# Catalogue Entry: NATP00220

## Newton's Waste Book (Part 1)

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

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

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

[28] $\begin{array}{l}\underset{_}{36}\hfill \\ 2025\\ \underset{_}{162\phantom{0}}\\ \phantom{0}405\end{array}$

$\begin{array}{l}\underset{_}{29376}\\ \underset{_}{162}\\ \underset{_}{29376}\\ \phantom{0}\underset{_}{7776}\\ \phantom{0}\underset{_}{729\phantom{0}}\\ \phantom{00}\underset{_}{486}\end{array}$ $\underset{\text{'}}{a}____________\underset{\text{'}}{b}_____\underset{\text{'}}{c}$ $\begin{array}{l}\underset{_}{32076×25dd}=801900dd\\ 160380\\ \underset{_}{64152}\end{array}$ $\begin{array}{}729\\ \phantom{0}7290\end{array}$ $\begin{array}{}134\\ \phantom{0}537\\ \phantom{0}\underset{_}{486}\\ \phantom{00}5184\\ \phantom{00}486\\ \phantom{000}324\end{array}$ $\begin{array}{l}\phantom{0}1296\\ \phantom{0}648\\ 962\end{array}\right\}=\begin{array}{c}\underset{_}{104976×9}\\ 944784\end{array}$ $\begin{array}{c}\phantom{0}900\\ \phantom{0}36\\ 1800\end{array}$ $\begin{array}{l}\phantom{0}405\\ \underset{_}{162\phantom{0}}\\ \underset{_}{2025×25dd}=50625\\ 10125\\ 4050\end{array}$ $\begin{array}{l}Q:\underset{_}{324}=104976\\ \phantom{0}1296\\ \phantom{0}648\\ 972\end{array}$ $\begin{array}{l}\phantom{0}216\\ \underset{_}{106}\\ \phantom{000}6\end{array}$ $\begin{array}{cc}36& \\ & \begin{array}{r}\phantom{0}216\\ \underset{_}{106}\phantom{0}\end{array}\\ & 1276\\ & 11484\end{array}$ $\begin{array}{rr}9×25×& 36\\ 3& 24\\ 16& 20\\ \underset{_}{64}& \underset{_}{8}\hfill \\ 81& 00\\ \underset{_}{18}& \underset{_}{00}\hfill \\ 99\end{array}$

$\begin{array}{r}36×324\\ \phantom{0}1944\\ \underset{_}{\phantom{0}972\phantom{0}}\\ 11664\end{array}$ $x+\frac{r}{q}xx=9$. $\frac{r}{2}+\frac{rxq}{q}=2$ {illeg}$x=2x-\frac{rx}{2}$ . $\frac{rx}{2}+2x=9$.
$x=3$. $rx=6$ $r=2$ $rx+4x=18$ $-\frac{qr}{2}+2q=rx$

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

[29] $cc=52c-117$. $\begin{array}{r}c=2 6-\sqrt{156}\phantom{\left(0}\\ \underset{_}{520}\phantom{\left(0}\\ 559\left(2\\ 4\\ 1\phantom{00\left(0}\end{array}$ $cc=80c-180$

[30] $\begin{array}{c}& \begin{array}{}1600\\ 14\\ 1420\left(3\\ \phantom{00}20\end{array}\\ \phantom{0}\\ \begin{array}{ccc}13& .& \\ 14& .& 192\\ 15& .& 225\\ 16& .& 5\\ 17& .& 289\\ 18& .& 324\\ 19& .& 361\\ 20& .& 400\end{array}\end{array}$

[31] {illeg}

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${x}^{3}+{y}^{3}-axy=0$. $v=\frac{3xxy-ayy}{ax-3yy}$ $x+\frac{3xx{y}^{2}-ayyz}{\sqrt{\begin{array}{c}9{x}^{4}yy-6axx{y}^{3}\\ +aa{y}^{4}+aaxxyy\\ -6{a}^{3}xyy+9{a}^{4}{y}^{4}\\ \end{array}}}$ for x. $x+\frac{3zxx-azy}{\sqrt{\begin{array}{c}9{x}^{4}-6ayxx+10aayy\\ +aaxx-6{a}^{3}x\end{array}}}$ for x . $y\frac{+3yyz-azx}{\sqrt{\begin{array}{c}9{x}^{4}-6ay{x}^{2}+10{a}^{2}{y}^{2}\\ +aa{x}^{2}-6{a}^{3}x\end{array}}}$ for y

$ab=a=2ad$ $ad=\frac{\sqrt{3aa}}{4}$

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{illeg}$=x$. $ce=y$. $af=q$. {illeg}$x-xx=yy$. $ed=z$. $ed=s$. {illeg}$c=q-x$. ${eb}^{2}=qq$. $eb=q$. {illeg}$:2qx-xx:zz:$ ${eh}^{2}$. $eh=\sqrt{\frac{2qxzz-xxzz}{qq}}$ {illeg}= $\sqrt{2qx-xx}-\frac{z}{q}\sqrt{\phantom{00000}xx}$ {illeg} $=2qx-xx+\frac{2zzqx-\phantom{000}}{qq}$ {illeg}$-4zx+\frac{2zxx}{q}$ {illeg}$=\frac{qz-xz}{q}$ {illeg}$-\frac{zx}{q}$

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[36] $\begin{array}{cccc}& dxqy& +& 2fxyy\\ -& 2pfxy& -& byx\\ -& 2ayyx& -& \frac{dexy-ddyyy}{f}\end{array}$

[37] {illeg} $\begin{array}{ccccc}xy& +& ay& =& 0\\ ax& +& yy\end{array}$

$ed=v$. $cd=x$. $ac=y$. $ab=s$. {illeg}$=z$. $eg=\xi$. $0=yy-2xy+ax-{x}^{2}$ $y=$ $\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2xx-ax}=ac$ $yy=\begin{array}{c}2xx\\ -ax\end{array}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}2\sqrt{2{x}^{4}-a{x}^{3}}$. ${eg}^{2}=\begin{array}{c}3{x}^{2}-60x-ax+ao\\ \phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2{x}^{4}-a{x}^{3}}\end{array}$. ${eg}^{2}=\begin{array}{c}3{z}^{2}\\ -az\end{array}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2{z}^{4}-a{z}^{3}}$ $yy-2xy+ax-xx=0$ $\frac{-2yy+ay-2xy}{-2y+2x}+x=v$ $x+y+\frac{ay}{2x-2y}=v$ $8{x}^{4}-4a{x}^{3}+4a{z}^{3}-8{z}^{4}$ $\begin{array}{r}{x}^{3}+4xxz+4xzz+4{z}^{3}\\ -2xxa-2axz-2a{z}^{2}\end{array}$ $\begin{array}{ccc}\sqrt{2xx-ax}& -& 2z\sqrt{2zz-az}\\ & +& 2z\sqrt{2xx-ax}\end{array}$

$ab=a$. $bd:be\colon\colon b:c$. $\frac{cx}{b}=be$ . $\frac{cϩ}{b}=eh$. ${fe}^{2}==\frac{ccϩϩ-bbϩϩ}{bb}$ $fe=\frac{cx-bz}{b}=\frac{ϩ\sqrt{cc-bb}}{b}$ $\frac{bz+ϩ\sqrt{cc-bb}}{c}=x$ $ed=\frac{x\sqrt{cc-bb}}{b}$ $ed=\frac{ϩcc-ϩbb+bz\sqrt{cc-bb}}{bc}$ $eh=\frac{-ϩb+z\sqrt{cc-bb}-ca}{b}$

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[39] $ϩ\sqrt{ec-dd}=ex$ .

[40] {illeg}$\begin{array}{}\phantom{0}\\ \phantom{0}\end{array}+4{d}^{3}{\varrho }^{3}ϩ\sqrt{ee-dd}+$

[41] {illeg}$\begin{array}{}\phantom{0}\\ \phantom{0}\\ \phantom{0}\end{array}{\varrho }^{5}+5{d}^{4}{\varrho }^{4}ϩ\sqrt{ee-dd}$

[42] {illeg}$\begin{array}{}\phantom{0}\\ \phantom{0}\end{array}-dϩ+\varrho \sqrt{ee-dd}=ey$.

[43] $\varrho \phantom{\rule{1em}{0ex}}\begin{array}{l}-2dd\varrho ϩ\\ +ee\varrho ϩ\end{array}\phantom{\rule{1em}{0ex}}\begin{array}{l}\phantom{0}\\ +ceϩ\sqrt{ee-dd}\\ \end{array}=eexy$.

[44] $\begin{array}{c}d{ϩ}^{3}\\ {ϩ}^{3}\end{array}\phantom{\rule{1em}{0ex}}+2ce\varrho ϩ\sqrt{ee-dd}\phantom{\rule{1em}{0ex}}\begin{array}{ll}+& 2dee\varrho \varrho ϩ\\ -& 3{d}^{3}\varrho \varrho ϩ\end{array}\right\}={e}^{3}xxy$

[45] {illeg} $x+ax+bb=0$ . {illeg} $+\frac{aa}{0}+bb$

[46] $ac=d$. {illeg}$=a$. $ad=b$. $ed=c$. $\frac{aa}{b}=ab$. $ae=e$ {$\frac{ae}{b}=eb$.} $bc=\frac{db-aa}{b}$ {illeg} ${c}^{2}=$ $cc$ . $\frac{ddbb-2aabd+{a}^{4}+aacc}{bb}=ee$.

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{illeg}$d=a$. $pg=b$. $pd=g$ $\frac{ea}{d}=dh$ . $\frac{fa}{d}=gh$ . $h=\frac{bd-fa}{d}$ . {illeg} $bdd-2bdfa+aaff+eeaa=bbgg$ {illeg} $\begin{array}{c}aee\\ aadd\end{array}=2bda\sqrt{ee-dd}+bbgg-bbdd$ . {illeg} $a=\frac{bd\sqrt{ee-dd}}{2ee-dd}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{-bbddee+2bbggee-bbggdd}{4{e}^{4}-4eedd+{d}^{4}}}$

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[55] September 1664

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[59] $\frac{d\varrho +sϩ}{e}=x$. $\frac{t\varrho -vϩ+ce}{e}=$ y

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$dg-eg-e\sqrt{dd-gg}+d\sqrt{dd-gg}=0$. $g=\sqrt{dd-gg}$. $dd=2gg$ or $d=e$ $ag-2cg-2c\sqrt{dd-gg}=0$ $\frac{ag}{2g+2\sqrt{dd-gg}}=c$. ${g}^{3}=-dd+gg\sqrt{dd-gg}$. $2gg=dd$. $-3ccdd\sqrt{dd-gg}=0=c$. $3ddg=3ee\sqrt{dd-gg}$ $d=e$. $-aegd$ $\begin{array}{l}-6cde\sqrt{dd-gg}=adeg=0\\ +add\hfill \end{array}$ $-3ddcc\sqrt{dd-gg}-addcg=0$ $9ccdd-9ccgg=aagg$ . ${d}^{4}gg=$ {${t}^{4}-$}$gg{e}^{4}$ $\begin{array}{llllll}& 36ccddee& -& 36ccggee& =& aaggee\\ +& aa{d}^{4}& -& aaddgg\\ -& 2cea{d}^{3}\end{array}$ $-r\sqrt{rx}-6x\sqrt{rx}$ $\begin{array}{ccccccc}3rx& +& 8xx& +& \frac{16{x}^{3}}{r}& +& \frac{rr}{4}\\ & +& 4xx\end{array}$

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Ian 20th 1664.

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[67] Noe motion is lost in reflection. For the circular motion being made by continuall reflection would decay.

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[69] Def 3d

[70] {illeg}s Axiome 4th.

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[72] p Axiom 4th.

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Of the seperation of body{s} after reflection

[74] r axiome 3d

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[78] The center of motions determinacon & velocity

[79] x axiom 14

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Of endeavor from the center

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[87] let this follow the 5t axiom

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[89] What force is required to beget or destroy equall velocity in unequall bodys

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What resistance in bodys

[91] What force Indeavor & Pression is

[92] What force or Motion is 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 the proportion of velocity & motion of given body moving ☞ through given spaces in given times.

[96] What force required to beget or destroy unequall celerity in equall bodys

[97] Of hindering and helping motion

[98] What celerity acquired or lost by equall forces in unequall bodys

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What velocity & motion gotten or lost by unequall forces in unequall bodys ☞ A Generall Theorem.

[100] Of the {illeg} force in reflected bodys

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Two bodys being uniformely moved in the same plaine their center of motion which describe a streight line

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{illeg} the {illeg} {as divers plaines}

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of the velocity of the center of motion 14

[105] The 28th & 30th proposition 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 hath the same velocity before & after reflection

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This ought to be proved by the 34th & 35t, & the 36t by this concerning the impresse of $\left(g\right)$ on $\left(qdp\right)$

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Of the Advantage of force in divers positions to some center.

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