac{1}{60} . & \frac{1}{100} \times \frac{1}{12} \end{matrix} & 30 \times 60 \times 60 & \begin{matrix} 1 \\ 60, 30, 30 \\ 54000 \end{matrix} & \begin{matrix} 1 \\ 60, 30, 60 \\ 108000 \end{matrix} & 1 c \\ \begin{matrix} 1 \\ \begin{matrix} 60000 \\ 12 \end{matrix} \end{matrix} & \begin{matrix} 1200 (40 \\ c \end{matrix} \end{matrix} \\ \begin{array}{l} 300, 12, 10 \\ 3000, 12 \\ 36000 \end{array} \\ \begin{array}{l} \frac{1}{60, 60, 60 .} & = & \frac{1}{300, 720} \end{array} \end{array}$ $\sqrt{9 a^{5}}$ $a$ $3 a \sqrt{a}$ $\begin{matrix} 0,051513 \\ 0,0132 \end{matrix}$ $(5$ {illeg} $b a a x - \sqrt{(9) a^{5}}$ $\frac{c}{d}) \frac{a}{b} (\frac{d a}{c b}$ $\begin{matrix} \begin{matrix} - 2 \\ - 2 \end{matrix} & \begin{matrix} \frac{3}{8} \times 9 \end{matrix} & \begin{array}{l} 14 \\ 56 \\ 14 \end{array} \end{matrix}$ $\begin{array}{l} 8 . \frac{1}{2} ∷ \frac{1}{2} . \frac{1}{32} = \frac{6}{16} = \frac{3}{8} . & \begin{array}{l} 9 & . & 196 \\ 9 & ) & 392 & ( & 43 \frac{5}{9} \end{array} \\ 2 . \frac{1}{4} ∷ \frac{1}{4} . \frac{1}{32} . \\ 96.7 ∷ 7 . \frac{49}{96} \end{array}$ $\begin{array}{l} 1 \frac{c}{d} ∷ \frac{a}{b} . \frac{d a}{c b} \\ \frac{c}{d} . 1 ∷ \frac{a}{b} . \frac{d a}{c b} \\ c . d ∷ \end{array}$ $1 .$
$\frac{c}{d} . 1 ∷ \frac{c}{d} in d . 1 \times d ∷ c . d ∷ \frac{c}{c} . \frac{d}{c} ∷ 1 . \frac{d}{c}$ $\frac{c}{d} 1 ∷ c . d ∷ 1 . \frac{d}{c} ∷ \frac{a}{b} . \frac{d a}{c b}$ . $\frac{c}{d} . 1 ∷ (\frac{c}{d} in d) c . (1 \times d) d ∷ (\frac{c}{c}) 1 . \frac{d}{c} ∷ \frac{a}{b} ∶ (\frac{d}{c} \times \frac{a}{b}) \frac{d a}{c b}$

$\frac{a + 3 x}{c} \sqrt{\frac{4 b b x^{4}}{81 a a}}$ $\frac{2 b x x}{9 a}$ $\frac{2 a b x x + 6 b x^{3}}{9 c a}$ $a a - 2 a y = a b - b b + b y$ $a a = a b - b b + b y + 2 a y$ $a a - a b + b b = b y - 2 a y$ $\frac{a a - a b + b b}{b - 2 a} = y$

$\begin{matrix} z = \frac{y y}{x} \\ 20 = a \\ 140 = b \end{matrix}$ $z z$ $x . y . \frac{y y}{x}$ . $y y = \frac{y y x}{x}$ $x + y + \frac{y y}{x} - a = 0$ $x x + y y + \frac{y^{4}}{x x} - b = 0$ $x + y + \frac{y y}{x} - a + x x + y y + \frac{y^{4}}{x x} - b = 0$

<1r>

To find a heavy bodys de{illeg}|s|cent in any given time, & y^e proportion of y^e pressure of y^e rays {illeg}|b|y gravity to |y^e| force by w^ch a{illeg} given body hath \any/ given motion; by this figure If y^e cilinders bc, df bee of glasse &c: to {kow} y^e proportion of their strength is knowne by y^e proportion of the{illeg} gravity of y^e circles a, e &c {illeg}|i|n respect of y^e axis in.

If a Staffe bee bended to find y^e crooked line w^ch it resembles.

If the motion of a line is knowne to find y^e crooked line w^ch y^t line toucheth continually.

If a stick ab revolve {sic} \w^th even velocity/ about y^e center a haveing y^e weight c fastened {in} it by {the} string bc , yⁿ shall y^e string bc bee a tangent to y^e circle bde.

But i{illeg}|t| may be inquired what line y^e weight $(c)$ would describe were y^e stick w^th uneven velocity, {illeg} or did y^e point b describe a Parabola or some other crooked line were y^e weight c in some other place as at y^e center { a } when y^e stick began to move.

If y^e ball b revolves about y^e center n y^e force by w^ch it endeavours from y^e center n would {beget} soe much { b } motion in a body \as ther {sic} is/ in y^e time y^t y^e body b moves y^e length {of y^e} {semidiamiter {sic}} bn . [as if b is moved \w^th one degree of {motion}/ through {illeg} \{illeg} {illeg}/e \ bn / in {a} seacond of an {hower} & bn is {os{illeg} {illeg}d{illeg}} yⁿ its force from y^e center {illeg} {illeg} being continually like y^e force of gravity impressed upon {illeg}|y^e| body during one second it will generate one degree of motion in {illeg}|y^t| body.] Or y^e force from {n} in one revolution is to y^e force of y^e body motion as rad ∷ $periph ∶ rad$ . Demonstracon. If { $ab = bc = dc =$ } $ef = fg = gh$ \ $ad .$ / $= he = 2 fa = 2 fb = 2 fb = 2 gc = 2 ed$ . & y^e globe {b} from a to b {yⁿ} $2 fa ∶$ {ak}∷ $∷ ab ∶ fa ∷$ force {or} pression of b {illeg}|u|pon fg {at} its reflecting ∶ for{illeg}|e| {sic} of b{'}s motion. therefore $4 ab = ab + bc + cd + da ∶$ {illeg}fa ∷ force of y^e reflection in one round (viz: in b , c , d , & a) ∶ force of b 's motion. by y^e sa{me} pro{illeg} y^e Globe b were reflected by each side of a circumscribed polygon of 6, 8, 12, 100, 1000 sides {illeg} y^e force of all y^e reflections is to y^e force of y^e bodys as y^e sum of those sides {illeg} r{adius} of y^e circle about w^ch they are circumscribed. {illeg} is if {illeg} And so if body were reflected by {illeg} the sides of an equilaterall {illeg} circum{}|s|cribed polygon of an infinite number of sides {illeg} by {illeg} {illeg} circle it selfe) y^e force of all y^e reflections are to y^e force of y^e bodys motion {as each} those sides ({illeg} y^e perimiter) to y^e radius.

If y^e body b moved in an Ellipsis y^e {sic} its force in each point (if its motion {in y^t point bee} g{illeg}) bee found by a tangent circle of equal crookednesse with y^e|t| {illeg}|p|oint of y^e Ellipsis.

If a body undulate in y^e circle bd all its undulations of any altitude are performed in y^e {same} time w^th y^e \same/ radius. Galileus.

As radi{illeg}|u|s ab to radius ac ∷ {illeg} so are y^e Squares of theire times in w^ch they undulate.

If c circulate in y^e circle { cgef } , to whose diamiter \{ ce ,}/ $ad = ab$ being perpendicular yⁿ will y^e body b undulate in y^e same time y^t c circulate.

And y^e {illeg} those body circulate in y^e same time whose {illeg} {illeg} from y^e {illeg} to y^e center d are equall

And $ad ∶ dc ∷$ force of gravity to y^e force of { c } to its center d . {illeg} {illeg} {illeg} {illeg} y^e motion of things falling were they not hindered by y^e {illeg} may very {illeg} {illeg} $cd ∶ ad ∷$ force form d ∶ force from a.

<1v>

^[1] $ag = x$ . $gh = y$ . $ah = c$ . $\begin{matrix} b x x & + & e x & + & c c & = & 0 & . \\ + & d x y & + & f y \\ + & g y y \end{matrix}$ . $dp = v$ {illeg}. $ad = \frac{x x - y y + c c}{2 c}$ ^[2] ${dg}^{2} =$ {illeg} $\frac{2 c c x^{2} + 2 x x y y + 2 c c y y - c - x^{4} - y^{4}}{4 a a} + v v = s s$ $v ∶ w ∷ \frac{v y}{x} ∶ \frac{w y}{x}$ . $\frac{v y}{x} = df$ . /fig 2^d. $\sqrt{x x - y y} - v ∶ y ∷ y$ . $ds = \frac{y y}{\sqrt{x x - y y} - v}$ \ $\begin{array}{l} - & 4 c^{4} & - & x^{4} & - & 2 b x^{4} & - & 2 e x^{3} & - & 2 d x^{3} & in & y \\ - & 2 f x x \\ - & 2 e c c x & - & b b x^{4} & - & 2 c c d x \\ - & 2 b c c x x & - & 2 e b x^{3} & - & 2 b c c x^{2} & - & 2 c c f \\ - & 2 c c e e x & - & e e x x \end{array}$ $y y ∶ + v \sqrt{x x - y y}$ $y y - x ∶ y$ $\frac{y^{3}}{x \sqrt{x x - y y} - v x}$ + {illeg} $\frac{+ y \sqrt{x x - y y}}{x} = df$ . $\frac{y^{3} - y^{3} + y x x - v y \sqrt{x x - y y}}{- v x + x \sqrt{x x - y y}}$ . ^[3] {illeg} $\frac{x x - v \sqrt{x x - y y}}{\sqrt{x x - y y} - v} = bd$ . $x x - v \sqrt{x x - y y} ∶ v \sqrt{x x - y y} ∷$ a {illeg}∶{illeg}f $ab = x$ . Parab $∠ mak = ∠ bad$ , ergo: ad $∠ mak = ∠ ade = ∠ bad$ . ergo, $ab = bd$ .

^[4] $∠ a = ∠ bad = ∠ adb - ∠ asb = ∠ sad$ ergo tri: anb, anm, sim: $ab = x$ . $bs = a$ . $eb = y$ . $ae = \sqrt{x x - y y}$ . $bo = v$ $ao = \sqrt{x x - 2 v y + v v}$ . $ab ∶ as ∷ bo ∶ os$ . $x ∶ \sqrt{x x - 2 a y + a a} ∷ v ∶ a - v$ . $∠ vat =$ {illeg}|∠ | $bao = ∠ oas$ . $ab = x$ . $as = y$ . $bo = v$ . $bs = a$ . $x ∶ y ∷ v ∶ a - v$ . $v y = a x - v x$ Hyperb.

^[5] $os = v$ . $sa = x$ . $oi = ae = y$ . $sa ∶ ae ∷ so ∶ oh = \frac{v y}{x}$ . $oi ∶ oh ∷ d ∶ e$ : ergo $e y = \frac{d v y}{x}$ . & $e x = d v$ . Hyp: $ab ∶ ae ∷ bc ∶ cn$ . $\frac{ae \times v}{x} = cn$ . $d ∶ e ∶ \frac{ae \times v}{x} . ae$ . $d x = e v$ .

^[6] $ab = y$ . $as = x$ . $bc = v$ . $∠ bac = ∠ cas$ , Ergo $y ∶ x ∷ v ∶ a - v$ . $bs = a$ . & $x v = a y - v y$ . Ellipsis.

^[7] $ab = x$ . $as = y$ . $∠ bac = tar = cas$ . Ergo, $ba ∶ as ∷ bc = v ∶ cs = a - v$ $a y - y v =$ $a x - v x = v y$ . Hyperb.

^[8]

The invention of Figures for refra|le|ctions. \at right angles/ at y^e point refra|le|cting ac y^e rad: reflected to y^e focus b. ag y^e radius reflected {illeg}|fr|o y^e focus b. aq a perpendic: to y^e ed y^e tangent of y^e crood|k|ed line shought. $ab = x$ . \ $ac = y$ , or,/ $bd = w$ . $ag = y$ . $bg = a$ , or, $bc = a$ . $bd = v$ , or $bq = a$ . fig: 1^st^[9]. $∠ eac = ∠ bad = ∠ adb$ . Ergo, $ab = bd$ , or $x = v$ . & $∠ caq = qab = aqb$ . e{illeg}|rg|o $ab =$ {illeg}qg. $x = v$ . fig: 2^d^[10]. $∠ eac = ∠ bad = ∠ adg$ . Ergo, $ab ∶ ag ∷ bd ∶ dg$ $a x - v x = v y$ . &, $ab ∶$ {illeg} $∶ ag ∷ qb ∶ bg$ . {illeg} Ergo $a x + v x = y$ y|v|. $v = bq$ . fig 3^d^[11]. $∠ eac = ∠ bad$ {illeg} Ergo $∠ caq = ∠ qab$ . Ergo, $ca ∶ ab ∷ cq ∶ qb$ . & $a x - v x = v y$ .

The invention of figures for refraction. b , & g y^e foci. ca y^e Rad: refracted to b . qa y^e Rad: refracted from b. bg y^e distance of y^e foci. qa y^e perpend: to de y^e tangent of y^e crood|k|ed line sought qr, $qh =$ perpendic:s to y^e Radiusi cg, fb.

$bg = a$ . $bq = v$ . $ba = x$ . $ag = y$ . fig: 1^st^[12]. $ab ∶ as ∷ bq ∶ qr = \frac{v y}{x}$ . $d ∶ e ∷ qr ∶ qh$ , Ergo. $d x = e v$ . fig: 2^d^[13]. $ab ∶ as ∷ bq ∶ qr = \frac{v y}{x}$ . $d ∶ e$ ∶ {sic} $qh ∶ qr$ , Ergo. {illeg}|e| $x = d v$ . fig 3^d^[14]. $ab ∶ as ∷ bq ∶ qr = \frac{v in as}{x}$ . $d ∶ e$ ∶ {sic} $qr ∶$ $ag ∶ as ∷ gq ∶ qh = \frac{a + v in as}{y}$ . $d ∶ e ∷$ $qh ∶ qr$ / $qr ∶ qh$ \: Ergo $e a x + e v x = d v y$ $d a x + d v x = e v y$ . fig {illeg} 4^th^[15]. $ab ∶ as ∷ bq ∶ qr = \frac{v in as}{x}$ . $ag ∶ as ∷ gq ∶ qh = \frac{v - a in as}{y}$ . $d ∶ e ∷ qr ∶ qh$ : $d v x - d a x = e v y$ . fig 5^t^[16]. $ab ∶ as ∷ bq ∶ qr = \frac{v in as}{x}$ . $ag ∶ as ∷ gq ∶ qh = \frac{a - v in as}{y}$ . $d ∶ e ∷ qh ∶ qr$ . $e a x - e v x = d v y$ . fig 6^t. $ab ∶ as ∷ bq ∶ qr = \frac{v in as}{x}$ . $ag ∶ as ∷ gq ∶ qh = \frac{a - v in as}{y}$ . $d ∶ e ∷ qr ∶ hq$ . e | d | $a x - d$ {illeg}| v | $x = e v y$ .

<2r>

$ad = a$ . $ae = x$ . $ed = y$ . $af = z$ . $fd = a - z$ .
$x x - z z = y y - z z + 2 a z - a a$ . $fg = v$ \2/ $ae ∶ ef ∷ ag ∶ gi$ . & $de ∶ ef ∷ dg ∶ gh$ . ^[17] $\frac{x x - y y + a a}{2 a} = af$ . $\frac{y y - x x + a a}{2 a} = fd$ . x {illeg} $∶ \frac{\sqrt{2 a a x x - x^{4} + 2 x x y y - y^{4} + 2 a a y y - a^{4}}}{4 a a} ∷ \frac{x x - y y + a a + 2 a v}{2 a} ∶ gi$ . gi \{illeg}/ $= \frac{\sqrt{2 a a x x + 2 x x y y + 2 a a y y - x^{4} - y^{4} - a^{4}} in \sqrt{+ 2 a a x x - 2 a a y y - 2 x x y y + x^{4} + y^{4} + a^{4} + 2 x x a v - 2 y y a v + 2 a^{3} v +}}{4 a a x}$ $gh = \sqrt{2 a a x x + 2 x x y y + 2 a a y y - x^{4} - y^{4} - a^{4}} in \sqrt{}$ $ef =$ {illeg}|m|. $ae = x ∶ m ∷ \frac{x x - y y + a a + 2 a v}{2 a}$ ∶ |=| $ag ∶ gi = \frac{m x x - m y y + m a a + 2 m a v}{2 a x}$ . $ed ∶ m ∷ gd ∶ \frac{m y y - m x x + m a a - 2 m a v}{2 a y} = gh$ . $gi ∶ gh ∷ d ∶ e$ : therefore $\frac{d y y - d x x + d a a - 2 d a v}{y} = \frac{e x x - e y y + e a a - 2 e a v}{x}$ . {illeg} $d = 2$ . e {illeg} $= 1$ . $\begin{matrix} y^{3} & + & 2 x y y & - & x x y & + & 2 a a x & = & 0 \\ - & a a y & - & 2 x^{3} \\ - & 2 a v y & - & 4 a x v \end{matrix}$ $\frac{y^{3} + 2 x y y - x x y - a a y + 2 a a x - 2 x^{3}}{2 a y + 4 a x} = v = \frac{x x - y y}{2 a} + \frac{2 a x - a y}{2 y + 4 x}$ $bf = \frac{2 a a x + 2 a a y + 2 x x y y - x^{4} - y^{4} - a^{4} in y + 2 x}{y^{3} + 2 x y y - x y y - a a y + 2 a a x - 2 x^{3} in 2 a}$ . $af = v$ $ae ∶$ {ef} $∷ ag ∶ gi$ . $de ∶ ef ∷ dg ∶ gh$ . $d ∶ e ∷ gi ∶ gh$ . $x ∶ m ∷ v ∶ \frac{m v}{x}$ . $y ∶ m ∷ a - v ∶ \frac{m a - m v}{y}$ . $d ∶ e ∷ \frac{m v}{x} ∶ \frac{m a - m v}{y}$ : & $a d x - d v x = e v y$ . $af = x$ . $fe = y$ . $ae = \sqrt{x x + y y}$ . $ed = \sqrt{x x - 2 a x + a a + y y}$ . $a d \sqrt{x x + y y} - d v \sqrt{x x + y y} = e v \sqrt{x x - 2 a x + a a + y y}$ $a a d d x x + a a d d y y + 2 a d d v x x + 2 a d d v y y + d d v v x x + d d v v y y = e e v v x x - 2 e e v v a x + e e v v a a + e e v v y y$ . $ef = o$ . $\begin{matrix} \begin{matrix} bc & = \end{matrix} & \begin{matrix} \begin{matrix} 2 a a x x y - x^{4} y - a^{4} y + 2 a a y^{3} + 2 x x y^{3} - y^{5} + 4 a^{2} x^{3} + 4 a a y y x + 4 y y x^{3} - 2 y^{4} x - 2 a^{4} x - 2 y^{4} x - x^{5} \\ - 2 a o y^{3} - 4 a o x y y + 2 a o x x y + 2 a^{3} o y - 4 a^{3} o x + 4 a o x^{3} \end{matrix} \\ 2 a y^{3} + 4 a x y y - 2 a x x y - 2 a^{3} y + 4 a^{3} x - 4 a x^{3} \end{matrix} \end{matrix}$ ^[18] $an = on$ $pg = a$ . $ab =$ {illeg}|y.| $ag =$ {illeg}|x.| $pg = b$ . $pm = c$ . $gb = z$ . $gq = v$ . $e z x - e v x = d v y$ . $gm = p$ . $gs =$ ^[19] $ab = x$ . $qb = v$ {illeg}. $v = x$ . $mp = a$ . {illeg} $aq = \sqrt{v v - x x}$ . $\frac{x x}{v} = bs$ . $sa = \sqrt{\frac{v v x x - x^{4}}{v v}}$ . $nb = y$ . $mn = \frac{y}{v} \sqrt{v v - x x}$ $nr = nb$ . $e ∶ d ∷ tb = nm ∶ rs = \frac{d y \sqrt{v v - x x}}{e v}$ . $rs ∶ nr ∷ mn ∶ np = \frac{\frac{y y}{v} \sqrt{v v - x x}}{\frac{d y}{e v} \sqrt{v v - x x}} = \frac{e y}{d} \sqrt{y y v v - y y x x + a a v v}$ $e e y y v v - d d y y v v + d d y y x x = d d a a v v$ . ${mn}^{2} = \frac{d d a a v v - d d a a x x}{e e z z - d d z z + d d x x}$ . ${np}^{2} = \frac{e e a a v v}{e e z z - d d z z + d d x x}$ . ${mb}^{2} = \frac{d d a a v v}{e e z z - d d z z + d d x x}$ {illeg} $= z z$ . $\frac{d d a a x x - d d x x z z}{e e z z - d d z z} = v v$ . $v = \frac{d x \sqrt{a a - z z}}{z \sqrt{e e - d d}}$ . ${sa}^{2}$ ${bs}^{2} = \frac{e e z z x x - d d z z x x}{d d a a - d d z z}$ . ${as}^{2} = \frac{d d a a x x - e e z z x x}{d d a a - d d z z}$ . $sb ∶ ab ∷ ab ∶$ \ $2$ / {bq}{bp}{illeg}sb 2 $x x =$ \ $2$ / {illeg}sb{illeg}b{illeg}sb. $sb = \sqrt{}$ . $sb = n$ : $x x - n n = {sa}^{2} = x$ −/+\ $n \times x - n$ . $x x$ {illeg} $x + 3 n$ {illeg} $x x - n n = n n$ {illeg}x $ab = x = bq = v$ . $as = y$ . $nb = z$ $nm = \frac{y z}{x}$ . $e ∶ d ∷$ {illeg} $z ∶ \frac{d z}{e} = np$ . {illeg} $e^{2} y y z z + a a x x e^{2} = d d z z x x$ ${nb}^{2} = \frac{e e a a x x}{d d x x - e e y y}$ . ${nm}^{2} = \frac{e e a a y y}{d d x x - e e y y}$ . ${mb}^{2} = \frac{e e a a x x - e e a a y y}{d d x x - e e y y}$ . {illeg} ${np}^{2} = \frac{a a d d x x}{d d x x - e e y y} =$ {illeg} $ξ^{2}$ . $x x$ {illeg} $y y$ {illeg} $x x$ {illeg} $y y$ ${as}^{2} = \frac{d d z z x x - e e a a x x}{e e z z} = \frac{- a a d d x x + d d x x ξ ξ}{e e ξ ξ}$ . $d d$ {illeg} $ξ ξ$ $- e e a a ξ ξ + a a d d z z = 0$ . ^[20] $ab =$ ^[21] $ab = bq = x$ . $pm = a$ . $as = y$ . $no = z$ : $\frac{z x}{y} = nb$ . $om = \frac{z z}{a} = mt$ . $po = \frac{a a - z z}{a}$ . ${pn}^{2} = \frac{a^{4} - a a z z + z^{4}}{a a} ∶ \frac{z z x x}{y y} ∷$ {p^t{illeg}} ${pt}^{2} = \frac{a^{4} + 2 a a z z + z^{4}}{a a}$ ${tb}^{2}$ {−} $z^{6} x x$ $b$ $x x = c y y$ . $x x - \frac{b x x}{c} = {sb}^{2}$ . $x \sqrt{\frac{c - b}{c}} - \sqrt{\frac{e e a a x x - \frac{e e a a b x x}{c}}{d d x x - \frac{e e b x x}{c}}} = \sqrt{}$ $x x$ | $y y$ | $= b x x + c$ {illeg} $x +$ {illeg}g. ${sb}^{2} = x x - b x x$ {illeg} $- c x - g$ . ${mb}^{2} = \frac{e e a a x x - e e a a b x x - e e a a c x - e e a a g}{d d x x - e e b x x - e e c x - e e g}$ . $qa = s$ . $qm = v$ . $v v + x x - b x x - c x - g$ . $+ \frac{e e a a x x - e e a a b x^{2} - e e a a c x - e e a g}{d d x x - e e b x x - e e c x - e e g}$ : $- 2 \sqrt{x x - b x x - c x - g} in \sqrt{\frac{c a a x x - c^{2} a^{2} b x^{2} - e^{2} a^{2} {illeg} - {illeg}}{d^{2} x^{2} - e^{2} b x^{2} - e^{2} c x - e {illeg}}}$ {illeg} $- 2 v \sqrt{x x - b x x - c x - g} + 2 v \sqrt{\frac{e^{2} a^{2} x^{2} - e^{2} a^{2} b x^{2} - e^{2} a^{2} c x - e^{2} a^{2} g}{d d x x - e e b x^{2} - e e c x - e e g}} = s s = v v + x x +$
$= 2 o x + 2 e e$ $mb = z$ . $sb = x$ . $sa = y$ . ${ab}^{2} = x x + y y$ . $z z = \frac{e e a a x x}{d d x x + d d y y - e e}$ $sm =$ {illeg}|x| {illeg} $sa = y$ . $mb = z$ . ${sb}^{2} = x x + 2 z x + z z$ . $x^{2} = b y y + c y + d$ . $y y$ {illeg} $= b x x + c x + g$ . $sm = x - \sqrt{\frac{e e a a x x}{d d x x + d d - e e in b x x + c x + y}}$ $v v - 2 v x + 2 v \sqrt{\frac{e e a a x x}{d d x x + d d - e e in b x x + c x + g}}$ {+} $x x \frac{+ e e a a x x}{d d x x + d d - e e in b x + c x + g} - 2 x \sqrt{\frac{e e a a x x}{+ d d in b b x x +}}$ $= s s = - 2 v o \begin{matrix} + 2 v \\ - 2 x \end{matrix} \sqrt{\frac{e e a a x x - 2 e e a a o x}{d d x x + 2 d d o x + d d - e e in b x x + 2 b o x + c x + c o + g}} +$ {illeg} $v v$ . $mb = z$ . $sb = x$ : $sa = y$ . { $mn =$ } $\frac{y z}{x}$ . ${ab}^{2} = x x + y y$ . $\frac{z z x x + z z y y}{x x} =$ { ${sb}^{2}$ }{illeg} $e e z z x x + e e z z y y - d d a a x x - d d y y z z = 0$ . $x^{2} = b z z +$ {illeg}. $y y = \frac{e}{d d}$ {illeg} $v v$ $- 2 v \sqrt{b z z + c z + g}$ $+ b z z + c z + g$ $+ \frac{e e z z - d d a a in b z x}{d d z z - e e z z}$ {illeg} { $4 v v b in$ } $b z^{2} + c z + g + \frac{z^{4} - 4 a a z^{2} + 4 a^{4} in Q: b z z + c z + g}{z^{4}}$ {illeg} $\frac{- 4 z^{4} v \sqrt{b z z + c z + g} in z z - 2 a a in b z z + c z + g}{z^{4}}$ {illeg} 4{illeg} $+ 16 o a^{4} in$ {illeg} $+ g g$ {illeg} $- 16 o v z^{2} \sqrt{b z z + c z + g} in z z - 2$ {illeg} $in b z$

<2v>

^[22] ^[23] Make y^e line ac to revolve about y^e point a : on y^e end c let y^e nut c bee fastened so {as} to t{illeg}|u|rne about its center. make $ab = ac$ & fastend another nut at y^e point b in y^e same manner. {illeg} make y^e line bc to slide through those two nuts soe y^t y^e △ abc will always be an isosceles. To y^e line cb fasten y^e line rstv at {illeg} right angles. {illeg} make y^e line kg w^th 2 nuts e & d at each end through w^ch y^e lines rs & tu must slide to keepe y^e line kg perpendicular to bc , in {illeg}|y^e| midst of kg fasten y^e nutt m so as it {illeg} may turne about its center & y^t y^e line ac may slit|d|e thro{illeg}|ugh| it then make y^t side of y^e line kg w^ch is next ab to be a file w^ch must be very smooth at the point m but must grow rougher towards y^e ends d & e . Then by turneing y^e line ac \to & from l & h / about its center & holding y^e file kg close to y^e plate hmflab , it shall fil{illeg}|e| it into y^e shape of a Parabola.

To describe y^e Parabola by points. {illeg}|Ma|ke $ca = \frac{r}{4}$ ; c y^e {illeg}|v|ertex; {illeg}|a | y^e focus; $ab = r$ . yⁿ w^th some radius as $ag = ae$ , describe y^e circle ge : & take $bd = 2 ga = 2$ {illeg} $ae =$ \ $2$ /de & y^e point e shall bee in y^e parabola, also if from e to g , a \streight/ line be drawne it shall touch y^e Parab: in e.

Or thus, take $ch = ca = \frac{r}{4}$ , $hd = 2 cg$ \or $da = 2 hg$ :/; & $ga = ae = de$ . &c:

Or thus, take $cm = gc$ , $dm = ma$ ; & $ga = ae = de$ ; &c.

Or thus take $cm = gc$ & raise me a perpendicular to ca, w^ch shall intersect y^e parab, & circle ge in y^e same point.

Or thus. {illeg}|U|pon y^e focus or center a describe y^e center {ef} make $ab = \frac{r}{4}$ . $bd = 2 bc =$ {illeg}|r|. $kb = bg$ . w^th y^e Rad bc describe bed . y^e circle.

^[24] Or thus take \ $\frac{r}{2} =$ / $ac = cn = Rad$ . Circle aen: $ab = \frac{r}{4}$ . $am = ap$ & produce mp indefinitely. Then take some point ad in y^e line an , & draw dg perpendic: to an y^t is soe y^t $dm = dg$ , yⁿ take $df = ae$ , & f shall be a point in y^e parabola afr.

<3r>

Banderon's addition to Ferrarius's Lexion \Geographicu/, y^e best for Geog. {T}{F}{illeg} \Ortelius/ Geogr. Lexicon. $\begin{matrix} M^{r} John Craige \\ D^{r} Archibald Pitcarne \end{matrix}} Scotch Mathematicians$

Experiments about \the resistance of/ things falling in water.

1. I filled to y^e top a {illeg}|woo|den vessel {illeg} 9 inches squa{illeg}|r|e within & 9 foot $4 \frac{1}{2}$ inche{illeg}|s| high \within/. And making balls of bees wax of several bignesses & w^th pieces of Lead stuck in them to give them weight: three balls each of w^ch weighed in the air $76 \frac{1}{2}$ grains & in y^e water $5 \frac{1}{16}$ grains, fell each of them in y^e water from y^e top to y^e bottom of y^e vessel in 15″ of time the motion of descent being (to {sence}) almost uniform almost from y^e top to y^e bottom. so then a globe equall to $71 \frac{7}{16}$ gr of water moving \uniformly/ 9 foot $4 \frac{1}{2}$ inches ({illeg}) in 15″ of time feels a resistance equal to $5 \frac{1}{16}$ gr of weight.

2. Two balls, each weighing in air $156 \frac{1}{4}$ gr in water 77 gr fell each of them the same height \of 9 foot $4 \frac{1}{2}$ {dig.}/ in 4″ of time. And these exp^ts seemed sufficiently accurate.

Corol. Ergo y^e resistance is as y^e square of the velocity.

3. Two balls weighing \each of them/ in air 245 gr, in water \{almost}{illeg}/ $1^{gr} \frac{1}{2}$ fell \each/ y^e same height in $44 \frac{1}{2}$ ″. But thi|e|se exp^ts w{illeg}|er|e not so accurate as f {sic} for{mers.}

The same two balls augmented w^th lead so as each of them to weigh in air $251 \frac{1}{2}$ in water $7^{gr} \frac{1}{8}$

Three balls

<4r>

^[25] $as = y$ . $sm = x$ . $mb = z$ . $mp = a$ . \ $qs = v$ / $mn = \frac{y z}{x + z}$ . ${ab}^{2} = x x + y y + 2 z x + z z$ . $\frac{z z y y + z z x x + 2 z^{3} x + z^{4}}{x x + 2 z x + z z} = {nb}^{2}$ . $\frac{e e z z y y + e e z z x x + 2 e e z^{3} x + e e z^{4}}{d d x x + 2 d d z x + d d z z} = {np}^{2} = a a + \frac{y y z z}{x x + 2 z x + z z}$ . $sb = z$ . $mn = \frac{z y - x y}{z}$ . ${ab}^{2} = {qb}^{2} = z z + y$ ${nb}^{2} = \frac{{\dot{z}}^{4} + y y z \dot{z} - 2 x {\dot{z}}^{3} - 2 x \dot{z} y y + \dot{x x z z} + \dot{x x y y}}{z z} = \frac{d d z z y \dot{y} - 2 d^{2} z x \dot{y} y + d d x \dot{x} y^{2} + a a d d z z}{e e z z}$ . $e^{2} = 1$ . $d^{2} =$ $\begin{matrix} z^{4} & - & 2 x z^{3} & + & x x z z & + & 2 x y y z & - & x x y y & = & 0 \\ - & y y \\ - & 2 a a \end{matrix}$ . Suppose. $y y = f x^{2} + 2 g x + h$ . |yⁿ,| $v = f x + g$ . $z + f x + g = ω$ . $\begin{array}{l} ω^{4} & - & 4 f x ω^{3} & + & 6 f f x x ω^{2} & - & 4 f^{3} x^{3} z & + & f^{4} x^{4} & = & 0 \\ - & 4 g & + & 12 f g x & - & 12 f f x x g & + & 4 f^{3} x^{3} g \\ - & 2 x & + & 6 g g & - & 12 f x g g & + & 6 f f x x g g \\ + & 5 f x x & - & 4 g^{3} & + & 4 f x g^{3} \\ + & 4 g x & - & 4 f f x^{3} & + & g^{4} \\ + & x x & + & 6 f g x^{2} & + & f^{3} x^{4} \\ - & 2 g g x & + & 2 f f x^{3} g \\ - & 2 a a & + & f x g g \\ - & h & + & 2 g x x \\ + & 2 f h x & - & f f x^{4} \\ + & 2 g h & - & 4 f g x^{3} \\ + & 4 f a a x & - & 3 g g x^{2} \\ + & 4 g a a & - & f f h x^{2} \\ + & 2 h x & - & 2 f g h x \\ - & g g h \\ - & 2 a a f f x x \\ - & 4 a a f g x \\ - & 2 a a g g \\ - & 2 f h x x \\ - & 2 g h x \\ - & f x^{4} \\ - & 2 g x^{3} \\ - & h x x \end{array}$ . & $ω = bq$ . $ab = ξ$ . ${sb}^{2} = ξ^{2} - y^{2}$ . $mb = \sqrt{ξ ξ - y y} - x$ . $\frac{\sqrt{ξ ξ - y y} - y x}{\sqrt{ξ ξ - y y}} = nm$ . $\frac{ξ \sqrt{ξ ξ - y y} - ξ x}{\sqrt{ξ ξ - y y}} = nb$ . | ${nb}^{2} =$ | $a \dot{a} d d + \frac{y y ξ ξ d \dot{d} - y^{4} d \dot{d} + y y x x d \dot{d} - 2 d^{2} y y x \sqrt{ξ^{2} - y^{2}}}{e e ξ ξ - y y e e}$ $= \frac{e e \dot{ξ^{4}} - e e ξ ξ y \dot{y} + e e ξ ξ \dot{x x} - 2 e e ξ ξ x \sqrt{ξ ξ} - y y}{e e ξ ξ - e e y y}$ . $d d = 2$ . $e e = 1$ . $\begin{array}{l} 2 a a ξ^{2} - 2 a a y y + 3 y y ξ^{2} - 2 y^{4} + 2 y y x x - ξ^{4} - ξ ξ x x \\ - 4 y y x \sqrt{ξ^{2} - y^{2}} + 2 ξ ξ x \sqrt{x x - y y} \end{array}} = 0$ .

Problems. 1 To find y^e axis, diameters, cente{illeg}|r|s, asymptotes \& vertices/ of lines

2 To compare their crookednesse w^th y^e crookednes of a gi{illeg}|v|en circle

3 To find y^e longest & shortest lines w^ch can {illeg}|b|e drawn w^th in & perpendicular to the line & to find a{illeg}|ll| such lines are {illeg}|per|pendicular at both ends to y^e given crooked line

4 To find where th{illeg}|ei|r greatest or least crokednesse is.

5 To find y^e areas, y^e l{illeg}|e|ngths, & centers of gravity {illeg}|o|f crooked lines \when it may b{e}/

^[26] 6 If y (one {illeg}|u|ndetermined quantity) moves perpendic{illeg}|u|larly to x (y^e other undetermin{ed} quantity. if $s = a secant = db$ . $v = dc$ . $y = bc$ . $x = ca$ . Then having y^e proportion of {illeg}|v| to {x}{s} to find y, or having y^e proportion of v to y to find x: when it may bee.

7 To reduce all kinds of equations, when it may bee

8 To find tangents to any crooked lines. Whither Geometricall or Mechanicall

9 To compare y^e superficies of one line w^th y^e area of another & to find y^e centers of gravity twixt two lines or sollids. 15

10 Ha{illeg}|v|eing y^e {illeg} respe position w^ch x must beare to y {illeg} (as if x is always in y^e same line, but y cutteth x at given angeles {sic}. or if x & y wheeling about 2 poles describe y^e lines by theire intersection &c) to find theire position in respect of y^e line soe y^e equation e{illeg}|x|pressing theire relation may bee as simple as may bee (as to find in w{illeg}|h|at line x is & w^t angles it maketh with y; or to find y^e distance of y^e 2 poles & in what line they must be, soe y^t y^e relation twix{t} x & y may bee had in a{illeg}|s| simple termes as may bee).

11 Of y^e description of lines.

12 Reasonings of gra{illeg}|v|ity & levity upon severall suppositions (as y^t y^e rays of gravity are parallel or verge towards a center; y^t they are reflected, refracted, or neith{er} by y^e weighty body &c.

13 Of y^e u{illeg}|s|e of line{illeg}|s|

< insertion from lower down f 4r >

14. To f{illeg}|i|nd such lines whose areas length or centers of gravity {illeg}|m|ay bee found.

15. To compare y^e areas, lenghs {sic}, gravity of lines \when it may bee./ & to find such lines whose lengths, {illeg}|ar|eas may be comp{illeg}

16. To doe y^e same to sollids in respect of theire areas, content, gravity &c w^ch was done to lines in respect of their are{illeg} lengths, areas, & gravity.

17. Of lines w^ch l{illeg}|y|e not in y^e same plane as tho{illeg}|se| made by y^e intersection of a cone & {sphæreides}.

18. Two equations given to {illeg}|k|now whither they expresse y^e same line or not.

19. Of y^e proportion w^ch y^e rootes of an equation beare to one another.

20 One line being to find other lines at {pleasure} of {illeg} {same length} {illeg}

21 How much doth any medium resist y^e motion of any given body.

22 To Determin maxima & minima in equation w^ch hath more then {sic} {illeg} unknowne quantitys.

To Determin max & min by numbers.

< text from higher up f 4r resumes >

<4v>

^[27] $cd = x$ . $gd = y$ . $r x + \frac{r}{q} x x = r x$ . $ac = a$ $de = \frac{1}{2} r + \frac{r}{q} x$ . $ag = \sqrt{a a + 2 a x + x x + r x + \frac{r x x}{q}}$ . $af = a + x + \frac{1}{2} x + \frac{r}{q}$ $\frac{a a + 2 a x + x x + a r + 2 \frac{a}{q} r x + \frac{r x x}{q} + \frac{r r x x}{q q} + \frac{1}{4} r r = {fa}^{2}, in r x + \frac{r}{q} x x}{a a + 2 a x + x x + r x + \frac{r x x}{q}} = {fo}^{2}$ . $fo ∶ fl ∷ 2.1$ . | $a a + a r + \frac{1}{4} r r = b b$ | | $x c = 2 a x + \frac{2 a r x}{q} + \frac{r r x}{q}$ | | ${fl}^{2} =$ | $\frac{a a r x + \frac{a a r}{q} x x + 2 a r x x + 2 a r x x x + r x^{3} + \frac{r}{q} x^{4} + a r r x + 3 \frac{a r r}{q} x x + \frac{1}{4} r^{3} x + \frac{r^{3} x x}{2 q} + \frac{2 a r r x^{3}}{q q} + \frac{2 r^{3} x^{3}}{q q} + \frac{r^{3} x^{4}}{q^{3}}}{2 a a + 4 a x + 2 x x + 2 r x + \frac{2 r x x}{q}}$ $a a r + a r r + \frac{1}{4} r^{3} = b b r$ . $a + \frac{1}{2} r = b$ . $\frac{b b r x + c r x x + g r x^{3} + \frac{r q q + r^{3} x^{4}}{q}}{2 a a + 4 b x + \frac{2 q + 2 r x x}{q}} = {fl}^{2}$ . $\frac{a a}{q} + 2 a + \frac{3 a r}{q} + \frac{r r}{2 q} = c$ . $\frac{2 a}{q} + 1 + \frac{2 a a}{q q} + \frac{2 r r}{q q} = g$ . {illeg} {illeg} $r =$ {illeg} / $r = 2$ $q = 6$ . $x = 3$ .\ $gd = r^{2} x + \frac{1}{3} x x = 3$ . {illeg} / $ac = 1$ \ $df = \frac{r}{2} + \frac{r}{q} x = 2$ . $ag = 5$ . $5 ∶ 3 ∷ af = 6 ∶ \frac{18}{5} = of$ . $fl = \frac{9}{5}$ . $d ∶ e ∷ of ∶ fl = \frac{18 e}{5 d}$ . $dk = \frac{9 \times 2}{9 - \frac{324 e e}{25 d d}} \overset{\cup}{O} \sqrt{\frac{\frac{18 \times 324 e e}{25 d d} + \frac{81 \times 324 e e}{25 d d} - 9 \times Q: \frac{324 e e}{25 d d} :}{81 - 18 \times \frac{324 e e}{25 d d} + Q: \frac{324 e e}{25 d d} :}}$ . $dk = \frac{450 d d}{225 d^{2} - 324 e e} \overset{\cup}{O} \sqrt{\frac{801900 d d e e - 944784 e^{4}}{50625 d d d d - 145800 d d e e + 104976 e^{4}}}$ $dk = \frac{50 d d}{25 d d - 36 e e} \overset{\cup}{O} \sqrt{\frac{9900 d d e e - 11664 e^{4}}{625 d^{4} - 1800 d d e e + 1296 e^{4}}}} = g$ . $de = a - 3$ . $g = p + \sqrt{p s}$ . $e = 1$ $\frac{9 a a e^{2} - 54 a e e + 81 e e - 18 a g e e + 54 g e e + 9 g g e e}{d d g g}$ $\frac{\begin{matrix} + a a e e - 6 a e e + 9 e e \\ - 2 a g e^{2} + 6 g e^{2} + g g e e \end{matrix}}{d d}$ $\frac{\begin{matrix} - 9 a a + 54 a - 81 \\ + 18 a g - 54 g - 9 g g \end{matrix}}{g g}$ $\begin{matrix} = e^{2} \\ = z^{2} \end{matrix}$ $\begin{array}{l} 9 a a e e & - & 54 a e e & + & 81 e e & - & 18 a p e e & - & 18 a e e \sqrt{q s} & + & 54 e e p & + & 54 e e \sqrt{q s} & + & 9 e e p p & + & 9 e e q s & - & 2 q s \sqrt{q s} \times e e a & = & z^{2} \\ + & e e p^{4} & + & 6 e e p p q s & + & e e q q r s & + & 2 a a e e p \sqrt{q s} & - & 2 a a e p^{3} & + & 18 e e p \sqrt{q s} & + & a a e e p p & + & a a e e q s & + & 6 e e q s \sqrt{q s} \\ - & 9 a a d d & + & 54 a d d & - & 81 d d & - & 12 a a e e p \sqrt{q s} & + & 6 e e p^{3} & + & 18 e e \sqrt{q s} & - & 6 a e e p p & - & 6 a e e q s \\ + & 18 a p d d & - & 54 p d d & - & 9 p p d d & - & 6 a a e e p \sqrt{q s} & - & 54 d^{2} \sqrt{q s} & + & 9 e e p p & + & 9 e e q s \\ + & 18 e e p p \sqrt{q s} & - & 18 p d d \sqrt{q s} & - & 6 a e e p q s \\ + & 4 e e p^{3} \sqrt{q s} & + & 18 e e p q s \\ + & 4 e e p q s \sqrt{q s} & - & 9 d d q s \\ + & 18 a d d \sqrt{q s} \end{array}$ $d d p p + 2 d d p \sqrt{q s} + d d q s$

^[28] $ac = a = ce$ . $cd = x$ . $dg = y$ . $bc = x$ . $df = \frac{1}{2} r$ . $r x = y y$ . $a a + 2 a x + r x + x x ∶ r x ∷ \begin{array}{l} a a + 2 a x + a r + r x \\ + x x + \frac{1}{4} r r \end{array} ∶ lf$ . $r x + \frac{a r + \frac{1}{4} r r in r x}{a a + 2 a x + r x + x x} = {fo}^{2}$ . ${fl}^{2} = \frac{e e r x}{d d} + \frac{4 e e a r r x + e e r^{3} x}{4 d d a a + 8 d d a x + 4 d d r x + 4 d d x x}$ . ${fl}^{2} = p p$ . $lk = r$ $ac = a$ . $ab = x$ . $a ∶ x ∷ x ∶ a - x$ . $x x = a a - a x$ . $x x = 4 - 2 x$ . $x = - 1 + \sqrt{5}$ $2 ∶ \sqrt{5} - 1 ∷ \sqrt{5} - 1 ∶ 3 - \sqrt{5}$ . $6 - 2 \sqrt{5} = 5 - 2 \sqrt{5} + 1$ . $x x = - a x + a a$ . $\begin{array}{l} 144 \\ \underline{18} \\ 324 \end{array}$ $\begin{array}{l} \underline{18 \times 25} = 450 \\ 90 \\ 36 \end{array}$ $\begin{array}{l} 144 \\ 10 \end{array}$ $\begin{array}{l} \underline{324 \times 18} = 5832 \\ 2592 \\ 324 \end{array}$ / $\begin{array}{l} 29160 \\ 11664 \end{array}} = 145800$ \ $\begin{array}{l} 324 \\ 2592 \end{array}} = \begin{array}{r} 26244 \\ 5832 \end{array} \begin{matrix} \begin{matrix} 32076 \end{matrix} \end{matrix}$ $x = - \frac{1}{a} + \sqrt{\frac{5 a a}{4}}$ $ei = a$ . $hk = z$ .h{illeg} $= \frac{e z}{d}$ \ $gd = y$ / $hi = \frac{e z}{d}$ . $he = \sqrt{\frac{e e z z - a a d d}{d d}}$ . $ek = \sqrt{\frac{d d z z - e e z z + a a d d}{d d}}$ . $ce = a$ . $fl = b$ $fk = c$ . $lk = \sqrt{c c - b b}$ . $z ∶ \sqrt{\frac{e e z z - a a d d}{d d}} ∷ c ∶ b$ {illeg}. $c = \frac{d b z}{e e z z - a a d d}$ . ${lk}^{2} = \frac{d d b b z z - b b e e z z + a a b b d d}{e e z z - a a d d}$ . $df = v$ $lk = \frac{b v + \frac{d b b z}{\sqrt{e e z z - a a d d}}}{y}$ {illeg} $= \frac{d b b y + b v \sqrt{e e z z - a a d d}}{y \sqrt{e e z z - a a d d}} = \frac{y \sqrt{d d b b z z - b b e e z z + a a b b d d}}{y \sqrt{e e z z - a a d d}}$ $d d b b y y + v v e e z z - v v a a d d - d d y y z z + e e y y z z - a a y y d d + 2 b d v y \sqrt{e e z^{2} - a a d d} = 0$ . $ab = q$ . $bc = x$ . $dc = y$ . $r = lat ∶ r x ∶ r x - \frac{r x x}{q} = y y$ . $ef = b$ . $fg = c$ . $eb = \frac{1}{2} q - b$ . $cf = x - \frac{1}{2} q$ . $ce = x - \frac{1}{2} q + b$ . $b ∶ c ∷ x - \frac{1}{2} q + b ∶ y$ . $\frac{c x - \frac{1}{2} q c + b c}{b} = y = \sqrt{r x - \frac{r x x}{q}}$ $\begin{matrix} c c x x & - & q c c x & + & 2 b c c x & + & \frac{1}{4} q q c c & - & q c c b & + & b b c c & = & 0 \\ + & \frac{b b r}{q} x x & - & b b r x \end{matrix}$ . $q = 3$ . $fg = c = 1$ {illeg} $b = 1$ . $x x + \frac{1}{3} x x = 3 x - 2 x + x - \frac{9}{4} + 3 - 1 = 0$ . $\frac{4 x x}{3} = 2 x - \frac{1}{4} = 0$ $x x = \frac{6}{4} x - \frac{3}{16}$ . $x = \frac{3}{4} \overset{\cup}{O} \sqrt{\frac{9 - 3}{16}}$ . $x = \frac{3}{4} \overset{\cup}{O} \sqrt{\frac{3}{8}}$ $r x + \frac{r x x}{q} = 9$ . $\frac{r}{2} + \frac{r x}{q} = 2$ / $\frac{1}{2} r + \frac{r x x}{q} = 7$ $\frac{r x}{2}$ {illeg} $= 7$ .\ $ac = a$ . $kd = b$ . $ad = c$ . $dc = d$ . {illeg} $c ∶ a ∷ b ∶ \frac{a b}{c}$ {illeg} $\frac{d a b}{e c} = cf$ . $kl = e$ . $x = 1$ . $r = \frac{1}{4}$ . $ad = a$ . $dl = b$ . $lk = c$ . $ac = g$ . $\frac{d g b}{e a} = ce$ . $\frac{d d g c}{e e a} = ef$ . ${of}^{2} = \frac{d^{4} g g c c - e^{4} c c a a}{e^{4} a a}$ . $e = 2$ . $d = 2$ . $+ b g g c$ $+ \frac{b g g c c}{a a} - c c = {of}^{2}$ . $g = c = 1 = a$ . $15 = {of}^{2}$ . $\frac{g g c c}{16 a a} - c c$ . $g =$ {illeg}|{ 8 }| {illeg} $= 8 a =$ {illeg} $2 c$ $3 = {of}^{2}$ . $ab = a$ . $bc = b$ . $cd = c$ . $be = x$ . $ce = x - b$ . $de = \sqrt{x x - 2 b x + b b - c c} = \frac{c x}{a}$ . $x x = \frac{2 b x - b b + c c in a a}{a a - c c}$ $x = \frac{a a b}{a a - c c} \overset{\cup}{O} \sqrt{\frac{a^{4} b b + a a b b c c + a^{4} c c - a a c^{4}}{a^{4} - 2 a a c c + c^{4}}}$ $bm = f$ . $be = g$ . $me = f - g$ . $ab = a$ . {illeg} $cd = c$ . $a ∶ g ∷ c ∶ \frac{g c}{a} = de$ . $g ∶ a ∶ f - g ∶ \frac{a f - a g}{g} = mr$ . $mn = z$ . $\frac{a a f f - 2 a a f g + a a g g}{g g} + f f - 2 f g + g g (= {re}^{2}) in \frac{e e}{d d} = {rn}^{2} = \frac{a a f f - 2 a a f g + a a g g}{g g} + z z$ .

<5r>

$eh = 8$ — − $ei = 6$ $hi = 10$ . $ek = 15$ . $\sqrt{\begin{matrix} 75 \\ 154 \\ 6 \end{matrix}} = \sqrt{289} = 17 = hk$ . { $gh =$ }{ $gk =$ } $de =$ {illeg}15. $dg = 16 = y$ . $cd = x$ . $df = a$ . $gk =$ {14} $34 ∶ 16 ∷ gk ∶ gd ∷ 17 ∶ 8 ∷ 30 - a ∶ \frac{240 - 8 a}{17} = fl$ . $fl ∶ fo ∷ 10 ∶ 17$ . $\frac{120 - 4 a}{5} = 24 - \frac{4 a}{5} = fo$ . $ad = b$ . $ag = \sqrt{b b} +$ {illeg} $256 ∶ b b + 256 ∷ 576 - \frac{192 a}{5} + \frac{16 a a}{25} ∶ \frac{576 b b - \frac{192}{5} a b b + \frac{16 a a b b}{25}}{256} + 576 - \frac{192 a}{5} + \frac{16 a a}{25} = {af}^{2} = a a + 2 a b +$ {illeg} $\frac{5 b b}{4} + \frac{3 a b b}{20} + \frac{a a b b}{400} - \frac{9 a a}{25} - 2 a b - \frac{192}{5} a + 576 = 0$ . $\begin{array}{l} 500 b b \\ + 60 a \\ + a a \end{array} \begin{matrix} = 200 a b + 144 a a + 15360 a - 2304 \end{matrix}$ {illeg} $y y = r x + \frac{r}{q} x x = 256$ . $\frac{r}{q} x + \frac{r}{2} = a$ . $256 - a x = \frac{1}{2} r x$ . $\frac{512}{x} - \frac{a}{2} = r$ $eh = \frac{5}{2}$ . {illeg} $ei = 6$ . $hi = \frac{13}{2}$ . $hk = \frac{2221}{20}$ . $ek = \frac{9 \sqrt{5149}}{20}$ $gh =$ {illeg} $\frac{221}{20} ∶ \frac{50}{20} ∷ \frac{221}{20} + c ∶ dg = \frac{50}{20} + \frac{}{221}$ $dg = \frac{5}{2} + \frac{50 c}{221}$ . $\frac{9 c \sqrt{5149}}{221} = de$ $fk = \frac{+ 9 \sqrt{5149}}{20}$ {illeg} $fk = \frac{9 c \sqrt{5149}}{221} +$ $ek = \frac{3 \sqrt{5149}}{20}$ . $kd = g$ . $\frac{5 g \times 20}{6 \sqrt{5149}} = \frac{50 g}{3 \sqrt{5149}} = dg$ . ${dg}^{2} = \frac{2500 g g}{46341}$ . ${gk}^{2} = 48841 g g$ . $fk = c - a$ . $\frac{50 c g - 50 a g}{3 \sqrt{5149} in 221} = fl = \frac{f
k \times dg}{gk}$ $\frac{50 c g - 50 a g}{390 \sqrt{5149}} = fo$ . $ae = p$ . {illeg} $d
e = g - \frac{3 \sqrt{5149}}{20}$ $ad = p - g + \frac{3 \sqrt{5149}}{20}$ . ${ag}^{2} = \begin{array}{l} p p - 2 p g + \frac{48841 g g}{46341} + \frac{46341}{400} \\ + \frac{3 p}{10} \sqrt{5149} - \frac{3 g}{10} \sqrt{5149} \end{array}$ ${ag}^{2} ∶ {gd}^{2} ∷ {af}^{2} ∶ {fo}^{2}$ .

$eh =$ {illeg}. {illeg} $eh = a$ . $ei = b$ . $hi = \sqrt{a a + b b}$ $hk = \frac{d}{e} \sqrt{a a + b b}$ . $ek = \sqrt{\frac{d d a a + d d b b - e e a a}{e}}$ $eh = c$ . $ek = \sqrt{\frac{d d c c + d d b b -}{e e}}$ $2 a b x = n$ . $a x x + a b b ∶ b x x + b a a ∷ f ∶ g$ . $\frac{a b b g - a a b f}{} - a x x g - b x x f = 0$ $\frac{n n}{4 a a b b} = x x$ . $a g n n - b f n n + 4 a^{3} b^{4} g - 4 a^{4} b^{3} f = 0$ . $f = 10$ . $g = 17$ . $b^{4}$ $a x x - a b b = 6$ . $x x = \frac{6 + a b b}{a}$ . $\frac{17 a b b - 10 a a b}{10 b - 17 a} = x x$ $\begin{matrix} 10 a^{3} b & - & 17 a a b b & + & 10 a b^{3} & + & 60 b & = & 0 \\ - & 17 a a b b & - & 102 a \end{matrix}$ {illeg} $\begin{matrix} a^{3} & - & \frac{34}{10} a a b & + & a b b & + & 6 & = & 0 \\ - & \frac{102 a}{10 b} \end{matrix}$ . $q a^{2} + q a + s \times a + c$ . $c$ {illeg} $= \frac{6}{s}$ . $r + \frac{6}{s} = \frac{34 b}{10}$ . $- \frac{10 r}{34} + \frac{60}{34 s} = b$ . $\begin{array}{l} a^{3} & + & r a a & + & s a & + & c s \\ + & c & + & c r \end{array}$ $r = - \frac{6}{s} - \frac{34 b}{10}$ . $- \frac{36}{s s} - \frac{204 b}{10} s + s = b b - \frac{102}{10 b}$ . $c c = 20 c - 45$ $\begin{array}{l} q a^{3} & + & r a a & + & s a & + & s c \\ + & q c & + & r c \end{array}$ . $b = 1$ $10 b a^{3}$ $10 a^{3} - 34 a a + 112 a + 60$ . $\begin{array}{l} 448 \\ \underline{336} \\ \underline{3808} \end{array}$ $\begin{matrix} \begin{array}{l} (2.2.2.2.2.7 . / 2.2.3.5 \\ \begin{matrix} \begin{matrix} 1904 \\ 952 \\ 476 \\ 238 \\ 119 \\ 17 \end{matrix} & \begin{array}{l} 3868 & . & (2.2.967 \\ 1934 \\ 967 \end{array} \end{matrix} \end{array} \\ \begin{matrix} \begin{array}{l} 1.2.4.6.10.3.5.15.30 . \\ 20 \end{array} \\ \begin{array}{l} 10 . . \\ \begin{matrix} \begin{matrix} 2.967.4.1934.3868 . \end{matrix} \\ 10 \end{matrix} \end{array} \end{matrix} \end{matrix}$ $\begin{array}{l} 36 \\ \underline{102} \\ 1056 = \end{array}$ $\frac{p q - r}{Div -}$ $- n n + 2 p n$ . $p - n = c$ \{illeg} $a a - 4 a$ / 4 $10 a^{3} - 34 a a - 92 a + 60$ $\begin{matrix} - \frac{34}{} & + & 2 & + & 4 \\ - & 2 & - \end{matrix}$ . $\begin{matrix} - 4 \end{matrix}$ $\begin{array}{r} 120 \\ \underline{- 34} \\ 86 \end{array}$ $\begin{array}{r} 150 \\ \underline{34} \\ 126 \end{array}$ $\begin{array}{l} 368 \\ \underline{27} \\ \underline{120} \\ 3168 \end{array}$ $(\begin{array}{l} 2.2 . \\ 1594.797 . \end{array}$ $\begin{matrix} \begin{array}{r} 372 \\ \underline{372} \\ 4094 \\ \underline{- 120} \\ 3974 \\ 1987 \end{array} & \begin{array}{l} 14 & c \\ \begin{array}{l} 56 \\ 14 \end{array} \\ \begin{matrix} (2 & \begin{array}{r} 96 \\ \underline{16} \\ 256 \end{array} \\ \begin{array}{r} 144 \\ \underline{18} \\ 324 \end{array} & 117 & 13 \end{matrix} \end{array} \end{matrix}$ $20 a^{3}$ $- 34 a a$ {illeg} $- 136 a a$ {illeg} $- 22 a$ \{illeg}/ + {illeg} 120 $\begin{matrix} \underline{49} & \begin{array}{r} 96 \\ \underline{48} \\ 576 \\ 49 \end{array} & \begin{array}{l} 625 \\ 2 \end{array} \end{matrix}$ $a^{2} - \frac{4}{10} a$ 6 {illeg} ^[29] $30 a^{3} - 306 a a + 168 a - 180$ . $\begin{array}{r} 1008 & (2.2.3.3.4.6.9.12.24.18.54.36 . \\ \underline{504} \\ \underline{- 180} \\ 51228 & . 25614.12807.4269.1423 \end{array}$ $\begin{array}{l} 136 & 1531 \\ 134 & 2 \\ 17 \end{array}$ 102 $\begin{matrix} 40 a^{3} & \begin{array}{r} - 204 \\ - 340 \end{array} a a & \begin{array}{r} + 640 \\ - 102 \end{array} a & + 240 . \\ 40 a^{3} & - 544 a a & - 538 a & + 240 . \end{matrix}$ $\begin{matrix} eh = 12 . & ei = 9 . \\ hi = 15 . & hk = 20 . & ek = 16 . \end{matrix}$ $\begin{array}{r} \underline{17} \\ 119 \\ \underline{17} \\ \underline{289} \\ 2083 \\ 289 \end{array}$ $\begin{array}{r} 4913 \\ \underline{- 1530} \\ 3 \\ 729 \end{array}$ $\begin{array}{r} 81 \\ \underline{648} \\ 6561 & \times 4 = \end{array}$ { 2 }620 {illeg}| 1 | {illeg}

^[30] d {illeg} $- 3 =$ $\frac{a x x - c}{a} = b b$ . $a x x + a b b ∶ b x x + b a a ∷ e ∶ d$ . $d a x x + d a b b = e b x x + e b a a$ . $\begin{array}{r} 25839 & ( \\ 6 \\ 239 \end{array}$ {illeg} $x x = \frac{a a b e - a b b d}{a d - e b} = \frac{10 a a b - 17 a b b}{17 a - 10 b}$ . $x x = \frac{17 a b b - 10 a a b}{10 b - 17 a}$ . $40 b - 17 a b b + 10 a a b$ . / $a = \frac{9}{17}$ . $b = 1$ . {illeg}\ $10 - \frac{90}{17}$ {illeg} $3, 4, 9$ . $5, 12, 13$ . $7, 24, 25$ . $12, 16, 20$ . $12, 9, 15$ . $x x = \frac{4 a b b - 3 a a b}{3 b - 4 a}$ . {3 } $\begin{matrix} x & \begin{matrix} - 4 a x x \\ - 4 a b b \end{matrix} & + 3 a a b \end{matrix}$ $a x \frac{2 x x + 2 b b}{3 b} \overset{\cup}{O} \sqrt{\frac{4 x^{4} + 8 b b x x + 4 b^{4} - 9 b b x x}{9 b b}}$ . $\sqrt{4 x^{4} - b b x x + 4 b^{4}}$ . $x = 4$ . $b = 1$ . {illeg} $\frac{+ 8}{6} \overset{\cup}{O} \frac{8}{6} = a = 3$ {illeg} {illeg} $= \frac{4}{3}$ $x =$ {illeg} $x =$ {illeg} {illeg} $+ \frac{1}{3} = a b$ {illeg} $4 + \frac{1}{9}$ {illeg}15 7 $4 x^{4} - b b x^{2} + 4 b^{4} = 4 b x^{4}$ ^[31] {illeg} $\frac{8}{3}$ $\frac{}{3}$ . $\frac{8 + 2}{3} = \frac{10}{3} = c b$ . $4 + \frac{4}{9}$ . $\frac{30}{9}$ . {illeg} $\frac{8}{3}$ {illeg} $b =$ {illeg} $\frac{4 x^{4} - c^{4}}{x^{5} +} = b b$ . {illeg}+{illeg}={illeg} $- 2 b c + c c$ . $\frac{c c + x x}{2 c} = b$ . $4 x^{4} - b b x x$ {illeg} $b^{4} = 4 b^{4} - b b x x + 6^{3}$ {illeg} {illeg} = {illeg} $c^{4} - 4 a^{2} b b$ {illeg} $3 a a b = 4 b b - 3 a b = 4 b b - 4$ {illeg} $c c$ . $\frac{}{4 c}$ {illeg} {illeg} $a a =$ {illeg} $- 4 a a$ {illeg} {illeg} $9 a c c -$ {illeg} $36 a$ {illeg} {illeg}

<5v>

^[32] $\sqrt{4 x^{4} - b b x x + 4 b^{4}} = \sqrt{4 b^{4} + 4 b b c c + c^{4}}$ $^{2}$ $\frac{4 x^{4} - c^{4}}{4 c c + x x} = b b = \frac{4 x^{6} + 16 c c x^{4} - 4 x x - 4 c^{6}}{4 c^{4} + 4 c c x x + x^{4}}$ . $h x^{2} + i n +$ {illeg} $a x^{3} + b x^{2} + c x + d$ . $\sqrt{4 x^{4} - b^{2} x^{2} + 4 b^{4}} = - b b + c x + d x x + e e$ . $4 x^{4} - b b x x = - 4 b b c x - 4 b b d x x - 4 b b e e + c c x x + 2 c d x^{3} + 2 c e e x + d d x^{4} + 2 d e e x x + e^{4}$ : $\frac{+ 4 x^{4} + d d x^{4} + 2 c d x^{3} + c c x x + 2 d e e x x + 2 c e e x + e^{4}}{- x x + 4 d x x + 4 c x + 4 e e} = b b$ . | $24 - \frac{50}{3}$ / $\frac{22}{36 + 45 - 20}$ \ $\frac{\frac{22}{3}}{12 + 15 - \frac{20}{3}}$ | $2 f x x + 2 g f x + f f$ . $2 e = f$ . $\frac{2 c}{e} = 2 g$ . $\frac{c c x x}{e} = - x x + 4 d x x$ $c c = - e e + 4 d e e$ . $d = \frac{c c + e e}{4 e e}$ $\frac{4 x^{4} : \frac{+ c^{4} + 2 c c e e + e^{4}}{16 e^{4}} in x^{4} : \frac{+ c^{3} + c e e}{2 e e} in x^{3} : + c c x x : \frac{+ e^{4} + c c e e}{2 e e} in x x : + 2 c e e x + e^{4} .}{- x x : \frac{+ c c + e e}{2 e e} in x x + 4 c x + 4 e e} = b b$ $\begin{matrix} h h x^{4} & + & 2 h i x^{3} & + & 2 h k x^{2} & + & 2 i k x & + & k k \\ + & i j x x \end{matrix}$ . $\frac{- 63 e^{4} + 2 c c e e + c^{4}}{16 e^{4}} = h k$ . {illeg} $\sqrt{}$ {illeg}e {illeg}k {illeg} $- e e$ . $2 f x x + 2 g f x + f f$ . $2 e = f$ . $\frac{2 c}{e} = 2 g$ . $\frac{c c x x}{e} = - x x + 4 d x x$ $c c = - e e + 4 d e e$ . $d = \frac{c c + e e}{4 e e}$ $\frac{- 4 x^{4} : \frac{+ c^{4} + 2 c c e e + e^{4}}{16 e^{4}} in x^{4} : \frac{+ c^{3} + c e e}{2 e e} in x^{3} : + \frac{3}{2} c c x x + \frac{e e}{2} x x + \frac{3}{2} c c x x + 2 c e e x + e^{4}}{\frac{c c x x}{e e} + 4 c x + 4 e e} = b b$ $k = e e$ . $2 i = 2 c$ . $i = c$ . {illeg} $h = \frac{c c + e e}{4 e e}$ . $\frac{c^{4} + 2 c c e e - 63 e^{4}}{16 e^{4}} = h k$ {illeg} $c^{4} + 2 c c e e - 63 e^{4} = c^{4} + 2 c c e e + e^{4}$ . $e e = k$ . $i = c$ . $d = h$ . $\frac{\frac{c^{4} + 2 c c e e + e^{4}}{e e} in x^{4} : \frac{+ c^{3} + c e e}{2 e e} in x^{3} : + \frac{3}{2} c c x x + \frac{e e}{2} x x + \frac{3}{2} c c x x + 2 c e e x + e^{4} + f f}{2 f} = x x$ .

^[33] $\frac{4 - 3 + \frac{3}{2} + \frac{1}{2} + 2 + 9}{2} =$ {illeg} 6 /{illeg}\ $\begin{array}{r} x^{3} \\ + y^{3} \end{array} \frac{+ 9 z z^{4} - 3 a z x^{2} y + y^{3} z - 3 a z x y y}{\sqrt{9 x^{4} - 6 a y x^{2} + 10 a a y^{2} + a a x^{2} - 6 a^{3} x}}$ $25 + 5 + 6 + \frac{1}{2} + 4 + 1 41 \frac{1}{2}$ $\frac{83}{4 f}$ $2 f f$ / 5 \ $\frac{25}{4} + \frac{5}{8} +$ $64 - 24 + 6 + 2$ {illeg} {illeg}|5| $\begin{matrix} d a b b - e b x x - e b a a + d a x x = 0 \\ d p q q - e q y y - e q p p + d p y y = 0 \end{matrix}$ . $s s - v^{2} + 2 v y - y y = \frac{a^{4}}{y y} - \frac{2 a^{4}}{2 y^{3}} + 2 y$ $2 a b = 2 e x x$ . $\frac{d a x x}{e a a - e x x} = b$ . $\frac{e b x x + e b a a}{a b b + a x x} = d = \frac{e q y y + e q p p}{p q q + p y y}$ . $\begin{matrix} p q q b x x + p q q b a a + p y y b x x + p y y b a a \\ - a b b q y y - a b b q p p - a x x q y y - a x x q p p \end{matrix} = 0$ $p q q a + p y y a = b q y y + b q p p$ . $a x x q y y + a x x q p p = p q q b x x + p y y b x x$ . $p q q a b + p y y a b = b b p p q + x x p p q$ . $a x x q y y + a b b q y y = p q q b x x + p y y b x x$ . $\frac{b b p q + x x p q}{b q q + b y y} = a = \frac{p q q b x x + p y y b x x}{q y y x x + q y y b b}$ . $\begin{matrix} a a b p y y & - & a b b q y y & + & b p y y x x & = & 0 \\ + & a a b p q q & - & a b b q p p & + & b p q q x x & = & 0 \\ - & a x x q y y \\ - & a x x q p p \end{matrix}$ . $a a + x x = \frac{a b b q y y + a b b q p p + a x x q y y + a x x q p p}{b p y y + b p q q}$ $\begin{matrix} a a b p y y + a a b p q q = a b b q y y + a x x q y y \\ a = x = b q q = y y = p p \end{matrix}$

< insertion from the bottom of the page >

$b^{3} + z^{3} = a b z$ . $b b - 2 b x + x x + y y -$ $\begin{matrix} s s & - & 2 v v & + & 2 v x & - & x x & = & \frac{r^{4}}{x x} \\ 0 & 0 & 1 & 2 & -2 \end{matrix}$ $- \frac{2 r^{4}}{2 x x x} + \frac{2 x x}{2 x} = v$ {illeg}

< text from f 5v resumes >

$\begin{matrix} h g x x & - & g i x & + & . \\ - & h k x & + & i k & . \end{matrix}$ $2 r x - x x + r r = y y$ . $v = r - x$ . $\begin{matrix} r x & + & \frac{r}{q} x x & - & y y & = & 0 \\ 1 & 2 & 0 \end{matrix}$ . $- r = \frac{2 r x}{q}$ $\begin{matrix} 2 r x + 2 y x + x x = y y \\ r + y + x \end{matrix}$ . $x x$ {illeg} $= - \frac{q}{2}$ $y y = - \frac{r q}{2} + \frac{r q}{4} x x \begin{matrix} + \\ - \end{matrix} y y$ $a x - x x = y y$ . $y y - \frac{r q}{4}$ . $2 r x - x x = y y$ . $v = r - x$ . $x =$ {illeg}r Symbol (taurus Operator with dot inside circle) in text $\sqrt{2 r r}$ ♉x $r z$ Symbol (inverted taurus Operator with dot outside circle) in text $z x$ for x {illeg}|wri|te $x + z - \frac{z x}{r}$ $\sqrt{+ r r}$ {illeg} for {illeg}| y | write $\sqrt{}$ $2 r x - x x - 4 z x + 2 \frac{z x x}{r} + 2 \frac{z z x}{r} - \frac{z z x x}{r r}$ {illeg} $\sqrt{x x}$ {illeg} r {illeg} x \♉/ $z \frac{\sqrt{2 r x - x x}}{r}$ for y ♉ $- r x$ {illeg} z Symbol (inverted taurus Operator with dot outside circle) in text $2 z x$ {illeg} x $- 2 x$ {illeg} Symbol (taurus Operator with dot inside circle) in text ² $z x$ ² $\frac{z x x}{r}$ $- \frac{2 z z x}{r} = + 2 r x x x + 2 r x - x x + \frac{2 x^{3}}{r} -$ /♉ $4 r x$ $\frac{2 x^{3}}{r} - 4 x x + \frac{2 x^{3}}{r}$ \ {illeg} Symbol (inverted taurus Operator with dot outside circle) in text {illeg} $x x +$ $x x z$ Symbol (taurus Operator with dot outside circle) in text $4 z x$ $\frac{2 z x x}{r}$ ♉ ♉² $z - 2$ {illeg} z $\frac{4 z x}{r}$ $2 z$ {illeg} $\frac{4 z x}{r} = 0$ {illeg}{illeg} $x x$ $- z z$ $+ \frac{2 z z x}{r}$ $- \frac{z x x}{r r}$ {illeg}−{illeg} $x x$

<6r>

^[34] $ag = x$ . $bg = y$ . $ag = x$ . $gd = d$ . $dc = e$ . $bo = s$ . $de = w$ . $gh = b$ . $y ∶ v ∷ y + b ∶ \frac{v y + v b}{y} =$ . \ $go = v$ ./ $if = c$ . $e ∶ w ∷ e + b ∶ \frac{e w}{e}$ $\frac{e w + b w}{e} = if$ . $y ∶ v ∷ y + b ∶ \frac{y v + b v}{y} = hf$ . $dc \times de$ {illeg} $dc + bw$ $de + hi + \frac{gh \times de}{dc} = go + \frac{gh \times go}{bg}$ . ag{illeg}
$ag = x$ . $bg = y$ . $r x = y y$ . $go = de = \frac{1}{2} r$ . $gd = o$ . $dc = \sqrt{r x + r o}$ . {illeg} $gh = z$ . $o + \frac{r z}{z \sqrt{r x + r o}} = \frac{r z}{z \sqrt{r x}}$ $2 o \sqrt{r x}$ {illeg} $4 o \sqrt{r r x x + r r o x} + 2 r z \sqrt{r x} = 2 r z \sqrt{r x + r o}$ . $4 o z \sqrt{r x^{3} + r o x x} = 4 z z r o$ . $x^{3} = z z r$ . $z = \sqrt{\frac{x^{3}}{r}}$ . $\sqrt{r x} ∶ \frac{r}{2} ∷ \sqrt{\frac{x^{3}}{r}} ∶ \frac{x \sqrt{r x}}{2 \sqrt{r x}} = \frac{x}{2}$ {illeg} $\frac{x + r}{2} = hf$ . $\frac{x^{3}}{r} + \frac{9 x x}{4} + \frac{3 r x}{2} + \frac{r r}{4} = {bf}^{2}$ . $r x + x x = y y$ {illeg}
^[35] $x y = r r$ . $v = - \frac{y y}{x} = \frac{- y^{3}}{r r} = - \frac{r^{4}}{x^{3}}$ . $gh = z$ . $dg = o$ . $de = \frac{+ r^{4}}{x^{3} + 3 o x x} ∶ dc = \frac{r r}{x + o} ∷ fi ∶ \frac{z x - r r +}{x + o}$ $r r$ $\frac{+ r r z x - r^{4} + r r z o}{x^{3} + 3 x x o} = fi$ . $\frac{- r^{4}}{x^{3}} ∶ \frac{r r}{x} ∷ fh ∶ \frac{z x - r r}{x}$ . $\frac{- r^{4} + z x r r - o x^{3}}{x^{3}} = fi$ $- r r$ {illeg} $x^{4} + r^{4} + x^{3}$ {illeg} $- r r z x^{4} + r^{4} x^{3} - r r z o x^{3} - r^{4} x^{3} - z r r x^{4} + o x^{6} + 3 r^{4} x x 0 - 3 z x^{3} r r o + 3 x^{5} o o -$ $+ 2 r r z x - x^{4} - 3 r^{4} = 0$ . $z = \frac{3 r r}{2 x} + \frac{x^{3}}{2 r r} = gh$ . $bh = \frac{r r}{2 x} + \frac{x^{3}}{2 r r}$ {illeg} $+ \frac{r^{4}}{x^{3}} + \frac{x}{2} = fh$ . $\frac{r^{8}}{x^{6}} + \frac{r^{4}}{x x} + \frac{x x}{4} + \frac{r^{4}}{4 x x} + \frac{x x}{2} + \frac{x^{6}}{4 r^{4}} = {bf}^{2}$ $- \frac{6 r^{8}}{x^{6}} - \frac{3 r^{4}}{2 x x} + \frac{3 x x}{4} + \frac{6 x^{6}}{2 r^{4}}$ . $6 x^{12} + 16 x^{8} r^{4} - 10 r^{8} x^{4} - 24 r^{12}$ . $6 r^{12} +$

$go ∶ gb ∷ fk + go ∶ hg$ . $de ∶ dc ∷ fi + de ∶ di$ . $dc \times fi + dc \times de + dc$ $\frac{dc \times fi + dc \times de}{de} = di = \frac{gb \times fi + gb \times dg + gb \times go}{go}$ . $\frac{\frac{r r z}{x + o} + \frac{r^{6}}{x^{4} + 4 x^{3} o}}{\frac{r^{4}}{x^{3} + 3 x^{2} o}} = \frac{\frac{r r z}{x} + \frac{r^{6}}{x^{4}} + \frac{r r o}{x}}{\frac{r^{4}}{x^{3}}}$ . $\frac{z x}{x + o} + \frac{r^{4} x}{x^{4} + 4 o x^{3}} = \frac{z x + 3 z o}{x} + \frac{r^{4} x + 3 r^{4} o}{x^{4}} + x o$ . $\frac{z x^{5} + 4 z x^{4} o + r^{4} x x + r^{4} o x}{x^{5} + 5 o x^{4}} = \frac{z x^{5} + 3 z o x^{4} + r^{4} x x + 3 r^{4} o x}{x^{5}}$ $4 z x^{9} + r^{4} x^{6} = 5 z x^{9} + 5 r^{4} x^{6} + 3 z x^{9} + 3 r^{4} x^{6}$ . $4 z x^{4} + 7 r^{4} = 0$ $\frac{r r z}{x + o} + \frac{r^{6}}{x^{4} + 4 o x^{3}} in \frac{r^{4}}{x^{3}} = \frac{r r z}{x} + \frac{r r o}{x} + \frac{r^{6}}{x^{4}} in \frac{r^{4}}{x^{3} + 3 o x^{2}}$ . $\frac{z}{x x + o x} \frac{+ r^{4}}{x^{5} + 4 o x^{4}} = \frac{z + o}{x x + 3 o x} \frac{+ r^{4}}{x^{5} + 3 o x^{4}}$ . $\frac{z x^{3} + 3 z o x x + r^{4}}{x^{5} + 4 o x^{4}} = \frac{z x^{4} + o x^{4} + 3 o z x^{3} + x r^{4} + 3 o r^{4}}{x^{6} + 6 o x^{5}}$ $\begin{matrix} \begin{matrix} z x^{5} & + & 3 z o x^{4} & + & r^{4} x x \\ 6 z o x^{4} & + & 6 r^{4} x o \end{matrix} & \begin{matrix} = & z x^{5} & + & 2 o x^{5} & + & 3 o z x^{4} & + & x x r^{4} \\ + & 4 o z x^{4} & + & 3 o x r^{4} \\ + & 4 o z r^{4} \end{matrix} \end{matrix}$ $9 z x^{4} + 6 x r^{4} = x^{5} + 7 z x^{4} + 7 x r^{4}$ . $2 z x^{4} - x r^{4} - x^{5} = 0$ . $z = \frac{r^{4}}{2 x^{3}} + \frac{x}{2}$ . $\frac{r^{4}}{x^{3}} ∶ \frac{r r}{x} ∷ \frac{r^{4} + x^{4}}{2 x^{3}} ∶ hb$ . $x x ∶ r r ∷ \frac{r^{4} + x^{4}}{2 x^{3}} \frac{r^{4} + x^{4}}{2 r r x} = bh$ . $\frac{r^{8} + 2 r^{4} x^{4} + x^{8}}{4 r^{4} x x} \frac{r^{8} + 2 r^{4} x^{4} + x^{8}}{4 x^{6}} = \frac{3 r^{8} x^{4} + 3 r^{4} x^{8} + x^{12} + r^{12}}{4 r^{4} x^{6}} = bf$ . $6 x^{12} + 6 r^{4} x^{8} - 6 r^{8} x^{4} - 6 x^{12} = 0$ . $x^{8} - r^{8} = 0$ . $x^{4} - r^{4} = 0$ . $x x - r r = 0$ . $x = r$ . therefore take $ag = gb = r$ , & y^e greatest crookedness of y^e line cb will be found at b . $bh = r = fh$ . $bf = r \sqrt{2}$ . $gh = w$ . $hf = σ$ . {illeg} $gb = y$ . $go = v$ . $w ∶ v + σ ∷ y ∶ v$ {illeg}. or, $y ∶ v ∷ w - y ∶ σ = \frac{v w - v y}{y}$ . $\frac{go \times gh - go \times bg}{bg} = fh$ . $\frac{y^{3} w}{r r} - \frac{y^{4}}{r r} = σ y$ . $\frac{2 y y w}{r r} - \frac{3 y y y}{r r} = 0$ . $w = \frac{3 y}{2}$ . $\frac{3 r^{4} + x^{4}}{2 r r x} = 0$ . $x^{3} = \frac{r^{6}}{y^{3}}$ . $\frac{3 r r}{2 x} + \frac{r^{4}}{2 y^{3}} = \frac{3 y}{2} + \frac{r^{4}}{2 y^{3}}$ . $bf = p$ . $al = q$ . $gb = y$ . $go = v$ . $ag = x$ . $bf = p$ . {illeg} $v ∶ y ∷ q + v - x ∶ \frac{q y + y v - y x}{v} = p$ . $\frac{gb \times al + gb \times go - gb \times ga}{go} =$ {illeg} $y q + \frac{y^{4}}{r r} - r r = \frac{p y^{3}}{r r}$ . $\frac{y^{4}}{r r} + 3 r r - 2 y q = 0$ . $q = \frac{y^{3}}{2 r r} + \frac{3 r r}{2 y}$ . as before. or $q = \frac{y y}{2 x} + \frac{3 x}{2} = \frac{y y + 3 x x}{2 x}$ $gl = \frac{y y + x x}{2 x}$ . $w = v + x = \frac{d y y + e y + 2 f x y + b x + d x^{2} + 2 a y x}{d + d x + 2 a y}$ $+ b x + d x w + 2 a y w + d y y + e y + 2 f x y + b x + d x^{2} + 2 a y x = 0$ . $\begin{array}{l} f x x & + & e x & + & c & = & 0 \\ + & d y x & + & b y \\ + & d y x \end{array}$ . $\begin{array}{l} w^{2} - 2 w x + x x \\ w^{2} - \frac{2 w c - 2 w b y - 2 w a y y - 2 w d y x}{f x + e} \end{array}$ $\frac{e x + d y x + c + b y + a y y}{f}$ . $\frac{+ f x x + e x + d y x + c + b y}{a}$ $x y = a a$ . $x = \frac{a a}{y}$ . $y = \frac{a a}{x}$ . $v = \frac{y y}{x}$ . $\frac{q a a}{x} + \frac{a a y y}{x x} - \frac{a^{4}}{x y} = 0$ . $\frac{q a a y}{x x} - \frac{2 a^{2} y^{3}}{x^{3}} - \frac{2 a^{4}}{x y}$ . $q =$ $\begin{array}{l} f x x & + & e x & + & c & = & 0 \\ + & d y x & + & b y \\ + & d y x \end{array}$ . $\frac{2 f x y + d y y + e y}{b + d x + 2 a y} = v$ . $q y + \frac{2 f x y y + d y^{3} + e y y - 2 p f x y - p d y y - p e y}{b + d x + 2 a y} - y x = 0$ ^[36] $\frac{2 b q y f + 4 a q y y f + 2 d y^{3} f + 2 e y y f - 2 p d y y f - 2 p e y f - 2 d y c - 2 d b y y - 2 d a y^{3}}{2 p f y + b y + 2 a y y + d e y + d d y y - d f q y - 2 f f y y} = x$ $x = \frac{- e - d y}{2 f} \overset{\cup}{O} \sqrt{\frac{\begin{matrix} e e + 2 d e y + d d y y \\ - 4 c f - 4 b f y - 4 a f y \end{matrix}}{4 f f}}$ {illeg} $e y f + 2 a e y y f + d e e y + d d y e f - d e f q y$ {illeg} {illeg} $\frac{b q y f + 2 a q y y f + d y^{3} f + e y y f - p d y y f - p e y f - d c y - d b y y - d a y^{3}}{2 p f f y + b f y + 2 a f y y + e d y + d d y y - d f q y - 2 f f y y} = x$ . $2 b q y f f + 4 a q y y f f + 2 d y^{3} f f + 2 f f e y - 2 p d f f y y - 2 p e f f y - 2 d c f y - d b f y y - 2 d a f y^{3}$ . $4 b e f y + 2 a e f y y + e d f y + c d d f y y - e d f q y + e d d y y + d^{2} y^{3} - d d f q y y$

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^[37] $y y$ {illeg} $v v + y y = s s = y y + 2 o y$ $\begin{matrix} \dot{v v} & + & \dot{x y} & + & \dot{a y} \\ + & \dot{a x} & + & \dot{y y} \end{matrix} = s s = \dot{v v} - 2 o v + \dot{x y} + x o + \dot{a x} + \dot{a y} + a o + \dot{y y} + \dot{2 o y}$ . $2 o v = o x + a o + 2 o y$ . $= v v - 2 o v + z y + z o + a z + o$ = {illeg} $v v + 2 o v + o o + r s + r a + a s + s s$ . {illeg} $v = \frac{\begin{matrix} x y & + & a x & + & a y & + & y y \\ - & r s & - & a r & - & a s & - & s s \end{matrix}}{z o}$ . $x y - r s$ {illeg}. $v = x y$
$y y - s s = 2 o$ {illeg} $x y + a x + a y + 2 y y + v v - 2 v y = s s = ξ z + a ξ + a z + 2 z z + v v - 2 v z$ $\frac{x y - ξ z + a x - a ξ}{2 y - 2 z} = b$ . $x y - ξ z - 2 b y + 2 b z = 0$ . $x y - 2 b y = 0$ $x = 2 b$ . $\frac{x x y y - ξ ξ z z}{a^{2} x - ξ a^{2}} = b$ . $\frac{1}{2} y + \frac{1}{2} x$ $\frac{\begin{matrix} 4 x^{2} & - & a x & \overset{\cup}{O} & 2 \sqrt{2 x^{4} - a x^{3}} \\ - & 4 z^{2} & + & a z & \underset{\cap}{O} & 2 z \sqrt{z z z - a z} \end{matrix}}{- 2 z + 2 x} = v$ $\frac{4 x^{2} - 4 z z}{2 x - 2 z} = b$ . $\begin{matrix} 2 x^{2} & - & 2 x^{3} & - & b x & - & b z \\ 2 & 0 & 1 & 0 \end{matrix}$ / $b = 4 x$ \ $2 x^{4} + 2 z^{4} - a x^{3} - a z^{3} - 2 z z \sqrt{4 x^{2} z^{2} - a x z^{2} + a a z x - 2 a z x x} = c c x x - 2 c^{2} z x + c c z z$ \ $\overset{\cup}{O} \sqrt{2 x^{4} - a x^{3}} \underset{\cap}{O} \sqrt{2 z^{4} - a z^{3}} = c x - c z$ /. $2 x + 2 z - \frac{1}{2} a \overset{\cup}{O} \sqrt{2 x x - a x} \underset{\cap}{O} \sqrt{\frac{1}{2} z z - \frac{a x z}{4 x}}$ \ $2 x x + 2 z x - a x$ / $2 x \overset{\cup}{O} \sqrt{2 x x - a x} \frac{+ a x \overset{\cup}{O} a \sqrt{2 x x - a x}}{\underset{\cap}{O} 2 \sqrt{x x - a x}}$ $\frac{- 4 x y + 2 a x - 2 x x + a y - 2 x y}{2 x - 2 y} = \frac{- x x \overset{\cup}{O} 6 x \sqrt{2 x x - a x} + 3 a x \underset{\cap}{O} x \sqrt{2 x x - a x}}{\underset{\cap}{O} 2 \sqrt{2 x x - a x}}$ $\frac{\begin{matrix} - 8 x x \overset{\cup}{O} 6 x \sqrt{2 x x - a x} \\ + 3 a x \underset{\cap}{O} a \sqrt{2 x x - a x} \end{matrix}}{\overset{\cup}{O} 2 \sqrt{2 x x - a x}}$ . $\frac{64 x^{3} - 48 a x x + 9 a a x}{4 x - 2 a}$ $2 x x - a x + \frac{\sqrt{4 z z x - 2 z z}}{2 x}$ $- x x + d y + y y = 0$ . $\frac{2 b ϩ z \sqrt{c c - b}}{c c} + \frac{d ϩ b}{c} + \frac{2 a b ϩ}{c} + \frac{2 b ϩ z \sqrt{c c - b}}{c c}$ . $- \frac{d}{2} = + a$ . $b = 0$ . $c =$ any finite line. as {illeg} $x = ϩ$ . $y = - \frac{1}{2} d + z$ . $ϩ^{2} + \frac{1}{4} d d \mp d z - z z$ $y^{3} = a x x + a a x$ . $y + a = ϱ$ . ϩ $ϱ^{3} - 3 a ϱ^{2} + 3 a a ϱ - a x x + a a x - a^{3} = 0$ $\begin{matrix} \begin{matrix} 450 & . \\ 225 \end{matrix} & 48 x x & \begin{array}{l} \underline{54 \times 16 \times 27} \\ 324 \\ \underline{54} \\ 864 \end{array} \\ \begin{array}{l} 240 \\ \underline{48} \\ 720 \end{array} \end{matrix}$ $- ϩ^{3} b^{3} = 0$ . $\frac{+ 6 ϩ a b c + c ϩ \sqrt{c c - b b} \times a - 3 a b c ϩ}{c c} = 0$ $\begin{matrix} \begin{matrix} 105 \\ \underline{3} \\ 2025 \end{matrix} & \begin{matrix} 75 \\ \underline{15} \\ 225 \end{matrix} & \begin{matrix} 1125 \\ \underline{225} \\ 3375 \end{matrix} \\ \begin{matrix} \underline{16} \\ 96 \\ \underline{16} \\ 256 \end{matrix} \end{matrix}$ $db = a$ . $ba = b$ . $ac = c$ . $ad = d$ . $\frac{d d}{b} = r$ , $lat rec : Parab dbe$ . $\frac{d d}{c} = s = lat : rec : Parab : dce$ . $\frac{c c}{b} = lat : rec : Par bgqc$ {illeg}. {illeg} $lq = a$ . $a x x = y^{3}$ . $y = x + z$ . $- a x x + x^{3} + 3 x^{2} z + 3 x z^{2} + z^{3} = 0$ . $\sqrt{c a x x} - x = z$ . $\frac{+ 2 a x y - 3 x x y - 6 x y y - 3 y^{3}}{3 y y + 6 y x + 3 x x} = v = \frac{2 a x y}{3 y y + 6 y x + 3 x x}$ $z^{3} + 3 x z^{2} + 3 x x z + x^{3} - d x x = 0$ $a = 8$ . $x = 1$ . {illeg} $- 7 + 3 z + 3 z z + z^{3} = 0$ $\begin{matrix} y^{3} & + & a y y & + & 2 a b & + & a b b \\ + & 2 b & + & b b \end{matrix}$ $\begin{array}{l} z z + 4 z + 7 . z = 2 \overset{\cup}{O} \sqrt{4 - 7} \\ + 448 + 192 z + 24 z z + z^{3} = 0 . \\ z z + 20 z + 11
2 \end{array}$ $\begin{matrix} \begin{matrix} 3 x = a + 2 b . & 3 x x = 2 a b + b b & b^{3} - d b b = a b b . & b - d = a \end{matrix} \\ \begin{matrix} 3 x - 2 b = a = \frac{3 x x - b b}{2 b} & - 6 b x + 3 b b + 3 x x = 0 . \\ x = a & a = b = x . b - a = d . \end{matrix} \end{matrix}$ $\begin{matrix} y + a \times y + b \times y + b \\ \begin{matrix} y y y & + & a y y & + & 2 a b y & a b b \\ + & 2 b y y & + & b b y \end{matrix} \end{matrix}$ $\begin{matrix} a x = y y . \\ - a x - y y + 2 a y + a a = 0 . \\ \sqrt{a x} = y + a . \end{matrix}$ $\begin{matrix} - & c a ϩ r & - & a b z c & + & d d c c & - & 2 d b c ϩ \\ + & 2 d c z r & + & c c z z & + & b b ϩ z & - & 2 a a b ϩ \\ - & 2 b ϩ z r & - & b b z z & + & 2 a z c d \\ + & 2 a a z r \end{matrix}$ . $\begin{matrix} - 2 b \sqrt{c c - b b} . b = c . \\ - 2 d b c - 2 a a b . \\ a a c^{4} - a a b b c c = 4 b b c c d d + 8 \\ 2 d c + 2 a c = 0 . d = \frac{a c}{- c} . d = - a . \end{matrix}$ $af = a$ . $ag = b$ . $ab = ϱ$ . $bc = z$ . $a ∶ b ∶ bh = \frac{a ϱ - a b}{b} ∶ ϱ - b = gb$ . $ch = ϩ b - a ϱ + a b$ . $fg = c$ . $ag = b$ . $fg = c$ . $fh = \frac{c ϩ}{b}$ . $af = \sqrt{c c - b b}$ . $\frac{ϱ - b in \sqrt{c c - b b}}{b} = bh$ . $\frac{b ϩ - ϱ + b \times \sqrt{c c - b b}}{b} = ch$ . $\frac{b ϩ - ϱ + b in \sqrt{c c - b b}}{c} = cd$ . $\frac{a b ϩ - a a ϱ + a a b + c c ϱ}{b c} = df$ . $c c =$ {illeg} $a a + b b$ . $\frac{ϩ b - a ϱ + a b}{\sqrt{a a + b b}} = cd = x$ . $\frac{a ϩ - a a + b ϩ}{\sqrt{a a + b b}} = df = y$ . $\frac{b ϩ + a b - x \sqrt{a a + b b}}{a} = ϱ = \frac{y \sqrt{a a + b b} - a a - a ϩ}{b}$ $b b ϩ + a a ϩ = a y \sqrt{a a + b b} - a^{3}$ {illeg} $- a b b + x \sqrt{a a + b b}$ . a {illeg} $ϩ = \frac{a y - a \sqrt{a a + b b} + b x}{\sqrt{a a + b b}}$ . Or $ϩ = \frac{a y - a c + x \sqrt{c c - a a}}{c}$ . / $ϱ = \frac{y \sqrt{c c - a a} - a x}{c}$ \ $ϱ = \frac{c y}{b} - \frac{a a y}{b c} - \frac{x a}{c}$ {illeg} $c = 5$ . $a = 3$ . $ϩ = \frac{3 y - 15 + 4 x}{5}$ . $ϱ = \frac{4 y - 3 x}{5}$ . {illeg} $y^{4}$ $ϩ^{4} = ϱ^{3}$ . |5 | $\begin{matrix} 27 y^{3} - 405 y y + 2025 y - 3375 & = & 256 y^{4} - 768 y^{3} x + 864 y y x x - 432 y x^{3} + 81 x^{4} \\ + 108 y y x - 1080 y x + 2700 x \\ 144 y x x - 720 x x \\ + 64 x^{3} \end{matrix}$ $ϱ ϱ ϩ = a^{3}$ . {illeg} $c = 5 r$ . $a = 4 r$ . $ϩ = \frac{4 y - 20 r + 3 x}{5}$ . $ϱ = \frac{3 y - 4 x}{5}$ {illeg} $= 2 r r$ . {illeg} $= r r$ .
$\frac{9 y y - 24 x y + 16 x x}{25}$ . $\begin{matrix} \begin{matrix} 36 y^{3} & - & 96 x y y & + & 64 x x y & + & 48 x^{3} & = & 225 a^{3} \\ + & 27 & - & 72 x x y \end{matrix} \\ - 180 r y y + 480 r x y - 360 r x x \end{matrix}$ $12 y^{3}$ {illeg}3 {illeg} $y y$ $\begin{matrix} \begin{matrix} 36 y^{3} & - & 69 x y y & - & 8 x x y & + & 48 x^{3} & = & 0 \end{matrix} \\ - 180 r y y + 480 r x y - 360 r x x \\ r r r \end{matrix}$ .

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$\begin{matrix} \begin{array}{l} 2 b q f f & + & 4 a q y f f \\ - & 2 q e f f \\ - & 2 d c f & + & d^{3} y y \\ - & d b y f \\ + & b e f & + & 2 a e f y \\ + & d e e & + & 2 d d e y \\ - & d e f q & - & d d f q y \end{array} & \begin{matrix} = & \begin{matrix} 2 p f f + b f + 2 a f y \\ + d e + d d y - d f q - 2 f f y \end{matrix} in \sqrt{e e + 2 e d y + d d y y - 4 a f - 4 b f y - 4 a} \end{matrix} \end{matrix}$ {illeg} $\begin{matrix} \begin{array}{l} 4 b b q q f^{4} & + & 16 a b q q f^{4} y & + & 6 b q f f d^{3} y y & + & 8 a q f f d^{3} y^{3} \\ - & 8 b q f^{3} d c & - & 4 b b q f^{3} d & - & 4 d^{4} c f & - & 2 d^{4} b f \\ + & 4 b b q f^{3} e & + & 8 a e f^{3} b q & + & 2 b e f d^{3} & + & 4 a e f d^{3} \\ + & 6 b q f f d d e & + & 4 d^{5} e \\ - & 4 b q q f^{3} d e & - & 4 b q q f^{3} d d & - & 2 d^{5} f q \\ + & 16 a a q q f^{4} \\ + & 4 d d c f f b & - & 8 a q f^{3} d b \\ + & 4 d d c c f f & - & 16 d c f^{3} a q & + & 16 a a q^{3} f^{3} & + & d^{6} y^{4} \\ - & 4 d c f f b e & - & 8 d c f f a e \\ - & 4 d d c f e e & - & 8 d^{3} f c e & - & 8 a q q f^{3} d d \\ + & 4 d d c f f e q & + & 4 d^{3} c f f q & + & d d b b f f \\ + & b b e e f f & + & 8 a q b e f^{3} & - & 4 a e d b f f \\ + & 2 b e^{3} f d & - & 2 b b e f f d & - & 4 d^{3} e b f \\ + & 2 b e e f f d q & + & 4 a b e e f f \\ + & d d e^{4} & + & 4 b d d e e f & + & 4 a a e e f f \\ - & 2 d e^{3} f q & + & 8 a e e d d f \\ + & d d e e f f q q \\ - & 2 d d e e b f \\ + & 4 a d e^{3} f & + & 6 d^{4} e e \\ + & 4 d^{3} e^{3} & - & 10 d^{4} e f q \\ + & d^{4} f f q q \\ - & 8 a d e f^{3} q q & + & 12 a e d d f q \\ + & 2 d d e f f q b \\ + & 4 a d e e f f q \\ - & 6 d^{3} e e f q \\ + & 2 d^{3} e f f q q \\ - & 8 p p f^{4} e d y \end{array} & \begin{array}{l} = & 4 p p f^{4} & + & 4 p b f^{3} \\ + & 4 b b f^{3} & + & 8 p a f^{3} q & + & 4 a a f f y y & in & \begin{matrix} e e + \\ - 4 b f - 4 c f \end{matrix} \\ + & 4 p d e f f & + & 4 p d d f f & + & 4 a f d d \\ + & 4 d f^{3} q q & + & 4 a b f f & + & d^{4} \\ - & 4 p d q f^{3} & + & 2 d d b f \\ - & 8 p q f^{4} & + & 4 a f d e \\ + & b b f f & - & 4 a d f f q \\ + & 2 d e b f & - & 8 a q f^{3} \\ - & 2 d b f f q & + & 2 e d^{3} \\ - & 4 b q f^{3} & - & 2 d^{3} f q \\ + & d d e e & - & 4 d d f f q \\ - & 2 d d e f q \\ - & 4 d e f f q \\ + & d d f f q q \\ + & f^{4} q q \end{array} \end{matrix}$

$bc = x$ . $cd = y$ . $ef = a$ . $ea = b$ . $ad = c$ . $\sqrt{x x + y y - c c} (= {ab}^{2}) + b = eb$ . $\frac{a x}{b + \sqrt{x x + y y - c c}} = ch$ . $a a + x x + y y - c c + b b + 2 b \sqrt{x x + y y - c c} = {bf}^{2}$ $\frac{c \sqrt{a a + x x + y y - c c + b b + 2 b \sqrt{x x + y y - c c}} + a x}{b + \sqrt{x x + y y - c c}} = y$ . $c c a a + c c x x + 2 c c y y - c^{4} + b b c c + 2 b c c \sqrt{x x + y y - c c} = b b y y \begin{array}{l} + 2 b \sqrt{y^{2} (&c)} \\ - 2 a \sqrt{x x} \end{array} + x x y y + y^{4}$ . {illeg} $y y + x x + b b - c c + 2 b \sqrt{x x + y y - c c} \frac{- c c a a - 2 a x y \sqrt{x x + y y} - a a x x - a x}{y y - c c}$
$ac = x$ . $bc = y$ . $af = z$ . $bf = \frac{z z}{a}$ . \{illeg}/ $df = \frac{a}{2}$ . $de = \frac{a a + 4 z z}{2 a}$ . $fe = \frac{2 z z}{a} ∶ z ∷ x ∶ \frac{a a + z z}{2 a}$ $2 a a x = 2 a a z + 4 a z^{3}$
$bf = z$ $af = \sqrt{a z}$ . $2 z$ $ad = \sqrt{\frac{a a}{4} + a z}$ $z ∶ \frac{a}{4} + z ∷ x x ∶ \frac{a a}{4} + a z + z z$ . $4 z^{3} + 4 a z z + a a z = a x x + 4 z x x$ .

☞ To know whi|e|ther y^e changing of y^e sines {sic} of an Equation change y^e nature of y^e crooked line signified by y^t Equation observe y^t

If y^e sines {sic} of every other terme (of y^t Equa{illeg}|t|ion ordered{illeg} according t{illeg}|o| {either} of y^e undetermined quanti{illeg}|t|ys) be changed y^e nature of y^e line is not changed. but if y^e signes of some signes bee changed but not in {illeg}|e|ve{ry} other termes (of it ordered according to one of y^e unknowne quantitys) y^e nature of y^{illeg}|e| line is changed.

If y^e knowne qua\n/titys are every where divers, & one of y^m be blotted {illeg} out y^t produceth a line, when one terme is already wanting

Those lines may bee defined y^e same whose natures {illeg} \may be/ expressed by y^e same equation although angles made by x & y are not y^e same.

In y^e {illeg}|H|yperbola y^e area of it beares y^e same respect to its Asymptote w^ch a logarithme {di}{illeg} number.

To make y^e equation $x^{3} - a x^{2} + a b x - a b c = 0$ . be divisible by $x - c = 0$ . suppose $c = x$ , yⁿ tis $c^{3} - a c c + 0 = 0$ $c = a$ . therefore write c in steade of a & it is $x^{3} - c x^{2} + c b x - b c c = 0$ . w^ch is divisible by $x - c$ To make y^e same Equation divisible by $x x - 2 a x + a c = 0$ Suppose it to bee divided by it & y^e ration will bee $\begin{matrix} x x - 2 a x + a b) & x^{3} & - & a x^{2} & + & a b x & - & a b c & = & 0 (x + a \\ - & x^{3} & + & 2 a x^{2} & - & a b x \\ 0 & + & a x^{2} & + & 0 & - & a b c \\ - & a x x & + & 2 a a x & - & a a b \\ 0 & + & 2 a a x & - & a a b & - & a b c \end{matrix}$ . The quote is $2 a a x - a b c - a a b$ w^ch have vanished therefore to m{illeg}|a|ke soe suppose each terme $= 0$ & the{illeg} will be $2 a a x = 0$ & $a b c + a b a = 0$ both include $a = 0$ . Which since it cannot happen y^e equation cannot be divided y^e one by y^e othe{r}

The rootes of {illeg}|t|wo divers equations may easily be added to substracted from multiplyed {&c} by one another while they are unknowne.

^[38] That y^e penultimate terme of y^e Equation $x^{3} * - a^{2} x + b^{3} = 0$ . bee wanting I multiply & then suppose x a knowne {illeg} quantity & y an unknowne {illeg} $x^{3}$ {illeg} $+ b^{3} y^{3}$ {illeg}{ $d^{3} - a y y + y^{3} = 0$ .} by this {having}{illeg} x {illeg} {illeg} {illeg}

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^[39] $d d ϱ^{2} + 2 d ϱ ϩ \sqrt{e e - d d} + e e ϩ ϩ - d d ϩ ϩ = e e x x$ . $d^{3} ϱ^{3} + 3 d d ϱ ϱ ϩ \sqrt{e e - d d} + 3 d ϱ ϩ^{2} e e - 3 d ϱ ϩ^{2} d d \begin{matrix} + e e \\ - d d \end{matrix} ϩ^{3} \sqrt{e e - d d} =$ { $e^{3} x^{3}$ } ^[40] $6 d d ϱ ϱ ϩ ϩ e e - 6 d^{4} ϱ ϱ ϩ ϩ \begin{matrix} + 4 e e \\ - 4 d d \end{matrix} d ϱ ϩ^{3} \sqrt{e e - d d} + e^{4} ϩ^{4} - 2 e e d d ϩ^{4} + d^{4} ϩ^{4} = e^{4} x^{4}$ ^[41] $\begin{matrix} + 10 e e \\ - 10 d d \end{matrix} d^{3} ϱ^{3} ϩ ϩ \begin{matrix} + 10 e e \\ - 10 d d \end{matrix} d d ϱ ϱ ϩ^{3} \sqrt{e e - d d} \begin{array}{l} + 5 e^{4} \\ - 10 e e d d \\ + 5 d^{4} \end{array} d ϱ ϩ^{4} \begin{array}{l} + e^{4} \\ - 2 e e d d \\ + d^{4} \end{array} ϩ^{5} \sqrt{e e - d d} = e^{5} x^{5}$ ^[42] $c c e e - 2 c e d ϩ + d d ϩ ϩ \begin{array}{l} + 2 c e ϱ \\ - 2 d ϩ ϱ \end{array} \sqrt{e e - d d} \begin{array}{l} + e e ϱ ϱ \\ - d d ϱ ϱ \end{array} = e e y y$ . $\begin{array}{l} + & c^{3} e^{3} & + & 3 c c e e ϱ & \sqrt{e e - d d} & + & 3 c e^{3} ϱ ϱ & + & e e & ϱ^{3} \sqrt{e e - d d} & = & e^{3} y^{3} \\ - & 3 c c e e d ϩ & - & 6 c e d ϩ ϱ & - & 3 e e d ϩ ϱ ϱ & - & d d \\ + & 3 c e d d ϩ ϩ & + & 3 d d ϩ ϩ ϱ & - & 3 c e d d ϱ ϱ \\ - & d^{3} ϩ^{3} & + & 3 d^{3} ϩ ϱ ϱ \end{array}$ . $\begin{array}{l} + & c^{4} e^{4} & + & 4 c^{3} e^{3} & ϱ \sqrt{e e - d d} & + & 6 c c e^{4} & ϱ ϱ & + & 4 c e^{3} & ϱ^{3} \sqrt{e e - d d} & + & e^{4} & ϱ^{4} & = & e^{4} y^{4} \\ - & 4 c^{3} e^{3} d ϩ & - & 12 c c e e d ϩ & - & 12 c e^{3} d ϩ & - & 4 c e d d & - & 2 e e d d \\ + & 6 c c e e d d ϩ ϩ & + & 12 c e d d ϩ ϩ & + & 6 e e d d ϩ ϩ & - & 4 e e d ϩ & + & d^{4} \\ - & 4 c e d^{3} ϩ^{3} & - & 4 d^{3} ϩ^{3} & - & 6 c c e e d d & + & 4 d^{3} ϩ \\ + & d^{4} ϩ^{4} & + & 12 c c e d^{3} ϩ \\ - & 6 d^{4} ϩ ϩ \end{array}$ . $\begin{array}{l} c^{5} e^{5} & + & 5 c^{4} e^{4} & ϱ \sqrt{e e - d d} & + & 10 c^{3} e^{5} & ϱ ϱ & + & 10 c c e^{4} & ϱ^{3} \sqrt{e e - d d} & + & 5 c e^{5} & ϱ^{4} & + & e^{4} & ϱ^{5} \sqrt{e e - d d} & = & e^{5} y^{5} \\ - & 5 c^{4} e^{4} d ϩ & - & 20 c^{3} e^{3} d ϩ & - & 30 c c e^{4} d ϩ & - & 20 c e^{3} d ϩ & - & 5 d ϩ e^{4} & - & 2 e e d d \\ + & 10 c^{3} e^{3} d d ϩ^{2} & + & 30 c c e e d d ϩ^{2} & + & 30 c e^{3} d d ϩ ϩ & + & 10 e e d d ϩ ϩ & - & 10 c e^{3} d d & + & d^{4} \\ - & 10 c c e e d^{3} ϩ^{3} & - & 20 c e d^{3} ϩ^{3} & - & 10 e e d^{3} ϩ^{3} & - & 10 c c e e d d & + & 10 d^{3} ϩ e e \\ + & 5 c e d^{4} ϩ^{4} & + & 5 d^{4} ϩ^{4} & - & 10 c^{3} e^{3} d d & + & 20 c e d^{3} ϩ & + & 5 c e d^{4} \\ - & d^{5} ϩ^{5} & + & 30 c c e e d^{3} ϩ & - & 10 d^{4} ϩ ϩ & - & 5 ϩ d^{5} \\ - & 30 c e d^{4} ϩ ϩ \\ + & 10 d^{5} ϩ^{3} \end{array}$ . ^[43] $\begin{array}{l} c c e e ϩ \sqrt{e e - d d} & + & 2 c e^{3} ϩ ϱ & + & e e ϩ ϱ ϱ \sqrt{e e - d d} & = & e^{3} x y y \\ - & 3 d d ϩ \\ + & d d ϩ^{3} & - & 4 c e d d ϩ ϱ \\ + & c c e e d ϱ \end{array}$ . ^[44] $e^{4} x x y y = \begin{array}{l} - & 2 c e^{3} d ϩ^{3} \\ + & 2 c e d^{3} ϩ^{3} \end{array} \begin{array}{l} - & 2 d e e ϱ ϩ^{3} & \sqrt{e e - d d} \\ + & 4 d^{3} ϱ ϩ^{3} \\ + & 2 c c d e e ϱ ϩ \end{array} \begin{array}{l} + & 4 c d e^{3} ϱ ϱ ϩ \\ - & 6 c d^{3} e \end{array} \begin{array}{l} + & 2 d e e ϱ^{3} ϩ \sqrt{e e - d d} \\ - & 4 d^{3} \end{array}$ . $\begin{array}{l} - & 6 d d c e ϱ ϱ ϩ \sqrt{e^{2} - d^{2}} & - & 5 d d e e ϱ^{3} ϩ & + & c^{3} e^{3} ϩ \sqrt{e e - d d} & = & e^{4} x y^{3} \\ - & 4 d^{4} ϱ ϩ^{3} & + & 3 c e^{3} ϱ ϱ ϩ \sqrt{e e - d d} & + & 4 d^{4} ϱ^{4} ϩ & + & c e d d ϩ^{3} \\ + & 3 c c e^{4} ϱ ϩ & - & 3 c e d d ϱ ϱ ϩ \sqrt{e e - d d} & + & e^{4} ϱ^{3} ϩ \\ + & 3 d d e e ϱ ϩ^{3} \end{array}$ . $\begin{array}{l} - & 8 d d c^{3} e^{3} ϱ ϩ & - & 18 d d c c e e ϱ ϱ ϩ \sqrt{e^{2} - d^{2}} & - & 2 d d c e^{3} ϱ^{3} ϩ & - & 6 d d e e ϱ^{4} ϩ \sqrt{e e - d d} & + & c^{4} e^{4} ϩ \sqrt{e e - d d} & = & e^{5} x y^{4} \\ - & 16 d^{4} c e ϱ ϩ^{3} & - & 10 d^{4} ϱ ϱ ϩ^{3} & + & 16 d^{4} c e ϱ^{3} ϩ & + & 5 d^{4} ϱ^{4} ϩ & + & 6 c c d d e e ϩ^{3} \\ + & 4 c^{3} e^{5} ϱ ϩ & + & 6 c c e^{4} ϱ ϱ ϩ & + & 4 c e^{5} ϱ^{3} ϩ & + & e^{4} ϱ^{4} ϩ & + & d^{4} ϱ^{5} \\ + & 12 c e^{3} d d ϱ ϩ^{3} & + & 6 d d e e ϱ ϱ ϩ^{3} \end{array}$ . $\begin{array}{l} c^{5} e^{5} ϩ \sqrt{e e - d d} & + & 5 c^{4} e^{6} ϱ ϩ & + & 10 c^{3} e^{5} ϱ ϱ ϩ \sqrt{e e - d d} & + & 10 c c e^{6} ϱ^{3} ϩ & + & 5 c e^{5} ϱ^{4} ϩ \sqrt{e e - d d} & + & e^{6} ϱ^{5} ϩ & = & e^{6} x y^{5} \\ + & 10 c^{3} e^{3} ϩ^{3} d d & - & 10 c^{4} e^{4} d d ϱ ϩ & + & 30 c e^{3} d d ϱ ϱ ϩ^{3} & - & 50 c c e^{4} d d ϱ^{3} ϩ & - & 30 c e^{3} d d ϱ^{4} ϩ & - & 8 e^{4} d d \\ + & 5 c e d^{4} ϩ^{5} & + & 30 c c e^{4} d d ϱ ϩ^{3} & - & 30 c^{3} e^{3} d d ϱ ϱ ϩ & + & 10 d d e^{4} ϱ^{3} ϩ^{3} & + & 25 c e d^{4} ϱ^{4} ϩ & + & 13 e e d^{4} \\ - & 40 c c e e d^{4} ϱ ϩ^{3} & - & 50 c e d^{4} ϱ ϱ ϩ^{3} & - & 30 e e d^{4} ϱ^{3} ϩ^{3} & - & 6 d^{6} \\ + & 5 e e d^{4} ϱ ϩ^{5} & + & 40 c c e e d^{4} ϱ^{3} ϩ^{3} \\ - & 6 d^{6} ϱ ϩ^{5} & + & 20 d^{6} ϱ^{3} ϩ^{3} \end{array}$ . $\begin{array}{l} - & 3 c^{2} e e e e d ϩ^{3} & - & 6 c e^{3} d ϩ^{3} ϱ \sqrt{e e - d d} & - & 3 e^{4} d ϱ ϱ ϩ^{3} & + & 6 c d e^{3} ϱ^{3} ϩ \sqrt{e e - d d} & + & e^{4} d ϱ^{4} ϩ & = & e^{5} x x y^{3} \\ - & d^{3} e e ϱ^{5} & + & 12 c e d^{3} ϱ ϩ^{3} & + & 12 e e d^{3} ϱ ϱ ϩ^{3} & - & 12 c d^{3} e \\ + & 3 c c e e d^{3} ϩ^{2} & + & 2 c^{3} e^{3} d ϱ ϩ & - & 10 d^{5} ϱ ϱ ϩ^{3} \\ + & d^{5} ϩ^{5} & + & 6 c c d e^{4} ϱ ϱ ϩ & - & 5 e e d^{3} \\ + & 3 c c e e d^{3} ϩ & + & 4 d^{5} \end{array}$ . $\begin{array}{l} - & 4 c^{3} e^{5} d ϩ^{3} & - & 12 c c e^{4} d ϩ^{3} ϱ \sqrt{e e - d d} & - & 12 c d e^{5} ϩ^{3} ϱ^{2} & - & 4 d e^{4} ϩ^{3} ϱ^{3} \sqrt{e e - d d} & + & 8 c e^{5} d ϩ ϱ^{4} & + & 2 d e^{4} ϱ ϩ \sqrt{e e - d d} & = & e^{6} x x y^{4} \\ - & 4 c e^{3} d^{3} ϩ^{5} & + & 24 c c e e d^{3} ϩ^{3} & - & 8 d^{3} e e ϱ^{5} ϩ \\ + & 4 c^{3} e^{3} d^{3} ϩ^{3} & - & 4 d^{3} e e ϩ^{5} & - & 4 c e d^{5} ϩ^{3} & + & 20 d^{5} ϩ^{3} & + & 20 c e d^{5} ϩ & + & 6 d^{5} ϱ^{5} ϩ \\ + & 4 c e d^{5} ϩ^{5} & + & 6 d^{5} ϩ^{5} & + & 8 c^{3} d e^{5} ϩ ϱ ϱ & + & 12 c c d e^{4} ϩ & - & 28 c e^{3} d^{3} ϩ \\ + & 2 c^{4} e^{4} d ϩ & + & 20 d^{3} e e ϩ^{3} \\ + & 48 c e^{3} d^{3} ϩ^{3} ϱ ϱ & - & 24 c c e e d^{3} ϩ \\ - & 12 c^{3} d^{3} e^{3} ϩ \end{array}$ . $\begin{array}{l} c e^{3} ϩ^{3} \sqrt{e e - d d} & + & 4 d^{4} ϱ ϩ^{3} & + & 3 c e d d ϩ ϱ ϱ \sqrt{e e - d d} & - & 4 d^{4} ϱ^{3} ϩ & = & e^{4} x^{3} y \\ - & c e e d d & - & 5 d d e e ϱ ϩ^{3} & + & 3 d d e e \\ + & e^{4} ϱ ϩ^{3} \end{array}$ . $\begin{array}{l} + & c c e^{4} ϩ^{3} \sqrt{e e - d d} & + & 8 c e d^{4} ϱ ϩ^{3} & + & 3 c c d d e e ϱ^{2} ϩ \sqrt{e e - d d} & - & 8 c e d^{4} ϱ^{3} ϩ & - & 5 d^{4} ϱ^{4} ϩ \sqrt{e e - d d} & = & e^{5} x^{3} y y \\ - & c c e e d d ϩ^{3} & - & 10 c e^{3} d d ϱ ϩ^{3} & + & 10 d^{4} ϱ ϱ ϩ^{3} & + & 6 c e^{3} d d ϱ^{3} ϩ & + & 3 d d e e ϱ^{4} ϩ \\ + & d d e e ϱ^{5} & + & 2 c e^{5} ϱ ϩ^{3} & - & 8 d d e e ϱ ϱ ϩ^{3} \\ - & d^{4} ϩ^{5} & + & e^{4} ϱ ϱ ϩ^{3} \end{array}$ . ^[45]

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^[46] $ϱ ϱ ϩ ϩ - ϱ^{2} g^{2} =$ { $b^{4}$ . } $a = r$ . $c = 2 r$ . {illeg} $\frac{y - 2 r + x \sqrt{x}}{2} = ϩ$ . $ϱ = \frac{- x + y \sqrt{3}}{2}$ $x x y y - 2 x y^{3}$ {illeg} $ϩ = \frac{a y - a c + x \sqrt{c c - a a}}{c}$ . $ϱ = \frac{y \sqrt{c c - a a} - a x}{c}$ $\begin{matrix} \begin{array}{l} a a c c y^{4} & - & 2 a a c^{3} y^{3} & + & 2 a c c x y^{3} \sqrt{c c - a a} & - & 4 a a c c x x y y \\ a^{4} y^{4} & - & 2 a^{4} c y^{3} & + & 2 a^{3} x y^{3} \sqrt{c c - a a} & + & 4 a^{4} x x y y \\ - & 2 a^{3} x y^{3} \sqrt{c c - a a} & + & a^{4} x x y y \\ + & 4 a^{3} c x y y \sqrt{c c - a a} & + & a a c^{4} y y \\ + & c^{4} x x y y \\ - & 2 a c^{3} x y^{2} \sqrt{c c - a a} & - & a a c c x x y y \\ + & a^{4} c c y y \end{array} & \begin{array}{l} ϩ ϩ & = & a a y y - 2 a a c y + a a c c + c c x x - a a x x \\ + 2 a y x \sqrt{c c - a a} - 2 a c x \sqrt{c c - a a} \\ ϱ ϱ & = & c c y^{2} - a a y y - 2 a x y \sqrt{c c - a a} + a a x x \\ \begin{matrix} - & 2 a a x y & + & a a c x & - & a x x \sqrt{c c - a a} \\ + & c c x y & + & a y y \sqrt{c c - a a} \\ - & a c y \sqrt{c c - a a} \end{matrix} \end{array} \end{matrix}$ $\begin{array}{l} a a c c x^{4} & + & 4 a^{3} x^{3} y \sqrt{c c - a a} & + & 6 a^{4} x x y y \\ - & a^{4} x^{4} & - & 2 a^{3} c x^{3} \sqrt{c c - a a} & - & 6 a a c c x x y y & + & 6 a^{3} c x y y \sqrt{} & + & a a c c y^{4} \\ - & 2 a c c x^{3} y \sqrt{c c - a a} & + & 4 a a c^{3} x x y & - & 4 a^{3} x y^{3} \sqrt{} & - & a^{4} y^{4} \\ - & 6 a^{4} c x x y & - & 2 a^{3} c c x y \sqrt{} & + & 2 a^{4} c y^{3} \\ - & 2 a a c^{3} y^{3} \\ + & a^{4} c c x x & + & 2 a c c x y^{3} \sqrt{} & + & a a c^{4} y y \\ + & c^{4} x x y y & - & 2 a c^{3} x y y \sqrt{} & - & a^{4} c c y y \end{array}$ $\begin{matrix} \begin{array}{l} \begin{array}{l} 3 x^{4} & - & 4 x^{3} y \sqrt{3} & - & 2 x x y y \\ - & 2 c x^{3} \sqrt{3} & + & 10 c x x y & + & 2 c x y y \sqrt{3} & + & 3 y^{4} & = & 0 \\ + & c c x x & - & 2 c c x y & - & 6 c y^{3} \\ + & 4 x y^{3} & + & 2 c c y y \\ - & g^{3} y \sqrt{3} \end{array} \\ \begin{array}{l} 3 x^{4} & - & 4 x^{3} y \sqrt{3} & + & 6 c x x y & + & 4 x y^{3} \sqrt{3} \\ - & 2 c x^{3} \sqrt{3} & - & 6 x x y y & - & 2 c c x y \sqrt{3} \\ + & 4 x x y y \\ + & 4 c x y y & + & 2 c x y y \sqrt{3} \\ + & c c x x \end{array} \end{array} & \begin{matrix} \begin{array}{l} + & y y \sqrt{3} & - & x x \sqrt{3} & + & 2 x y & . \\ + & c x & - & c y \sqrt{3} \end{array} \\ \begin{array}{l} x x y y & - & 2 a x x y & + & a a x x & - & a a x x & - & a^{4} & = & 0 \end{array} \\ \begin{array}{l} x x y y & + & 4 b x y y & + & 4 b b y y & - & 2 a x x y & - & 8 a b x y \\ - & a^{4} & - & 8 a b b y & - \end{array} \end{matrix} \end{matrix}$

^[47] $bc = x$ . $cd = y$ . $bf = c$ . $bp = ϱ$ . $pd = ϩ$ . $bc = d$ . $bg =$ {illeg} $fe ∶ fg ∷ d ∶ e$ . $fe ∶ eg ∷ d ∶ f$ . bf $hp =$ {illeg} z. $fg = \frac{e x}{d}$ . $eg = \frac{f x}{d}$ . $pd ∶ dh ∷ r ∶ s$ . $dh = \frac{s ϩ}{r}$ . $r ∶ t ∷ pd ∶ ph = \frac{t ϩ}{r}$ . $d ∶ e ∷ dh ∶ dg = \frac{e s ϩ}{d r}$ . $gh = \frac{f s ϩ}{d r}$ $fg = ϱ + \frac{t ϩ}{d r} + \frac{f s ϩ}{d r} = \frac{e x}{d}$ . $\frac{d r ϱ + d t ϩ + f s ϩ}{e r} = x$ . $\frac{r f x + c d r - e s ϩ}{d r} = y$ . $\frac{f d r ϱ + d t f ϩ + f f s ϩ + c d e r - e e s x}{d e r} = y = \frac{f r ϱ + t f ϩ - d s ϩ + c e r}{e r}$ $ec = c$ . $bc = x$ . $dc = y$ . $pd = ϩ$ . $pf = ϱ$ . $fg = \frac{e x}{d}$ . $eg = \frac{f x}{d}$ . $pg = \frac{s ϩ}{r}$ . $gd = \frac{t ϩ}{r}$ . $fg = \frac{ϱ r + s ϩ}{r} = \frac{e x}{d}$ . {illeg} $x = \frac{d r ϱ + d s ϩ}{r e}$ . $eg + ec - gd = y = \frac{f r ϱ + f s ϩ + r e c - t e ϩ}{r e}$ .

^[48] $fe = bc = x$ . $cd = y$ . $fp = ϱ$ . $pd = ϩ$ . $fe ∶ fg ∷ d ∶ e ∷ dh ∶ dg$ . $fe ∶ eg ∷ d ∶ f ∷ hd ∶ hg$ . $pd ∷ dh ∷ r ∶ s$ . $pd ∶ ph ∶ r ∶ ht$ $\frac{d r ϱ - d t ϩ + f s ϩ}{e r} = x$ . $\frac{f r ϱ - f t ϩ - d s ϩ + c e r}{e r} = y$ . $d = r$ . $f = \sqrt{e e - d d}$ . $s = \sqrt{r r - t t}$ . $s = \sqrt{d d - t t}$ . $\frac{d d ϱ - d t ϩ + ϩ \sqrt{e e d d - e e t t + d d t t - d^{4}}}{e d} = x$ . $\frac{d ϱ \sqrt{e e - d d} - t ϩ \sqrt{e e - d d} - d ϩ \sqrt{d d - t t} + c e d}{e d} = y$ . Lastly $dp = - dt + \sqrt{e e d d - e e t t + d d t t - d^{4}}$ . $n = \sqrt{e e - d d}$ . $dq = t \sqrt{e e - d d} + d \sqrt{d d - t t}$ . & Therefore $x = \frac{d ϱ + p ϩ}{e}$ . & $y = \frac{n ϱ - q ϩ + c e}{e}$ .

^[49] $bf = c$ . $fa = z$ . $fk = v$ . $bc = x$ . $cd = y$ . $ap = ϱ$ . $dp = ϩ$ . $v v + z z ∶ v v ∷ ϩ ϩ ∷ \frac{v v ϩ ϩ}{v v + z z} = {pg}^{2}$ . $d ∶ e ∶ \frac{v ϩ}{\sqrt{v v + z z}} ∶ \frac{e v ϩ}{d \sqrt{v v + z z}} = po$ . $ac = x - z ∶ ao = \frac{e x - e z}{d} ∷ d ∶ e$ . $\frac{e x - e z}{d} \frac{- e v ϩ}{d \sqrt{v v + z z}} = ϱ$ . $d ϱ \sqrt{v v + z z}$ .

^[50] $bf = c$ . $af = z$ . $bc = x$ . $fc = x - c = an$ . $fk = v$ . {illeg} $d ∶ e ∷ an ∶ no = \frac{e x - e c}{d}$ . $oc = \frac{e x - e c + d z}{d}$ . $ak = \sqrt{v v + z z} ∶ z ∷ ed = ϩ ∶ \frac{z ϩ}{\sqrt{v v + z z}} = gd$ . $\frac{v ϩ}{\sqrt{v v + z z}} = eg$ . $go = \frac{e v ϩ}{d \sqrt{v v + z z}}$ . $\frac{e x - e c + d z}{d} \frac{- d z ϩ - e v ϩ}{d \sqrt{v v + z z}} = y$

^[51]

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$fd = x$ . $db = y$ . $dc = v$ . ed{illeg} $ab = z$ . $v v + y y ∶ v v ∷ z z \frac{v v z z}{v v + y y} = {ed}^{2}$ . $fe =$ {illeg} x {illeg} / y \ $fe = x \frac{\overset{\cup}{O} v z}{\sqrt{y y + v v}}$ . $ae = y \frac{\underset{\cap}{O} y z}{\sqrt{y y + v v}}$ . in y^e 1^st case.^[52] $fe = x \frac{\overset{\cup}{O} v z}{\sqrt{y y + v v}}$ . $ae = - y \overset{\cup}{O} \frac{y z}{\sqrt{y y + v v}}$ . in y^e 2^d {illeg}|c|ase.^[53] $fe = \overset{\cup}{O} x + \frac{v z}{\sqrt{y y + v v}}$ . $ae = y$ .

^[54] Have{illeg}|in|g y^e nature of a crooked line expressed in Algebr: termes to find its {illeg}|a|xes, to det{illeg}|e|rmin it & describe it Geometrically &c

^[55] If $fd = x$ . $db = y$ . & y being pe{illeg}|r|pendicular to x describes y^e crooked line y^e crook y^e line w^th its one of its extremes. Then reduce y^e Equation (expressing y^e nature of y^e line in w^ch x & y onely are undetermin{illeg}|ed|) to one side soe y^t it be $= 0$ . ☞ Then Then {sic} find y^e perpendicular bc {illeg} w^ch is done by findind|g| $dc = v$ . for $v v + y y = {bc}^{2}$ (In finding $dc = v$ {illeg} {sic}\obse{illeg}|r|v{illeg}|s| this rule./ Multiply {illeg} y^e each terme of y^e Equat: by so many units as x hath dimensions in y^t terme, divide it by x & multiply it by y ^{illeg} for a Numerator. Againe multiply each terme of y^e Equation by soe many units as y hath dimensions in each terme and divide by $- y$ f{illeg}|o|r a denom: in y^e val{illeg}|o|r of v. ^[56] Example, $- r x + \frac{r x x}{q} + y y = 0$ . $\frac{- 1 r x + \frac{2 r x x}{q} + 0 y y in y}{x} = - r y + \frac{2 r x y}{q}$ . $\frac{- 0 r x + \frac{0 r x x}{q} + 2 y y}{- y} = - 2 y$ . t{illeg}|h|erefore $\frac{r y + \frac{2 r x y}{q}}{- 2 y} = v = + \frac{1}{2} r - \frac{r x}{q}$ . Also if $\begin{matrix} x^{3} & - & b x x & + & y y x & - & y^{3} & = 0 \\ + & y x x \end{matrix}$ . $\begin{matrix} \frac{\begin{matrix} 3 x^{3} & - & 2 b x x & + & y y x & in & y \\ + & 2 y x x \end{matrix}}{x} \\ \frac{\begin{matrix} - & 3 y^{3} & + & 2 y x x & + & y x x \end{matrix}}{- y} \end{matrix} \begin{matrix} = \\ = \end{matrix} \begin{matrix} \begin{matrix} 3 x x y - 2 b x y + 2 x y y + y^{3} \\ 3 y y - 2 y x + x x \end{matrix} & = & v \end{matrix}$ ^[57] And if $x^{4} - y y x x + a a y x - y^{4} = 0$ . then $\begin{matrix} \begin{matrix} 4 y x^{3} - 2 y^{3} x + a a y y \\ 4 y^{3} - a a x + 2 y x x \end{matrix} & = & v \end{matrix}$ . &c) Then make $ab = z$ . $fe = x \frac{+ v z}{\sqrt{y y + v v}}$ . $ae = y \frac{- v z}{\sqrt{y y + v v}}$ . & substitute this valor of $(fe)$ into y^e place of x {illeg} & this valor of $(ae)$ into y^e place of y in y^e Equation \& th{illeg}|e|r{illeg}|e| {illeg}|take| {{illeg}|in|} a 2^d equation/. {illeg}|t|hen by multiplication or by some other meanes take a{illeg}|w|ay y^e irrational quantity $\sqrt{y y + v v}$ & lastly take awa{y} y or x {illeg}|b|y y^e helpe of these 2 Equations, soe y^t you have a {illeg} \3^rd/ equation in w^ch there is either x onely, or y onely & supposeing it to have 2 equall roots multiply each terme by {illeg}|soe| many units as {illeg}|y^e| unknowne quanti{illeg}|t|y hath dimensions in y^t terme {illeg} w^ch {illeg}|p|roduct is a 3^rd equation it according to Huddenius his Method for a 3^rd /4^th\ Equation & by y^e helpe of y^e 3^rd & 4^th equation take away {illeg} y^e in w^ch & there will result a 5^t Equation in w^ch there b{illeg} one unknowne quantity viz: either x or y. & there will result a 5^t Equation in w^ch is neither x nor nor {sic} y. & by w^ch the valor of z may be found. one \The greatest/ of whose valors signifies y^e longest, another \the least {illeg}/ of y^m y^e shortest of all y^e perpendicular lines ab. & if it have other rootes they signifie other lines $(ab)$ w^ch are perpendicular to y^e crooked line at both ends, a & b; & some of these must signifie y^e axes of y^e line if it bee of an elliptical nature.

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^[58] $ac = x$ . $ch = y$ . $bf =$ {illeg} \z {illeg}/. $fh =$ {illeg} \{illeg} ϩ/ $cd = a$ . $db ∶ be ∷ fh ∶ eh ∷ b ∶ c$ . $\frac{c x}{b} = be$ . $\frac{c ϩ}{b} = eh$ . {illeg} $\frac{ϩ \sqrt{c c - b b}}{b} = fe = \frac{c x - b z}{b}$ . $\frac{ϩ \sqrt{c c - b b} + b z}{c} = x$ . ed{illeg} $= \frac{x \sqrt{c c - b b}}{b} = \frac{ϩ c c - ϩ b b + b z \sqrt{c c - b b}}{b c}$ $hc = \frac{- ϩ b b + b z \sqrt{c c - b b} + a b c}{b c} = \frac{a c - ϩ b + z \sqrt{c c - b b}}{c} = y$ .
$\begin{matrix} \begin{matrix} for \end{matrix} & { & \begin{matrix} x \\ x x \\ x^{3} \\ x^{4} \\ y \\ y y \\ y^{3} \\ y^{4} \\ x y \\ x y y \\ x y^{3} \\ x x y y \\ x x y \\ x^{3} y \end{matrix} & } & \begin{matrix} = \\ = \\ write \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \end{matrix} & { & \begin{array}{l} \frac{+ ϩ \sqrt{c c - b b}}{c} \\ \frac{2 b ϩ z \sqrt{c c - b b}}{c c} \\ \frac{ϩ^{3} c c \sqrt{c c - b b}}{c^{3}} - \frac{ϩ^{3} b b \sqrt{c c - b b}}{c^{3}} + \frac{3 b b z z ϩ \sqrt{c c - b b}}{c^{3}} \\ \frac{4 ϩ^{3} c c \sqrt{c c - b b} \times b z - 4 y^{3} b^{3} z \sqrt{c c - b b} + 4 ϩ b^{3} z^{3} \sqrt{c c - b b}}{c^{4}} \\ - \frac{ϩ b}{c} \\ \frac{- 2 ϩ a b c - 2 ϩ b z \sqrt{c c - b b}}{c c} \\ - ϩ^{3} b^{3} - 3 ϩ b a a c c - 3 ϩ b c c z z + 3 ϩ b^{3} z z - 6 ϩ a b c z \sqrt{c c - b b} \\ \frac{\begin{matrix} \begin{matrix} - 4 ϩ^{3} b^{3} a c - 4 ϩ^{3} b^{3} z \sqrt{c c - b b} - 4 ϩ a^{3} b c^{3} - 12 ϩ a a b c c z \sqrt{c c - b b} - 12 ϩ a b b^{3} z z + 12 ϩ a b^{3} c z^{2} \end{matrix} & \begin{array}{l} + ϩ z^{3} b^{3} \\ - ϩ z^{3} b c c \sqrt{c c - b b} \end{array} \end{matrix}}{c^{4}} \\ \frac{a c ϩ \sqrt{c c - b b} + ϩ z c c - 2 ϩ z b b}{c c} \\ \frac{- 4 ϩ a b b c z - 3 ϩ b b z \sqrt{c c - b b} \begin{array}{l} + a a c c ϩ \\ + z z c c \end{array} \sqrt{c c - b b} + 2 a c^{3} z ϩ + b^{2} ϩ^{3} \sqrt{c c - b b}}{c^{3}} \\ \frac{- 3 b b c c ϩ^{3} z - 4 b^{4} z ϩ^{3} + 3 a b b c ϩ^{3} \sqrt{c c - b b} + z^{3} c^{4} ϩ + 4 z^{3} b^{4} ϩ - 5 z^{3} b b c c ϩ \begin{matrix} - 9 a b c c \\ + 3 a c^{3} \end{matrix} z z ϩ \sqrt{c c - b b} \begin{matrix} - a a b b c c \\ + 3 a a c^{4} \end{matrix} z ϩ + a^{3} c^{3} \sqrt{c c - b b}}{c^{4}} \\ \frac{2 a b^{3} c ϩ^{3} - 2 a b c^{3} ϩ^{3} \begin{matrix} + 4 b^{3} \\ - 2 b c c \end{matrix} z ϩ^{3} \sqrt{c c - b b} \begin{matrix} + 2 b c c \\ - 4 b^{3} \end{matrix} z^{3} ϩ \sqrt{c c - b b} + 2 a a b b c c z ϩ \sqrt{c c - b b} + 4 a b c^{3} z z ϩ - 6 a b^{3} c z z ϩ}{c^{4}} \\ \frac{b^{3} ϩ^{3} - b c c ϩ^{3} + 2 b c c z z ϩ - 3 b^{3} z z ϩ + 2 a b c z \sqrt{c c - b b}}{c^{3}} \\ \frac{4 b^{4} z ϩ^{3} - 5 b b c c z ϩ^{3} + c^{4} z ϩ^{3} \begin{matrix} + a c^{3} \\ - a b b c \end{matrix} ϩ^{3} \sqrt{c c - b b} + 3 b c c z z ϩ - 4 b^{3} z z ϩ + 3 a b c z ϩ \sqrt{c c - b b}}{c^{4}} \end{array} \end{matrix}$

Haveing therefore an equation expressing y^e nature of a{illeg} crooked line To find its axis. \Supposeing $c =$ some quantity most frequent in y^e equation/ Subroga{illeg}|t|e $\frac{b z + ϩ \sqrt{c c - b b}}{c}$ into y^e roome of x ; & $\frac{a c - ϩ b + z \sqrt{c c - b b}}{c}$ into y^e roome of y: Order y^e Equation according to {illeg} ϩ, make every terme $= 0$ , in w^ch ϩ is of {illeg} \one/ dimensios {sic} Order every terme in the|i|se 2^dary Equations according to y^e dimensions of z. & supposeing every terme of each of y^m $= 0$ , by y^e helpe of the{illeg}|s|e {illeg}|E|quations (in w^ch is neither x , y , z or ϩ ) may be found y^e valors of a & b . Then perpendicular to ac from y^e point a draw $ab = a$ . & from y^e point d | b | draw $bk = b$ , & parallel to ac. from y^e point k draw $mk = \sqrt{c c - b b}$ , & perpendicular to bk . & through y^e points b , {illeg} m {illeg} draw bl y^e axis of y^e line hgn . & y^t y^e relation twixt $bf =$ {illeg}z. & $fh = ϩ$ may bee had, write y^e valors of a , b , c now found in their stead in y^e 2^dary equation.

Example $d d + d y + x y - y y = 0$ . Then makeing $d = c$ I write $\frac{b z + ϩ \sqrt{c c - b b}}{c}$ , or $\frac{b z + ϩ \sqrt{d d - b b}}{d}$ for x & its square for $x x$ &c. & $\frac{a d - b ϩ + z \sqrt{d d - b b}}{d}$ for y , & its square for $y y$ . & soe I have this equation, $0 = d d + a d - b ϩ + z \sqrt{d d - b b} \begin{matrix} \frac{+ a d b z - 2 b b ϩ z \begin{matrix} - b ϩ ϩ \\ + b z z \\ + a d ϩ \\ + 2 a z d \\ + 2 b ϩ z \end{matrix} \sqrt{d d - b b} + ϩ z d d - a a d d + 2 a d b ϩ - b b ϩ ϩ \begin{matrix} - d d z z \\ + b b z z \end{matrix}}{d d} \end{matrix}$ or by ordering it according to ϩ , $\begin{matrix} \begin{matrix} b b ϩ ϩ & + & b ϩ ϩ \sqrt{d d - b b} & + & d d b ϩ & - & d a ϩ \sqrt{d d - b b} & - & d^{4} \\ + & 2 b b z ϩ & - & 2 b z ϩ \sqrt{d d - b b} & \begin{matrix} - \\ + \end{matrix} & \begin{matrix} a b d z \\ a a d d \end{matrix} \\ - & d d z & + & d d z z \\ - & 2 a b d & - & b b z z \end{matrix} & \begin{matrix} + & 2 a d z \\ - & b z z \\ - & d d z \end{matrix} \sqrt{d d - b b} \end{matrix} \begin{matrix} = & 0 \end{matrix}$ Then by makeing those quantitys in y^e last terme save one {illeg} {illeg} $= 0$ . I have this Equation $2 b b z - d d z - 2 b z \sqrt{d d - b b} + d d b - 2 a b d - d a \sqrt{d d - b b} = 0$ . {illeg}|W|hich I divide into 2 pts makeing those termes $= 0$ in w^ch z is not, & those $= 0$ in w^ch z is of one dimension. & then I have these 2 equation{s} $2 b b - d d - 2 b \sqrt{d d - b b} = 0$ . & $d b - 2 a b - a \sqrt{d d - b b} = 0$ . by y^e f{illeg}|irs|t $4 b^{4} - 4 b b d d + d^{4} = 4 b b d d - 4 b^{4}$ . Or $8 b^{4} - 8 b b d d + d^{4} = 0$ . That is $b b = \frac{d d}{2} \overset{\cup}{O} \sqrt{\frac{d^{4}}{8}} = \frac{d d}{2} \overset{\cup}{O} \frac{d d}{2 \sqrt{2}} = \frac{d d \sqrt{2} \overset{\cup}{O} d d}{2 \sqrt{2}}$ . by y^e 2^d Equation I find $d d b b - 4 a d b b + 4 a a b b = a a d d - a a b b$ . or $\begin{matrix} 5 a a b b & - & a a d d & = & 0 \\ - & 4 a d b b \\ + & d d b b \end{matrix}$ . & by writing y^e valor of $d d$ w^ch was found before I have $\frac{5 a a d d \sqrt{2} \overset{\cup}{O} 5 a a d d - 4 a d^{3} \sqrt{2} \underset{\cap}{O} 4 a d^{3} + d^{3} \sqrt{2} \overset{\cup}{O} d^{4} - 2 a a d d \sqrt{2}}{2 \sqrt{2}} = 0$ . Or $\begin{matrix} 3 a a \sqrt{2} & - & 4 a d \sqrt{2} & + & d d \sqrt{2} & = & 0 \\ \overset{\cup}{O} & 5 a a & \overset{\cup}{O} & 4 a d & \overset{\cup}{O} & d d \end{matrix}$ . Or $a a = \frac{4 a d \sqrt{2} \underset{\cap}{O} 4 a d - d d \sqrt{2} \overset{\cup}{O} d d}{3 \sqrt{2} \underset{\cap}{O} 5}$ . & $a = \frac{d \sqrt{8} \underset{\cap}{O} 2 d}{18 \underset{\cap}{O} 5} \overset{\cup}{O} \sqrt{\frac{\begin{matrix} 8 d d \underset{\cap}{O} 4 d d \sqrt{8} + 4 d d - 5 d d \\ - d d \sqrt{36} \overset{\cup}{O} 5 d d \sqrt{2} \overset{\cup}{O} d d \sqrt{18} \end{matrix}}{18 \underset{\cap}{O} 5 \sqrt{18} + 25}}$ $a = \frac{d \sqrt{8} \underset{\cap}{O} 2 d}{\sqrt{18} \underset{\cap}{O} 5} \begin{matrix} \overset{\cup}{O} \\ + \end{matrix} \sqrt{\frac{d d}{18 \underset{\cap}{O} 5 \sqrt{18} + 25}}$ or $\frac{2 d \sqrt{2} - 2 d}{3 \sqrt{2} - 5} \begin{matrix} \overset{\cup}{O} \\ + \end{matrix} \sqrt{\frac{d d}{43 - 15 \sqrt{2}}} = a$ . & $\frac{d d \sqrt{2} \overset{\cup}{O} d d}{2 \sqrt{2}} = b b$ . $a = \frac{2 d \sqrt{2} \overset{\cup}{O} 2 d \overset{\cup}{O} d}{3 \sqrt{2} \overset{\cup}{O} 5}$

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^[59] $\frac{d d ϱ^{2} + 2 d s ϱ ϩ + s s ϩ ϩ}{e e} = x x$ . $d^{3} ϱ^{3} + 3 d d ϱ ϱ s ϩ + 3 d ϱ ϱ s ϩ ϩ + s^{3} ϩ^{3} = e^{3} x^{3}$ . |{d }. $a + y$ | $d^{4} ϱ^{4} + 4 d^{3} ϱ^{3} s ϩ + 6 d d ϱ ϱ s s ϩ ϩ + 4 d ϱ s^{3} ϩ^{3} + s^{4} ϩ^{4} = e^{4} x^{4}$ . $d^{5} ϱ^{5} + 5 d^{4} ϱ^{4} s ϩ + 10 d^{3} ϱ^{3} s s ϩ ϩ + 10 d d s^{3} ϱ ϱ ϩ^{3} + 5 d ϱ s^{4} ϩ^{4} + s^{5} ϩ^{4} = e^{4} x^{5}$ $\begin{matrix} t t ϱ ϱ & - & 2 t ϱ v ϩ & + & v v ϩ ϩ & = & y y \\ + & 2 t ϱ c e & - & 2 c e v ϩ \\ + & c c e e \end{matrix}$ . $\begin{array}{l} t^{3} ϱ^{3} & - & 3 t t ϱ ϱ v ϩ & + & 3 t ϱ v v ϩ ϩ & - & v^{3} ϩ^{3} & = & e^{3} y^{3} \\ + & 3 t t ϱ ϱ c e & - & 6 t ϱ c e & + & 3 c e \\ + & 3 t ϱ c c e e & - & 3 c c e e \\ + & c^{3} e^{3} \end{array}$ . $\begin{array}{l} t^{4} ϱ^{4} & - & 4 t^{3} ϱ^{3} v ϩ & + & 6 t t ϱ ϱ v v ϩ ϩ & - & 4 t ϱ v^{3} ϩ^{3} & + & v^{4} ϩ^{4} & = & e^{4} y^{4} \\ + & 4 t^{3} ϱ^{3} c e & - & 12 t t ϱ ϱ c e & + & 12 t ϱ c e & - & 4 c e \\ + & 6 t t ϱ ϱ c c e e & - & 12 t ϱ c c e e & + & 6 c c e e \\ + & 4 t ϱ c^{3} e^{3} & - & 4 c^{3} e^{3} \\ + & c^{4} e^{4} \end{array}$ . $\begin{array}{l} t^{5} ϱ^{5} & - & 5 t^{4} ϱ^{4} v ϩ & + & 10 t^{3} ϱ^{3} v v ϩ ϩ & - & 10 t t ϱ ϱ v^{3} ϩ^{3} & + & 5 t ϱ v^{4} ϩ^{4} & - & v^{5} ϩ^{5} & = & e^{5} y^{5} \\ + & 5 t^{4} ϱ^{4} c e & - & 20 t^{3} ϱ^{3} c e & + & 30 t t ϱ ϱ c e & - & 20 t ϱ c e & + & 5 c e \\ + & 10 t^{3} ϱ^{3} c c e e & - & 30 t t ϱ ϱ c c e e & + & 30 t ϱ c c e e \\ + & 10 t t ϱ ϱ c^{3} e^{3} & - & 20 t ϱ c^{3} e^{3} & + & 10 c^{3} e^{3} \\ + & 5 t ϱ c^{4} e^{4} & - & 5 c^{4} e^{4} \\ + & c^{5} e^{5} \end{array}$

^[60] $bc = x$ . $dc = y$ . $df = z$ . $bh = c$ . $hk ∶ ke ∷ d ∶ e$ . $ke = \frac{e x}{d}$ . $df ∶ de ∷ d ∶ f$ . $\frac{f z}{d} = de$ . $\frac{c d + e x - f z}{d} = y$ . $fm = ϱ$ . $mh = \frac{d ϱ}{e}$ . $mk = \frac{x e - d ϱ}{e}$ . $pe = \frac{x e - d ϱ}{d} =$ $p = n$ . $fp = \sqrt{ϩ^{2} - n n}$ . $ep = \frac{e \sqrt{ϩ^{2} - n n} + d n}{d} = \frac{e x}{d}$ $e e ϩ ϩ = e e n n + d d n n - 2 d n e x + e e x x$ . $fd ∶ fp ∷$ {illeg} $f ∶ g$ . $\frac{ϩ g}{f} = fp$ . $\frac{f ϱ + ϩ g}{f} = x$ . $fp ∶ pe ∷ d ∶ e$ . $\frac{e ϩ g}{d f} =$ {illeg}. $pd = \frac{ϩ \sqrt{f f - g g}}{f}$ $fm = \frac{e ϱ}{d}$ . $hm = ϱ$ . $\frac{e ϱ + c d}{d} - ϩ \frac{\sqrt{f f - g g}}{f} = y$ . $f = d$ . &c $\frac{d ϱ + g ϩ}{d} = x$ . $\frac{e ϱ + c d - ϩ \sqrt{f f - g g}}{d} = y$ .

^[61] If any crooked line be revolved about its \owne/ {illeg}|a|xis it generates a Sollid intersected by any plaine \not perpendicular to y^e axis/ produceth another line ${\begin{cases} not more compound y^{n} \\ of the same kind with \end{cases}$ y^e forme{illeg}|r|. But if it bee revolved by any other line it generates a Sollid which intersected by any plaine \not perpendicular to y^e axis/ produceth another whose composition is {n}ot ${\begin{cases} lesse y^{n} equall \\ more y^{n} double \end{cases}}$ to y^e formere

In the △ adb if $ab = a$ , & $db = b$ are definite, but $ad = v$ , & $bc = x$ indefined. Then y^e Equation is $b b - v v + a a - 2 a x = 0$ . But in this case y^e maximu or minimum of either v or x cannot bee found according {illeg}|t|o Cartes or Hudde{nius} method, by reason y^t ${\begin{matrix} v \\ x \end{matrix}}$ hath not 2 divers valors when ${\begin{matrix} x \\ v \end{matrix}}$ is determined, w^ch become equall when ${\begin{matrix} x \\ v \end{matrix}}$ is y^e least or greatest y^t may be. But if cd might heve {sic} bee used inste{ad} of cb &c. There be other instances of this Nature against Huddenius h{illeg}|i|s assertion.

These points a/,\b/,\c/,\ being given {illeg}|a| circle may {illeg}|b|e described (w^ch shall pass through a point them all) by an instrument whose angle $edf = ∠ abc$ . And soe y^e sides ed & df being moved close by y^e points a & c, y^e point $(d)$ shall describe y^e a{illeg}|r|ch abc

To w{illeg}|o|rke mechanic{illeg}|a|lly & exactly b{illeg}|y| a {{illeg}|s|cal{illeg}|e|} it may be{illeg}|e| better divided according to y^e fassion {illeg} {illeg}|r|epresented by y^e figure A, yⁿ by that at B.

To {illeg}|m|ake a {plated} superficies exactly: Take three plates A, B, D. & {g{illeg}|r|ind} them together A & B, A & D, B & D; pressing y^e uppermost pla{illeg}|t|e onely in the middle at {illeg}|c| that it may not weare {move} a{illeg}|w|ay in the edges yⁿ in the midst & move it to & fro w^th but sma{illeg}|ll| vibrations. Soe shall the 3 {illeg}|f|iduciall sides of y^e 3 plates bee {ground} exactly plane.

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^[62] Probleme. Of Usury. $ab = a =$ the princi{illeg}|p|le. bc{illeg}={illeg} $x =$ the time of the m{illeg}|o|ney lent $ce = \frac{d x}{e} =$ the use due for y^e principle {illeg}|fo|r y^e time bc. $de =$ the use upon {illeg}|u|se. {illeg} cd y^e $cd = y =$ y^e use{illeg} for y^e principle & use during y^e time bc. $a x ∶ \frac{d x}{e} ∷ a + y in fc ∶ y$ . ab y^e sume of princi df a t{illeg}|a|ngent to y^e line bd. dg a perpendicular. $cg = v$ . Then as y^e sume $(ab)$ or principle drawne into y^e time bc, is to $(ce)$ y^e use for it in y^t time ^[63] So is $ab + dc$ the sume $ab + dc$ drawne into y^e time fc, to y^e use $(dc)$ for it in y^e time fc. {illeg} therefore $a x ∶ \frac{d x}{e} ∷ a + y in {\begin{matrix} h \\ fc \end{matrix} ∶ y$ {illeg}. $a x y = \frac{a h d x + y h d x}{e}$ . $\frac{e a y}{a d + y d} = h ∶ y$ {illeg} $∷ y ∶ v$ . $e a v = a d y + d y y$ . $v = \frac{a d y + d y y}{e a}$ . $d = e$ . $\frac{a y + y y}{a} = v$

Of Reflect{illeg}|i|ons.

Suppoe {sic} y^t {illeg}|t|he bodys a, b, have noe vis elastica to reflect y^e one from y^e other but at their occursion conjoine & keepe together as i{illeg}|f| they were one body. Then 1^st if Theire bulke \& motion/ be equall yⁿ at theire {illeg} meeteing they shall rest. 2 If $(b)$ have more motion yⁿ $(a)$ all {illeg} y^e motion of $(a)$ shall be lost & soe much of $(b)$ s as $(a)$ had & they shall both move towards c shareing y^e differencde of y^e motion \proportionally/ twixt y^m. Demon: {illeg}|S|uppose y^e motion of $ehf =$ motion of a
2 If a rest & b hit it they shall both move towards c w^th {illeg} shareing y^e motion \of $(b)$ / twixt them.

^[64] Of Reflection.

Suppose y^e Bodys a, b doe not reflect one another b{illeg}ut conjoyne either moveing or resting together of at theire meeting & soe move or rest to{illeg}|g|ether. $a =$ y^e body a; $b =$ y^e body fec; $c =$ body ced; $d =$ body fedc. $m =$ motion of $a$ , $n =$ motion of $b$ , $p =$ motion of $c$ , $q =$ motion of $d = n + p$ , before reflection. $e =$ motion of $a$ , $f =$ motion of $b$ , $g =$ moti {sic} of $c$ , $h =$ motion of $d$ {illeg}|a|fter reflection. $r =$ swiftnesse of $a$ , $s =$ motion of $b$ , $c$ , or $d$ , before reflection $t =$ swift{illeg}nesse of $a$ , $v =$ swiftnes $b$ , $c$ , or $d$ ref occursion. ⊙ y^e point of theire occursion.

Axiome 1^st

^[65] Two bodys \ $(b, c)$ / being alike swift {illeg} y^e motion of $b ∶$ mot: $c ∷ b ∶ c$ . for equall pts have equall motion. Therefore $b ∶ c ∷$ all y^e parts of $b ∶$ all such pts of $c ∷$ mot: $b ∶$ the motion of c.

Prop: 1^st. If before y^e occursion \of a & d/ a rest yⁿ shall $e ∶ q ∷ a ∶$ {illeg} $e + h = q$ . & since $t = v$ , tis {alsoe} $e ∶ q ∷ a ∶ a + d$ . Or $e = \frac{a q}{a + d}$ also $h = q - \frac{a q}{a + d} = \frac{d q}{a + d}$ .

Prop: 2^d. If a meete d, & have lesse motion yⁿ it, then, $q - m = e + h$ . for suppo{illeg}|s|e, $m = n$ . yⁿ should {illeg}|a| & b rest after occursion did not $p = q - m$ {illeg} force y^m towards k.

Prop 3 suppose i y^e center of gravity in d, y in a. z & f y^e in w^ch y^e bodys a & d touch in theire meeting. ⊙ y^e point of theire meeting. $(a)$ y^e magnitude of \y^e body/ $(a)$ , $(d)$ y^e magnitude of d. $m =$ motion of a before meeting, $n =$ mot: {illeg}|d| before me{illeg}|et|ing. y^e time in w^ch a or d moves to $⊙ =$ time in w^ch they both move to y. $p =$ motion of a, {illeg} $q =$ moti{illeg}|o|n of d after occursion{illeg} $m + n = p + q$ . $a ∶ d ∷ p ∶ q$ . or $a + d ∶ d ∷ m + n (= p + q) ∶ q =$ $q m + q n$ $\frac{d m + d n}{a + d}$ . $\frac{a m + a n}{a + d} = p$ . $m ∶ p ∷ z⊙ ∶ zy$ . $\frac{a m \times z⊙ + a n \times z⊙}{a m + d m} = ⊙y$ .

$a =$ magnitude of y^e body a, $d =$ mag: of y^e body d. {illeg} $o =$ {illeg} y^e point of concourse: $z, f =$ y^e points of contac{illeg}|t|, at o. $zo =$ {illeg}|b|. $fo = c$ . $op = e$ . $t =$ time in w^ch y^e bodys move from z & f to o. $v =$ time in w^ch they move from o to p. $m =$ motion of a before occursion $n =$ motion of d after occursion $d m$ {illeg}{illeg} $= \frac{d n + a n}{a}$ or $c d m + b a m = b d n + b a n$ . {illeg}|&| $\frac{c d m + b a m}{b d + b a} = n$ {illeg} $∷ m ∶ n$ . $t m ∶ v n ∷$ {illeg} $b ∶ e ∷ t b d + t b a ∶ v c d + v b a$ . {illeg} $c d$ {illeg} $v = \frac{d e t + a e t}{c d + a b}$ . . {illeg}{illeg}{&} d {illeg} meete {illeg} {illeg} y^t c must be negative y^t is $\frac{a b v -}{n t}$ {illeg}e be negative y^e point {illeg} must be t{illeg}on y^e same side {illeg}

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Definitions.

^[66] {illeg} \1^st/ When a body \{illeg}|Q|uan{illeg}|ti|ty/ ${\begin{cases} is translated \\ passeth \end{cases}}$ from one pte of Extension to another it is saide to mo{ve.}

2 One body \Quantity/ is soe much swifter yⁿ another, as y^e distance th{illeg}|r|ough w^{illeg}|c|h it passeth is greater yⁿ y^e distance through w^ch y^e other passeth in y^e same time.

One body /quantity\ {illeg}|h|ath soe much more motion yⁿ another, as y^e summe of y^e spaces th{illeg}|r|ough w^ch each of its pts moveth i{illeg}|s| to y^e summe of y^e spaces through which each of y^e pts of y^e other \quantity/ body moveth supp in y^e same time, suppi|o|seing each of the{illeg} pts in both bodys /Quantitys\ to be equall & alike to one another \& moved in y^e same position./ {illeg}.

3 One Quantity hath so much more motion yⁿ another, as y^e distance through w^ch it moveth drawne into {illeg}|it|s quantity, is to y^e distance {illeg}|t|rough w^ch y^e other moveth in y^e same time d{illeg}|raw|ne into its quantity. As if y^e line ab move y^e length of bc & ef y^e length of eh in y^e same time, y^e motion of ab is to y^e motion of cd, as $ab \times bc = abcd$ , to $ef \times eh = iehk$ . Alsoe if \y^e cube/ $lmqyz = 8$ move y^e length of $op = 5$ ; & the piramis $tvwx = 7$ move y^e length of $rs = 3$ in y^e same tim{illeg}|e|; y^{illeg}|n|, as, $op \times lmqyz ∶ rs \times tvwx ∷ 40 ∶ 21 ∷$ y^e motion {illeg}|of| lmqyz to y^e motion tvwx. Or th{illeg}|e| motio{illeg}|n| of one quantity to another is a{illeg}|s| their powers to /persever in that state\

Those bodys \Qua{illeg}|n|titys/ are said to have y^e same determination of their motion w^ch move y^e same way, {illeg}|&| those have divers w^ch move divers ways.

^[67] 5 A body /quantity\ is reflected when meeting w^th another quantity it looseth y^e determination of its motion by rebounding from i{illeg}|t|. As if y^e bodys a, b meete one A quantity is said to bee refracted another in y^e point c they are parted either by some springing motion in y selves or of \in/ y^e matter {crouded} bet{illeg}|w|ixt y^m. & as {illeg} y^e spring is more dull or {illeg}ety \{illeg}|vi|gor{illeg}|ous|/ /quick\ s{illeg}|o|e y^e bodys {illeg}|w|ill bee reflected w^th w^th more \or lesse/ force; a{illeg}|s| if it endeavour to get liberty \to inlarge it selfe/ w^th as greate strength & vigor as y^e bodys $a, b$ , pressed it together, y^e y^e {sic} motion of y^e body{illeg} a from b will bee as greate after as before y^{illeg}|e| reflection. but if y^e spring have but halfe y^t vigor, yⁿ y^e distance twixt a & b,|at| {illeg}|t|he minute after y^e reflection shall bee halfe as much as it was befor at y^e minute beef{ore} y^e reflecti{illeg}on.

7 Refraction is when {illeg} y^e body \{illeg}/ c passing \obliquely/ through y^e surface ed at y^e point b meets w^th more or lesse oppo{illeg}|s|i{illeg}|t|ion on one side of y^e surface yⁿ on y^e other & soe looseth its determinacon; as if it turne towards a.

^[68] 9 Force ii|s| y^e pressure or erouding of one body {illeg} upon an{illeg}|o|ther,

10 The center of Gravity \{ Motion }/ \in y^e same body/ is such a point w^thin a quantity \{illeg}w^ch rests when a body is moved w^th \any/ ◯^lar but noe progres\sive motion//; y^t if it \be considered to as at a/ rest & y^e quantity & some quantity line as $(mn)$ be drawne through it: about w^ch (as \about/ an axis) y^e f{illeg} quantity $(dklp)$ revolving. {illeg}|t|here shall bee y^e same quantity of motion on both sides any {illeg}|p|laine {illeg}|w|^th w^ch $(mn)$ is coincident; also y^e line in n \drawne throug{illeg}|h| it/ is called an axis of gravity \motion/.

The center of motion in \2/ divers bodys is {illeg}|a| point soe placed {illeg}|t|wixt those bodys y^t (if it bee conceived to rest {&}) if {illeg} the bodys bee moved about it {illeg} w^th circular motion they shall both have an equall quantity {o}f motion, the line about w^ch they move is y^e axis of motion.

12 A Body is said to move toward another body either when all its pts move towar{d}s it or else {illeg}|w|hen some of its pts have more motion towards it y^{illeg}|n| others have from it. Otherwise /not\

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13 Bodys are more or lesse distant as y^e distances of {illeg} their pts \centres of motion/ are more or lesse. or as their distances might bee acquired w^th more or less motion

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Axiomes. & Propositions.

1 If a quantity once move it will never res{illeg}|t| unlesse hindered by some externall caus{e.}

2 A quantity will always move on in y^e same streight line (not changing y^e determination \{nor} celerity/ of its motion) unlesse some externall cause divert it.

3 There is exactly required so much \& noe more/ force to reduce a body to rest {illeg} /as\ there {illeg} was {to} put it {illeg}n motion: et e contra.

4 S{illeg} much {illeg}{illeg}s is required to d{illeg}p{illeg}e {illeg}|d|estroy any quantity of motion in a body soe {illeg} to generate it; & soe much as is required to generate it soe mu{illeg}ived to destroy it.

^[69]6 {illeg}ove 2 unequall bodys \(a & b)/ y^e swiftness{illeg}|e| of {illeg} \on{illeg}e/ body $(a)$ is to y^e s{illeg} is to a . {(1)} & therefore y^e motion of both bodys shall bee equall.

5 If {illeg}{b}ee moved by unequall forces, as y^e force moveing $(b)$ is to y^e force {illeg} motion {illeg}|o|f b. to y^e m{illeg} of c, so is y^e swiftnes of b, to y^t of c.

^[70]7 If two body {illeg}{illeg} \{illeg}way to{illeg}s {illeg}{illeg}r{illeg}cking {illeg}/ {illeg}e of theire motion shall be lost. f{illeg} pres{illeg}{illeg}{illeg}{illeg} y^e motion of $(b)$ shall {illeg}{illeg}

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^[71] 8 If two quantitys (a & b) move d{illeg} towards one another & meete in {illeg} o, Then y^e difference of theire motion shall not bee lost nor {illeg}|l|oose its determinacon. For at their occursion they presse equally uppon one another, & (p)^[72] therefore one must loose noe more motion yⁿ y^e other doth; soe y^t y^e difference of their motions cannot be destroyed.

9. If one body $(a)$ overtake ano{illeg}|t|her body $(b)$ they both moveing towards o then they shall always move together. v
If y^e body $(c)$ move against an imm{illeg}|o|veable quanty {sic} $(d)$ it shall not bee rebounded for c having urged d w^th

^[73] 9 If the body t{illeg}|w|o \equall & equally swift/ bodys (d & {illeg}|c|) meete one another they shall bee reflected, s{illeg}|o|e as to move as swiftly frome {illeg}|o|ne another a{illeg}|f|ter y^r reflection as they did to one another before i{illeg}|t|. For {illeg}ng For firs{illeg}|t| suppose y^e {illeg}|sphæricall| bodys $e, f$ to have a springing or elastic{illeg}|k| for{illeg}|c|e soe y^t {illeg}|m|eeting one another they will relent & be pressed into a sphæri|o|idicall figure, {illeg}|{&}| in y^t moment in w^ch there is a period put to the{illeg}|ir|e motion towards one another theire figure will be most sphæroidicall & theire pression one upon the other |i|s at y^e greatest, & if th{illeg}|{e}| bodys endeavour to restore theire \◯^call/ figure w^{illeg}h \bee/ as much vigor|ous| & force|i||ble| as it was destroyed by, & as theire pressure upon one another wa{illeg}|s| to destroy it they will gaine theire whole /as much\ motion from one another \thei{illeg}|r| {illeg}|p|arting/ as their|y| \had/ towards one another {illeg} theire reflection. Secondly suppose they be sphæricall & absolute\ly/ sollid then at the period of theire motion towards one another (y^t is at y^e moment of theire meeting) theire pression is at y^e greatest or rather tis \done with/ the whole for{illeg}|c|e by w^ch theire motion is stopt, \for theire whole motion was stoped b{illeg}|y| y^e force of theire pressure upon one another in y^sone mome\nt// for /&\ there cannot $\begin{matrix} bee \\ succeede \end{matrix}$ divers degrees of pressure twixt two bodys in one moment) Now i{illeg}|f| /so long as\ neither of these 2 bodys yeild to one another they will {illeg} always retaine y^e same forcible pressure towards one another: that is soe much force as deprived y^e bodys {illeg} of th{illeg}|ei|r motion (when it moved towards h \towards one another soe much/ doth now {illeg} it towards g, & therefore (r) y^{illeg} urge them from one ano{illeg}|t|her, & therefore (r)^[74] they shall move from one another as much as they did towards one another before theire reflection.

10. There is {illeg}|y|^e same reason wh{illeg}|e|n unequall & unequally moved \bodys/ reflect, y^t they should sepera{illeg}|t|e from{illeg} one another w^th as much m{illeg}|{ot}|ion as they they came together.

^[75] ^[76] 11. If a line \ $(df)$ / bee moved not w^th a Circular \Progressive/ but onely a Circular motion i{illeg}|t|s middle point \ $(n)$ / shall rest. For if it move let it move towards r soe y^t, when y^e point $(d)$ is moved in $(p)$ & f in $(q)$ , then $(n)$ shall be moved to $(s)$ :

11 If {illeg}|a| line $(ce)$ be bisected in $(a)$ about w^ch y^e line $(ce)$ doth circulate {illeg}|&| y^t point bee fixed. yⁿ y^e whole line hath noe progressive motion. For makeing $ab = ad$ , bf, ag, & dh bee parallel, & perpendic to fh, yⁿ is $vb = dp$ . & $vf + ph = bf + dh = 2 ag$ . Wherefore y^e point c moveing to{illeg}|w|ards n y^e point {illeg}|d| shall move soe much towards y^e line fh as y^e point b doth from it, & all y^e points in {illeg} $(ac)$ or y^e line ac move as mue|c|h from to y^e line fh as \all/ y^e points in $(ae)$ or y^e line $(ae)$ moves from it soe y^t y^e whole line ce stays in equilibrio neither moveing to nor from fh, by y^e 12^th Defin{illeg}|.|

12 H{illeg}|e|nce when \y^e center of a line/ $(a)$ is not in y^e center \midst/ of a line $(me)$ y^e whole line moves y^e same way w^ch y^e longest pte doth. for supposeing $ca = ae$ y^{illeg}|n| y^e line ce in equilibrio (ꝑ ax:1{illeg}|1|) but if $(mc)$ moves towards $(fh)$ & be added to $(ce)$ yⁿ $(me)$ moves towards ce (by def {12}

13. When $(ce)$ moves circularly but maketh noe progression it{illeg}|s| midle {sic} point shall rest & is therefore y^e center of its motion, for if y^e middle point move y^e whole line let it bee to r from a soe y^t y^e line ec bee moved into y^e place { $(wt)$ } yⁿ let y^e { $(wt)$ } move about y^e fixed center r into y^e place xs, yⁿ {xs} & {wt} are equally distant from fh (by def: 13 & ax: 11) & alsoe $(ln)$ & $(ce)$ are equally distant from y^e {illeg}{am} f{illeg} but xs is further from $(fh)$ yⁿ $(ln)$ there & ln are not equally distant from f{illeg}{illeg}ore neither are wt & ce {illeg}|e|qually distant f{illeg}|r|om fh. & therefore y^e line {illeg} pro{gre}ssive motion when it {pressed} into {wt}

14 By y^e same reason y^e midle {sic} point {illeg}ogram, parallelipipedon, prisme cilinder, circle, sphære, {illeg}|e|lip{illeg}|s|is, sphæroides {illeg} of theire motion

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^[77] 15 A Body moves y^t way \hath y^e same determinacon of its motion/ w^ch the center of its motion hath. As if y^e line ac move into y^e place gh, y^e center of its motio{illeg}|n| b moveing into y^e place d. th{illeg}|e|n let gh move about y^e center d untitill {sic} it be parallel to ac, as into ef \soe y^t y^e point a fall into y^e {illeg}|p|oint e/. Now since {illeg} gh by turning about y^e center d hath noe progressive motion (by Def: 10) tis plaine y^t gh & ef {illeg} have determinacon from ac but ef hath y^e same determic|n|acon from ec w^ch d hath from {illeg}|b| (for if ac be understood to move parallell to its selfe into y^e place ef, all its points describe parallel lines & therefore have y^e s{illeg}|a|me deter{illeg}|m|inacon one w^th another & each with y^e whole body (by axiom 14.) therefore gh hath y^e same determinacon from ac w^ch y^e point d hath from y^e point b

14 A Body being moved parallell to its selfe all it{illeg}|s| points describe ∥ lines, \& each{illeg} y have y^e same determ \& velocity/ w^th y^e body./ for (by axiome 2^d) they {illeg}|m|ust {illeg}|a|ll bee streight ones w^ch if they intersect y^e body will not be \moved/ ∥ to its selfe. &c.

16 If a body move forward \& circularly/ its center of motion shall allways bee in y^e same streight line. For y^e body hath allways y^e same determinacon (ax 2^d) & y^e cent' of its motion hath y^e same determinacon w^th y^e body (ax 15) therefore it hath always y^e same determinacon, & soe will move continually in y^e same streight line.

^[78] 17 If a body move \streight/ forward & circularly its cente{illeg}|r| of motion shall have y^e same determinacon & velocity y^t {illeg}have had did y^e body move Pa∥ll to it selfe \the body hath/. For {suppose} ac wh to be moved into y^e place gh & its center of motion b into y^e place d, yⁿ let it turne about y^e center d into y^e place ef ∥ to ac s{illeg}|o|e y^t y^e point w^ch was in $(a)$ bee now in $(e)$ . N{illeg}|o|w since { $(gf)$ } by moveing into y^e place $(fe)$ makes noe progressiv motion (def: 10) it follows y^t {illeg}parallell to it selfe into y^e place ef it would {illeg} y^e same determinon {sic} & quantity of motion y^e same quantity \(Or since $gh = ef = ac$ ) y^e same velocity/ & \(axiom 10)/ determinacon of motion \in y^e same time/ would translate ac parallell to it selfe into y^e place ef y^t w{illeg}ld {sic} translate it into y^e pla{illeg}|ce| gh , had it bo{illeg}|t|h progressive & circular motion. But y^e point d {illeg} hath y^e f|s|ame velocity & determinacon w^ch y^e line ef hath when moved ∥ to its selfe (x)^[79] therefore y^e point d hath y^e same determinacon & velocity w^ch y^e line gh hath when moved w^th both ◯^lar & progressive motion \vide ax 37/.

18 If a body move progressively in some crooked line & alsoe circularly {illeg} its center of motion shall have y^e same determinacon & velocity w^ch y^e body {hath} for in {illeg} {illeg} any point of y^e crooked line its determinaco {sic} is in y^e tangent \(ax 17) th{illeg}|i|s is trew when its {illeg}|m|otion is in a strei{illeg}|gh|t lin{illeg}|e| but a crooked line may/ bee conceived to consist of an infinite number of streight lin{illeg}es. Or else in any p{illeg}|oi|nt of y^e croked {sic} line y^e motion may bee conceived to be on in y^e tangent.

^[80] 19 I {sic} 2 bodys circu make y^e same number of circulations w^th y^e same dista{nce} from y^e center c : yⁿ as y^e Radij of y^e circles w^ch they|ir| \centers of motion/ {illeg}|de|scribe are to one another soe are y^e perimeters one to another soe are theire velocitys one to another (ax: 10, def: 2), & their motions are to {illeg}|o|ne another as theire bulkes drawne into y^e Radij of those circle {sic} (w^ch theire centers of motion describe) are to one another (def 3). As: $ec = eo ∷$ velocity of $eb ∶$ velocity of ac. & $eb \times ec ∶ ao \times co ∷$ mot $eb ∶$ mot of ao.

^[81] 20 If a s{illeg}|ph|ære ⊙c move w^th in \be compelled by/ /move w^th in\ {illeg}|y|^e concave sphæcall {sic} or cilindricall surface of y^e body edf to move circularly abou{illeg}|t| y^e center m it shall press upon y^e body def for when it is in c (suposeing y^e $⊙$ bhc to be described by it {sic} center of motion {illeg} & y^e line cg a tangent to y^t $⊙$ at ⊙) it moves it moves towards g or y^e determination of its motion is towards c therefore if at y^t moment y^e body edf should cease to check it it would continually move in y^e {illeg}|l|ine cg (ax 1. 2.) obliq{illeg}|l|y from y^e center m , but if y^e body def oppose it selfe to this indeavour in ever keeping it equidistant from m , that is done by a continued \checki{illeg}|n|g or/ reflection of it from y^e tangent line in every point of y^e ◯ chb , but y^e body edf cannot check & curbe y^e determinacon of y^e body c⊙ unless they continually presse upon one another. |The same may {illeg}|b|e understood if y^e body adb bee restrained into ◯^lar motion by y^e thred $(om)$ |

21. Hence it appear{illeg}|e|s y^t all {illeg}|b|odys moved ◯larly have an endeavour from y^e {illeg}|c|enter abot|u|t w^ch they move, otherwise y^e b{illeg}|o|dy ⊙c would not continually presse upon edf .

2{illeg}|2| The whole force by w^ch a body c⊙ indevours from y^e center {illeg}| m | in halfe a revolution {i}s \more yⁿ/ double to the force w^ch is able to generate or destroy its motion for supposeing it have moved from $(c)$ by $(h)$ to $(b)$ then i|y|^e resistance of y^e body ef {illeg} (w^ch |is| equall to its pressure upon def ) is able to destroy its force of moveing {illeg}{illeg} & to generate in it as much force of moveing from $(b)$ to $(h)$ the qu{illeg}g {illeg}|w|ay.

^[82] 2{{illeg}|5|}{{illeg}|3|} {illeg}|H|aving {illeg}of motion of y^e 2 bodys ob & dc to find y^e common center of both {in}{illeg}draw a line ⊙e from the centers of theire motions ⊙ & e & divi{illeg}oe y^t {illeg}|th|e body ob is to y^e body de as the line ae t{illeg}|o| y^e line oa : y^t is soe{illeg} $ae \times de$ . For th{illeg}|e|n if they move about y^e center {illeg}{illeg}{illeg} they have equall motion (ax 19^th) & consequen{tly} {illeg} <12r> have an equall endeavour from y^e center a (ax 24|3|) soe y^t if they bee joyned tig{illeg}th by the line to center $(a)$ by y^e lines {illeg} ae & ao they shall not mo {illeg}|t|he one h{illeg}|i|ndereth y^e other from forcing y^e center a any way soe y^t it shall stand in equilibrio betwixt them & (by def 10) is therefore their center of \motion/

^[83] 24 If two bodys ( cb & de ) move about a center $(a)$ yⁿ as y^e motion o makeing $bc = a$ , $de = b$ , $ac = c$ , $ae = d$ , y^e time in w^ch bc makes halfe a revolution call e, y^t in w^ch de doth y^e sam{illeg}|e| call f , y^e pressure of cb from y^e center a in halfe a revolution call q , {illeg}|&| y^t of de call h ; y^e motion of cb in halfe a revoluco k & y^t of de call l . yⁿ $k ∶ l ∷$

24 If two bodys ( cb & de move about a center $(a)$ yⁿ ∼ ∼ ∼ ∼ ∼ ∼ The \whole/ force by w^ch y^e body cb tends from y^e center a \in one revolutio {sic}/ being equall to {{illeg}|6|}{{illeg}|61|} times y^e force by w^ch y^{illeg}|t| body is moved \(ax 22)/ is to y^e motion of y^t body {illeg} one revolucon as y^e \whole/ force by w^ch y^e body de tends from y^e center a in one revolution (w^ch is equall to {illeg}|6|{+} times y^e force by w^ch y^e de is moved, or w^ch is able to stop i{illeg}|t|s motion (ax 22) ) {illeg}|is| to y^e motion of y^e body de . Vide Axioma 23ũ.

^[84] {illeg}|2|6 If y^{illeg}|e| body a move through y^e space ab \ $= b$ / in y^e time d \ $= be$ /, & y^e body c {illeg}|thr|ough y^e space cd | $= e$ | in y^e time f . then y^e velocity of a is to y^e velocity of {illeg} c as $(ab \times dc)$ to $cd \times$ {illeg} be . for supposeing $cb = gp$ then y^e velocity of a is to y^e velocity of c as $(ab)$ to $(cp)$ (def 2^d) or as $ab \times gp ∶ cp \times eb ∷ ab \times cd ∶ ad \times eb$ . for since \Then is/ $cp \times fd = cd \times gp = cd \times eb$ . Or {illeg} $cp \times fd \times ab \times {\begin{matrix} eb \\ gp \end{matrix} = cd \times eb \times ab \times gp$ y^t is $ab \times gp ∶ cp \times eb ∷ ab \times fd ∶ cd \times eb ∷ ab ∶ cp ∷$ velocity of $a ∶$ velocity of c velocity of $a ∶$ to y^e velocity of $c ∷ ab ∶ cp$ {illeg} For supposeing y^t $gp = eb$ . yⁿ is $cp = \frac{eb \times cd}{fd}$ And (by def: 2) y^e velocity of $a ∶$ is to y^e velocity of $c ∷ ab ∶ cp ∷ ab ∶ \frac{eb \times cd}{fd} ∷ ab \times fd ∶ eb \times cd$

^[85] Alsoe y^e motion of a is to y^e motion of c (by def 3^d) $∷ a \times ab ∶ c \times cp ∷$ $a \times ab ∶$ \ $c \times c$ / $a \times ab ∶ c \times cp ∷ a \times ab ∶ \frac{c \times eb \times cd}{fd} ∷ a \times ab \times fd ∶ c \times eb \times cd ∷ a \times its velocity ∶ c \times its velocity$ .

Note y^t when y^e motion is uniforme y^t is when a body moves over y^e same sp{illeg}|a|ce in y^e same time (w^ch will ever bee when y^e motion of {illeg}|y|^t body is neit{illeg}|h|er helped nor hindred) yⁿ in a right angled triangle a b may designe y^e space through a body moveth {illeg} in y^e ti{illeg}|m|e eb. Otherwise when tis not uniforme y^e proportion of y^e time in w^ch a body moves to y^e {illeg} distance through w^ch it moves may be designed by lines drawne to a crooked line, as y^e time by gf y^e dis & ih , y^e distance by gh or fi , y^e velocity by y^e proportion of nh to hi , ni being tangent to y^e crooked line at i . &c.

^[86] 23 If 2 bodys be moved w^th equall or uneq
If y^e body bace is moved \acquire y^e motion q / by y^e force {illeg}| d | {illeg}|&| y^e body f \y^e motion p/ {illeg}|b|y y^e force g . yⁿ {illeg} $d ∶ q ∷ g ∶ p$ . for suppose y^e body $rscb = f$ , yⁿ ({illeg} {illeg}) to {illeg} acquire y^e motion w by y^e force d , yⁿ (ax: 5)^[87] $d ∶ g ∷ w ∶ p$ . but $q = w$ (by ax: 4) therefore $d ∶ g ∷ q ∶ p$ .

Ax: {illeg}|{1}|00 Every thing doth naturall{illeg}|y| persevere in y^t state in w^ch i{illeg}|t| is unlesse it bee interrupted by some externall cause, hence axiome 1^st, & 2^d, & {γ}, A body once moved will always keepe y^e same cele{illeg}|r|ity, quantity & determinacon of its motion.

^[88] If 2 equall bodys (bcqp & r) meete one another w^th equall motion \celerity/ (unlesse they could pass through one y^e other by penera|tr|acon of dimensions) they must mutually hinder their perseverance {illeg} in their{illeg} states, & (since y^e one hath no{illeg}|e| advantage more yⁿ /over\ y^e other they must) equally hinder y^e one y^e o{illeg}|t|hers haveing both of them an equall power to persever in theire state |celerity power to persevering in its state| |perseverance in its state| likewise if y^e body aocb be = & equivelox w^th r they have a like power of persevering &{illeg} |{illeg}ing {illeg} equally {illeg}|hin|der or op{illeg}|p|ose y^e one y^t {offers} progression or perseverance in their states| therefore y^e power of y^e body aopq (wh{illeg}|e|n tis equivelox w^th r) is double to y^e power y^t r hath to persever i{illeg}|n| its state. y^t is y^e e{illeg}|ff|ic{illeg}|a|cy force or power \of y^e cause/ w^ch can reduce aopq to rest must bee double to y^e power & efficacy of y^e cause w^ch can reduce r to rest, or y^e power w^ch ca{illeg}|n| move y^e one must {illeg}|b|e double to y^e power w^ch can move y^e other soe y^t they bee made equivelox.

Hence in equivelox bodys y^e powers of persevering in their states are proportionall to their quantitys.

101 Hence may bee perceived what is meant. Supposeing y^e bodys aobc & cbqp to be equall & equivelox: Then {illeg}|t|hat cause \hindrence, impedim^nt resist{ance}/ or opposi{illeg}|ti|on w^ch can \onely/ deprive cbqp of its \whole velocity &/ motion by hindering its p{er}severance can {illeg}|a|l{illeg}|s|o \onely/ deprive aocb of its \whole {whole velocity &}/ moti{illeg}|o|n {illeg} y^t cau{illeg}|s| hath y^e same {illeg}{illeg} over both y^e bodys. Now if {illeg} add y^e opposition $(a)$ w^ch can \{being} {illeg}ive of its {illeg}/ {illeg}|r|educe cbpq to {illeg}{illeg}ion $(b)$ w^ch can reduce aob {illeg} \{illeg}/ y^e whole opposition ( $a + b = 2 a = 2 b$ ) {illeg} \{illeg}/ both {illeg} bodys $aobc + bcpq = aopq$ ) {illeg} \{illeg}/ motion when they are joyned into one $(aopq)$ for $a ∶ cbpq ∷ b ∶ aobc ∷ a + b ∶ cbpq + aobc ∷ 2 a ∶ aopq$ Also neither {illeg} a or b {illeg} aopq of {illeg} motion for {illeg} <12v> pte ( a or b ) would be equall to y^e whole ( $a + b = 2 a = 2 b$ ). By y^e same r{illeg}|e|ason | aopq {illeg}|&| cbqp loosing equall velocity y^e resistance /impediment\ of aopq must be double to y^e opposition of cbpq .|

^[89] 102 By they {sic} same reason y^t \Since/ beacuse aopq {illeg} is double to cbpq & both of y^m equivelo{x} therfore y^e opposition w^ch can deprive aopq of its motion must be double to y^t w^ch can deprive cbpq of its motion; by y^e same reason it will follow y^t in equivelox bodys as one body{illeg} \ $(a)$ / {illeg}|i|s to another \ $(b)$ / {illeg}|s|oe must y^e resistance w^ch can deprive y^t body $(a)$ of its ${\begin{cases} velocity \\ motion \end{cases}$ bee to y^e resistance w^ch can deprive $(b)$ of its whole ${\begin{cases} velocity \\ motion \end{cases}$ so is y^e resistance w^ch can deprive $(a)$ of some pte of its velocity, to y^e restance {sic} w^ch can deprive $(b)$ of y^e same quantity of velocity, soe y^t a & b bee still equivelox.

Now {illeg}|i|t may be perceived how & why {illeg} amongst bodys moved some require a greater dome a lesse opposition to deprive y^m of theire whole velocity or of some pt{e} of it w^ch {illeg}p

103 By y^e same reason alsoe If two bodys rest or bee {illeg}|e|quivelox: yⁿ as y^e body $(a)$ is to y^e body $(b)$ soe must y^e power orf efficacy \vigor strength/ or virtue of y^e cause w^ch begets new velocity in $(a)$ {illeg}|b|ee to y^e power virtue or efficacy of y^e cause w^ch begets y^e = same quantity of velocity in b , soe y^t a & b {illeg} bee still equivelox.

104 Hence it appeares how & why amongst bodys moved som{illeg}|e| require a more potent or efficacious cause others {a lesse} to hinder or helpe their velocity. And y^e power of this cause is usually called force. And as this cause \useth or/ apply|eth| its power or force to hinder {illeg} or helpe \or change/ y^e {illeg}g perseverance of bodys in theire state, it is said to Indeavour {illeg}|t|o change their {illeg} perseverance.

^[90] 105 If y^e equall & equivelox bodys a & b meete (unlesse they could passe y^t one through y^e other by penetra{illeg}|t|ing its dimen{illeg}|tio|ns) they must necessarily hinder y^e one y^e others progression, & since these bodys have noe advantage y^e one over y^e other y^e hindrance on both pts will be equall, likewise if y^e bodys $d + a$ & $b + c$ bee equall & equivelox they must equally hinder one anothers progression in its s{illeg}|t|ate But y^e body $(b)$ (being lesse yⁿ y^e body $(b + c)$ ) & equivelox w^th $(d + a)$ ) canot equally hinder y^e progression of y^e body $d + a$ soe much as y^e body $(b + c)$ ca{illeg}|n|; for {yⁿ} the power of $(b)$ being part of y^e power of y^e body $b + c$ would bee equall to y^e {illeg} {illeg}|w|hole power of $b + c$ therefore y^t $b + c$ & $d + a$ being equivelox d{illeg}|o|e equally hinder y^e one y^e others progression tis required y^t they be equall.

^[91] 106 Now if y^e {illeg} bodys a & b meete one another y^e cause w^ch hindereth y^e progression of a is y^e power w^ch b hath to persever in its state velocity \or state/ & is usually called y^e force of y^e body b & {illeg}|i|s this power or force are said to |soe y^t a body is {so} to be moved w^th more or lesse force w^ch meeting w^th another body can cause a greater or lesse mutation in its state, or w^ch requireth more or l{illeg}|e|ss for{illeg}|e| {sic} to destroy its motion.| & as {one} body b useth or applyeth this force to stop y^e progression of a it is said to {illeg} Indeavour ty|o|^e hinder y^e {illeg} progression of a \w^ch indeavour in body is performed by pressure/ & by y^e same reason y^e body b may {illeg}|b|ee said to endeavor to helpe y^e motion of a if it should apply its force to move it forward: soe y^t it is e{illeg}|v|ident w^t y^e Force & indeavor of \i{illeg}|n|/ bodys are.

^[92] 107 If y^e bodys b & c be equiv{illeg}|e|lox yⁿ as {illeg} $b ∶ c ∷$ y^e force of \w^th w^ch/ b \is moved/ (or y^e power of b to persever in its velocity \or to \{keepe}{helpe} {illeg}/ hinder another body f{illeg}|r|o persevering in its velocity/ to y^e force of c . For let there be 2 other bodys a & d equivelox to th{illeg}|e|m soe y^t a meeting b , & d meeting c they would eqaully hinder one others progression yⁿ is $a = b$ , & $c = d$ (ax 105) & $a (= b) ∶ d (= c) ∷$ yⁿ force w^ch can stop $a$ (= to y^e force of b ) to y^e force w^ch can stop d (= to y^e force of c ). |(vide ax: 102.|

^[93] 108 Tis knowne by y^e light of nature y^t equall forces shall effect an {illeg}|e|quall ch{illeg}|a|nge in equall bodys. Therefore if y^e forces g, h, k, m, be equall, & y^e bodys a, b, equall {illeg}|&| rest, then let $(a)$ bee moved by y^e force g ; & b by h , a & b shall be equiv{illeg}|e|lox: Also (since tis noe greater change for $(a)$ to acquire another part of motion now it hath one yⁿ for it to acquire y^{illeg}|t| one when it {illeg} had none) if $(a)$ bee againe moved forward by y^e force k , its velocity shall be double to y^e velocity of b , & if it bee againe moved forward by y^e force m its velocity shall be triple to y^t of b . &c. Whence as y^e force moving $(a)$ is to y^e force moving $(b)$ soe is |y^e| velocity acquired in $(a)$ to y^e velocity acquired in $(b)$ \{{illeg}|b|y that force}/

109 By y^e same reason if \ $a = b$ &/ y^e velocity of a be t{illeg}|r|i{illeg}|p|le to y^e velocity of b , y^{illeg}|t| force {illeg} w^ch can deprive a of its velocity. must be \w^ch is/ 3^ple to y^e force w^ch can deprive b of its velocity. Or in gener{illeg}|a|ll {\2/} {illeg} \so/ \is/ y^e \lost/ velocity of a {illeg} to y^e \lost/ velocity of b soe is \As/ y^e force w^ch one deprives a of \some or all of/ its velocity, to y^e force w^ch can deprives b of \some or all of/ its velocity & so is y^e force \{That}/ w^ch can deprive a of

{illeg} y^e force {illeg}|w|^th{illeg} is {illeg}\or preserve it selfe in its {illeg}/ is to y^e force w^th w^{illeg}|c|h b is moved {illeg} y^e velocity of a {illeg} velocity of b otherwise it {illeg} not be {illeg} y^e {illeg} Ax 5^t

The force w^ch y^e body $(a)$ hath to preserve it selfe in its state shall bee equall to y^e force w^c{h}{illeg}{illeg} y^t state; not greater for y^e effect cannot exceede the {illeg} for {illeg}{illeg}{illeg} y^e {illeg} w^ch was not in y^e cause \nor lesse for/ y^e cause only {illeg}to its effect{illeg}no reason why its {illeg}{illeg}

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^[94] \112/ A body is saide to have more or lesse motion as {illeg} it is moved w^th more or lesse force, y^t is as ther{illeg}|e| is more or lesse force required to generate or destroy its \whole/ motion.

^[95] 11{illeg}|3| If a body $(a)$ move through y^e space ab \ $= r$ / in y^e time c . & {illeg}|t|he body {illeg}|f | through gf \ $= v$ / in the time h then, time $c ∶$ time $h ∷$ line ab \ $= r$ / $∶ \frac{h \times ab}{c} =$ fk ak . & y^e body a would move through y^e space ak in y^e same tim $(h)$ in w^ch y^e body f moves through y^e space fg . Therefore y^e velocity of a is to y^e velocity of f as y^e line $ak = \frac{h \times ab}{c} ∶$ line fg {illeg} $∷ h \times ab ∶ c \times fg$ {illeg} \(def 2)/ Then I ad {sic} y^e body r to f soe y^t $r + f = a$ . since f & r are equivelox, \(ax: 107)/ as $f ∶ f + r = a ∷ m =$ force or motion of f , to {illeg} $\frac{am}{f} =$ force of $f + r$ . againe sine {sic} $a = f + r$ , & they move \(ax: 111)/ as y^e velocity of a ; to y^e velocity of $f + r ∷ h \times ab ∶ c \times fg ∷$ $∷ n =$ y^e force or motion of a , to {illeg} the $\frac{n \times c \times fg}{h \times ab} = \frac{a \times m}{f} =$ to y^e force of $f + r$ . Soe y^t, $n \times f \times c \times fg = m \times a \times h \times ab$ . So{illeg}|e| y^t haveing any 7 of t{illeg}|h|ese y^e 8^th m{illeg}|a|y bee found. {illeg} but suppose y^e bodys moved in equall ti{illeg}|m|es y^t is if $c = h$ , yⁿ y^e rest of y^e termes may bee found by, $m \times a \times ab = n \times f \times fg$ . &c. y^t i{illeg}|s| as $f \times fg$ is to $a \times ab$ soe is y^e motion $(m)$ of y^e body f to y^e motion $(n)$ of y^e body a . &c.

^[96] 110. If y^e body {sic} ( a & b ) bee equall & y^e celerity of a ti|r|iple to y^t of b , yⁿ if y^e force {illeg} {illeg}| d | can deprive {illeg}| b | of its motion, y^e force $3 d$ may can deprive a of its motion. But if there bee lesse force \ $(3 d - p)$ / it cannot deprive a of its motion for soe y^e pte $(3 d - p)$ would be = to y^e whole $3 d$ ; if there be more force $(3 d + p)$ it will doe more yⁿ deprive the body a of its motion (i.e. move it y^e contrary way) otherwise y^e pte $(3 d)$ would be equall to y^e wh{illeg}|o|le $(3 d + p)$ . |Therefore y^e |If y^e body a bee equall to y^e body b .
force which can deprive a of its motion must bee 3^ple to y^e force w^ch can deprive b of its motion & consequently ({illeg}|d|ef 106) y^e for{illeg}|c|e wherw^th a is moved is 3^ple to y^e force wherew^th b is moved

111 By y^e same reason as y^e celerity of y^e body $a (= b)$ is to y^e celerity of b so is y^e for{illeg}|c|e wherew^th a moveth to y^e force wherew^th b moveth.

^[97] 114 There is required soe much & noe more force to reduce a body to rest y^{illeg}|n| there is to move it: et e contra. And

115 Soe much force a is required to generate any quantity of motion in a body so{illeg} much is required to destroy it, & e contra. For d \in/ loose|i|{illeg}|n| {sic} or to get|ting| y^e same {illeg}|q|uanty {sic} of motion a body suffers y^e same quantity of mutacon in its state, & in y^e same body equall forces will effect a equall change

^[98] 116 If y^e bodys $a = 3 b$ , & a & b {illeg} are moved w^th y^e same force $(d)$ yⁿ y^e celerity of b {illeg} is tri{illeg}|p|le to y^e celerity of {illeg} a . for if a be moved by suppose $b ∶ a ∷ d ∶ \frac{a d}{b}$ & let $(a)$ bee moved by $(\frac{a d}{b})$ & $(b)$ by {illeg} for if $a (= 3 b)$ be moved by $3 d$ , & $(b)$ be {illeg} moved by $(d)$ , ( a & b ) shall {illeg} }{illeg} \moved by/{ for $3 b$ moved by $3 d$ is equivelox to b moved by d , but since $3 b = a$ , therfore a moved {illeg}|b|y $3 d$ is equivelox to b moved by d . And (ax 108) as y^e ce{illeg}|l|erity of a moved by {illeg}| d | is to y^e celerity of a moved by $3 d$ , soe is 1 to 3 , soe is y^e celerity of a moved by d to y^e celerity of b moved by d .

By y^e \same/ reason, Any bodys f & g being moved by y^e same force as $(f)$ is to $(g)$ , soe is y^e celerity of $(g)$ to y^e celerity of $(f)$ acquired by y^t force. tis axiome y^e 4^th And (by ax: 113) y^e bodys will have equall motion.

^[99] 118 If y^e body p , be moved by y^e forc{illeg}|e| q , & r by y^e force s , \to find {illeg} $(v)$ y^e celerity of p & $(w)$ y^t of r ,/ I add $(t)$ to p , soe y^t $p + t = r$ , & y^t $(p)$ , & $(p + t)$ are moved w^th equall force, yⁿ $p + t (= r) ∶ p ∷ v ∶ \frac{p v}{r}$ y^e celerity of $p + t$ , (ax 117) alsoe, (ax 108) $s ∶ q ∷ w ∶ \frac{q w}{s} = \frac{p v}{r}$ . Or $q r w = p v s$ . that is y^e celerity of p is to y^e celerity of r as $q r$ is to $p s$ . And by ax 113 y^e motion of p is to y^e motion of r as y^e force of p to y^e force of r . And by y^e same reason if y^e motion of p & r bee hindered by y^e force q & s , {illeg}y^e motion lost in p is to y^e motion lost in r , as q is to s . or if y^e motion of p be increased by y^e force q , by|u|t y^e motion of r hindered by y^e force s ; the {illeg} as q , to s ∷ so is y^e increase of motion in p , to y^e decrease of it in r (ax 111

^[100] 1{1}9 \121/ If 2 bodys p & r p {illeg}{nest} y^e one y^e other, y^e resistance {illeg}|i|n both is y^e same for soe much as p presseth upon r so much r presseth on p . And therefore they must both suffer an equall mutacon in the {illeg} motion.

11{illeg}|9| {illeg} If r presseth p towards w then p presseth r towards w . {illeg} w^thout {illeg}

120 A body must move y^t way w^ch it is pressed.

122 Therefore if y^e body p comes from c & y^e body r from d soe much as { p {illeg} } motion is changed towards w soe much y^e motion of {illeg} changed {illeg}{illeg}

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^[101] \27/ If two bodys {illeg}| b | & c move from $(o)$ their center of gravity they shall have equal motion For suppose b moved into y^e place d ; then putting, $c ∶ b ∷ do ∶ \frac{b \times do}{c} = oe$ (ax 25) y^e body c must be the{illeg}|n| moved into y^e place {illeg} e . Alsoe $c ∶ b ∷ bo ∶ \frac{b \times bo}{c} = oc$ . (ax 25) therefore $ec = \frac{b \times bo - b \times do}{c} = \frac{b \times bd}{c}$ . that is {illeg} $c \times ec = b \times bd$ . But (ax $\{\begin{matrix} 113 \\ 26 \end{matrix}\}$ or) $c \times ec ∶ bd \times b ∷$ y^e motion of c , to y^e motion of d , & therefore c , & d have equall motion towards o

If y^e body b move through y^e places d , f , r , & y^e body c through y^e places {illeg}| e | g h & their center of o , p , q , r y^e line opqr shall be a {illeg}|s|treight line For nameing y^e lines $bc = g$ . $cr = h$ . $br = k$ . $bd = x$ . $d ∶ e ∷ x ∶ ce$ . Then Then $ce = \frac{e x}{d}$ . $k ∶ g ∷ x ∶ \frac{g x}{k} = cv$ & supposeing $bc ∥ dv ∥ ps$ . Therefore {illeg} $ve = \frac{e x}{d} = \frac{g x}{k}$ & then $k ∶ g ∷ k - x ∶ \frac{g k - g x}{k} = dv$ . $b + c ∶ c ∷ de ∶ pe ∷ dv ∶ ps ∷ ve ∶ es$ also $k ∶ h ∷ x ∶ \frac{h x}{k} = cv$ , & $\frac{e x}{d} - \frac{h x}{k} = ve$ therefore $ps = \frac{c g k - c g x}{b k + c k}$ . & $es = \frac{c e k x - c d h x}{b d k + c d k}$ . $cr - ce = er = h - \frac{e x}{d}$ . $er + es = sr = \frac{c e k x - c d h x}{b d k + d c k} + \frac{d h - e x}{d}$ . Or $rs = \frac{b d h k + c d h k - b k e x - c d h x}{b d k + d c k}$ . {illeg} $b + c ∶ c ∷ bc ∶ co ∷ g ∶ \frac{c g}{b + c} = co$ . $cr ∶ co ∷ h ∶ \frac{c g}{b + c} ∷ b h + c h ∶ c g ∷ rs ∶ sp ∷ b d h k + c d h k - b k e x - d h c x ∶ c d g k - c d g x$ . Whence $c d g k \dot{b h} - c d g x b h + c d g k \dot{c h} - c d g x \dot{c h} = b d h k \dot{c g} + c d h k \dot{c g} + k e x b c g - d h c x \dot{c g}$

^[102] 28. To fin If two bodys b & c mo{illeg}|v| {illeg} through in y^e lines br & cr . The body c moveing through y^e space cg in y^e ti{illeg}|m|e vs , & though g h in y^e time nv , & through kr in y^e time nr . & y^e velocity of y^e body b is to y^e velocity of c as d , t{illeg}|o| e , & a{illeg}|s| y^e line cg to y^e line be , or as ck to br , then {illeg} when y^e body c is in y^e place g , b will bee in e , & when c is in k , b will be in r . to find y^e line w^ch the center of their motion describes, viz dfo . {illeg}|T|hen nameing y^e quantitys $br = a$ , $cr = f$ . $bc = g$ . {illeg} $e ∶ d ∷ a ∶ \frac{d a}{e} = ck$ . $kr = \frac{e f - d a}{e}$ . If o be y^e center of gravity motion of y^e bodys at k {illeg}|&| r ; yⁿ, $b + c ∶ b ∷ \frac{e f - d a}{e} ∶ \frac{b e f - a b d}{e b + e c} = or$ . And y^e line df must passe through o . againe making $gk = v$ . yⁿ $d ∶ e ∷ v ∶ \frac{e v}{d} = er$ . & if $bc ∥ fi ∥ em$ , yⁿ $a ∶ g ∷ \frac{e v}{d} = er ∶ \frac{g e v}{a d} = em$ . also since f is y^e &, $a ∶ f ∷ \frac{e v}{d} = er ∶ \frac{f e v}{a d} = mr$ $gr = gk + kr = v + \frac{e f - a d}{e}$ . $gm = gr - mr = \frac{e v + e f - a d}{e} - \frac{f e v}{a d} = \frac{a d e v + a d e f - a a d d - f e e v}{a d e}$ . since f is y^e center of motion in y^e bodys at g & e tis, $b + c ∶ c ∷$ {illeg} $ge ∶ fg$ . {illeg} $b + c ∶ c ∷ eg ∶ fg ∷ em = \frac{g e v}{a d} ∶ \frac{c g e v}{a b d + a c d} = fi ∷ gm = &c ∶ gi = \frac{c a d e v + c a d e f - c a a d d - c f e e v}{b a d e + c a d e}$ . $gr = gk + kr = \frac{e v + e f - a d}{e}$ . $go = gr - or = \frac{e v + e f - a d}{e} + \frac{a b d - b e f}{e b + e c} = \frac{b e v + c e v + c e f - c a d}{e b + e c}$ . $go - gi = io = \frac{a d b e v + c f e e v}{b a d e + c a d e} = \frac{a b d v + c e f v}{a b d + a c d}$ . $co = cr - or = \frac{f e c + a b d}{e b + e c}$ . $b + c ∶ c ∷ g ∶ \frac{c g}{b + c} = cd$ . ^[103] {illeg}|N|ow if the lines $oi ∶ if ∷ oc ∶ cd$ . Then y^e line od must be a streight line. but $oi ∶ if ∷ a d b + c f e ∶ c g e ∷ oc ∶ cd ∷ \frac{f e c + a b d}{e} ∶ c g ∷ f e c + a b d ∶ e c g$ . therefore y^e line do is a streight line, w^th w^ch may bee found by y^e two points d & o . |The demonstracon is y^e same if y^e body b moved from a to b |

29 If two bodys q & c be moved in divers plines {sic}, then find y^e shortest {illeg} line $(pr)$ w^ch can bee drawne frome one line $(cr)$ {illeg}|to| y^e other line $(qp)$ in w^ch those bodys are moved, & y^t line pr shall bee perpendicular to both y^e lines cr & qp , viz $∠ qpr =$ $∠ rps = prc = recto$ . then draw qb equall & $∥ pr$ & draw $br = qp$ . Then shall y^e plaine qbrp be perpendicular to y^e plaine bcr . Suppose also y^e body c moves over y^e space $\begin{matrix} cg \\ gk \\ kr \end{matrix}\} in the time \{\begin{matrix} vw \\ wt \\ tr \end{matrix}$ {illeg}|&| y^t y^e body q moves over y^e space qa in y^e time vw , & ap in y^e time wt . Also suppose another body $b (= q)$ & equivelox to { $(q)$ } y^t is to move over y^e space $be = qa$ in y^e time vw {illeg}

<14r>

Soe y^t when $(c)$ is in $(g)$ or $(k)$ {illeg}|,| $(b)$ will bee in $(e)$ or $(r)$ & $(q)$ in $(a)$ or $(p)$ . {illeg} Then \Then drawing {the} streight lines qc , ag , bk / if $b + c ∶ c ∷ bc ∶ cd ∷ eg ∶ fg ∷ rk ∶ ok$ ;|,| the points d , f , & o , shall be y^e centers of gravity motion of y^e bodys b & c , when they are in y^e places b & c , e & g , r & k . & (prop 28) therefore y^e line ( dfo {illeg}|)| in a streight line. Likewise if it bee {illeg} $q + c ∶ c ∷ qc ∶ lc ∷ ag ∶ mg ∷ pk ∶ nk$ , then y^e points l , m , n are centers of motion to y^e bodys ( q & c ) being in y^e places $(q)$ & $(c)$ , a & g , p & k . Then drawing y^e lines ld , mf , no , f{illeg} y^e (twixt y^e neighbouring centers of motion) since $b + c ∶ c ∷$ {illeg} $q + c ∶ c ∷ bc ∶ cd ∷ q ∶$ {bc }. therefore $∠ qbc = ∠ ldc$ & by y^e same reason $∠ gfm = ∠ gea$ {illeg}|&| $∠ krp = ∠ kon$ . Wherefore |all the lines qb , ae , pr , ld , ml , no are parallell t{illeg}|o| one another. And| $b + ∶ c ∷ bc ∶ dc ∷ qb ∶ ld ∷ eg ∶ fg ∷ ea (= qb) ∶ mf ∷ kr ∶ ko ∷ pr (= bq) ∶ no$ , soe y^t $ld = mf = no$ . & since these line line {sic} ld , mf , no , are parallell, equall, in y^e same plaine ldon , & stand upon y^e same streigh{illeg}|t| line do , their other ends \(the ce{illeg}|n|ters of motion of c & q )/ l , m , n , must bee in y^e same streight line lmn , w^ch line y^e line \ $(lmn)$ / in w^ch their other ends l , m , n , are are terminated (i.e. in w^ch /are\ all y^e centers of gravity motion of y^e bodys ( c & q )) {illeg} must bee a streight line.

The demonstracon is the same if $(q)$ moved from $(p)$ to $(q)$ .

^[104] 30. Suppose y^e bodys b , & c moved towards, r ; so y^n|t| when b is in $\{\begin{matrix} b \\ e \\ p \end{matrix}\}$ then c is in $\{\begin{matrix} c \\ g \\ k \end{matrix}\}$ . & theire centers of motion describe y^e line dq . Then y^e motion of the|i|re centers of motion shall be uniforme. For $pr ∶ pw ∷ er ∶ ey$ \if $pw ∥ nt ∥ ey ∥ fs ∥ bc$ / $pr ∶ er ∷ pw ∶ ey ∷ nt ∶ fs ∷ nq ∶ fq$ . y^t is $pr ∶ ep (= er - pr) ∷ nq ∶ fn = (fq - nq)$ & therfor since y^e motion of y^e body in epr is uniforme, y^e motion of theire centers of motion in y^e line fnq , must b{illeg}|ee| uniforme, y^t is have allway alike {illeg}|v|elocity |The demonstracon is y^e same in all other cases.|

^[105] 28 \& 30/ Supposeing y^e thing{illeg}|s| suppose {sic} in y^e 28^th prot{illeg}|p| {sic} by {illeg}|sc|hem {sic} 38^th it may be thus done. $pr ∶$ {illeg} $er ∷$ {illeg} $er ∶ br ∷ ey ∶ bc ∷ fs ∶ dc$ . Also $ep ∶ eb ∷ cg ∶ gk$ . $∷ cy ∶ yw$ {illeg} & $cg ∶ ck ∷ cy ∶ cw ∷ gy (= cy - cg) ∶ hw (= cw - ck) ∷ gs ∶ kt ∷ sy ∶ tw ∷ be ∶ bp$ $∷ cs ∶ ct ∷ dc - fs ∶ dc - nt$ &c. Makeing $fs ∥ dc ∥ nt$ . & $mf ∥ qn ∥ et$ , yⁿ is $mf = cs$ & $qn = ct$ . & $fs = mc$ , & $nt = qc = is$ . Then $be ∶ bp ∷ cy ∶ cw ∷$ \ $cg ∶$ /ck (so is y^e velocity of b to y^e velocity of c ) $∷ gy (= cy - cg) ∶ hw (= cw - ck) ∷ sy ∶ tw$ (for $c + b ∶ b ∷ eg ∶ ef$ $∷ gy ∶ sy ∷ kp ∶ np ∷ kw ∶ tw$ .) {illeg} $∷ mf (= cy - sy) ∶$ {illeg} \ qn / $(= cw - tw)$ {illeg}. Againe $br ∶ er ∷ bc ∶ ey$ {illeg} $∷ dc ∶ fs$ \ $= mc$ ,/ (for $b + c ∶ c ∷ bc ∶ dc ∷$ fg $eg ∶ fg ∷ ey ∶ fs$ ) {illeg} \{illeg}/ Also Whence $be (= br - er) ∶ br ∷ dm (= dc - mc) ∶ dc$ . Also $br ∶ pr ∷ bc ∶ pw ∷ dc ∶ nt (= qc)$ , whence Therefore $be ∶ bp (= br - pr) ∶ dm ∷ dq (= dc - qc)$ . That is $be ∶ bp ∷ dm ∶ dq ∷ mf ∶ qn$ . & consequently y^e points d f n are in one streight line. & since y^e motion of $(b)$ is uniforme &, $be ∶ bp ∷ dm ∶ dq ∷ df ∶ fn$ , the motion of y^e center d is uniforme.

^[106]28 & 30^th. The bd|od|ys \( b & c )/ being in b & c , e & g , r & k , {illeg}|i|n y^e same times, & d n being described by their centers of motion. Also making $de ∥ fs ∥ ey$ . {illeg}|&| $mf ∥ cn$ . Then $be ∶ br ∷ cy ∶ cr ∷ cg ∶ ck$ (for y^e motions of b & c are u{illeg}|n|ifor) $∷ gy (= cy - cg) ∶ kr (= cr - ck) ∷ gs ∶ kn$ (for $b + c ∶ c ∷ ge ∶ gf ∷ gy ∶ gs ∷ kr ∶ kn$ ) {illeg} $mf (= cs = cg + gs) ∶ cn (= ck + kn)$ . Againe $br ∶ er ∷ bc ∶ ey ∷ dc ∶ fs = mc$ (for $b + c ∶ c ∷ ge ∶ gf ∷ ey ∶ fs ∷ bc ∶ dc$ (prop 25)). Therefore $be ∶ br ∷ dm ∶ dc ∷ mf ∶ cn$ & consequently y^e points d f n are in one streight line. als{illeg}|o| since $be ∶ br ∷ df ∶ dn$ y^e cente{illeg}|r| of motions motion {sic} {illeg}|m|ust bee uniforme.

^[107] 31 If two bodys \( b & c )/ meete & reflect one another \at/ their center of motion shall bee in y^e same line \ $(kp)$ / after reflection in w^ch it was before it. For y^e motion of {illeg}| b | towards d {illeg} \y^e {center} of their motion/ is equall to y^e motion of c towards d , by {illeg}|p|rop 25. then drawing {illeg} { $bk ⊥ kp$ } & { $cm ⊥ bp$ }. {illeg} yⁿ $cd ∶ bd ∷ cm ∶ bk$ . therefore y^e bodys b & c have equall towards y^e the points k & { c }. y^t is towards y^e line kp . And \{illeg}/ {illeg} reflection so much as $(c)$ presseth $(b)$ from y^e {illeg} after reflection y^t is $gp ⊥$ {illeg} {illeg} $⊥ kp$ . { $(e)$ } & $(g)$ {illeg} $(n)$ $(p)$ {illeg} tis {illeg} equall {illeg} y^e {illeg} <14v> must therefore be y^e center of motion of y^e bodys b & c when they are in y^e places g & e . {illeg}|&| i{illeg}|t| is in y^e line kp . The Demonstracon is same in all cases.

^[108] 32 If y^e bodys ( b & c ) reflect at q to e & g , & {illeg}|y^e| centers of their motion describe y^e line kdop . y^e velocity of y^t center $(o)$ after reflection shall bee equall to y^t|e| center d {illeg} velocity of y^t center $(d)$ before reflection. For from y^e centers {illeg} & $(d)$ draw y^e lines af {illeg} perpendicularly to kp . s|a|lso draw $ba ∥ fc ∥ eh ∥ rg ∥ kp$ & su{illeg}|p|pose y^e line af to have y^e same celerity w^ch (y^e point d ) y^e center of motion hath before reflection, soe y^t when y^e bodys (after reflection are in e & q , y^e line af may {illeg}|b|ee in kr . Then \Also/ draw in y^e ab {illeg} $∥ kp$ $ab ∥ fc ∥ eh ∥ rg ∥ kp$ . {illeg} Then since d is y^e center of motion in $(b)$ & $(c)$ , y^e bodys $(b)$ & $(c)$ have equall motion towards d , but, $bd ∶ ba ∷ dc ∶ fc$ . Therefore, y^e bodys $(b)$ & $(c)$ have equall motion towards a & f y^t is {illeg}|t|owards y^e line adf . Now when y^e bodys reflect, so much as y^e body b presseth y^e body c from y^e line af (or sf ) \or towards p / soe much y^e body c presseth y^e body b from y^e same line, or towards k , (by {illeg}x ax: 119) theref{illeg}|o|re {illeg}|t|he bodys b , & c , have equall motion from y^e line af , after reflection \(by ax 121)/ y^t is when are at e & g they doe equally move from y^e points h & r ; then drawing eg , tis $eh ∶ eo ∷ rg ∶ go$ . Therefore y^e bodys doe equally move from y^e point $(o)$ {illeg} w^ch (by ax: 25) must bee their center of motion, & since y^e motion of y^e line ( af or hr ) is uniforme ({illeg}|b|y supposition) & y^e point o is in y^e line hr , & also in kp (by ax|prop| 31.) its motion must be uniforme.

Note y^t by this, \&/ y^e 31^th p{illeg}|r|o{illeg}|p| I can find y^e center of motion of {illeg}|t|wo bodys at any given time; & by prop: 9, or def: 5^th, I can find their distance, & by prop 25, their distance from their center of motion. & consequently \that is/ y^e 2 spheres in whose perimeters , they are {illeg}ly be found; There wants therefore onely their determinacon to \bee/ knowne y^t the{illeg}ir places \in y^e spære {illeg}/ may bee found.

^[109] 33 Suppose y^e body dcgk immoveable, y^e surface dcg being plaine. Also let y^e body shæricall {sic} body amn bee moved in y^e ⊥ ch . so a{illeg}|s| to be reflected in c Then since y^e side am hath as much force to w{illeg}eigh or presse towards $(d)$ as y^e side an to presse towards e \by reason y^t a y^e center of its motion is in y^e line ch / ( ) y^e body{illeg} {illeg}|m|ust be in equilibrio neither pressed towards d nor e but reflected back in y^e line ch . The same may be said of any line bodys whose motion center is in y^e perpendicular to y^e reflecting point.

^[110] 34 Take $an = 2 bn = 4 cn = 8 dn$ . \soe y^t $ad ∶ dn ∷ 7 ∶ 1$ . Then draw y^e ⊥s eb , fc , gd , hm . And/ Set a body $(aem)$ upon y^e points a & m & let $(efk)$ {illeg} stand on y^e points e & k w^ch are in y^e ⊥s eb , & hkm , & fh , on y^e points f , & h & lay a Globe $(g)$ let y^e same be {illeg} same be {sic} supposed of hkm . Then suppose (for distinc{illeg}|t|ions sake $(g)$ ha{illeg}|v|e 8 pts of {illeg} force, w^th w^ch it pr{illeg}esseth fh . also Then {illeg}|m|ust fh presse w^th 8 pts of {illeg} force, y^t is (since $fg = gh$ , or $cd = dn$ ) w^th 4 pts on y^e point h , & w^th 4 on y^e point f upon y^e body efk , so y^t efk must presse w^th 4 {illeg}|p|ts of force \viz:/ (since it presseth equally on y^e y^e {sic} points e & k ) it presseth on k w^th 2 pts of force & {illeg} {illeg} w^th y^e other 2 a{illeg}|t| e on y^e body aem so y^e {sic} aem presseth w^th 2 pts of force, {illeg}|v|iz: w^th one on y^e point { n }{ m } w^th y^e other on y^e point a ; Soe y^t y^e body $(hkm)$ hath 7 pts of pression upon y^e point n , 4 at h , 2 at k , & one at m , \& since y^e bodys p{illeg}|re|ssure of all y^e point h k & m is directly towards { n } {illeg} n will be pressed by it all./ but y^e Glo{illeg}|b|e causeth but one part of pression upon a . Now if these bodys fh , ek , aem , {illeg}|u|nderstood continually to diminish & come nearer to \y^e line ad n / y^e pressure of y^e body g upon a & n will still bee y^e same y^t is as 1 is to 7 , & so is y^e line dn to da . By y^e same reason it may be generally pronounced $ad ∶ dn ∷$ pressure of y^e body {illeg}| g | upon $n ∶$ to y^e pressure of it upon g a

35 Or if y^e bodys a & {illeg}| n | bee supposed united by y^e line ( and ) & another by hitting on body $(g)$ moving towards y^m {illeg} hit \perpendicularly/ upon y^e line an at d ; as {illeg} dn to ad so is y^e pressure of g upon { a } to its press{illeg}|u|re upon {illeg}| n |, so is y^e motion in $(a)$ to y^e motion in $(n)$ w^ch is generated in y^m by those pressures of g , y^t is w^ch they received from {illeg}| g |, at y^e moment of reflection, & w^ch they would /might\ continually injoy \as in fig 6^t./^[111] did not their union by y^e line adn hinder

By y^e same r{illeg}|e|as{illeg}|o|n if g reflect {nor} twixt y^e bodys, yⁿ $ad ∶ dn ∷$ pression of $(n)$ tow{illeg}|a|rds $s ∶$ to y^e pression of a towards $r ∷$ motion of n to { $a ∶$ } motion of a to {illeg}| r |.

Note y^t {illeg} & n are taken for y^e centers of motion in a & n . & adn for {illeg}

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^[112] All things put as before \That a & n are loose, then,/ $qd ∶ pd ∷$ motion of $n ∶$ motion of a received from g . (pr: 34) Therefore (by prop 113), $qd \times a ∶ qd \times n ∷$ {illeg} velocity of $a ∶$ velocity of n . $n \times pd ∶ a \times qd ∷$ velocity of $(a) ∶$ velocity of $(n)$ . Or supposeing y^t y^e bodys a {illeg} & n move through y^e spac distances qr & ps in y^e same time yⁿ, $n \times pd ∶ a \times qd ∷ qr ∶ ps$ . & producing hd to (t in) y^e line rs , since $qr ∥ dt ∥ ps$ , $qp = ri ∶ is = ps - qr ∷ qd = rv ∶ \frac{ps \times qd - qr \times dq}{qp}$ /{illeg}|&c|\ $ps$ {illeg} $= \frac{a \times qd \times qr}{n \times pd}$ {illeg}. Therefore & $dt = \frac{ps \times qd - qr \times qd + qp \times qr}{qp}$ . Therefore $dt = \frac{a \times qr \times qd \times qd}{n \times pd \times qp} - \frac{qr \times qd}{qp} + qr$ . Now {i{illeg}|f|}{if} in y^e same time y^t y^e body g moved from $(h)$ to $(d)$ in y^e same time suppose y^t a & n move to r & s , & y^e point d to y^e point, t & y^e body g to y^e point e , then must $et = dh$ (by ax: 9^th) Also Li|e|t {illeg}|y^e| whole motion of g (at h towards d ) be called m . y^e \whole/ motion of a & & {sic} n to {illeg} r & s be called {illeg}| y |: yⁿ y^e motion of g after reflection $= m - y$ . & $m ∶ m - y ∷ hd ∶ de$ . or $de = \frac{hd \times m - hd \times y}{m}$ . & since $et = hd$ , tis $dt = \frac{2 hd \times m - hd \times y}{m} = \frac{a \times qr \times qd \times qd}{n \times pd \times pq} - \frac{qr \times qd}{qp} + qr$ . And since y^e whole motion of a & n to {illeg} r & s is equall to $a \times qr + n \times ps =$ $a \times qr + \frac{a \times qd \times qr}{pd} = y$ {illeg} or, $\frac{pd \times y}{a \times pq} = qr$ . tis $dt = \frac{2 hd \times m - hd \times y}{m} = \frac{qd \times qd \times y}{n \times pq \times pq} - \frac{qd \times pd \times y}{a \times pq \times pq} + \frac{pd \times y}{a \times pq}$ &c Or thus.

^[113] 3. {sic}6 If y^e $\{\begin{matrix} bodys \\ Globes \end{matrix}\}$ a & n doe rest, but soe y^t y^e body $(g)$ moveing \perpendicularly/ \to qp / to them & refleting on y^e line qp (w^ch is supported by \but not fastened to/ y^e bodys a & n \& ought \now/ to be conceived a line onely./) doth move them by communicating its whole motion to y^m w^ch it selfe looseth, Tis required what motion a & n shall have receive from g . Suppose y^t in y^e same \so much/ time \as/ g moveth to $(d)$ before reflection in so much time it moveth from d {illeg} to e after reflection & in so much time y^e bodys a & n move y^e one to r y^e other to s ; & y^t yⁿ y^e point of reflection $(d)$ is moved to $(t)$ . Then (by ax 9^th) $et = gd$ . & since bq re
flection noe motion is lost or gotten $g \times gd = a \times ar = n \times ns = g \times de$ . And (by prop 113) as y^e velocit And (by prop 34) y^e motion of n is to { v} Then na{illeg}|m|e|i|ng y^e given quantitys $gd = b$ . $qd = c$ . $qp = e$ . $dp = e - c = f$ . $qr = z$ . y^e whole motion of $(a)$ & $(n)$ to $(r)$ & $(s)$ call $x x$ , y^t is $a \times qr + n \times ps = x x$ , or if $a ∶ n ∷ pw ∶ wq$ , (yⁿ w is y^e center of y^e bodys motion {illeg}|)| ) & if $wl ∥ dt ∥ qr ∥ ps$ , then $(lw)$ is a line described by their center of motion & consequently \Then is/ {illeg}|(| ) $x x = \overset{—}{a + n} \times \overset{—}{lw}$ .^[114] Now, as y^e motion of a to y^e motion of n (prop 34) so is $(dp)$ to $(dq)$ ; Therefore (ax 113) as y^e velocity of $(a)$ to y^e velocity of $n ∷$ wf $n \times pd ∶ a \times qd ∷ qr ∶ ps$ . y^t is $f n ∶ c a ∷ z ∶ \frac{c a z}{f n} = ps$ . but $\frac{x x - a z}{{illeg}}$ $a z + n \times ps = x x$ ;y^t is $a z + \frac{c a z}{f} = x x$ ; or $\frac{f x x}{a f + a c} = z = \frac{f x x}{a e} = qr$ . & $ps = \frac{c a z}{f n} = \frac{c x x}{e n}$ . $is = \frac{f x x}{a e} - \frac{c x x}{n e} = \frac{f x x x - c a x x}{a n e}$ . $ri ∶ is ∷ ri ∶ tv$ . therefore, $tv = \frac{c f n x x - c c a x x}{a n e e}$ . & $td qr - tv = \frac{e f n x x - c f n x x + c c a x x}{a n e e} =$ $\frac{f f n x x + c c a x x}{a n e e} = td$ . Also (by ax: 9^th) $gd = et = b$ . Therefore $ed = \frac{f f n x x + c c a x x - b a n e e}{a n e e}$ . Now since g hath soe much motion before reflection as all three bodys $(g + a + n)$ have after=ward {sic}, therefore (ax 113) $g b = x x + g \times de$ ; Or, $ed = \frac{g b - x x}{g} = \frac{f f n x x + c c a x x - b a n e e}{a n e e}$ . That is $2 a b e e g n = a e e n x x + g f f n x x + c c g a x x$ . Or, $x x = \frac{2 a b e e g n}{a e e n + g f f n + c c g a} = \overset{—}{a + n} \times \overset{—}{lw}$ . Or, calling $a + n = d$ ; then $\frac{2 a b e e g n}{a d e e n + g d f f n + c c g a d} = wl$ . Soe y^t by this Equation y^e point l twixt the bodys r & s being then their center of motion may bee always found. Note y^{illeg}|t| y^e lines qr & ps must be described by y^e \centers of motion of/ 2 bodys on divers sides of y^e point d y^t is ar by center of motion of {illeg}|y^e| body $(a)$ or $(ad)$ , & ps by y^{illeg}|t| {sic} center of $(n)$ or $(dn)$

37 Also Now when a & n are united or ad & dn are united together (as in y^e 2^d fig) they cannot seperate y^e one from y^e other, & therefore since \(wⁿ th{illeg}|e|y are not equivelox)/ rs is longer yⁿ pq , the one cannot be at s when y^e other is a{illeg}|t| r , but they will check {illeg} y^e one y^e others {illeg} motio soe as their centers of motion shall \not/ describe streight lines (as $(qr)$ or $(ps)$ ) but crooked ones ({illeg}|as| perhaps Trochoides as qmk , pht). yet y^e comon center of their motion w or l shall retaine both y^e same determinacon & velocity that it would did y^e bodys move par{illeg}|a|llell to y^m selves or were the{illeg}|y| n{illeg}|o|t united (by ax {7}{17}). Soe y^t if y^e conjoyned bodys (fig 2^d) move to m & h {illeg}|i|n y^e same time y^t they would have moved to r & s were they th{illeg}|e|ir center of motion $(l)$ when they ar{illeg}|e| at m & h is {y^e} same y^t it would be we{illeg}|r| they at {illeg} & therefore may be found by y^e {illeg} rule, viz; $a e e d n + g d f f n + g c c a d ∶ 2 a e e g n ∷ b ∶ wl ∷$ y^e {veloci}ty of y^e point l ; to y^e velocity of y^e body {illeg} befor{e}{illeg}{r}eflection. Vide

^[1] {Se}{p}t 1664.

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{illeg} $2 x + \frac{1}{3} x x = {gd}^{2}$ . $x x + 4 x + 1 = {ag}^{2}$ .

^[28] 64800 $\begin{array}{l} \underline{36} \\ 2025 \\ \underline{162} \\ 405 \end{array}$

$\begin{array}{l} \underline{29376} \\ \underline{162} \\ \underline{29376} \\ \underline{7776} \\ \underline{729} \\ \underline{486} \end{array}$ $\underset{'}{a}____________\underset{'}{b}_____\underset{'}{c}$ $\begin{array}{l} \underline{32076 \times 25 d d} = 801900 d d \\ 160380 \\ \underline{64152} \end{array}$ $\begin{array}{l} 729 \\ 7290 \end{array}$ $\begin{array}{l} 134 \\ 537 \\ \underline{486} \\ 5184 \\ 486 \\ 324 \end{array}$ $\begin{array}{l} 1296 \\ 648 \\ 962 \end{array}} = \begin{matrix} \underline{104976 \times 9} \\ 944784 \end{matrix}$ $\begin{matrix} 900 \\ 36 \\ 1800 \end{matrix}$ $\begin{array}{l} 405 \\ \underline{162} \\ \underline{2025 \times 25 d d} = 50625 \\ 10125 \\ 4050 \end{array}$ $\begin{array}{l} Q: \underline{324} = 104976 \\ 1296 \\ 648 \\ 972 \end{array}$ $\begin{array}{l} 216 \\ \underline{106} \\ 6 \end{array}$ 1800 $\begin{matrix} 36 \\ \begin{array}{r} 216 \\ \underline{106} \end{array} \\ 1276 \\ 11484 \end{matrix}$ $\begin{array}{r} 9 \times 25 \times & 36 \\ 3 & 24 \\ 16 & 20 \\ \underline{64} & \underline{8} \\ 81 & 00 \\ \underline{18} & \underline{00} \\ 99 \end{array}$

$\begin{array}{r} 36 \times 324 \\ 1944 \\ \underline{972} \\ 11664 \end{array}$ $x + \frac{r}{q} x x = 9$ . $\frac{r}{2} + \frac{r x q}{q} = 2$ {illeg} $x = 2 x - \frac{r x}{2}$ . $\frac{r x}{2} + 2 x = 9$ .
$x = 3$ . $r x = 6$ $r = 2$ $r x + 4 x = 18$ {illeg} $- \frac{q r}{2} + 2 q = r x$

$2 q - \frac{q r}{2} + 4 x = 18$ $\frac{q r}{2} + \frac{72}{r + 4} - 18 = 0$

^[29] $c c = 52 c - 117$ . {illeg} $\begin{array}{r} c = 2
6 - \sqrt{156} \\ \underline{520} \\ 559 (2 \\ 4 \\ 1 \end{array}$ $c c = 80 c - 180$

^[30] $\begin{matrix} \begin{array}{l} 1600 \\ 14 \\ 1420 & (3 \\ 20 \end{array} \\ \begin{matrix} 13 & . \\ 14 & . & 192 \\ 15 & . & 225 \\ 16 & . & 5 \\ 17 & . & 289 \\ 18 & . & 324 \\ 19 & . & 361 \\ 20 & . & 400 \end{matrix} \end{matrix}$

^[31] {illeg}

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$x^{3} + y^{3} - a x y = 0$ . $v = \frac{3 x x y - a y y}{a x - 3 y y}$ $x + \frac{3 x x y^{2} - a y y z}{\sqrt{\begin{matrix} 9 x^{4} y y - 6 a x x y^{3} \\ + a a y^{4} + a a x x y y \\ - 6 a^{3} x y y + 9 a^{4} y^{4} \end{matrix}}}$ for x. {illeg} $x + \frac{3 z x x - a z y}{\sqrt{\begin{matrix} 9 x^{4} - 6 a y x x + 10 a a y y \\ + a a x x - 6 a^{3} x \end{matrix}}}$ |for x |. $y \frac{+ 3 y y z - a z x}{\sqrt{\begin{matrix} 9 x^{4} - 6 a y x^{2} + 10 a^{2} y^{2} \\ + a a x^{2} - 6 a^{3} x \end{matrix}}}$ |for y| {illeg}

$a b = a = 2 a d$ $a d = \frac{\sqrt{3 a a}}{4}$

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{illeg} $= x$ . $ce = y$ . $af = q$ . {illeg} $x - x x = y y$ . $ed = z$ . $ed = s$ . {illeg} $c = q - x$ . ${eb}^{2} = q q$ . $eb = q$ . {illeg} $: 2 q x - x x : z z :$ {illeg} / ${eh}^{2}$ .\ $eh = \sqrt{\frac{2 q x z z - x x z z}{q q}}$ {illeg}= {illeg} $\sqrt{2 q x - x x} - \frac{z}{q} \sqrt{x x}$ {illeg} $= 2 q x - x x + \frac{2 z z q x -}{q q}$ {illeg} $- 4 z x + \frac{2 z x x}{q}$ {illeg} $= \frac{q z - x z}{q}$ {illeg} $- \frac{z x}{q}$

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^[36] $\begin{matrix} d x q y & + & 2 f x y y \\ - & 2 p f x y & - & b y x \\ - & 2 a y y x & - & \frac{d e x y - d d y y y}{f} \end{matrix}$

^[37] {illeg} $\begin{matrix} x y & + & a y & = & 0 \\ a x & + & y y \end{matrix}$

$ed = v$ . $cd = x$ . $ac = y$ . $ab = s$ . {illeg} $= z$ . $eg = ξ$ . $0 = y y - 2 x y + a x - x^{2}$ $y =$ {illeg} $\overset{\cup}{O} \sqrt{2 x x - a x} = ac$ $y y = \begin{matrix} 2 x x \\ - a x \end{matrix} \overset{\cup}{O} 2 \sqrt{2 x^{4} - a x^{3}}$ . ${eg}^{2} = \begin{matrix} 3 x^{2} - 60 x - a x + a o \\ \overset{\cup}{O} \sqrt{2 x^{4} - a x^{3}} \end{matrix}$ . ${eg}^{2} = \begin{matrix} 3 z^{2} \\ - a z \end{matrix} \overset{\cup}{O} \sqrt{2 z^{4} - a z^{3}}$ $y y - 2 x y + a x - x x = 0$ $\frac{- 2 y y + a y - 2 x y}{- 2 y + 2 x} + x = v$ $x + y + \frac{a y}{2 x - 2 y} = v$ $8 x^{4} - 4 a x^{3} + 4 a z^{3} - 8 z^{4}$ $\begin{array}{r} x^{3} + 4 x x z + 4 x z z + 4 z^{3} \\ - 2 x x a - 2 a x z - 2 a z^{2} \end{array}$ $x x -$ {illeg} {illeg} $\begin{matrix} \sqrt{2 x x - a x} & - & 2 z \sqrt{2 z z - a z} \\ + & 2 z \sqrt{2 x x - a x} \end{matrix}$

$ab = a$ . $bd ∶ be ∷ b ∶ c$ . $\frac{c x}{b} = be$ . $\frac{c ϩ}{b} = eh$ . {illeg} ${fe}^{2} = = \frac{c c ϩ ϩ - b b ϩ ϩ}{b b}$ $fe = \frac{c x - b z}{b} = \frac{ϩ \sqrt{c c - b b}}{b}$ $\frac{b z + ϩ \sqrt{c c - b b}}{c} = x$ $ed = \frac{x \sqrt{c c - b b}}{b}$ {illeg} $ed = \frac{ϩ c c - ϩ b b + b z \sqrt{c c - b b}}{b c}$ {illeg} $eh = \frac{- ϩ b + z \sqrt{c c - b b} - c a}{b}$

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^[39] $ϩ \sqrt{e c - d d} = e x$ .

^[40] {illeg} $\begin{matrix} \end{matrix} + 4 d^{3} ϱ^{3} ϩ \sqrt{e e - d d} +$

^[41] {illeg} $\begin{matrix} \end{matrix} ϱ^{5} + 5 d^{4} ϱ^{4} ϩ \sqrt{e e - d d}$

^[42] {illeg} $\begin{matrix} \end{matrix} - d ϩ + ϱ \sqrt{e e - d d} = e y$ .

^[43] $ϱ \begin{array}{l} - 2 d d ϱ ϩ \\ + e e ϱ ϩ \end{array} \begin{array}{l} + c e ϩ \sqrt{e e - d d} \end{array} = e e x y$ .

^[44] $\begin{matrix} d ϩ^{3} \\ ϩ^{3} \end{matrix} + 2 c e ϱ ϩ \sqrt{e e - d d} \begin{array}{l} + & 2 d e e ϱ ϱ ϩ \\ - & 3 d^{3} ϱ ϱ ϩ \end{array}} = e^{3} x x y$

^[45] {illeg} $x + a x + b b = 0$ . {illeg} $+ \frac{a a}{0} + b b$

^[46] $ac = d$ . {illeg} $= a$ . $ad = b$ . $ed = c$ . $\frac{a a}{b} = ab$ . $ae = e$ { $\frac{a e}{b} = eb$ .} {illeg} $bc = \frac{d b - a a}{b}$ {illeg} $c^{2} =$ $\frac{a^{4} + a a}{b b}$ $c c$ . $\frac{d d b b - 2 a a b d + a^{4} + a a c c}{b b} = e e$ .

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{illeg} $d = a$ . $pg = b$ . $pd = g$ $\frac{e a}{d} = dh$ . $\frac{f a}{d} = gh$ . $h = \frac{b d - f a}{d}$ . {illeg} $b d d - 2 b d f a + a a f f + e e a a = b b g g$ {illeg} $\begin{matrix} a e e \\ a a d d \end{matrix} = 2 b d a \sqrt{e e - d d} + b b g g - b b d d$ . {illeg} $a = \frac{b d \sqrt{e e - d d}}{2 e e - d d} \overset{\cup}{O} \sqrt{\frac{- b b d d e e + 2 b b g g e e - b b g g d d}{4 e^{4} - 4 e e d d + d^{4}}}$

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^[55] September 1664

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^[59] $\frac{d ϱ + s ϩ}{e} = x$ . $\frac{t ϱ - v ϩ + c e}{e} =$ {illeg}y

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$d g - e g - e \sqrt{d d - g g} + d \sqrt{d d - g g} = 0$ . $g = \sqrt{d d - g g}$ . $d d = 2 g g$ or $d = e$ $a g - 2 c g - 2 c \sqrt{d d - g g} = 0$ $\frac{a g}{2 g + 2 \sqrt{d d - g g}} = c$ . $g^{3} = - d d + g g \sqrt{d d - g g}$ . $2 g g = d d$ . $- 3 c c d d \sqrt{d d - g g} = 0 = c$ . $3 d d g = 3 e e \sqrt{d d - g g}$ $d = e$ . $- a e g d$ $\begin{array}{l} - 6 c d e \sqrt{d d - g g} = a d e g = 0 \\ + a d d \end{array}$ $- 3 d d c c \sqrt{d d - g g} - a d d c g = 0$ $9 c c d d - 9 c c g g = a a g g$ . $d^{4} g g =$ {illeg}{ $t^{4} -$ }{illeg} $g g e^{4}$ $\begin{array}{l} 36 c c d d e e & - & 36 c c g g e e & = & a a g g e e \\ + & a a d^{4} & - & a a d d g g \\ - & 2 c e a d^{3} \end{array}$ $- r \sqrt{r x} - 6 x \sqrt{r x}$ $\begin{matrix} 3 r x & + & 8 x x & + & \frac{16 x^{3}}{r} & + & \frac{r r}{4} \\ + & 4 x x \end{matrix}$

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Ian 20^th 1664.

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^[67] N{illeg}|o|e motion is lost in reflection. For y^{illeg}|e| circular motion being ma{illeg}|d|e by continuall reflection would not bee decay.

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^[69] q^{illeg} Def 3^d

^[70] {illeg}s Axiome 4^th.

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^[72] p Axiom 4^th.

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Of y^e seperacon of body{s} after reflection

^[74] r axi{illeg}|o|me 3^d

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^[78] The center of motions {illeg}|de|ter & velocity

^[79] x axiom 14

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Of endeavor from y^e center

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^[87] let this follow the 5^t ax

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^[89] What force is required to beget or destroy equall velocit{illeg}|y| in {illeg} unequall bodys

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What resistance in bodys

^[91] What force Indeavor & Pression is

^[92] What force or Motion i{illeg}|s| in equivelox bodys

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|What| velocity acquired or lost in equall bodys by unequall forces

^[94] What motion in bodys

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A generall Theorem of y^e proportion of velocity & motion of given body moving ☞ through given spaces in given times.

^[96] {illeg}|What| force required to beget or destroy unequall s|c|elerity in equall bodys

^[97] Of hindering and helping motion

^[98] What celerity acquired or lost by {illeg}|e|quall forces in unequall bodys

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What \velocity &/ motion gotten or lost by uneq{illeg}|u|all forces in unequall bodys ☞ A Generall Theorem.

^[100] Of y^e {illeg} force in reflected bodys

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Two bodys being uniformely moved in y^e same plaine their centers of motion w^ch describe a str{illeg}|e|ight li{illeg}|n|e

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{illeg} y^e {illeg} {as divers plaines}

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of y^{illeg}|e| velocity of y^e center of motio 14

^[105] The 28^th & 30^th prop done otherwise

^[106] Or thus

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The {illeg} of motion is {illeg} before after {illeg}

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The center of motion in finite bodys {illeg}|h|ath t{illeg}|h|e same velocity before & after reflection

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This ought to be proved by y^e 34^th & 35^t, & y^e 36^t by this concernig y^e imp impresse of $(g)$ on $(qdp)$

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Of y^e Advantage of force in divers positions to some center.

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