<ii(r)>

{$+6=xx$}{$+\delta =xx$}

<ii(v)>

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

$\sqrt{aa}×${$\sqrt{3:{a}^{3}}$ } = {illeg} $\sqrt{ab}×\sqrt{3:cde}=\sqrt{6:{a}^{3}{b}^{3}}×\sqrt{6:ccddee}=\sqrt{6:{a}^{3}{b}^{3}ccddee}$. to find ye proportion of two irrationall rootes. to free ye Numerato{r} or denom from {illeg}surde q{illeg} $\frac{a}{b+\sqrt{c}}=x=\frac{a}{b+\sqrt{c}}×\frac{b-\sqrt{c}}{b-\sqrt{c}}=\frac{ab-a\sqrt{c}}{bb-c}$.

<iii(r)>

$\begin{array}{c}\begin{array}{ccc}\frac{2ax}{c}& -& \frac{\sqrt{{a}^{3}}}{c}\\ 3a& +& \frac{\sqrt{abb}}{c}\end{array}\\ \begin{array}{ccc}6aax& -& 3a\sqrt{{a}^{3}}\\ & +& \frac{2ax}{c}\sqrt{\frac{abb}{cc}}& -& \frac{\sqrt{{a}^{4}bb}}{cc}\end{array}\\ \begin{array}{c}6aax-3a\sqrt{{a}^{3}}+2ax\sqrt{abb}-\frac{aab}{c}\\ \text{error tantum 1′ in metallo 2″ in radio}\end{array}\end{array}$ $\begin{array}{l}\begin{array}{cccccc}\frac{1}{10}& \begin{array}{cc}\frac{1}{60}\text{.}& \frac{1}{100}×\frac{1}{12}\end{array}& 30×60×60& \begin{array}{c}1\\ 60,30,30\\ 54000\end{array}& \begin{array}{c}1\\ 60,30,60\\ 108000\end{array}& 1c\\ & \begin{array}{c}1\\ \begin{array}{c}60000\\ 12\end{array}\end{array}& \begin{array}{c}1200\left(40\\ c\end{array}\end{array}\\ \phantom{0}\\ \phantom{0}\\ \begin{array}{l}300,12,10\\ 3000,12\hfill \\ 36000\hfill \end{array}\\ \phantom{0}\\ \phantom{0}\\ \begin{array}{lll}\frac{1}{60,60,60.}& =& \frac{1}{300,720}\end{array}\end{array}$ $\sqrt{9{a}^{5}}$$a$ $3a\sqrt{a}$ $\begin{array}{c}0,051513\\ 0,0132\hfill \end{array}$ $\left(5$ {illeg} $baax-\sqrt{\left(9\right){a}^{5}}$ $\frac{c}{d}\right)\frac{a}{b}\left(\frac{da}{cb}$ $\begin{array}{ccccc}\begin{array}{c}-2\\ -2\end{array}& \phantom{\rule{0.5em}{0ex}}& \begin{array}{c}\frac{3}{8}×9\\ \end{array}& \phantom{\rule{0.5em}{0ex}}& \begin{array}{l}\phantom{0}14\\ \phantom{0}56\\ 14\end{array}\end{array}$ $\begin{array}{lll}8.\frac{1}{2}\colon\colon \frac{1}{2}.\frac{1}{32}=\frac{6}{16}=\frac{3}{8}.& \phantom{\rule{0.5em}{0ex}}& \begin{array}{lll}9& .& 196\\ 9& \right)& 392& \left(& 43⁤\frac{5}{9}\end{array}\\ 2.\frac{1}{4}\colon\colon \frac{1}{4}.\frac{1}{32}.\\ 96.7\colon\colon 7.\frac{49}{96}\end{array}$ $\begin{array}{l}1\phantom{.}\frac{c}{d}\colon\colon \frac{a}{b}.\frac{da}{cb}\\ \frac{c}{d}.1\colon\colon \frac{a}{b}.\frac{da}{cb}\\ c.d\colon\colon \end{array}$ $1.$
$\frac{c}{d}.1\colon\colon \frac{c}{d}ind.1×d\colon\colon c.d\colon\colon \frac{c}{c}.\frac{d}{c}\colon\colon 1.\frac{d}{c}$ $\frac{c}{d}\phantom{.}1\colon\colon c.d\colon\colon 1.\frac{d}{c}\colon\colon \frac{a}{b}.\frac{da}{cb}$. $\frac{c}{d}.1\colon\colon \left(\frac{c}{d}ind\right)c.\left(1×d\right)d\colon\colon \left(\frac{c}{c}\right)1.\frac{d}{c}\colon\colon \frac{a}{b}:\left(\frac{d}{c}×\frac{a}{b}\right)\frac{da}{cb}$

$\frac{a+3x}{c}\sqrt{\frac{4bb{x}^{4}}{81aa}}$ $\frac{2bxx}{9a}$ $\frac{2abxx+6b{x}^{3}}{9ca}$ $aa-2ay=ab-bb+by$ $aa=ab-bb+by+2ay$ $aa-ab+bb=by-2ay$ $\frac{aa-ab+bb}{b-2a}=y$

$\begin{array}{c}z=\frac{yy}{x}\\ 20=a\\ 140=b\end{array}$ $zz$ $x.y.\frac{yy}{x}$. $yy=\frac{yyx}{x}$ $x+y+\frac{yy}{x}-a=0$ $xx+yy+\frac{{y}^{4}}{xx}-b=0$ $x+y+\frac{yy}{x}-a+xx+yy+\frac{{y}^{4}}{xx}-b=0$

<1r>

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

If a Staffe bee bended to find ye crooked line wch it resembles.

If the motion of a line is knowne to find ye crooked line wch yt line toucheth continually.

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

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

If ye ball b revolves about ye center n ye force by wch it endeavours from ye center n would {beget} soe much { b } motion in a body \as ther {sic} is/ in ye time yt ye body b moves ye length {of ye} {semidiamiter {sic}} bn . [as if b is moved \wth 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 ye center {illeg} {illeg} being continually like ye force of gravity impressed upon {illeg}|ye| body during one second it will generate one degree of motion in {illeg}|yt| body.] Or ye force from {n} in one revolution is to ye force of ye body motion as rad$periph:rad$ . Demonstracon. If {$ab=bc=dc=$ } $ef=fg=gh$ \ $ad\text{.}$ / $=he=2fa=2fb=2fb=2gc=2ed$. & ye globe {b} from a to b {yn} $2fa:${ak}$\colon\colon ab:fa\colon\colon$ force {or} pression of b {illeg}|u|pon fg {at} its reflecting ∶ for{illeg}|e| {sic} of b{'}s motion. therefore $4ab=ab+bc+cd+da:$ {illeg}fa ∷ force of ye reflection in one round (viz: in b , c , d , & a) ∶ force of b 's motion. by ye sa{me} pro{illeg} ye Globe b were reflected by each side of a circumscribed polygon of 6, 8, 12, 100, 1000 sides {illeg} ye force of all ye reflections is to ye force of ye bodys as ye sum of those sides {illeg} r{adius} of ye circle about wch 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) ye force of all ye reflections are to ye force of ye bodys motion {as each} those sides ({illeg} ye perimiter) to ye radius.

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

If a body undulate in ye circle bd all its undulations of any altitude are performed in ye {same} time wth ye \same/ radius. Galileus.

As radi{illeg}|u|s ab to radius ac ∷ {illeg} so are ye Squares of theire times in wch they undulate.

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

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

And $ad:dc\colon\colon$ force of gravity to ye force of { c } to its center d . {illeg} {illeg} {illeg} {illeg} ye motion of things falling were they not hindered by ye {illeg} may very {illeg} {illeg} $cd:ad\colon\colon$ force form d ∶ force from a.

<1v>

 $ag=x$. $gh=y$. $ah=c$. $\begin{array}{cccccccc}bxx& +& ex& +& cc& =& 0& \text{.}\\ & +& dxy& +& fy\\ & & & +& gyy\end{array}$. $dp=v${illeg}. $ad=\frac{xx-yy+cc}{2c}$  ${dg}^{2}=${illeg}$\frac{2cc{x}^{2}+2xxyy+2ccyy-c-{x}^{4}-{y}^{4}}{4aa}+vv=ss$ $v:w\colon\colon \frac{vy}{x}:\frac{wy}{x}$. $\frac{vy}{x}=df$. /fig 2d. $\sqrt{xx-yy}-v:y\colon\colon y$. $ds=\frac{yy}{\sqrt{xx-yy}-v}$\ $\begin{array}{llllllllllllll}& & -& 4{c}^{4}& -& {x}^{4}& -& 2b{x}^{4}& -& 2e{x}^{3}& -& 2d{x}^{3}& in& y\\ & & & & & & & & & & -& 2fxx\\ & & & & -& 2eccx& -& bb{x}^{4}& & & -& 2ccdx\\ -& 2bccxx& & & -& 2eb{x}^{3}& -& 2bcc{x}^{2}& & & -& 2ccf\\ -& 2cceex& & & & & -& eexx\end{array}$ $yy:+v\sqrt{xx-yy}$ $yy-x:y$ $\frac{{y}^{3}}{x\sqrt{xx-yy}-vx}$ + {illeg}$\frac{+y\sqrt{xx-yy}}{x}=df$. $\frac{{y}^{3}-{y}^{3}+yxx-vy\sqrt{xx-yy}}{-vx+x\sqrt{xx-yy}}$.  {illeg} $\frac{xx-v\sqrt{xx-yy}}{\sqrt{xx-yy}-v}=bd$. $xx-v\sqrt{xx-yy}:v\sqrt{xx-yy}\colon\colon$a {illeg}{illeg}f $ab=x$. Parab $\angle mak=\angle bad$ , ergo: ad $\angle mak=\angle ade=\angle bad$. ergo, $ab=bd$.

 $\angle a=\angle bad=\angle adb-\angle asb=\angle sad$ ergo tri: anb, anm, sim: $ab=x$. $bs=a$. $eb=y$. $ae=\sqrt{xx-yy}$. $bo=v$ $ao=\sqrt{xx-2vy+vv}$. $ab:as\colon\colon bo:os$. $x:\sqrt{xx-2ay+aa}\colon\colon v:a-v$. $\angle vat=${illeg}||$bao=\angle oas$. $ab=x$. $as=y$. $bo=v$. $bs=a$. $x:y\colon\colon v:a-v$. $vy=ax-vx$ Hyperb.

 $os=v$. $sa=x$. $oi=ae=y$. $sa:ae\colon\colon so:oh=\frac{vy}{x}$. $oi:oh\colon\colon d:e$: ergo $ey=\frac{dvy}{x}$. & $ex=dv$. Hyp: $ab:ae\colon\colon bc:cn$. $\frac{ae×v}{x}=cn$. $d:e:\frac{ae×v}{x}.ae$. $dx=ev$.

 $ab=y$. $as=x$. $bc=v$. $\angle bac=\angle cas$, Ergo $y:x\colon\colon v:a-v$. $bs=a$. & $xv=ay-vy$. Ellipsis.

 $ab=x$. $as=y$. $\angle bac=tar=cas$. Ergo, $ba:as\colon\colon bc=v:cs=a-v$ $ay-yv=$ $ax-vx=vy$. Hyperb.



The invention of Figures for refra|le|ctions. \at right angles/ at ye point refra|le|cting ac ye rad: reflected to ye focus b. ag ye radius reflected {illeg}|fr|o ye focus b. aq a perpendic: to ye ed ye tangent of ye 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: 1st. $\angle eac=\angle bad=\angle adb$. Ergo, $ab=bd$, or $x=v$. & $\angle caq=qab=aqb$. e{illeg}|rg|o $ab=${illeg}qg. $x=v$. fig: 2d. $\angle eac=\angle bad=\angle adg$. Ergo, $ab:ag\colon\colon bd:dg$ $ax-vx=vy$. &, $ab:$ {illeg}$:ag\colon\colon qb:bg$. {illeg} Ergo $ax+vx=y$y|v|. $v=bq$. fig 3d. $\angle eac=\angle bad$ {illeg} Ergo $\angle caq=\angle qab$ . Ergo, $ca:ab\colon\colon cq:qb$. & $ax-vx=vy$.

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

$bg=a$. $bq=v$. $ba=x$. $ag=y$. fig: 1st. $ab:as\colon\colon bq:qr=\frac{vy}{x}$. $d:e\colon\colon qr:qh$, Ergo. $dx=ev$. fig: 2d. $ab:as\colon\colon bq:qr=\frac{vy}{x}$. $d:e$∶  {sic}$qh:qr$, Ergo. {illeg}|e|$x=dv$. fig 3d. $ab:as\colon\colon bq:qr=\frac{v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{x}$. $d:e$∶  {sic}$qr:$ $ag:as\colon\colon gq:qh=\frac{a+v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{y}$. $d:e\colon\colon$$qh:qr$ /$qr:qh$\: Ergo $eax+evx=dvy$ $dax+dvx=evy$. fig {illeg} 4th. $ab:as\colon\colon bq:qr=\frac{v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{x}$. $ag:as\colon\colon gq:qh=\frac{v-a\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{y}$. $d:e\colon\colon qr:qh$: $dvx-dax=evy$. fig 5t. $ab:as\colon\colon bq:qr=\frac{v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{x}$. $ag:as\colon\colon gq:qh=\frac{a-v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{y}$. $d:e\colon\colon qh:qr$. $eax-evx=dvy$. fig 6t. $ab:as\colon\colon bq:qr=\frac{v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{x}$. $ag:as\colon\colon gq:qh=\frac{a-v\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}as}{y}$. $d:e\colon\colon qr:hq$. e | d |$ax-d${illeg}| v |$x=evy$.

<2r>

$ad=a$. $ae=x$. $ed=y$. $af=z$. $fd=a-z$.
$xx-zz=yy-zz+2az-aa$. $fg=v$ \2/ $ae:ef\colon\colon ag:gi$. & $de:ef\colon\colon dg:gh$.  $\frac{xx-yy+aa}{2a}=af$. $\frac{yy-xx+aa}{2a}=fd$. x {illeg} $:\frac{\sqrt{2aaxx-{x}^{4}+2xxyy-{y}^{4}+2aayy-{a}^{4}}}{4aa}\colon\colon \frac{xx-yy+aa+2av}{2a}:gi$. gi \{illeg}/ $=\frac{\sqrt{2aaxx+2xxyy+2aayy-{x}^{4}-{y}^{4}-{a}^{4}}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}\sqrt{+2aaxx-2aayy-2xxyy+{x}^{4}+{y}^{4}+{a}^{4}+2xxav-2yyav+2{a}^{3}v+}}{4aax}$ $gh=\sqrt{2aaxx+2xxyy+2aayy-{x}^{4}-{y}^{4}-{a}^{4}}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}\sqrt{\phantom{0000000000000000000}}$ $ef=${illeg}|m|. $ae=x:m\colon\colon \frac{xx-yy+aa+2av}{2a}$|=|$ag:gi=\frac{mxx-myy+maa+2mav}{2ax}$. $ed:m\colon\colon gd:\frac{myy-mxx+maa-2mav}{2ay}=gh$. $gi:gh\colon\colon d:e$: therefore $\frac{dyy-dxx+daa-2dav}{y}=\frac{exx-eyy+eaa-2eav}{x}$. {illeg} $d=2$. e {illeg}$=1$ . $\begin{array}{cccccccccccc}{y}^{3}& +& 2xyy& -& xxy& +& 2aax& \phantom{\rule{1em}{0ex}}& =& \phantom{\rule{1em}{0ex}}& 0& \\ & & & -& aay& -& 2{x}^{3}\hfill \\ & & & -& 2avy& -& 4axv\hfill \end{array}$ $\frac{{y}^{3}+2xyy-xxy-aay+2aax-2{x}^{3}}{2ay+4ax}=v=\frac{xx-yy}{2a}+\frac{2ax-ay}{2y+4x}$ $bf=\frac{2aax+2aay+2xxyy-{x}^{4}-{y}^{4}-{a}^{4}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}y+2x}{{y}^{3}+2xyy-xyy-aay+2aax-2{x}^{3}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}2a}$. $af=v$ $ae:$ {ef}$\colon\colon ag:gi$. $de:ef\colon\colon dg:gh$. $d:e\colon\colon gi:gh$. $x:m\colon\colon v:\frac{mv}{x}$. $y:m\colon\colon a-v:\frac{ma-mv}{y}$. $d:e\colon\colon \frac{mv}{x}:\frac{ma-mv}{y}$: & $adx-dvx=evy$. $af=x$. $fe=y$. $ae=\sqrt{xx+yy}$. $ed=\sqrt{xx-2ax+aa+yy}$. $ad\sqrt{xx+yy}-dv\sqrt{xx+yy}=ev\sqrt{xx-2ax+aa+yy}$ $aaddxx+aaddyy+2addvxx+2addvyy+ddvvxx+ddvvyy=eevvxx-2eevvax+eevvaa+eevvyy$. $ef=o$. $\begin{array}{cc}\begin{array}{cc}bc& =\end{array}& \begin{array}{c}\begin{array}{c}2aaxxy-{x}^{4}y-{a}^{4}y+2aa{y}^{3}+2xx{y}^{3}-{y}^{5}+4{a}^{2}{x}^{3}+4aayyx+4yy{x}^{3}-2{y}^{4}x-2{a}^{4}x-2{y}^{4}x-{x}^{5}\\ -2ao{y}^{3}-4aoxyy+2aoxxy+2{a}^{3}oy-4{a}^{3}ox+4ao{x}^{3}\hfill \end{array}\\ \phantom{+}2a{y}^{3}+4axyy-2axxy-2{a}^{3}y+4{a}^{3}x-4a{x}^{3}\end{array}\end{array}$  $an=on$ $pg=a$. $ab=${illeg}|y.| $ag=${illeg}|x.| $pg=b$. $pm=c$. $gb=z$. $gq=v$. $ezx-evx=dvy$. $gm=p$. $gs=$  $ab=x$. $qb=v$ {illeg}. $v=x$. $mp=a$. {illeg} $aq=\sqrt{vv-xx}$. $\frac{xx}{v}=bs$. $sa=\sqrt{\frac{vvxx-{x}^{4}}{vv}}$. $nb=y$. $mn=\frac{y}{v}\sqrt{vv-xx}$ $nr=nb$. $e:d\colon\colon tb=nm:rs=\frac{dy\sqrt{vv-xx}}{ev}$. $rs:nr\colon\colon mn:np=\frac{\frac{yy}{v}\sqrt{vv-xx}}{\frac{dy}{ev}\sqrt{vv-xx}}=\frac{ey}{d}\sqrt{yyvv-yyxx+aavv}$ $eeyyvv-ddyyvv+ddyyxx=ddaavv$. ${mn}^{2}=\frac{ddaavv-ddaaxx}{eezz-ddzz+ddxx}$. ${np}^{2}=\frac{eeaavv}{eezz-ddzz+ddxx}$. ${mb}^{2}=\frac{ddaavv}{eezz-ddzz+ddxx}$ {illeg} $=zz$. $\frac{ddaaxx-ddxxzz}{eezz-ddzz}=vv$. $v=\frac{dx\sqrt{aa-zz}}{z\sqrt{ee-dd}}$. ${sa}^{2}$ ${bs}^{2}=\frac{eezzxx-ddzzxx}{ddaa-ddzz}$. ${as}^{2}=\frac{ddaaxx-eezzxx}{ddaa-ddzz}$. $sb:ab\colon\colon ab:$ \$2$/ {bq}{bp}{illeg}sb 2 $xx=$ \$2$/ {illeg}sb{illeg}b{illeg}sb. $sb=\sqrt{}$. $sb=n$: $xx-nn={sa}^{2}=x$/+\$n×x-n$. $xx${illeg}$x+3n${illeg} $xx-nn=nn${illeg}x $ab=x=bq=v$ . $as=y$. $nb=z$ $nm=\frac{yz}{x}$. $e:d\colon\colon${illeg}$z:\frac{dz}{e}=np$. {illeg} ${e}^{2}yyzz+aaxx{e}^{2}=ddzzxx$ ${nb}^{2}=\frac{eeaaxx}{ddxx-eeyy}$. ${nm}^{2}=\frac{eeaayy}{ddxx-eeyy}$. ${mb}^{2}=\frac{eeaaxx-eeaayy}{ddxx-eeyy}$. {illeg} ${np}^{2}=\frac{aaddxx}{ddxx-eeyy}=${illeg}${\xi }^{2}$. $xx${illeg}$yy${illeg}$xx${illeg}$yy$ ${as}^{2}=\frac{ddzzxx-eeaaxx}{eezz}=\frac{-aaddxx+ddxx\xi \xi }{ee\xi \xi }$. $dd${illeg}$\xi \xi$$-eeaa\xi \xi +aaddzz=0$.  $ab=$  $ab=bq=x$. $pm=a$. $as=y$. $no=z$: $\frac{zx}{y}=nb$ . $om=\frac{zz}{a}=mt$. $po=\frac{aa-zz}{a}$. ${pn}^{2}=\frac{{a}^{4}-aazz+{z}^{4}}{aa}:\frac{zzxx}{yy}\colon\colon${pt{illeg}} ${pt}^{2}=\frac{{a}^{4}+2aazz+{z}^{4}}{aa}$ ${tb}^{2}${}${z}^{6}xx$ $b$$xx=cyy$. $xx-\frac{bxx}{c}={sb}^{2}$. $x\sqrt{\frac{c-b}{c}}-\sqrt{\frac{eeaaxx\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\frac{eeaabxx}{c}}{ddxx\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\frac{eebxx}{c}}}=\sqrt{}$ $xx$|$yy$|$=bxx+c${illeg}$x+$ {illeg}g. ${sb}^{2}=xx-bxx${illeg}$-cx-g$. ${mb}^{2}=\frac{eeaaxx-eeaabxx-eeaacx-eeaag}{ddxx-eebxx-eecx-eeg}$ . $qa=s$. $qm=v$. $vv+xx-bxx-cx-g$. $+\frac{eeaaxx-eeaab{x}^{2}-eeaacx-eeag}{ddxx-eebxx-eecx-eeg}$: $-2\sqrt{xx-bxx-cx-g}in\sqrt{\frac{c\phantom{?}aaxx-{c}^{2}{a}^{2}b{x}^{2}-{e}^{2}{a}^{2}\left\{illeg\right\}-\left\{illeg\right\}}{{d}^{2}{x}^{2}-{e}^{2}b{x}^{2}-{e}^{2}cx-e\left\{illeg\right\}}}${illeg} $-2v\sqrt{xx-bxx-cx-g}+2v\sqrt{\frac{{e}^{2}{a}^{2}{x}^{2}-{e}^{2}{a}^{2}b{x}^{2}-{e}^{2}{a}^{2}cx-{e}^{2}{a}^{2}g}{ddxx-eeb{x}^{2}-eecx-eeg}}=ss=vv+xx+$
$=2ox+2ee$ $mb=z$. $sb=x$. $sa=y$. ${ab}^{2}=xx+yy$. $zz=\frac{eeaaxx}{ddxx+ddyy-ee}$ $sm=${illeg}|x| {illeg} $sa=y$. $mb=z$. ${sb}^{2}=xx+2zx+zz$. ${x}^{2}=byy+cy+d$. $yy${illeg}$=bxx+cx+g$. $sm=x-\sqrt{\frac{eeaaxx}{ddxx+dd-ee\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bxx+cx+y}}$ $vv-2vx+2v\sqrt{\frac{eeaaxx}{ddxx+dd-ee\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bxx+cx+g}}${+}$xx\frac{+eeaaxx}{ddxx+dd-ee\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bx+cx+g}-2x\sqrt{\frac{eeaaxx}{\phantom{????}+dd\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bbxx+}}$ $=ss=-2vo\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill +2v\\ -2x\end{array}\sqrt{\frac{eeaaxx-2eeaaox}{ddxx+2ddox+dd-ee\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bxx+2box+cx+co+g}}+${illeg} $vv$. $mb=z$. $sb=x$: $sa=y$. {$mn=$}$\frac{yz}{x}$. ${ab}^{2}=xx+yy$. $\frac{zzxx+zzyy}{xx}=${${sb}^{2}$ }{illeg} $eezzxx+eezzyy-ddaaxx-ddyyzz=0$. ${x}^{2}=bzz+${illeg}. $yy=\frac{e\phantom{\text{?????}}}{dd}${illeg} $vv$$-2v\sqrt{bzz+cz+g}$$+bzz+cz+g$$+\frac{eezz-ddaa\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bzx}{ddzz-eezz}${illeg} {$4vvbin$}$b{z}^{2}+cz+g+\frac{{z}^{4}-4aa{z}^{2}+4{a}^{4}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}Q:\phantom{\rule{0.5em}{0ex}}bzz+cz+g}{{z}^{4}}${illeg} $\frac{-4{z}^{4}v\sqrt{bzz+cz+g}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}zz-2aa\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}bzz+cz+g}{{z}^{4}}${illeg} 4{illeg} $+16o{a}^{4}in${illeg}$+gg${illeg} $-16ov{z}^{2}\sqrt{bzz+cz+g}inzz-2${illeg}$inbz$

<2v> Make ye line ac to revolve about ye point a : on ye end c let ye nut c bee fastened so {as} to t{illeg}|u|rne about its center. make $ab=ac$ & fastend another nut at ye point b in ye same manner. {illeg} make ye line bc to slide through those two nuts soe yt ye △ abc will always be an isosceles. To ye line cb fasten ye line rstv at {illeg} right angles. {illeg} make ye line kg wth 2 nuts e & d at each end through wch ye lines rs & tu must slide to keepe ye line kg perpendicular to bc , in {illeg}|ye| midst of kg fasten ye nutt m so as it {illeg} may turne about its center & yt ye line ac may slit|d|e thro{illeg}|ugh| it then make yt side of ye line kg wch is next ab to be a file wch must be very smooth at the point m but must grow rougher towards ye ends d & e . Then by turneing ye line ac \to & from l & h / about its center & holding ye file kg close to ye plate hmflab , it shall fil{illeg}|e| it into ye shape of a Parabola. To describe ye Parabola by points. {illeg}|Ma|ke $ca=\frac{r}{4}$; c ye {illeg}|v|ertex; {illeg}|a | ye focus; $ab=r$. yn wth some radius as $ag=ae$, describe ye circle ge : & take $bd=2ga=2${illeg}$ae=$\$2$/de & ye point e shall bee in ye parabola, also if from e to g , a \streight/ line be drawne it shall touch ye Parab: in e.

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

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

Or thus take $cm=gc$ & raise me a perpendicular to ca, wch shall intersect ye parab, & circle ge in ye same point. Or thus. {illeg}|U|pon ye focus or center a describe ye center {ef} make $ab=\frac{r}{4}$. $bd=2bc=$ {illeg}|r|. $kb=bg$. wth ye Rad bc describe bed . ye circle.

 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 ye line an , & draw dg perpendic: to an yt is soe yt $dm=dg$, yn take $df=ae$, & f shall be a point in ye parabola afr.

<3r>

Banderon's addition to Ferrarius's Lexion \Geographicu/, ye best for Geog. {T}{F}{illeg} \Ortelius/ Geogr. Lexicon. $\begin{array}{c}{\text{M}}^{\text{r}}\phantom{\rule{1em}{0ex}}\text{John Craige}\\ {\text{D}}^{\text{r}}\phantom{\rule{1em}{0ex}}\text{Archibald Pitcarne}\end{array}\phantom{\rule{1em}{0ex}}\right\}\phantom{\rule{1em}{0ex}}\text{Scotch Mathematicians}$

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

1. I filled to ye 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 & wth pieces of Lead stuck in them to give them weight: three balls each of wch weighed in the air $76⁤\frac{1}{2}$ grains & in ye water $5⁤\frac{1}{16}$ grains, fell each of them in ye water from ye top to ye bottom of ye vessel in 15″ of time the motion of descent being (to {sence}) almost uniform almost from ye top to ye 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 expts seemed sufficiently accurate.

Corol. Ergo ye resistance is as ye 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/ ye same height in $44⁤\frac{1}{2}$″. But thi|e|se expts w{illeg}|er|e not so accurate as {sic} for{mers.}

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

Three balls

<4r>

 $as=y$. $sm=x$. $mb=z$. $mp=a$. \$qs=v$/ $mn=\frac{yz}{x+z}$. ${ab}^{2}=xx+yy+2zx+zz$. $\frac{zzyy+zzxx+2{z}^{3}x+{z}^{4}}{xx+2zx+zz}={nb}^{2}$. $\frac{eezzyy+eezzxx+2ee{z}^{3}x+ee{z}^{4}}{ddxx+2ddzx+ddzz}={np}^{2}=aa+\frac{yyzz}{xx+2zx+zz}$. $sb=z$. $mn=\frac{zy-xy}{z}$. ${ab}^{2}={qb}^{2}=zz+y$ ${nb}^{2}=\frac{{\stackrel{\text{.}}{z}}^{4}+yyz\stackrel{\text{.}}{z}-2x{\stackrel{\text{.}}{z}}^{3}-2x\stackrel{\text{.}}{z}yy+\stackrel{\text{.}}{xxzz}+\stackrel{\text{.}}{xxyy}}{zz}=\frac{ddzzy\stackrel{\text{.}}{y}-2{d}^{2}zx\stackrel{\text{.}}{y}y+ddx\stackrel{\text{.}}{x}{y}^{2}+aaddzz}{eezz}$. ${e}^{2}=1$. ${d}^{2}=$ $\begin{array}{ccccccccccc}{z}^{4}& -& 2x{z}^{3}& +& xxzz& +& 2xyyz& -& xxyy& =& 0\\ & & & -& yy\hfill \\ & & & -& 2aa\hfill \end{array}$. Suppose. $yy=f{x}^{2}+2gx+h$. |yn,| $v=fx+g$. $z+fx+g=\omega$. $\begin{array}{lllllllllll}{\omega }^{4}& -& 4fx\phantom{\rule{0.5em}{0ex}}{\omega }^{3}& +& 6ffxx\phantom{\rule{0.5em}{0ex}}{\omega }^{2}& -& 4{f}^{3}{x}^{3}\phantom{0000}z& +& {f}^{4}{x}^{4}& =& 0\\ & -& 4g& +& 12fgx& -& 12ffxxg& +& 4{f}^{3}{x}^{3}g\\ & -& 2x& +& 6gg& -& 12fxgg& +& 6ffxxgg\\ & & & +& 5fxx& -& 4{g}^{3}& +& 4fx{g}^{3}\\ & & & +& 4gx& -& 4ff{x}^{3}& +& {g}^{4}\\ & & & +& xx& +& 6fg{x}^{2}& +& {f}^{3}{x}^{4}\\ & & & & & -& 2ggx& +& 2ff{x}^{3}g\\ & & & -& 2aa& & & +& fxgg\\ & & & -& h& +& 2gxx& & \\ & & & & & +& 2fhx& -& ff{x}^{4}\\ & & & & & +& 2gh& -& 4fg{x}^{3}\\ & & & & & +& 4faax& -& 3gg{x}^{2}\\ & & & & & +& 4gaa& -& ffh{x}^{2}\\ & & & & & +& 2hx& -& 2fghx\\ & & & & & & & -& ggh\\ & & & & & & & -& 2aaffxx\\ & & & & & & & -& 4aafgx\\ & & & & & & & -& 2aagg\\ & & & & & & & -& 2fhxx\\ & & & & & & & -& 2ghx\\ & & & & & & & -& f{x}^{4}\\ & & & & & & & -& 2g{x}^{3}\\ & & & & & & & -& hxx\end{array}$. & $\omega =bq$. $ab=\xi$. ${sb}^{2}={\xi }^{2}-{y}^{2}$. $mb=\sqrt{\xi \xi -yy}-x$. $\frac{\sqrt{\xi \xi -yy}-yx}{\sqrt{\xi \xi -yy}}=nm$. $\frac{\xi \sqrt{\xi \xi -yy}-\xi x}{\sqrt{\xi \xi -yy}}=nb$. |${nb}^{2}=$ |$a\stackrel{\text{.}}{a}dd+\frac{yy\xi \xi d\stackrel{\text{.}}{d}-{y}^{4}d\stackrel{\text{.}}{d}+yyxxd\stackrel{\text{.}}{d}-2{d}^{2}yyx\sqrt{{\xi }^{2}-{y}^{2}}}{ee\xi \xi -yyee}$ $=\frac{ee\stackrel{\text{.}}{{\xi }^{4}}-ee\xi \xi y\stackrel{\text{.}}{y}+ee\xi \xi \stackrel{\text{.}}{xx}-2ee\xi \xi x\sqrt{\xi \xi }-yy}{ee\xi \xi -eeyy}$. $dd=2$. $ee=1$. $\begin{array}{l}2aa{\xi }^{2}-2aayy+3yy{\xi }^{2}-2{y}^{4}+2yyxx-{\xi }^{4}-\xi \xi xx\\ -4yyx\sqrt{{\xi }^{2}-{y}^{2}}+2\xi \xi x\sqrt{xx-yy}\end{array}\phantom{\rule{0.5em}{0ex}}\right\}\phantom{\rule{0.5em}{0ex}}=0$.

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

2 To compare their crookednesse wth ye crookednes of a gi{illeg}|v|en circle

3 To find ye longest & shortest lines wch can {illeg}|b|e drawn wth in & perpendicular to the line & to find a{illeg}|ll| such lines are {illeg}|per|pendicular at both ends to ye given crooked line

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

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

 6 If y (one {illeg}|u|ndetermined quantity) moves perpendic{illeg}|u|larly to x (ye other undetermin{ed} quantity. if $s=\text{a secant}=db$. $v=dc$. $y=bc$. $x=ca$. Then having ye proportion of {illeg}|v| to {x}{s} to find y, or having ye 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 ye superficies of one line wth ye area of another & to find ye centers of gravity twixt two lines or sollids. 15

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

11 Of ye description of lines.

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

13 Of ye 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 ye 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 ye same to sollids in respect of theire areas, content, gravity &c wch was done to lines in respect of their are{illeg} lengths, areas, & gravity.

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

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

19. Of ye proportion wch ye 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 ye motion of any given body.

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

To Determin max & min by numbers.

< text from higher up f 4r resumes >
<4v>

 $cd=x$. $gd=y$. $rx+\frac{r}{q}xx=rx$. $ac=a$ $de=\frac{1}{2}r+\frac{r}{q}x$. $ag=\sqrt{aa+2ax+xx+rx+\frac{rxx}{q}}$. $af=a+x+\frac{1}{2}x+\frac{r}{q}$ $\frac{aa+2ax+xx+ar+2\frac{a}{q}rx+\frac{rxx}{q}+\frac{rrxx}{qq}+\frac{1}{4}rr={fa}^{2}\text{,}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}rx+\frac{r}{q}xx}{aa+2ax+xx+rx+\frac{rxx}{q}}={fo}^{2}$. $fo:fl\colon\colon 2.1$. |$aa+ar+\frac{1}{4}rr=bb$| |$xc=2ax+\frac{2arx}{q}+\frac{rrx}{q}$| |${fl}^{2}=$| $\frac{aarx+\frac{aar}{q}xx+2arxx+2arxxx+r{x}^{3}+\frac{r}{q}{x}^{4}+arrx+3\frac{arr}{q}xx+\frac{1}{4}{r}^{3}x+\frac{{r}^{3}xx}{2q}+\frac{2arr{x}^{3}}{qq}+\frac{2{r}^{3}{x}^{3}}{qq}+\frac{{r}^{3}{x}^{4}}{{q}^{3}}}{2aa+4ax+2xx+2rx+\frac{2rxx}{q}}$ $aar+arr+\frac{1}{4}{r}^{3}=bbr$. $a+\frac{1}{2}r=b$. $\frac{bbrx+crxx+gr{x}^{3}+\frac{rqq+{r}^{3}{x}^{4}}{q}}{2aa+4bx+\frac{2q+2rxx}{q}}={fl}^{2}$. $\frac{aa}{q}+2a+\frac{3ar}{q}+\frac{rr}{2q}=c$. $\frac{2a}{q}+1+\frac{2aa}{qq}+\frac{2rr}{qq}=g$ . {illeg} {illeg} $r=$ {illeg} /$r=2$ $q=6$. $x=3$.\ $gd={r}^{2}x+\frac{1}{3}xx=3$. {illeg} /$ac=1$\ $df=\frac{r}{2}+\frac{r}{q}x=2$. $ag=5$. $5:3\colon\colon af=6:\frac{18}{5}=of$ . $fl=\frac{9}{5}$. $d:e\colon\colon of:fl=\frac{18e}{5d}$. $dk=\frac{9×2}{9-\frac{324ee}{25dd}}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{\frac{18×324ee}{25dd}+\frac{81×324ee}{25dd}-9×Q:\frac{324ee}{25dd}:}{81-18×\frac{324ee}{25dd}+Q:\frac{324ee}{25dd}:}}$. $dk=\frac{450dd}{225{d}^{2}-324ee}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{801900ddee-944784{e}^{4}}{50625dddd-145800ddee+104976{e}^{4}}}$ $dk=\frac{50dd}{25dd-36ee}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{9900ddee-11664{e}^{4}}{625{d}^{4}-1800ddee+1296{e}^{4}}}\right\}=g$. $de=a-3$. $g=p+\sqrt{ps}$ . $e=1$ $\frac{9aa{e}^{2}-54aee+81ee-18agee+54gee+9ggee}{ddgg}$ $\frac{\begin{array}{c}+aaee-6aee+9ee\\ -2ag{e}^{2}+6g{e}^{2}+ggee\end{array}}{dd}$ $\frac{\begin{array}{c}-9aa+54a-81\\ +18ag-54g-9gg\end{array}}{gg}$ $\begin{array}{c}={e}^{2}\\ ={z}^{2}\end{array}$ $\begin{array}{llllllllllllllllllllll}\phantom{+}& 9aaee& -& 54aee& +& 81ee& -& 18apee& -& 18aee\sqrt{qs}& +& 54eep& +& 54ee\sqrt{qs}& +& 9eepp& +& 9eeqs& -& 2qs\sqrt{qs}×eea& =& {z}^{2}\\ +& ee{p}^{4}& +& 6eeppqs& & & +& eeqqrs& +& 2aaeep\sqrt{qs}& -& 2aae{p}^{3}& +& 18eep\sqrt{qs}& +& aaeepp& +& aaeeqs& +& 6eeqs\sqrt{qs}\\ -& 9aadd& +& 54add& & & -& 81dd& -& 12aaeep\sqrt{qs}& +& 6ee{p}^{3}& +& 18ee\sqrt{qs}& -& 6aeepp& -& 6aeeqs\\ +& 18apdd& -& 54pdd& & & -& 9ppdd& -& 6aaeep\sqrt{qs}& & & -& 54{d}^{2}\sqrt{qs}& +& 9eepp& +& 9eeqs\\ & & & & & & & & +& 18eepp\sqrt{qs}& & & -& 18pdd\sqrt{qs}& & & -& 6aeepqs\\ & & & & & & & & +& 4ee{p}^{3}\sqrt{qs}& & & & & & & +& 18eepqs\\ & & & & & & & & +& 4eepqs\sqrt{qs}& & & & & & & -& 9ddqs\\ & & & & & & & & +& 18add\sqrt{qs}\end{array}$ $ddpp+2ddp\sqrt{qs}+ddqs$

 $ac=a=ce$ . $cd=x$. $dg=y$. $bc=x$. $df=\frac{1}{2}r$. $rx=yy$. $aa+2ax+rx+xx:rx\colon\colon \begin{array}{l}aa+2ax+ar+rx\\ +xx+\frac{1}{4}rr\end{array}:lf$. $rx+\frac{ar+\frac{1}{4}rr\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}rx}{aa+2ax+rx+xx}={fo}^{2}$. ${fl}^{2}=\frac{eerx}{dd}+\frac{4eearrx+ee{r}^{3}x}{4ddaa+8ddax+4ddrx+4ddxx}$. ${fl}^{2}=pp$. $lk=r$ $ac=a$. $ab=x$. $a:x\colon\colon x:a-x$. $xx=aa-ax$. $xx=4-2x$. $x=-1+\sqrt{5}$ $2:\sqrt{5}-1\colon\colon \sqrt{5}-1:3-\sqrt{5}$. $6-2\sqrt{5}=5-2\sqrt{5}+1$. $xx=-ax+aa$. $\begin{array}{l}144\\ \underset{_}{18}\\ 324\end{array}$ $\begin{array}{l}\underset{_}{18×25}=450\\ \phantom{0}90\\ 36\end{array}$ $\begin{array}{l}144\\ 10\end{array}$ $\begin{array}{l}\underset{_}{324×18}=5832\\ 2592\\ 324\end{array}$ / $\begin{array}{l}\phantom{0}29160\\ 11664\end{array}\right\}=145800$\ $\begin{array}{l}\phantom{00}324\\ 2592\end{array}\right\}=\begin{array}{r}26244\\ 5832\end{array}\begin{array}{c}\phantom{0}\\ \phantom{0}\\ \begin{array}{||}\hline 32076\\ \hline\end{array}\end{array}$ $x=-\frac{1}{a}+\sqrt{\frac{5aa}{4}}$ $ei=a$. $hk=z$.h{illeg}$=\frac{ez}{d}$ \$gd=y$/ $hi=\frac{ez}{d}$. $he=\sqrt{\frac{eezz-aadd}{dd}}$. $ek=\sqrt{\frac{ddzz-eezz+aadd}{dd}}$. $ce=a$. $fl=b$ $fk=c$. $lk=\sqrt{cc-bb}$. $z:\sqrt{\frac{eezz-aadd}{dd}}\colon\colon c:b${illeg}. $c=\frac{dbz}{eezz-aadd}$. ${lk}^{2}=\frac{ddbbzz-bbeezz+aabbdd}{eezz-aadd}$. $df=v$ $lk=\frac{bv+\frac{dbbz}{\sqrt{eezz-aadd}}}{y}$ {illeg} $=\frac{dbby+bv\sqrt{eezz-aadd}}{y\sqrt{eezz-aadd}}=\frac{y\sqrt{ddbbzz-bbeezz+aabbdd}}{y\sqrt{eezz-aadd}}$ $ddbbyy+vveezz-vvaadd-ddyyzz+eeyyzz-aayydd+2bdvy\sqrt{ee{z}^{2}-aadd}=0$. $ab=q$. $bc=x$. $dc=y$. $r=\text{lat}:rx:rx-\frac{rxx}{q}=yy$. $ef=b$. $fg=c$. $eb=\frac{1}{2}q-b$. $cf=x-\frac{1}{2}q$. $ce=x-\frac{1}{2}q+b$. $b:c\colon\colon x-\frac{1}{2}q+b:y$. $\frac{cx-\frac{1}{2}qc+bc}{b}=y=\sqrt{rx-\frac{rxx}{q}}$ $\begin{array}{cccccccccccccc}& ccxx& -& qccx& +& 2bccx& +& \frac{1}{4}qqcc& -& qccb& +& bbcc& =& 0\\ +& \frac{bbr}{q}xx& & & -& bbrx\hfill \end{array}$. $q=3$. $fg=c=1$ {illeg} $b=1$. $xx+\frac{1}{3}xx=3x-2x+x-\frac{9}{4}+3-1=0$. $\frac{4xx}{3}=2x-\frac{1}{4}=0$ $xx=\frac{6}{4}x-\frac{3}{16}$. $x=\frac{3}{4}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{9-3}{16}}$. $x=\frac{3}{4}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{3}{8}}$ $rx+\frac{rxx}{q}=9$. $\frac{r}{2}+\frac{rx}{q}=2$ /$\frac{1}{2}r+\frac{rxx}{q}=7$ $\frac{rx}{2}$ {illeg}$=7$.\ $ac=a$ . $kd=b$. $ad=c$. $dc=d$. {illeg} $c:a\colon\colon b:\frac{ab}{c}$ {illeg} $\frac{dab}{ec}=cf$. $kl=e$. $x=1$. $r=\frac{1}{4}$. $ad=a$. $dl=b$. $lk=c$. $ac=g$. $\frac{dgb}{ea}=ce$. $\frac{ddgc}{eea}=ef$. ${of}^{2}=\frac{{d}^{4}ggcc-{e}^{4}ccaa}{{e}^{4}aa}$. $e=2$. $d=2$. $+bggc$ $+\frac{bggcc}{aa}-cc={of}^{2}$. $g=c=1=a$. $15={of}^{2}$. $\frac{ggcc}{16aa}-cc$. $g=${illeg}|{ 8 }| {illeg} $=8a=${illeg}$2c$ $3={of}^{2}$ . $ab=a$. $bc=b$. $cd=c$. $be=x$. $ce=x-b$. $de=\sqrt{xx-2bx+bb-cc}=\frac{cx}{a}$ . $xx=\frac{2bx-bb+cc\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}aa}{aa-cc}$ $x=\frac{aab}{aa-cc}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{{a}^{4}bb+aabbcc+{a}^{4}cc-aa{c}^{4}}{{a}^{4}-2aacc+{c}^{4}}}$ $bm=f$. $be=g$. $me=f-g$. $ab=a$. {illeg} $cd=c$. $a:g\colon\colon c:\frac{gc}{a}=de$. $g:a:f-g:\frac{af-ag}{g}=mr$. $mn=z$. $\frac{aaff-2aafg+aagg}{gg}+ff-2fg+gg\left(={re}^{2}\right)in\frac{ee}{dd}={rn}^{2}=\frac{aaff-2aafg+aagg}{gg}+zz$.

<5r>

$eh=8$ — − $ei=6$ $hi=10$. $ek=15$. $\sqrt{\begin{array}{c}\phantom{0}75\\ 154\\ 6\end{array}}=\sqrt{289}=17=hk$. {$gh=$ }{$gk=$ } $de=${illeg}15. $dg=16=y$. $cd=x$. $df=a$. $gk=${14} $34:16\colon\colon gk:gd\colon\colon 17:8\colon\colon 30-a:\frac{240-8a}{17}=fl$. $fl:fo\colon\colon 10:17$. $\frac{120-4a}{5}=24-\frac{4a}{5}=fo$ . $ad=b$. $ag=\sqrt{bb}+${illeg} $256:bb+256\colon\colon 576-\frac{192a}{5}+\frac{16aa}{25}:\frac{576bb-\frac{192}{5}abb+\frac{16aabb}{25}}{256}+576-\frac{192a}{5}+\frac{16aa}{25}={af}^{2}=aa+2ab+${illeg} $\frac{5bb}{4}+\frac{3abb}{20}+\frac{aabb}{400}-\frac{9aa}{25}-2ab-\frac{192}{5}a+576=0$. $\begin{array}{l}\phantom{+}500bb\\ +60a\\ +aa\end{array}\begin{array}{c}=200ab+144aa+15360a-2304\\ \phantom{0}\\ \phantom{0}\end{array}$ {illeg} $yy=rx+\frac{r}{q}xx=256$. $\frac{r}{q}x+\frac{r}{2}=a$. $256-ax=\frac{1}{2}rx$. $\frac{512}{x}-\frac{a}{2}=r$ $eh=\frac{5}{2}$ . {illeg} $ei=6$. $hi=\frac{13}{2}$. $hk=\frac{2221}{20}$. $ek=\frac{9\sqrt{5149}}{20}$$gh=${illeg} $\frac{221}{20}:\frac{50}{20}\colon\colon \frac{221}{20}+c:dg=\frac{50}{20}+\frac{}{221}$ $dg=\frac{5}{2}+\frac{50c}{221}$ . $\frac{9c\sqrt{5149}}{221}=de$ $fk=\frac{\phantom{00}+9\sqrt{5149}}{20}$ {illeg} $fk=\frac{9c\sqrt{5149}}{221}+$ $ek=\frac{3\sqrt{5149}}{20}$. $kd=g$. $\frac{5g×20}{6\sqrt{5149}}=\frac{50g}{3\sqrt{5149}}=dg$. ${dg}^{2}=\frac{2500gg}{46341}$. ${gk}^{2}=48841gg$. $fk=c-a$. $\frac{50cg-50ag}{3\sqrt{5149}\phantom{\rule{1em}{0ex}}in\phantom{\rule{1em}{0ex}}221}=fl=\frac{f k×dg}{gk}$ $\frac{50cg-50ag}{390\sqrt{5149}}=fo$. $ae=p$. {illeg} $d e=g-\frac{3\sqrt{5149}}{20}$ $ad=p-g+\frac{3\sqrt{5149}}{20}$. ${ag}^{2}=\begin{array}{l}pp-2pg+\frac{48841gg}{46341}+\frac{46341}{400}\\ +\frac{3p}{10}\sqrt{5149}-\frac{3g}{10}\sqrt{5149}\end{array}$ ${ag}^{2}:{gd}^{2}\colon\colon {af}^{2}:{fo}^{2}$.

$eh=${illeg}. {illeg} $eh=a$. $ei=b$. $hi=\sqrt{aa+bb}$ $hk=\frac{d}{e}\sqrt{aa+bb}$. $ek=\sqrt{\frac{ddaa+ddbb-eeaa}{e}}$ $eh=c$. $ek=\sqrt{\frac{ddcc+ddbb-\phantom{000}}{ee}}$ $2abx=n$. $axx+abb:bxx+baa\colon\colon f:g$. $\frac{abbg-aabf}{}-axxg-bxxf=0$ $\frac{nn}{4aabb}=xx$. $agnn-bfnn+4{a}^{3}{b}^{4}g-4{a}^{4}{b}^{3}f=0$. $f=10$. $g=17$. ${b}^{4}$ $axx-abb=6$. $xx=\frac{6+abb}{a}$. $\frac{17abb-10aab}{10b-17a}=xx$ $\begin{array}{ccccccccc}10{a}^{3}b& -& 17aabb& +& 10a{b}^{3}& +& 60b& =& 0\\ & -& 17aabb& -& 102a\end{array}$ {illeg} $\begin{array}{ccccccccc}{a}^{3}& -& \frac{34}{10}aab& +& abb& +& 6& =& 0\\ & & & -& \frac{102a}{10b}\hfill \end{array}$. $q{a}^{2}+qa+s×a+c$. $c${illeg}$=\frac{6}{s}$. $r+\frac{6}{s}=\frac{34b}{10}$. $-\frac{10r}{34}+\frac{60}{34s}=b$. $\begin{array}{lllllll}{a}^{3}& +& raa& +& sa& +& cs\\ & +& c& +& cr\end{array}$ $r=-\frac{6}{s}-\frac{34b}{10}$. $-\frac{36}{ss}-\frac{204b}{10}s+s=bb-\frac{102}{10b}$. $cc=20c-45$ $\begin{array}{lllllll}q{a}^{3}& +& raa& +& sa& +& sc\\ & +& qc& +& rc\end{array}$. $b=1$ $10b{a}^{3}$ $10{a}^{3}-34aa+112a+60$. $\begin{array}{l}\phantom{0}448\\ \underset{_}{336\phantom{0}}\\ \underset{_}{3808}\end{array}$ $\begin{array}{c}\begin{array}{l}\left(2.2.2.2.2.7./2.2.3.5\\ \begin{array}{cc}\begin{array}{c}1904\\ 952\\ 476\\ 238\\ 119\\ 17\end{array}& \begin{array}{llll}\phantom{0}& 3868& .& \left(2.2.967\\ \phantom{0}& 1934& & \phantom{\left(}\\ \phantom{0}& \phantom{0}967& & \phantom{\left(}\end{array}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{l}1.2.4.6.10.3.5.15.30.\\ 20\end{array}\\ \begin{array}{l}\hfill 10.\phantom{0}.\\ \phantom{0}\\ \begin{array}{c}\begin{array}{c}2.967.4.1934.3868.\end{array}\\ 10\end{array}\end{array}\end{array}\end{array}$ $\begin{array}{l}\phantom{00}36\phantom{=}\\ \underset{_}{102\phantom{0}\phantom{=}}\\ 1056=\end{array}$ $\frac{pq-r}{\text{Div}-}$ $-nn+2pn$ . $p-n=c$ \{illeg} $aa-4a$/ 4 $10{a}^{3}-34aa-92a+60$ $\begin{array}{ccccc}-\frac{34}{}& +& 2& +& 4\\ & -& 2& -\end{array}$. $\begin{array}{||}\hline -4\\ \hline\end{array}$ $\begin{array}{r}120\\ \underset{_}{-34}\\ 86\end{array}$ $\begin{array}{r}150\\ \underset{_}{34}\\ 126\end{array}$ $\begin{array}{l}\phantom{0}368\\ \underset{_}{27\phantom{00}}\\ \underset{_}{\phantom{0}120}\\ 3168\end{array}$ $\left(\begin{array}{l}2.2.\\ 1594.797.\end{array}$ $\begin{array}{cc}\begin{array}{r}372\\ \underset{_}{372\phantom{0}}\\ 4094\\ \underset{_}{-120}\\ 3974\\ 1987\end{array}& \begin{array}{ll}\phantom{0}14& c\\ \begin{array}{l}\phantom{0}56\\ 14\end{array}\\ \begin{array}{ccc}\left(2& & \begin{array}{r}96\\ \underset{_}{16\phantom{0}}\\ 256\end{array}\\ \begin{array}{r}144\\ \underset{_}{18\phantom{0}}\\ 324\end{array}& 117& 13\end{array}\end{array}\end{array}$ $20{a}^{3}$$-34aa${illeg}$-136aa$ {illeg}$-22a$ \{illeg}/ + {illeg} 120 $\begin{array}{ccc}\underset{_}{49}& \begin{array}{r}96\\ \underset{_}{48\phantom{0}}\\ 576\\ 49\end{array}& \begin{array}{l}625\\ 2\end{array}\end{array}$ ${a}^{2}-\frac{4}{10}a$ 6 {illeg}  $30{a}^{3}-306aa+168a-180$. $\begin{array}{rr}1008& \left(2.2.3.3.4.6.9.12.24.18.54.36.\\ \underset{_}{504\phantom{00}}\\ \underset{_}{\phantom{0}-180}\\ 51228& .25614.12807.4269.1423\hfill \end{array}$ $\begin{array}{ll}136& 1531\\ 134& 2\hfill \\ 17\end{array}$ 102 $\begin{array}{cccc}40{a}^{3}& \begin{array}{r}-204\\ -340\end{array}aa& \begin{array}{r}+640\\ -102\end{array}a& +240\text{.}\\ \phantom{0}\\ 40{a}^{3}& \hfill -544aa& \hfill -538a& +240\text{.}\end{array}$ $\begin{array}{cccc}& & eh=12\text{.}& ei=9\text{.}\\ hi=15\text{.}& hk=20\text{.}& ek=16\text{.}\end{array}$ $\begin{array}{r}\underset{_}{17}\\ 119\\ \underset{_}{17\phantom{0}}\\ \underset{_}{\phantom{0}289}\\ 2083\\ 289\phantom{0}\end{array}$ $\begin{array}{r}\phantom{-}4913\\ \underset{_}{-1530}\\ 3\\ \hfill 729\hfill \end{array}$ $\begin{array}{r}81\\ \underset{_}{648\phantom{0}}\\ 6561& ×4=\end{array}$ { 2 }620 {illeg}| 1 | {illeg}

 d {illeg}$-3=$ $\frac{axx-c}{a}=bb$. $axx+abb:bxx+baa\colon\colon e:d$. $daxx+dabb=ebxx+ebaa$. $\begin{array}{rr}25839& \left(\hfill \\ 6\phantom{00}\\ 239\end{array}${illeg} $xx=\frac{aabe-abbd}{ad-eb}=\frac{10aab-17abb}{17a-10b}$. $xx=\frac{17abb-10aab}{10b-17a}$. $40b-17abb+10aab$. /$a=\frac{9}{17}$. $b=1$. {illeg}\ $10-\frac{90}{17}${illeg} $3,4,9$. $5,12,13$. $7,24,25$. $12,16,20$. $12,9,15$. $xx=\frac{4abb-3aab}{3b-4a}$. {3 } $\begin{array}{cccc}x& \begin{array}{c}-4axx\\ -4abb\end{array}& & +3aab\end{array}$ $ax\frac{2xx+2bb}{3b}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{4{x}^{4}+8bbxx+4{b}^{4}-9bbxx}{9bb}}$. $\sqrt{4{x}^{4}-bbxx+4{b}^{4}}$. $x=4$ . $b=1$ . {illeg}$\frac{+8}{6}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\frac{8}{6}=a=3${illeg} {illeg}$=\frac{4}{3}$ $x=${illeg} $x=${illeg} {illeg}$+\frac{1}{3}=ab${illeg}$4+\frac{1}{9}${illeg}15 7 $4{x}^{4}-bb{x}^{2}+4{b}^{4}=4b{x}^{4}$  {illeg}$\frac{8}{3}$ $\frac{}{3}$. $\frac{8+2}{3}=\frac{10}{3}=cb$. $4+\frac{4}{9}$. $\frac{30}{9}$ . {illeg} $\frac{8}{3}$ {illeg} $b=${illeg} $\frac{4{x}^{4}-{c}^{4}}{{x}^{5}+}=bb$. {illeg}+{illeg}={illeg}$-2bc+cc$. $\frac{cc+xx}{2c}=b$. $4{x}^{4}-bbxx$ {illeg} ${b}^{4}=4{b}^{4}-bbxx+6{}^{3}$ {illeg} {illeg} = {illeg}${c}^{4}-4{a}^{2}bb${illeg}$3aab=4bb-3ab=4bb-4${illeg}$cc$. $\frac{}{4c}${illeg} {illeg}$aa=${illeg}$-4aa${illeg} {illeg}$9acc-${illeg}$36a${illeg} {illeg}

<5v>

 $\sqrt{4{x}^{4}-bbxx+4{b}^{4}}=\sqrt{4{b}^{4}+4bbcc+{c}^{4}}$ ${}^{2}$ $\frac{4{x}^{4}-{c}^{4}}{4cc+xx}=bb=\frac{4{x}^{6}+16cc{x}^{4}-4xx-4{c}^{6}}{4{c}^{4}+4ccxx+{x}^{4}}$. $h{x}^{2}+in+${illeg} $a{x}^{3}+b{x}^{2}+cx+d$. $\sqrt{4{x}^{4}-{b}^{2}{x}^{2}+4{b}^{4}}=-bb+cx+dxx+ee$. $4{x}^{4}-bbxx=-4bbcx-4bbdxx-4bbee+ccxx+2cd{x}^{3}+2ceex+dd{x}^{4}+2deexx+{e}^{4}$: $\frac{+4{x}^{4}+dd{x}^{4}+2cd{x}^{3}+ccxx+2deexx+2ceex+{e}^{4}}{-xx+4dxx+4cx+4ee}=bb$. |$24-\frac{50}{3}$ /$\frac{22}{36+45-20}$\ $\frac{\frac{22}{3}}{12+15-\frac{20}{3}}$| $2fxx+2gfx+ff$. $2e=f$. $\frac{2c}{e}=2g$. $\frac{ccxx}{e}=-xx+4dxx$ $cc=-ee+4dee$. $d=\frac{cc+ee}{4ee}$ $\frac{4{x}^{4}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{c}^{4}+2ccee+{e}^{4}}{16{e}^{4}}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{4}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{c}^{3}+cee}{2ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{3}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}ccxx\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{e}^{4}+ccee}{2ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}xx\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}2ceex\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{e}^{4}.}{-xx\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+cc+ee}{2ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}xx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}4cx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}4ee}=bb$ $\begin{array}{ccccccccc}hh{x}^{4}& +& 2hi{x}^{3}& +& 2hk{x}^{2}& +& 2ikx& +& kk\\ & & & +& ijxx\end{array}$. $\frac{-63{e}^{4}+2ccee+{c}^{4}}{16{e}^{4}}=hk$. {illeg}$\sqrt{}$ {illeg}e {illeg}k {illeg}$-ee$. $2fxx+2gfx+ff$. $2e=f$. $\frac{2c}{e}=2g$. $\frac{ccxx}{e}=-xx+4dxx$ $cc=-ee+4dee$. $d=\frac{cc+ee}{4ee}$ $\frac{-4{x}^{4}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{c}^{4}+2ccee+{e}^{4}}{16{e}^{4}}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{4}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{c}^{3}+cee}{2ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{3}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{3}{2}ccxx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{ee}{2}xx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{3}{2}ccxx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}2ceex\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{e}^{4}}{\frac{ccxx}{ee}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}4cx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}4ee}=bb$ $k=ee$. $2i=2c$. $i=c$. {illeg} $h=\frac{cc+ee}{4ee}$. $\frac{{c}^{4}+2ccee-63{e}^{4}}{16{e}^{4}}=hk$ {illeg} ${c}^{4}+2ccee-63{e}^{4}={c}^{4}+2ccee+{e}^{4}$. $ee=k$. $i=c$. $d=h$. $\frac{\frac{{c}^{4}+2ccee+{e}^{4}}{ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{4}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}\frac{+{c}^{3}+cee}{2ee}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}{x}^{3}\phantom{\rule{0.5em}{0ex}}:\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{3}{2}ccxx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{ee}{2}xx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{3}{2}ccxx\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}2ceex\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{e}^{4}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}ff}{2f}=xx$.

 $\frac{4-3+\frac{3}{2}+\frac{1}{2}+2+9}{2}=$ {illeg} 6 /{illeg}\ $\begin{array}{r}{x}^{3}\\ +{y}^{3}\end{array}\phantom{\rule{1em}{0ex}}\frac{+9z{z}^{4}-3az{x}^{2}y+{y}^{3}z-3azxyy}{\sqrt{9{x}^{4}-6ay{x}^{2}+10aa{y}^{2}+aa{x}^{2}-6{a}^{3}x}}$ $25+5+6+\frac{1}{2}+4+1\phantom{\rule{2em}{0ex}}41⁤\frac{1}{2}$ $\frac{83}{4f}$ $2ff$ / 5 \ $\frac{25}{4}+\frac{5}{8}+$ $64-24+6+2${illeg} {illeg}|5| $\begin{array}{c}dabb-ebxx-ebaa+daxx=0\\ dpqq-eqyy-eqpp+dpyy=0\end{array}$. $ss-{v}^{2}+2vy-yy=\frac{{a}^{4}}{yy}-\frac{2{a}^{4}}{2{y}^{3}}+2y$ $2ab=2exx$. $\frac{daxx}{eaa-exx}=b$. $\frac{ebxx+ebaa}{abb+axx}=d=\frac{eqyy+eqpp}{pqq+pyy}$. $\begin{array}{c}pqqbxx+pqqbaa+pyybxx+pyybaa\\ -abbqyy-abbqpp-axxqyy-axxqpp\end{array}=0$ $pqqa+pyya=bqyy+bqpp$. $axxqyy+axxqpp=pqqbxx+pyybxx$. $pqqab+pyyab=bbppq+xxppq$. $axxqyy+abbqyy=pqqbxx+pyybxx$. $\frac{bbpq+xxpq}{bqq+byy}=a=\frac{pqqbxx+pyybxx}{qyyxx+qyybb}$. $\begin{array}{cccccccc}& aabpyy& -& abbqyy& +& bpyyxx& =& 0\\ +& aabpqq& -& abbqpp& +& bpqqxx& =& 0\\ & & -& axxqyy\\ & & -& axxqpp\end{array}$. $aa+xx=\frac{abbqyy+abbqpp+axxqyy+axxqpp}{bpyy+bpqq}$ $\begin{array}{c}aabpyy+aabpqq=abbqyy+axxqyy\\ a=x=b\phantom{\rule{3em}{0ex}}qq=yy=pp\end{array}$

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${b}^{3}+{z}^{3}=abz$ . $bb-2bx+xx+yy-$ $\begin{array}{ccccccccc}ss& -& 2vv& +& 2vx& -& xx& =& \frac{{r}^{4}}{xx}\\ 0& & 0& & 1& & 2& & -2& \end{array}$ $-\frac{2{r}^{4}}{2xxx}+\frac{2xx}{2x}=v$ {illeg}

< text from f 5v resumes >

$\begin{array}{cccccc}hgxx& -& gix& +& & \text{.}\\ & -& hkx& +& ik& \text{.}\end{array}$ $2rx-xx+rr=yy$. $v=r-x$. $\begin{array}{ccccccc}rx& +& \frac{r}{q}xx& -& yy& =& 0\\ 1& & 2& & 0\end{array}$. $-r=\frac{2rx}{q}$ $\begin{array}{c}2rx+2yx+xx=yy\\ r+y+x\end{array}$. $xx$ {illeg} $=-\frac{q}{2}$ $yy=-\frac{rq}{2}+\frac{rq}{4}xx\phantom{\rule{0.5em}{0ex}}\begin{array}{c}+\\ -\end{array}\phantom{\rule{0.5em}{0ex}}yy$ $ax-xx=yy$. $yy-\frac{rq}{4}$. $2rx-xx=yy$. $v=r-x$. $x=${illeg}r $\sqrt{2rr}$ x $rz$ $zx$ for x {illeg}|wri|te $x+z-\frac{zx}{r}$ $\sqrt{\phantom{00000}+rr}${illeg} for {illeg}| y | write $\sqrt{}$ $2rx-xx-4zx+2\frac{zxx}{r}+2\frac{zzx}{r}-\frac{zzxx}{rr}$ {illeg}$\sqrt{\phantom{00000}xx}$ {illeg} r {illeg} x \/ $z\frac{\sqrt{2rx-xx}}{r}$ for y $-rx$ {illeg} z $2zx$ {illeg} x $-2x$ {illeg} 2 $zx$ 2 $\frac{zxx}{r}$ $-\frac{2zzx}{r}=+2rxxx+2rx-xx+\frac{2{x}^{3}}{r}-$ /$4rx$ $\frac{2{x}^{3}}{r}-4xx+\frac{2{x}^{3}}{r}$\ {illeg} {illeg} $xx+$ $xxz$ $4zx$ $\frac{2zxx}{r}$ 2 $z-2$ {illeg} z $\frac{4zx}{r}$ $2z${illeg} $\frac{4zx}{r}=0$ {illeg}{illeg}$xx$ $-zz$$+\frac{2zzx}{r}$$-\frac{zxx}{rr}$ {illeg}{illeg}$xx$

<6r>

 $ag=x$. $bg=y$. $ag=x$. $gd=d$. $dc=e$. $bo=s$. $de=w$. $gh=b$. $y:v\colon\colon y+b:\frac{vy+vb}{y}=$. \$go=v$./ $if=c$. $e:w\colon\colon e+b:\frac{ew\phantom{000}}{e}$ $\frac{ew+bw}{e}=if$. $y:v\colon\colon y+b:\frac{yv+bv}{y}=hf$. $dc×de${illeg}$dc+bw$ $de+hi+\frac{gh×de}{dc}=go+\frac{gh×go}{bg}$. ag{illeg}
$ag=x$. $bg=y$. $rx=yy$. $go=de=\frac{1}{2}r$. $gd=o$. $dc=\sqrt{rx+ro}$. {illeg} $gh=z$. $o+\frac{rz}{z\sqrt{rx+ro}}=\frac{rz}{z\sqrt{rx}}$ $2o\sqrt{rx}${illeg} $4o\sqrt{rrxx+rrox}+2rz\sqrt{rx}=2rz\sqrt{rx+ro}$. $4oz\sqrt{r{x}^{3}+roxx}=4zzro$. ${x}^{3}=zzr$. $z=\sqrt{\frac{{x}^{3}}{r}}$. $\sqrt{rx}:\frac{r}{2}\colon\colon \sqrt{\frac{{x}^{3}}{r}}:\frac{x\sqrt{rx}}{2\sqrt{rx}}=\frac{x}{2}$ {illeg} $\frac{x+r}{2}=hf$. $\frac{{x}^{3}}{r}+\frac{9xx}{4}+\frac{3rx}{2}+\frac{rr}{4}={bf}^{2}$. $rx+xx=yy${illeg}
 $xy=rr$. $v=-\frac{yy}{x}=\frac{-{y}^{3}}{rr}=-\frac{{r}^{4}}{{x}^{3}}$. $gh=z$. $dg=o$. $de=\frac{+{r}^{4}}{{x}^{3}+3oxx}:dc=\frac{rr}{x+o}\colon\colon fi:\frac{zx-rr+\phantom{000}}{x+o}$ $rr$ $\frac{+rrzx-{r}^{4}+rrzo}{{x}^{3}+3xxo}=fi$. $\frac{-{r}^{4}}{{x}^{3}}:\frac{rr}{x}\colon\colon fh:\frac{zx-rr}{x}$ . $\frac{-{r}^{4}+zxrr-o{x}^{3}}{{x}^{3}}=fi$ $-rr$ {illeg}${x}^{4}+{r}^{4}+{x}^{3}${illeg} $-rrz{x}^{4}+{r}^{4}{x}^{3}-rrzo{x}^{3}-{r}^{4}{x}^{3}-zrr{x}^{4}+o{x}^{6}+3{r}^{4}xx0-3z{x}^{3}rro+3{x}^{5}oo-$ $+2rrzx-{x}^{4}-3{r}^{4}=0$. $z=\frac{3rr}{2x}+\frac{{x}^{3}}{2rr}=gh$. $bh=\frac{rr}{2x}+\frac{{x}^{3}}{2rr}$ {illeg} $+\frac{{r}^{4}}{{x}^{3}}+\frac{x}{2}=fh$. $\frac{{r}^{8}}{{x}^{6}}+\frac{{r}^{4}}{xx}+\frac{xx}{4}+\frac{{r}^{4}}{4xx}+\frac{xx}{2}+\frac{{x}^{6}}{4{r}^{4}}={bf}^{2}$ $-\frac{6{r}^{8}}{{x}^{6}}-\frac{3{r}^{4}}{2xx}+\frac{3xx}{4}+\frac{6{x}^{6}}{2{r}^{4}}$. $6{x}^{12}+16{x}^{8}{r}^{4}-10{r}^{8}{x}^{4}-24{r}^{12}$. $6{r}^{12}+$

$go:gb\colon\colon fk+go:hg$ . $de:dc\colon\colon fi+de:di$ . $dc×fi+dc×de+dc$ $\frac{dc×fi+dc×de}{de}=di=\frac{gb×fi+gb×dg+gb×go}{go}$. $\frac{\frac{rrz}{x+o}+\frac{{r}^{6}}{{x}^{4}+4{x}^{3}o}}{\frac{{r}^{4}}{{x}^{3}+3{x}^{2}o}}=\frac{\frac{rrz}{x}+\frac{{r}^{6}}{{x}^{4}}+\frac{rro}{x}}{\frac{{r}^{4}}{{x}^{3}}}$. $\frac{zx}{x+o}+\frac{{r}^{4}x}{{x}^{4}+4o{x}^{3}}=\frac{zx+3zo}{x}+\frac{{r}^{4}x+3{r}^{4}o}{{x}^{4}}+xo$. $\frac{z{x}^{5}+4z{x}^{4}o+{r}^{4}xx+{r}^{4}ox}{{x}^{5}+5o{x}^{4}}=\frac{z{x}^{5}+3zo{x}^{4}+{r}^{4}xx+3{r}^{4}ox}{{x}^{5}}$ $4z{x}^{9}+{r}^{4}{x}^{6}=5z{x}^{9}+5{r}^{4}{x}^{6}+3z{x}^{9}+3{r}^{4}{x}^{6}$. $4z{x}^{4}+7{r}^{4}=0$ $\frac{rrz}{x+o}+\frac{{r}^{6}}{{x}^{4}+4o{x}^{3}}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\frac{{r}^{4}}{{x}^{3}}=\frac{rrz}{x}+\frac{rro}{x}+\frac{{r}^{6}}{{x}^{4}}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\frac{{r}^{4}}{{x}^{3}+3o{x}^{2}}$ . $\frac{z}{xx+ox}\phantom{\rule{1em}{0ex}}\frac{+{r}^{4}}{{x}^{5}+4o{x}^{4}}=\frac{z+o}{xx+3ox}\phantom{\rule{1em}{0ex}}\frac{+{r}^{4}}{{x}^{5}+3o{x}^{4}}$. $\frac{z{x}^{3}+3zoxx+{r}^{4}}{{x}^{5}+4o{x}^{4}}=\frac{z{x}^{4}+o{x}^{4}+3oz{x}^{3}+x{r}^{4}+3o{r}^{4}}{{x}^{6}+6o{x}^{5}}$ $\begin{array}{cc}\begin{array}{ccccc}z{x}^{5}& +& 3zo{x}^{4}& +& {r}^{4}xx\\ & & 6zo{x}^{4}& +& 6{r}^{4}xo\end{array}& \begin{array}{cccccccc}=& z{x}^{5}& +& 2o{x}^{5}& +& 3oz{x}^{4}& +& xx{r}^{4}\\ & & & & +& 4oz{x}^{4}& +& 3ox{r}^{4}\\ & & & & & & +& 4oz{r}^{4}\end{array}\end{array}$ $9z{x}^{4}+6x{r}^{4}={x}^{5}+7z{x}^{4}+7x{r}^{4}$. $2z{x}^{4}-x{r}^{4}-{x}^{5}=0$. $z=\frac{{r}^{4}}{2{x}^{3}}+\frac{x}{2}$. $\frac{{r}^{4}}{{x}^{3}}:\frac{rr}{x}\colon\colon \frac{{r}^{4}+{x}^{4}}{2{x}^{3}}:hb$. $xx:rr\colon\colon \frac{{r}^{4}+{x}^{4}}{2{x}^{3}}\phantom{\rule{1em}{0ex}}\frac{{r}^{4}+{x}^{4}}{2rrx}=bh$. $\frac{{r}^{8}+2{r}^{4}{x}^{4}+{x}^{8}}{4{r}^{4}xx}\phantom{\rule{1em}{0ex}}\frac{{r}^{8}+2{r}^{4}{x}^{4}+{x}^{8}}{4{x}^{6}}=\frac{3{r}^{8}{x}^{4}+3{r}^{4}{x}^{8}+{x}^{12}+{r}^{12}}{4{r}^{4}{x}^{6}}=bf$. $6{x}^{12}+6{r}^{4}{x}^{8}-6{r}^{8}{x}^{4}-6{x}^{12}=0$. ${x}^{8}-{r}^{8}=0$ . ${x}^{4}-{r}^{4}=0$ . $xx-rr=0$ . $x=r$. therefore take $ag=gb=r$ , & ye greatest crookedness of ye line cb will be found at b . $bh=r=fh$ . $bf=r\sqrt{2}$ . $gh=w$ . $hf=\sigma$ . {illeg} $gb=y$ . $go=v$ . $w:v+\sigma \colon\colon y:v$ {illeg}. or, $y:v\colon\colon w-y:\sigma =\frac{vw-vy}{y}$. $\frac{go×gh-go×bg}{bg}=fh$ . $\frac{{y}^{3}w}{rr}-\frac{{y}^{4}}{rr}=\sigma y$ . $\frac{2yyw}{rr}-\frac{3yyy}{rr}=0$ . $w=\frac{3y}{2}$ . $\frac{3{r}^{4}+{x}^{4}}{2rrx}=0$ . ${x}^{3}=\frac{{r}^{6}}{{y}^{3}}$ . $\frac{3rr}{2x}+\frac{{r}^{4}}{2{y}^{3}}=\frac{3y}{2}+\frac{{r}^{4}}{2{y}^{3}}$. $bf=p$ . $al=q$ . $gb=y$ . $go=v$ . $ag=x$. $bf=p$. {illeg} $v:y\colon\colon q+v-x:\frac{qy+yv-yx}{v}=p$. $\frac{gb×al+gb×go-gb×ga}{go}=$ {illeg} $yq+\frac{{y}^{4}}{rr}-rr=\frac{p{y}^{3}}{rr}$ . $\frac{{y}^{4}}{rr}+3rr-2yq=0$. $q=\frac{{y}^{3}}{2rr}+\frac{3rr}{2y}$ . as before. or $q=\frac{yy}{2x}+\frac{3x}{2}=\frac{yy+3xx}{2x}$ $gl=\frac{yy+xx}{2x}$. $w=v+x=\frac{dyy+ey+2fxy+bx+d{x}^{2}+2ayx}{d+dx+2ay}$ $+bx+dxw+2ayw+dyy+ey+2fxy+bx+d{x}^{2}+2ayx=0$. $\begin{array}{lllllll}fxx& +& ex& +& c& =& 0\\ & +& dyx& +& by\\ & & & +& dyx\end{array}$. $\begin{array}{l}{w}^{2}-2wx+xx\\ {w}^{2}-\frac{2wc-2wby-2wayy-2wdyx}{fx+e}\end{array}$ $\frac{ex+dyx+c+by+ayy}{f}$. $\frac{+fxx+ex+dyx+c+by}{a}$ $xy=aa$. $x=\frac{aa}{y}$. $y=\frac{aa}{x}$. $v=\frac{yy}{x}$. $\frac{qaa}{x}+\frac{aayy}{xx}-\frac{{a}^{4}}{xy}=0$ . $\frac{qaay}{xx}-\frac{2{a}^{2}{y}^{3}}{{x}^{3}}-\frac{2{a}^{4}}{xy}$. $q=$ $\begin{array}{lllllll}fxx& +& ex& +& c& =& 0\\ & +& dyx& +& by\\ & & & +& dyx\end{array}$. $\frac{2fxy+dyy+ey}{b+dx+2ay}=v$. $qy+\frac{2fxyy+d{y}^{3}+eyy-2pfxy-pdyy-pey}{b+dx+2ay}-yx=0$  $\frac{2bqyf+4aqyyf+2d{y}^{3}f+2eyyf-2pdyyf-2peyf-2dyc-2dbyy-2da{y}^{3}}{2pfy+by+2ayy+dey+ddyy-dfqy-2ffyy}=x$ $x=\frac{-e-dy}{2f}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{\begin{array}{c}ee+2dey+ddyy\\ -4cf-4bfy-4afy\end{array}}{4ff}}$ {illeg} $eyf+2aeyyf+deey+ddyef-defqy$ {illeg} {illeg} $\frac{bqyf+2aqyyf+d{y}^{3}f+eyyf-pdyyf-peyf-dcy-dbyy-da{y}^{3}}{2pffy+bfy+2afyy+edy+ddyy-dfqy-2ffyy}=x$. $2bqyff+4aqyyff+2d{y}^{3}ff+2ffey-2pdffyy-2peffy-2dcfy-dbfyy-2daf{y}^{3}$. $4befy+2aefyy+edfy+cddfyy-edfqy+eddyy+{d}^{2}{y}^{3}-ddfqyy$

<6v>

 $yy${illeg} $vv+yy=ss=yy+2oy$ $\begin{array}{ccccc}\stackrel{\text{.}}{vv}& +& \stackrel{\text{.}}{xy}& +& \stackrel{\text{.}}{ay}\\ & +& \stackrel{\text{.}}{ax}& +& \stackrel{\text{.}}{yy}\end{array}=ss=\stackrel{\text{.}}{vv}-2ov+\stackrel{\text{.}}{xy}+xo+\stackrel{\text{.}}{ax}+\stackrel{\text{.}}{ay}+ao+\stackrel{\text{.}}{yy}+\stackrel{\text{.}}{2oy}$. $2ov=ox+ao+2oy$ . $=vv-2ov+zy+zo+az+o$ = {illeg} $vv+2ov+oo+rs+ra+as+ss$. {illeg} $v=\frac{\begin{array}{cccccccc}& xy& +& ax& +& ay& +& yy\\ -& rs& -& ar& -& as& -& ss\end{array}}{zo}$. $xy-rs${illeg}. $v=xy$
$yy-ss=2o$ {illeg} $xy+ax+ay+2yy+vv-2vy=ss=\xi z+a\xi +az+2zz+vv-2vz$ $\frac{xy-\xi z+ax-a\xi }{2y-2z}=b$. $xy-\xi z-2by+2bz=0$ . $xy-2by=0$ $x=2b$ . $\frac{xxyy-\xi \xi zz}{{a}^{2}x-\xi {a}^{2}}=b$ . $\frac{1}{2}y+\frac{1}{2}x$ $\frac{\begin{array}{cccccc}& 4{x}^{2}& -& ax& \phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}& 2\sqrt{2{x}^{4}-a{x}^{3}}\\ -& 4{z}^{2}& +& az& \phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}& 2z\sqrt{zzz-az}\end{array}}{-2z+2x}=v$ $\frac{4{x}^{2}-4zz}{2x-2z}=b$. $\begin{array}{ccccccc}2{x}^{2}& -& 2{x}^{3}& -& bx& -& bz\\ 2& & 0& & 1& & 0\end{array}$ /$b=4x$\ $2{x}^{4}+2{z}^{4}-a{x}^{3}-a{z}^{3}-2zz\sqrt{4{x}^{2}{z}^{2}-ax{z}^{2}+aazx-2azxx}=ccxx-2{c}^{2}zx+cczz$ \ $\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2{x}^{4}-a{x}^{3}}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2{z}^{4}-a{z}^{3}}=cx-cz$/. $2x+2z-\frac{1}{2}a\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2xx-ax}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{1}{2}zz-\frac{axz}{4x}}$ \$2xx+2zx-ax$/ $2x\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{2xx-ax}\phantom{\rule{1em}{0ex}}\frac{+ax\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}a\sqrt{2xx-ax}}{\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}2\sqrt{xx-ax}}$ $\frac{-4xy+2ax-2xx+ay-2xy}{2x-2y}=\frac{-xx\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}6x\sqrt{2xx-ax}+3ax\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}x\sqrt{2xx-ax}}{\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}2\sqrt{2xx-ax}}$ $\frac{\begin{array}{c}-8xx\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}6x\sqrt{2xx-ax}\\ +3ax\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}a\sqrt{2xx-ax}\end{array}}{\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}2\sqrt{2xx-ax}}$. $\frac{64{x}^{3}-48axx+9aax}{4x-2a}$ $2xx-ax+\frac{\sqrt{4zzx-2zz}}{2x}$ $-xx+dy+yy=0$ . $\frac{2bϩz\sqrt{cc-b}}{cc}+\frac{dϩb}{c}+\frac{2abϩ}{c}+\frac{2bϩz\sqrt{cc-b}}{cc}$. $-\frac{d}{2}=+a$ . $b=0$ . $c=$ any finite line. as {illeg} $x=ϩ$. $y=-\frac{1}{2}d+z$ . ${ϩ}^{2}+\frac{1}{4}dd\mp dz-zz$ ${y}^{3}=axx+aax$ . $y+a=\varrho$ . ϩ ${\varrho }^{3}-3a{\varrho }^{2}+3aa\varrho -axx+aax-{a}^{3}=0$ $\begin{array}{ccc}\begin{array}{c}\phantom{0}450& \text{.}\\ 225\phantom{0}\end{array}& 48xx& \phantom{\rule{2em}{0ex}}\begin{array}{l}\underset{_}{54×16×27}\\ 324\\ \underset{_}{54\phantom{0}}\\ 864\end{array}\\ & \begin{array}{l}240\\ \underset{_}{48\phantom{0}}\\ 720\end{array}\end{array}$ $-{ϩ}^{3}{b}^{3}=0$ . $\frac{+6ϩabc+cϩ\sqrt{cc-bb}×a-3abcϩ}{cc}=0$ $\begin{array}{ccc}\begin{array}{c}105\\ \underset{_}{3\phantom{00}}\\ 2025\end{array}& \begin{array}{c}\phantom{0}75\\ \underset{_}{15\phantom{0}}\\ 225\end{array}& \begin{array}{c}1125\\ \underset{_}{225\phantom{0}}\\ 3375\end{array}\\ & & & \begin{array}{c}\underset{_}{16}\\ 96\\ \underset{_}{16\phantom{0}}\\ 256\end{array}\end{array}$ $db=a$. $ba=b$ . $ac=c$ . $ad=d$ . $\frac{dd}{b}=r$ , $lat\phantom{\rule{1em}{0ex}}rec:Parab\phantom{\rule{1em}{0ex}}dbe$ . $\frac{dd}{c}=s=lat:rec:Parab:dce$ . $\frac{cc}{b}=lat:rec:Par\phantom{\rule{1em}{0ex}}bgqc$ {illeg}. {illeg} $lq=a$ . $axx={y}^{3}$. $y=x+z$ . $-axx+{x}^{3}+3{x}^{2}z+3x{z}^{2}+{z}^{3}=0$ . $\sqrt{caxx}-x=z$. $\frac{+2axy-3xxy-6xyy-3{y}^{3}}{3yy+6yx+3xx}=v=\frac{2axy}{3yy+6yx+3xx}$ ${z}^{3}+3x{z}^{2}+3xxz+{x}^{3}-dxx=0$ $a=8$ . $x=1$ . {illeg} $-7+3z+3zz+{z}^{3}=0$ $\begin{array}{ccccccc}{y}^{3}& +& ayy& +& 2ab& +& abb\\ & +& 2b& +& bb\end{array}$ $\begin{array}{l}zz+4z+7\text{.}\phantom{\rule{2em}{0ex}}z=2\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{4-7}\\ +448+192z+24zz+{z}^{3}=0\text{.}\\ zz+20z+11 2\end{array}$ $\begin{array}{c}\begin{array}{cccc}3x=a+2b\text{.}& 3xx=2ab+bb& {b}^{3}-dbb=abb\text{.}& b-d=a\end{array}\\ \begin{array}{cc}3x-2b=a=\frac{3xx-bb}{2b}& -6bx+3bb+3xx=0\text{.}\\ x=a& a=b=x\text{.}\phantom{\rule{3em}{0ex}}b-a=d\text{.}\end{array}\end{array}$ $\begin{array}{c}y+a×y+b×y+b\\ \begin{array}{ccccccc}yyy& +& ayy& +& 2aby& \phantom{\rule{0.5em}{0ex}}& abb\\ & +& 2byy& +& bby\end{array}\end{array}$ $\begin{array}{c}ax=yy\text{.}\\ -ax-yy+2ay+aa=0\text{.}\\ \sqrt{ax}=y+a\text{.}\end{array}$ $\begin{array}{cccccccc}-& caϩr& -& abzc& +& ddcc& -& 2dbcϩ\\ +& 2dczr& +& cczz& +& bbϩz& -& 2aabϩ\\ -& 2bϩzr& -& bbzz& +& 2azcd\\ +& 2aazr\end{array}$. $\begin{array}{c}-2b\sqrt{cc-bb}\text{.}\phantom{\rule{2em}{0ex}}b=c\text{.}\\ -2dbc-2aab\text{.}\\ aa{c}^{4}-aabbcc=4bbccdd+8\\ 2dc+2ac=0\text{.}\phantom{\rule{2em}{0ex}}d=\frac{ac}{-c}\text{.}\phantom{\rule{2em}{0ex}}d=-a\text{.}\end{array}$ $af=a$ . $ag=b$ . $ab=\varrho$ . $bc=z$ . $a:b:bh=\frac{a\varrho -ab}{b}:\varrho -b=gb$ . $ch=ϩb-a\varrho +ab$ . $fg=c$ . $ag=b$ . $fg=c$ . $fh=\frac{cϩ}{b}$ . $af=\sqrt{cc-bb}$ . $\frac{\varrho -b\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\sqrt{cc-bb}}{b}=bh$ . $\frac{bϩ-\varrho +b×\sqrt{cc-bb}}{b}=ch$ . $\frac{bϩ-\varrho +b\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\sqrt{cc-bb}}{c}=cd$ . $\frac{abϩ-aa\varrho +aab+cc\varrho }{bc}=df$ . $cc=$ {illeg} $aa+bb$ . $\frac{ϩb-a\varrho +ab}{\sqrt{aa+bb}}=cd=x$. $\frac{aϩ-aa+bϩ}{\sqrt{aa+bb}}=df=y$. $\frac{bϩ+ab-x\sqrt{aa+bb}}{a}=\varrho =\frac{y\sqrt{aa+bb}-aa-aϩ}{b}$ $bbϩ+aaϩ=ay\sqrt{aa+bb}-{a}^{3}${illeg} $-abb+x\sqrt{aa+bb}$ . a {illeg} $ϩ=\frac{ay-a\sqrt{aa+bb}+bx}{\sqrt{aa+bb}}$ . Or $ϩ=\frac{ay-ac+x\sqrt{cc-aa}}{c}$ . /$\varrho =\frac{y\sqrt{cc-aa}-ax}{c}$ \ $\varrho =\frac{cy}{b}-\frac{aay}{bc}-\frac{xa}{c}$ {illeg} $c=5$ . $a=3$ . $ϩ=\frac{3y-15+4x}{5}$ . $\varrho =\frac{4y-3x}{5}$ . {illeg}${y}^{4}$ ${ϩ}^{4}={\varrho }^{3}$ . |5 | $\begin{array}{ccc}27{y}^{3}-405yy+2025y-3375& =& 256{y}^{4}-768{y}^{3}x+864yyxx-432y{x}^{3}+81{x}^{4}\\ +108yyx-1080yx+2700x\\ 144yxx-720xx\\ +64{x}^{3}\end{array}$ $\varrho \varrho ϩ={a}^{3}$ . {illeg} $c=5r$ . $a=4r$ . $ϩ=\frac{4y-20r+3x}{5}$ . $\varrho =\frac{3y-4x}{5}$ {illeg} $=2rr$ . {illeg} $=rr$ .
$\frac{9yy-24xy+16xx}{25}$ . $\begin{array}{c}\begin{array}{ccccccccc}36{y}^{3}& -& 96xyy& +& 64xxy& +& 48{x}^{3}& =& 225{a}^{3}\\ & +& 27\hfill & -& 72xxy\end{array}\\ -180ryy+480rxy-360rxx\end{array}$ $12{y}^{3}$ {illeg}3 {illeg}$yy$ $\begin{array}{c}\begin{array}{ccccccccc}36{y}^{3}& -& 69xyy& -& 8xxy& +& 48{x}^{3}& =& 0\end{array}\\ -180ryy+480rxy-360rxx\\ rrr\end{array}$.

<7r>

$\begin{array}{cc}\begin{array}{llllll}& 2bqff& +& 4aqyff& & \\ -& 2qeff& & & & \\ -& 2dcf& & & +& {d}^{3}yy\\ & & -& dbyf\\ +& bef& +& 2aefy\\ +& dee& +& 2ddey\\ -& defq& -& ddfqy\end{array}& \begin{array}{cc}=& \begin{array}{c}2pff+bf+2afy\\ +de+ddy-dfq-2ffy\end{array}in\sqrt{ee+2edy+ddyy-4af-4bfy-4a}\end{array}\end{array}${illeg} $\begin{array}{cc}\begin{array}{llllllll}& 4bbqq{f}^{4}& +& 16abqq{f}^{4}y& +& 6bqff{d}^{3}yy& +& 8aqff{d}^{3}{y}^{3}\\ -& 8bq{f}^{3}dc& -& 4bbq{f}^{3}d& -& 4{d}^{4}cf& -& 2{d}^{4}bf\\ +& 4bbq{f}^{3}e& +& 8ae{f}^{3}bq& +& 2bef{d}^{3}& +& 4aef{d}^{3}\\ & & +& 6bqffdde& & & +& 4{d}^{5}e\\ -& 4bqq{f}^{3}de& -& 4bqq{f}^{3}dd& & & -& 2{d}^{5}fq\\ & & & & +& 16aaqq{f}^{4}\\ & & +& 4ddcffb& -& 8aq{f}^{3}db\\ +& 4ddccff& -& 16dc{f}^{3}aq& +& 16aa{q}^{3}{f}^{3}& +& {d}^{6}{y}^{4}\\ -& 4dcffbe& -& 8dcffae& & \\ -& 4ddcfee& -& 8{d}^{3}fce& -& 8aqq{f}^{3}dd\\ +& 4ddcffeq& +& 4{d}^{3}cffq& +& ddbbff\\ +& bbeeff& +& 8aqbe{f}^{3}& -& 4aedbff\\ +& 2b{e}^{3}fd& -& 2bbeffd& -& 4{d}^{3}ebf\\ +& 2beeffdq& +& 4abeeff& & \\ +& dd{e}^{4}& +& 4bddeef& +& 4aaeeff\\ -& 2d{e}^{3}fq& & & +& 8aeeddf\\ +& ddeeffqq& & & & \\ & & -& 2ddeebf\\ & & +& 4ad{e}^{3}f& +& 6{d}^{4}ee\\ & & +& 4{d}^{3}{e}^{3}& -& 10{d}^{4}efq\\ & & & & +& {d}^{4}ffqq\\ & & -& 8ade{f}^{3}qq& +& 12aeddfq\\ & & +& 2ddeffqb\\ & & +& 4adeeffq\\ & & -& 6{d}^{3}eefq\\ & & +& 2{d}^{3}effqq\\ & & -& 8pp{f}^{4}edy\end{array}& \begin{array}{lllllll}=& & 4pp{f}^{4}& +& 4pb{f}^{3}& & \\ & +& 4bb{f}^{3}& +& 8pa{f}^{3}q& +& 4aaffyy& in& \begin{array}{c}ee\phantom{\rule{2em}{0ex}}+\\ -4bf-4cf\end{array}\\ & +& 4pdeff& +& 4pddff& +& 4afdd\\ & +& 4d{f}^{3}qq& +& 4abff& +& {d}^{4}\\ & -& 4pdq{f}^{3}& +& 2ddbf\\ & -& 8pq{f}^{4}& +& 4afde\\ & +& bbff& -& 4adffq\\ & +& 2debf& -& 8aq{f}^{3}\\ & -& 2dbffq& +& 2e{d}^{3}\\ & -& 4bq{f}^{3}& -& 2{d}^{3}fq\\ & +& ddee& -& 4ddffq\\ & -& 2ddefq\\ & -& 4deffq\\ & +& ddffqq\\ & +& {f}^{4}qq\end{array}\end{array}$ $bc=x$. $cd=y$. $ef=a$. $ea=b$. $ad=c$. $\sqrt{xx+yy-cc}\phantom{\rule{0.5}{0ex}}\left(={ab}^{2}\right)+b=eb$ . $\frac{ax}{b+\sqrt{xx+yy-cc}}=ch$ . $aa+xx+yy-cc+bb+2b\sqrt{xx+yy-cc}={bf}^{2}$ $\frac{c\sqrt{aa+xx+yy-cc+bb+2b\sqrt{xx+yy-cc}}+ax}{b+\sqrt{xx+yy-cc}}=y$. $ccaa+ccxx+2ccyy-{c}^{4}+bbcc+2bcc\sqrt{xx+yy-cc}=bbyy\phantom{\rule{0.5em}{0ex}}\begin{array}{l}+2b\sqrt{{y}^{2}\phantom{\rule{0.5em}{0ex}}\text{(&c)}}\\ -2a\sqrt{xx}\end{array}\phantom{\rule{0.5em}{0ex}}+xxyy+{y}^{4}$. {illeg} $yy+xx+bb-cc+2b\sqrt{xx+yy-cc}\phantom{\rule{1em}{0ex}}\frac{-ccaa-2axy\sqrt{xx+yy}-aaxx-ax}{yy-cc}$ $ac=x$. $bc=y$. $af=z$. $bf=\frac{zz}{a}$. \{illeg}/ $df=\frac{a}{2}$ . $de=\frac{aa+4zz}{2a}$ . $fe=\frac{2zz}{a}:z\colon\colon x:\frac{aa+zz}{2a}$ $2aax=2aaz+4a{z}^{3}$
$bf=z$ $af=\sqrt{az}$ . $2z$ $ad=\sqrt{\frac{aa}{4}+az}$ $z:\frac{a}{4}+z\colon\colon xx:\frac{aa}{4}+az+zz$ . $4{z}^{3}+4azz+aaz=axx+4zxx$ .

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

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

If ye knowne qua\n/titys are every where divers, & one of ym be blotted {illeg} out yt produceth a line, when one terme is already wanting

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

In ye {illeg}|H|yperbola ye area of it beares ye same respect to its Asymptote wch a logarithme {di}{illeg} number.

To make ye equation ${x}^{3}-a{x}^{2}+abx-abc=0$ . be divisible by $x-c=0$ . suppose $c=x$ , yn tis ${c}^{3}-acc+0=0$ $c=a$ . therefore write c in steade of a & it is ${x}^{3}-c{x}^{2}+cbx-bcc=0$ . wch is divisible by $x-c$ To make ye same Equation divisible by $xx-2ax+ac=0$ Suppose it to bee divided by it & ye ration will bee $\begin{array}{ccccccccccc}xx-2ax+ab\right)& & {x}^{3}& -& a{x}^{2}& +& abx& -& abc& =& 0\left(x+a\\ & -& {x}^{3}& +& 2a{x}^{2}& -& abx\\ & & 0& +& a{x}^{2}& +& 0& -& abc\\ & & & -& axx& +& 2aax& -& aab\\ & & & & 0& +& 2aax& -& aab& -& abc\hfill \end{array}$. The quote is $2aax-abc-aab$ wch have vanished therefore to m{illeg}|a|ke soe suppose each terme $=0$ & the{illeg} will be $2aax=0$ & $abc+aba=0$ both include $a=0$ . Which since it cannot happen ye equation cannot be divided ye one by ye 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.

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

<7v>

 $dd{\varrho }^{2}+2d\varrho ϩ\sqrt{ee-dd}+eeϩϩ-ddϩϩ=eexx$. ${d}^{3}{\varrho }^{3}+3dd\varrho \varrho ϩ\sqrt{ee-dd}+3d\varrho {ϩ}^{2}ee-3d\varrho {ϩ}^{2}dd\phantom{\rule{0.5em}{0ex}}\begin{array}{c}+ee\\ -dd\end{array}{ϩ}^{3}\sqrt{ee-dd}=${${e}^{3}{x}^{3}$}  $6dd\varrho \varrho ϩϩee-6{d}^{4}\varrho \varrho ϩϩ\phantom{\rule{0.5em}{0ex}}\begin{array}{c}+4ee\\ -4dd\end{array}d\varrho {ϩ}^{3}\sqrt{ee-dd}+{e}^{4}{ϩ}^{4}-2eedd{ϩ}^{4}+{d}^{4}{ϩ}^{4}={e}^{4}{x}^{4}$  $\begin{array}{c}+10ee\\ -10dd\end{array}{d}^{3}{\varrho }^{3}ϩϩ\phantom{\rule{1em}{0ex}}\begin{array}{c}+10ee\\ -10dd\end{array}dd\varrho \varrho {ϩ}^{3}\sqrt{ee-dd}\phantom{\rule{1em}{0ex}}\begin{array}{l}+5{e}^{4}\\ -10eedd\\ +5{d}^{4}\end{array}d\varrho {ϩ}^{4}\phantom{\rule{1em}{0ex}}\begin{array}{l}+{e}^{4}\\ -2eedd\\ +{d}^{4}\end{array}{ϩ}^{5}\sqrt{ee-dd}={e}^{5}{x}^{5}$  $ccee-2cedϩ+ddϩϩ\phantom{\rule{1em}{0ex}}\begin{array}{l}+2ce\varrho \\ -2dϩ\varrho \end{array}\sqrt{ee-dd}\phantom{\rule{1em}{0ex}}\begin{array}{l}+ee\varrho \varrho \\ -dd\varrho \varrho \end{array}=eeyy$. $\begin{array}{llllllllllll}+& {c}^{3}{e}^{3}& +& 3ccee\varrho & \sqrt{ee-dd}& +& 3c{e}^{3}\varrho \varrho & +& ee& {\varrho }^{3}\sqrt{ee-dd}& =& {e}^{3}{y}^{3}\\ -& 3cceedϩ& -& 6cedϩ\varrho & & -& 3eedϩ\varrho \varrho & -& dd\\ +& 3ceddϩϩ& +& 3ddϩϩ\varrho & & -& 3cedd\varrho \varrho \\ -& {d}^{3}{ϩ}^{3}& & & & +& 3{d}^{3}ϩ\varrho \varrho \end{array}$. $\begin{array}{llllllllllllllll}+& {c}^{4}{e}^{4}& +& 4{c}^{3}{e}^{3}& \varrho \sqrt{ee-dd}& +& 6cc{e}^{4}& \varrho \varrho & +& 4c{e}^{3}& {\varrho }^{3}\sqrt{ee-dd}& +& {e}^{4}& {\varrho }^{4}& =& {e}^{4}{y}^{4}\\ -& 4{c}^{3}{e}^{3}dϩ& -& 12cceedϩ& & -& 12c{e}^{3}dϩ& & -& 4cedd& & -& 2eedd\\ +& 6cceeddϩϩ& +& 12ceddϩϩ& & +& 6eeddϩϩ& & -& 4eedϩ& & +& {d}^{4}\\ -& 4ce{d}^{3}{ϩ}^{3}& -& 4{d}^{3}{ϩ}^{3}& & -& 6cceedd& & +& 4{d}^{3}ϩ\\ +& {d}^{4}{ϩ}^{4}& & & & +& 12cce{d}^{3}ϩ\\ & & & & & -& 6{d}^{4}ϩϩ\end{array}$. $\begin{array}{lllllllllllllllllll}& {c}^{5}{e}^{5}& +& 5{c}^{4}{e}^{4}& \varrho \sqrt{ee-dd}& +& 10{c}^{3}{e}^{5}& \varrho \varrho & +& 10cc{e}^{4}& {\varrho }^{3}\sqrt{ee-dd}& +& 5c{e}^{5}& {\varrho }^{4}& +& {e}^{4}& {\varrho }^{5}\sqrt{ee-dd}& =& {e}^{5}{y}^{5}\\ -& 5{c}^{4}{e}^{4}dϩ& -& 20{c}^{3}{e}^{3}dϩ& & -& 30cc{e}^{4}dϩ& & -& 20c{e}^{3}dϩ& & -& 5dϩ{e}^{4}& & -& 2eedd\\ +& 10{c}^{3}{e}^{3}dd{ϩ}^{2}& +& 30cceedd{ϩ}^{2}& & +& 30c{e}^{3}ddϩϩ& & +& 10eeddϩϩ& & -& 10c{e}^{3}dd& & +& {d}^{4}\\ -& 10ccee{d}^{3}{ϩ}^{3}& -& 20ce{d}^{3}{ϩ}^{3}& & -& 10ee{d}^{3}{ϩ}^{3}& & -& 10cceedd& & +& 10{d}^{3}ϩee\\ +& 5ce{d}^{4}{ϩ}^{4}& +& 5{d}^{4}{ϩ}^{4}& & -& 10{c}^{3}{e}^{3}dd& & +& 20ce{d}^{3}ϩ& & +& 5ce{d}^{4}\\ -& {d}^{5}{ϩ}^{5}& & & & +& 30ccee{d}^{3}ϩ& & -& 10{d}^{4}ϩϩ& & -& 5ϩ{d}^{5}\\ & & & & & -& 30ce{d}^{4}ϩϩ\\ & & & & & +& 10{d}^{5}{ϩ}^{3}\end{array}$.  $\begin{array}{llllllll}& cceeϩ\sqrt{ee-dd}& +& 2c{e}^{3}ϩ\varrho & +& eeϩ\varrho \varrho \sqrt{ee-dd}& =& {e}^{3}xyy\\ & & & & -& 3ddϩ\\ +& dd{ϩ}^{3}& -& 4ceddϩ\varrho & & \\ & & & \\ & & +& cceed\varrho \end{array}$.  ${e}^{4}xxyy=\begin{array}{ll}-& 2c{e}^{3}d{ϩ}^{3}\\ +& 2ce{d}^{3}{ϩ}^{3}\end{array}\phantom{\rule{1em}{0ex}}\begin{array}{lll}-& 2dee\varrho {ϩ}^{3}& \sqrt{ee-dd}\\ +& 4{d}^{3}\varrho {ϩ}^{3}\\ +& 2ccdee\varrho ϩ\\ & \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{ll}+& 4cd{e}^{3}\varrho \varrho ϩ\\ -& 6c{d}^{3}e\end{array}\phantom{\rule{1em}{0ex}}\begin{array}{ll}+& 2dee{\varrho }^{3}ϩ\sqrt{ee-dd}\\ -& 4{d}^{3}\end{array}$. $\begin{array}{l}\\ & & -& 6ddce\varrho \varrho ϩ\sqrt{{e}^{2}-{d}^{2}}& -& 5ddee{\varrho }^{3}ϩ& +& {c}^{3}{e}^{3}ϩ\sqrt{ee-dd}& =& {e}^{4}x{y}^{3}\\ -& 4{d}^{4}\varrho {ϩ}^{3}& +& 3c{e}^{3}\varrho \varrho ϩ\sqrt{ee-dd}& +& 4{d}^{4}{\varrho }^{4}ϩ& +& cedd{ϩ}^{3}\\ +& 3cc{e}^{4}\varrho ϩ& -& 3cedd\varrho \varrho ϩ\sqrt{ee-dd}& +& {e}^{4}{\varrho }^{3}ϩ\\ +& 3ddee\varrho {ϩ}^{3}\end{array}$ . $\begin{array}{llllllllllll}-& 8dd{c}^{3}{e}^{3}\varrho ϩ& -& 18ddccee\varrho \varrho ϩ\sqrt{{e}^{2}-{d}^{2}}& -& 2ddc{e}^{3}{\varrho }^{3}ϩ& -& 6ddee{\varrho }^{4}ϩ\sqrt{ee-dd}& +& {c}^{4}{e}^{4}ϩ\sqrt{ee-dd}& =& {e}^{5}x{y}^{4}\\ -& 16{d}^{4}ce\varrho {ϩ}^{3}& -& 10{d}^{4}\varrho \varrho {ϩ}^{3}& +& 16{d}^{4}ce{\varrho }^{3}ϩ& +& 5{d}^{4}{\varrho }^{4}ϩ& +& 6ccddee{ϩ}^{3}\\ +& 4{c}^{3}{e}^{5}\varrho ϩ& +& 6cc{e}^{4}\varrho \varrho ϩ& +& 4c{e}^{5}{\varrho }^{3}ϩ& +& {e}^{4}{\varrho }^{4}ϩ& +& {d}^{4}{\varrho }^{5}\\ +& 12c{e}^{3}dd\varrho {ϩ}^{3}& +& 6ddee\varrho \varrho {ϩ}^{3}\end{array}$. $\begin{array}{llllllllllllll}& {c}^{5}{e}^{5}ϩ\sqrt{ee-dd}& +& 5{c}^{4}{e}^{6}\varrho ϩ& +& 10{c}^{3}{e}^{5}\varrho \varrho ϩ\sqrt{ee-dd}& +& 10cc{e}^{6}{\varrho }^{3}ϩ& +& 5c{e}^{5}{\varrho }^{4}ϩ\sqrt{ee-dd}& +& {e}^{6}{\varrho }^{5}ϩ& =& {e}^{6}x{y}^{5}\\ +& 10{c}^{3}{e}^{3}{ϩ}^{3}dd& -& 10{c}^{4}{e}^{4}dd\varrho ϩ& +& 30c{e}^{3}dd\varrho \varrho {ϩ}^{3}& -& 50cc{e}^{4}dd{\varrho }^{3}ϩ& -& 30c{e}^{3}dd{\varrho }^{4}ϩ& -& 8{e}^{4}dd\\ +& 5ce{d}^{4}{ϩ}^{5}& +& 30cc{e}^{4}dd\varrho {ϩ}^{3}& -& 30{c}^{3}{e}^{3}dd\varrho \varrho ϩ& +& 10dd{e}^{4}{\varrho }^{3}{ϩ}^{3}& +& 25ce{d}^{4}{\varrho }^{4}ϩ& +& 13ee{d}^{4}\\ & & -& 40ccee{d}^{4}\varrho {ϩ}^{3}& -& 50ce{d}^{4}\varrho \varrho {ϩ}^{3}& -& 30ee{d}^{4}{\varrho }^{3}{ϩ}^{3}& & & -& 6{d}^{6}\\ & & +& 5ee{d}^{4}\varrho {ϩ}^{5}& & & +& 40ccee{d}^{4}{\varrho }^{3}{ϩ}^{3}\\ & & -& 6{d}^{6}\varrho {ϩ}^{5}& & & +& 20{d}^{6}{\varrho }^{3}{ϩ}^{3}\end{array}$. $\begin{array}{llllllllllll}-& 3{c}^{2}eeeed{ϩ}^{3}& -& 6c{e}^{3}d{ϩ}^{3}\varrho \sqrt{ee-dd}& -& 3{e}^{4}d\varrho \varrho {ϩ}^{3}& +& 6cd{e}^{3}{\varrho }^{3}ϩ\sqrt{ee-dd}& +& {e}^{4}d{\varrho }^{4}ϩ& =& {e}^{5}xx{y}^{3}\\ -& {d}^{3}ee{\varrho }^{5}& +& 12ce{d}^{3}\varrho {ϩ}^{3}& +& 12ee{d}^{3}\varrho \varrho {ϩ}^{3}& -& 12c{d}^{3}e& & \\ +& 3ccee{d}^{3}{ϩ}^{2}& +& 2{c}^{3}{e}^{3}d\varrho ϩ& -& 10{d}^{5}\varrho \varrho {ϩ}^{3}& & \\ +& {d}^{5}{ϩ}^{5}& & & +& 6ccd{e}^{4}\varrho \varrho ϩ& & & -& 5ee{d}^{3}\\ & & & & +& 3ccee{d}^{3}ϩ& & & +& 4{d}^{5}\\ & & & & & \end{array}$. $\begin{array}{llllllllllllll}-& 4{c}^{3}{e}^{5}d{ϩ}^{3}& -& 12cc{e}^{4}d{ϩ}^{3}\varrho \sqrt{ee-dd}& -& 12cd{e}^{5}{ϩ}^{3}{\varrho }^{2}& -& 4d{e}^{4}{ϩ}^{3}{\varrho }^{3}\sqrt{ee-dd}& +& 8c{e}^{5}dϩ{\varrho }^{4}& +& 2d{e}^{4}\varrho ϩ\sqrt{ee-dd}& =& {e}^{6}xx{y}^{4}\\ -& 4c{e}^{3}{d}^{3}{ϩ}^{5}& +& 24ccee{d}^{3}{ϩ}^{3}& & & & & & & -& 8{d}^{3}ee{\varrho }^{5}ϩ\\ +& 4{c}^{3}{e}^{3}{d}^{3}{ϩ}^{3}& -& 4{d}^{3}ee{ϩ}^{5}& -& 4ce{d}^{5}{ϩ}^{3}& +& 20{d}^{5}{ϩ}^{3}& +& 20ce{d}^{5}ϩ& +& 6{d}^{5}{\varrho }^{5}ϩ\\ +& 4ce{d}^{5}{ϩ}^{5}& +& 6{d}^{5}{ϩ}^{5}& +& 8{c}^{3}d{e}^{5}ϩ\varrho \varrho & +& 12ccd{e}^{4}ϩ& -& 28c{e}^{3}{d}^{3}ϩ\\ & & +& 2{c}^{4}{e}^{4}dϩ& & & +& 20{d}^{3}ee{ϩ}^{3}\\ & & & & +& 48c{e}^{3}{d}^{3}{ϩ}^{3}\varrho \varrho & -& 24ccee{d}^{3}ϩ\\ & & & & -& 12{c}^{3}{d}^{3}{e}^{3}ϩ& & \end{array}$. $\begin{array}{llllllllll}& c{e}^{3}{ϩ}^{3}\sqrt{ee-dd}& +& 4{d}^{4}\varrho {ϩ}^{3}& +& 3ceddϩ\varrho \varrho \sqrt{ee-dd}& -& 4{d}^{4}{\varrho }^{3}ϩ& =& {e}^{4}{x}^{3}y\\ -& ceedd& -& 5ddee\varrho {ϩ}^{3}& & & +& 3ddee\\ & & +& {e}^{4}\varrho {ϩ}^{3}\end{array}$. $\begin{array}{llllllllllll}+& cc{e}^{4}{ϩ}^{3}\sqrt{ee-dd}& +& 8ce{d}^{4}\varrho {ϩ}^{3}& +& 3ccddee{\varrho }^{2}ϩ\sqrt{ee-dd}& -& 8ce{d}^{4}{\varrho }^{3}ϩ& -& 5{d}^{4}{\varrho }^{4}ϩ\sqrt{ee-dd}& =& {e}^{5}{x}^{3}yy\\ -& cceedd{ϩ}^{3}& -& 10c{e}^{3}dd\varrho {ϩ}^{3}& +& 10{d}^{4}\varrho \varrho {ϩ}^{3}& +& 6c{e}^{3}dd{\varrho }^{3}ϩ& +& 3ddee{\varrho }^{4}ϩ\\ +& ddee{\varrho }^{5}& +& 2c{e}^{5}\varrho {ϩ}^{3}& -& 8ddee\varrho \varrho {ϩ}^{3}\\ -& {d}^{4}{ϩ}^{5}& & & +& {e}^{4}\varrho \varrho {ϩ}^{3}\end{array}$. 

<8r>

 $\varrho \varrho ϩϩ-{\varrho }^{2}{g}^{2}=$ { ${b}^{4}$ . } $a=r$. $c=2r$. {illeg} $\frac{y-2r+x\sqrt{x}}{2}=ϩ$ . $\varrho =\frac{-x+y\sqrt{3}}{2}$ $xxyy-2x{y}^{3}$ {illeg} $ϩ=\frac{ay-ac+x\sqrt{cc-aa}}{c}$ . $\varrho =\frac{y\sqrt{cc-aa}-ax}{c}$ $\begin{array}{}\begin{array}{llllllll}& aacc{y}^{4}& -& 2aa{c}^{3}{y}^{3}& +& 2accx{y}^{3}\sqrt{cc-aa}& -& 4aaccxxyy\\ & {a}^{4}{y}^{4}& -& 2{a}^{4}c{y}^{3}& +& 2{a}^{3}x{y}^{3}\sqrt{cc-aa}& +& 4{a}^{4}xxyy\\ & & & & -& 2{a}^{3}x{y}^{3}\sqrt{cc-aa}& +& {a}^{4}xxyy\\ & & & & +& 4{a}^{3}cxyy\sqrt{cc-aa}& +& aa{c}^{4}yy\\ & & & & & & +& {c}^{4}xxyy\\ & & & & -& 2a{c}^{3}x{y}^{2}\sqrt{cc-aa}& -& aaccxxyy\\ & & & & & & +& {a}^{4}ccyy\end{array}\begin{array}{lll}ϩϩ& =& aayy-2aacy+aacc+ccxx-aaxx\\ & & +2ayx\sqrt{cc-aa}-2acx\sqrt{cc-aa}\\ \phantom{0}\\ \varrho \varrho & =& cc{y}^{2}-aayy-2axy\sqrt{cc-aa}+aaxx\\ \phantom{0}\\ & & \begin{array}{cccccc}-& 2aaxy& +& aacx& -& axx\sqrt{cc-aa}\\ +& ccxy& & & +& ayy\sqrt{cc-aa}\\ & & & & -& acy\sqrt{cc-aa}\end{array}\end{array}\end{array}$ $\begin{array}{llllll}& aacc{x}^{4}& +& 4{a}^{3}{x}^{3}y\sqrt{cc-aa}& +& 6{a}^{4}xxyy\\ -& {a}^{4}{x}^{4}& -& 2{a}^{3}c{x}^{3}\sqrt{cc-aa}& -& 6aaccxxyy& +& 6{a}^{3}cxyy\sqrt{\phantom{000}}& +& aacc{y}^{4}\\ & & -& 2acc{x}^{3}y\sqrt{cc-aa}& +& 4aa{c}^{3}xxy& -& 4{a}^{3}x{y}^{3}\sqrt{\phantom{000}}& -& {a}^{4}{y}^{4}\\ & & & & -& 6{a}^{4}cxxy& -& 2{a}^{3}ccxy\sqrt{\phantom{000}}& +& 2{a}^{4}c{y}^{3}\\ & & & & & & & & -& 2aa{c}^{3}{y}^{3}\\ & & & & +& {a}^{4}ccxx& +& 2accx{y}^{3}\sqrt{\phantom{000}}& +& aa{c}^{4}yy\\ & & & & +& {c}^{4}xxyy& -& 2a{c}^{3}xyy\sqrt{\phantom{000}}& -& {a}^{4}ccyy\end{array}$ $\begin{array}{cc}\begin{array}{l}\begin{array}{lllll}3{x}^{4}& -& 4{x}^{3}y\sqrt{3}& -& 2xxyy\\ & -& 2c{x}^{3}\sqrt{3}& +& 10cxxy& +& 2cxyy\sqrt{3}& +& 3{y}^{4}& =& 0\\ & & & +& ccxx& -& 2ccxy& -& 6c{y}^{3}\\ & & & & & +& 4x{y}^{3}& +& 2ccyy\\ & & & & & & & -& {g}^{3}y\sqrt{3}\end{array}\\ \phantom{0}\\ \begin{array}{lllllll}3{x}^{4}& -& 4{x}^{3}y\sqrt{3}& +& 6cxxy& +& 4x{y}^{3}\sqrt{3}\\ & -& 2c{x}^{3}\sqrt{3}& -& 6xxyy& -& 2ccxy\sqrt{3}\\ & & & +& 4xxyy\\ & & & +& 4cxyy& +& 2cxyy\sqrt{3}\\ & & & +& ccxx\end{array}\end{array}& \begin{array}{c}\begin{array}{llllllllllll}& +& yy\sqrt{3}& & & & & -& xx\sqrt{3}& +& 2xy& \text{.}\\ & & & +& cx& -& cy\sqrt{3}\end{array}\\ \phantom{0}\\ \begin{array}{lllllllllll}xxyy& -& 2axxy& +& aaxx& -& aaxx& -& {a}^{4}& =& 0\end{array}\\ \phantom{0}\\ \begin{array}{lllllllll}xxyy& +& 4bxyy& +& 4bbyy& -& 2axxy& -& 8abxy\\ & & & & & -& {a}^{4}& -& 8abby& -\end{array}\end{array}\end{array}$

 $bc=x$. $cd=y$. $bf=c$. $bp=\varrho$. $pd=ϩ$. $bc=d$. $bg=${illeg} $fe:fg\colon\colon d:e$ . $fe:eg\colon\colon d:f$. bf $hp=${illeg} z. $fg=\frac{ex}{d}$ . $eg=\frac{fx}{d}$ . $pd:dh\colon\colon r:s$. $dh=\frac{sϩ}{r}$ . $r:t\colon\colon pd:ph=\frac{tϩ}{r}$. $d:e\colon\colon dh:dg=\frac{esϩ}{dr}$. $gh=\frac{fsϩ}{dr}$ $fg=\varrho +\frac{tϩ}{dr}+\frac{fsϩ}{dr}=\frac{ex}{d}$ . $\frac{dr\varrho +dtϩ+fsϩ}{er}=x$ . $\frac{rfx+cdr-esϩ}{dr}=y$ . $\frac{fdr\varrho +dtfϩ+ffsϩ+cder-eesx}{der}=y=\frac{fr\varrho +tfϩ-dsϩ+cer}{er}$ $ec=c$. $bc=x$. $dc=y$. $pd=ϩ$. $pf=\varrho$. $fg=\frac{ex}{d}$ . $eg=\frac{fx}{d}$ . $pg=\frac{sϩ}{r}$ . $gd=\frac{tϩ}{r}$ . $fg=\frac{\varrho r+sϩ}{r}=\frac{ex}{d}$ . {illeg} $x=\frac{dr\varrho +dsϩ}{re}$. $eg+ec-gd=y=\frac{fr\varrho +fsϩ+rec-teϩ}{re}$.

 $fe=bc=x$. $cd=y$. $fp=\varrho$. $pd=ϩ$. $fe:fg\colon\colon d:e\colon\colon dh:dg$. $fe:eg\colon\colon d:f\colon\colon hd:hg$. $pd\colon\colon dh\colon\colon r:s$. $pd:ph:r:ht$ $\frac{dr\varrho -dtϩ+fsϩ}{er}=x$. $\frac{fr\varrho -ftϩ-dsϩ+cer}{er}=y$. $d=r$ . $f=\sqrt{ee-dd}$ . $s=\sqrt{rr-tt}$ . $s=\sqrt{dd-tt}$ . $\frac{dd\varrho -dtϩ+ϩ\sqrt{eedd-eett+ddtt-{d}^{4}}}{ed}=x$. $\frac{d\varrho \sqrt{ee-dd}-tϩ\sqrt{ee-dd}-dϩ\sqrt{dd-tt}+ced}{ed}=y$. Lastly $dp=-dt+\sqrt{eedd-eett+ddtt-{d}^{4}}$ . $n=\sqrt{ee-dd}$ . $dq=t\sqrt{ee-dd}+d\sqrt{dd-tt}$ . & Therefore $x=\frac{d\varrho +pϩ}{e}$. & $y=\frac{n\varrho -qϩ+ce}{e}$.

 $bf=c$. $fa=z$. $fk=v$. $bc=x$. $cd=y$. $ap=\varrho$. $dp=ϩ$. $vv+zz:vv\colon\colon ϩϩ\colon\colon \frac{vvϩϩ}{vv+zz}={pg}^{2}$. $d:e:\frac{vϩ}{\sqrt{vv+zz}}:\frac{evϩ}{d\sqrt{vv+zz}}=po$. $ac=x-z:ao=\frac{ex-ez}{d}\colon\colon d:e$. $\frac{ex-ez}{d}\phantom{\rule{1em}{0ex}}\frac{-evϩ}{d\sqrt{vv+zz}}=\varrho$ . $d\varrho \sqrt{vv+zz}$ .

 $bf=c$. $af=z$. $bc=x$. $fc=x-c=an$. $fk=v$. {illeg} $d:e\colon\colon an:no=\frac{ex-ec}{d}$. $oc=\frac{ex-ec+dz}{d}$ . $ak=\sqrt{vv+zz}:z\colon\colon ed=ϩ:\frac{zϩ}{\sqrt{vv+zz}}=gd$. $\frac{vϩ}{\sqrt{vv+zz}}=eg$ . $go=\frac{evϩ}{d\sqrt{vv+zz}}$. $\frac{ex-ec+dz}{d}\phantom{\rule{1em}{0ex}}\frac{-dzϩ-evϩ}{d\sqrt{vv+zz}}=y$



<8v>

$fd=x$. $db=y$. $dc=v$. ed{illeg} $ab=z$. $vv+yy:vv\colon\colon zz\phantom{\rule{1em}{0ex}}\frac{vvzz}{vv+yy}={ed}^{2}$. $fe=$ {illeg} x {illeg} / y \ $fe=x\phantom{\rule{0.5em}{0ex}}\frac{\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}vz}{\sqrt{yy+vv}}$. $ae=y\phantom{\rule{0.5em}{0ex}}\frac{\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}yz}{\sqrt{yy+vv}}$. in ye 1st case. $fe=x\phantom{\rule{0.5em}{0ex}}\frac{\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}vz}{\sqrt{yy+vv}}$. $ae=-y\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\frac{yz}{\sqrt{yy+vv}}$. in ye 2d {illeg}|c|ase. $fe=\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}x+\frac{vz}{\sqrt{yy+vv}}$. $ae=y$.

 Have{illeg}|in|g ye 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

 If $fd=x$. $db=y$. & y being pe{illeg}|r|pendicular to x describes ye crooked line ye crook ye line wth its one of its extremes. Then reduce ye Equation (expressing ye nature of ye line in wch x & y onely are undetermin{illeg}|ed|) to one side soe yt it be $=0$. ☞ Then Then {sic} find ye perpendicular bc {illeg} wch is done by findind|g| $dc=v$. for $vv+yy={bc}^{2}$ (In finding $dc=v$ {illeg}  {sic}\obse{illeg}|r|v{illeg}|s| this rule./ Multiply {illeg} ye each terme of ye Equat: by so many units as x hath dimensions in yt terme, divide it by x & multiply it by y {illeg} for a Numerator. Againe multiply each terme of ye Equation by soe many units as y hath dimensions in each terme and divide by $-y$ f{illeg}|o|r a denom: in ye val{illeg}|o|r of v.  Example, $-rx+\frac{rxx}{q}+yy=0$. $\frac{-1rx+\frac{2rxx}{q}+0yy\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}y}{x}=-ry+\frac{2rxy}{q}$. $\frac{-0rx+\frac{0rxx}{q}+2yy}{-y}=-2y$. t{illeg}|h|erefore $\frac{ry+\frac{2rxy}{q}}{-2y}=v=+\frac{1}{2}r-\frac{rx}{q}$. Also if $\begin{array}{cccccccc}{x}^{3}& -& bxx& +& yyx& -& {y}^{3}& =0\\ & +& yxx\end{array}$. $\begin{array}{c}\frac{\begin{array}{cccccccc}\phantom{+}& 3{x}^{3}& -& 2bxx& +& yyx& in& y\\ & & +& 2yxx\end{array}}{x}\\ \frac{\begin{array}{cccccc}-& 3{y}^{3}& +& 2yxx& +& yxx\end{array}}{-y}\end{array}\begin{array}{c}\phantom{\frac{\begin{array}{c}\phantom{0}\\ \phantom{0}\end{array}}{\phantom{0}}}=\\ \phantom{\frac{\begin{array}{c}\phantom{0}\end{array}}{\phantom{0}}}=\end{array}\phantom{\rule{0.5em}{0ex}}\begin{array}{cc}\begin{array}{c}\phantom{\frac{\begin{array}{c}\phantom{0}\\ \phantom{0}\end{array}}{\phantom{0}}}3xxy-2bxy+2xyy+{y}^{3}\phantom{\frac{\begin{array}{c}\phantom{0}\\ \phantom{0}\end{array}}{\phantom{0}}}\\ \phantom{\frac{\begin{array}{c}\phantom{0}\end{array}}{\phantom{0}}}3yy-2yx+xx\phantom{\frac{\begin{array}{c}\phantom{0}\end{array}}{\phantom{0}}}\end{array}=& v\end{array}$  And if ${x}^{4}-yyxx+aayx-{y}^{4}=0$. then $\begin{array}{cc}\begin{array}{c}4y{x}^{3}-2{y}^{3}x+aayy\\ 4{y}^{3}-aax+2yxx\end{array}=& v\end{array}$. &c) Then make $ab=z$. $fe=x\frac{+vz}{\sqrt{yy+vv}}$. $ae=y\frac{-vz}{\sqrt{yy+vv}}$. & substitute this valor of $\left(fe\right)$ into ye place of x {illeg} & this valor of $\left(ae\right)$ into ye place of y in ye Equation \& th{illeg}|e|r{illeg}|e| {illeg}|take| {{illeg}|in|} a 2d equation/. {illeg}|t|hen by multiplication or by some other meanes take a{illeg}|w|ay ye irrational quantity $\sqrt{yy+vv}$ & lastly take awa{y} y or x {illeg}|b|y ye helpe of these 2 Equations, soe yt you have a {illeg} \3rd/ equation in wch 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}|ye| unknowne quanti{illeg}|t|y hath dimensions in yt terme {illeg} wch {illeg}|p|roduct is a 3rd equation it according to Huddenius his Method for a 3rd /4th\ Equation & by ye helpe of ye 3rd & 4th equation take away {illeg} ye in wch & there will result a 5t Equation in wch there b{illeg} one unknowne quantity viz: either x or y. & there will result a 5t Equation in wch is neither x nor nor {sic} y. & by wch the valor of z may be found. one \The greatest/ of whose valors signifies ye longest, another \the least {illeg}/ of ym ye shortest of all ye perpendicular lines ab. & if it have other rootes they signifie other lines $\left(ab\right)$ wch are perpendicular to ye crooked line at both ends, a & b; & some of these must signifie ye axes of ye line if it bee of an elliptical nature.

<9r>

 $ac=x$. $ch=y$. $bf=${illeg} \z {illeg}/. $fh=${illeg} \{illeg} ϩ/ $cd=a$. $db:be\colon\colon fh:eh\colon\colon b:c$. $\frac{cx}{b}=be$. $\frac{cϩ}{b}=eh$. {illeg} $\frac{ϩ\sqrt{cc-bb}}{b}=fe=\frac{cx-bz}{b}$. $\frac{ϩ\sqrt{cc-bb}+bz}{c}=x$. ed{illeg}$=\frac{x\sqrt{cc-bb}}{b}=\frac{ϩcc-ϩbb+bz\sqrt{cc-bb}}{bc}$ $hc=\frac{-ϩbb+bz\sqrt{cc-bb}+abc}{bc}=\frac{ac-ϩb+z\sqrt{cc-bb}}{c}=y$.
$\begin{array}{ccccccc}\begin{array}{c}\phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\text{for}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\phantom{=}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\end{array}& \left\{& \begin{array}{}x\phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ xx& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ {x}^{3}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ {x}^{4}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ y& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ yy& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ {y}^{3}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ {y}^{4}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ xy& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ xyy& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ x{y}^{3}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ xxyy& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ xxy& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ {x}^{3}y& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\end{array}& \right\}& \begin{array}{c}\phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\text{write}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}=& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\end{array}& \left\{& \begin{array}{l}\frac{+ϩ\sqrt{cc-bb}}{c}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{2bϩz\sqrt{cc-bb}}{cc}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{{ϩ}^{3}cc\sqrt{cc-bb}}{{c}^{3}}-\frac{{ϩ}^{3}bb\sqrt{cc-bb}}{{c}^{3}}+\frac{3bbzzϩ\sqrt{cc-bb}}{{c}^{3}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{4{ϩ}^{3}cc\sqrt{cc-bb}×bz-4{y}^{3}{b}^{3}z\sqrt{cc-bb}+4ϩ{b}^{3}{z}^{3}\sqrt{cc-bb}}{{c}^{4}}\phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ -\frac{ϩb}{c}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{-2ϩabc-2ϩbz\sqrt{cc-bb}}{cc}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ -{ϩ}^{3}{b}^{3}-3ϩbaacc-3ϩbcczz+3ϩ{b}^{3}zz-6ϩabcz\sqrt{cc-bb}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{\begin{array}{cc}\begin{array}{c}-4{ϩ}^{3}{b}^{3}ac-4{ϩ}^{3}{b}^{3}z\sqrt{cc-bb}-4ϩ{a}^{3}b{c}^{3}-12ϩaabccz\sqrt{cc-bb}-12ϩab{b}^{3}zz+12ϩa{b}^{3}c{z}^{2}\end{array}& \begin{array}{l}+ϩ{z}^{3}{b}^{3}\\ -ϩ{z}^{3}bcc\sqrt{cc-bb}\end{array}\end{array}}{{c}^{4}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{acϩ\sqrt{cc-bb}+ϩzcc-2ϩzbb}{cc}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{-4ϩabbcz-3ϩbbz\sqrt{cc-bb}\begin{array}{l}+aaccϩ\\ +zzcc\end{array}\sqrt{cc-bb}+2a{c}^{3}zϩ+{b}^{2}{ϩ}^{3}\sqrt{cc-bb}}{{c}^{3}}\phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{-3bbcc{ϩ}^{3}z-4{b}^{4}z{ϩ}^{3}+3abbc{ϩ}^{3}\sqrt{cc-bb}+{z}^{3}{c}^{4}ϩ+4{z}^{3}{b}^{4}ϩ-5{z}^{3}bbccϩ\begin{array}{c}-9abcc\\ +3a{c}^{3}\end{array}zzϩ\sqrt{cc-bb}\begin{array}{c}-aabbcc\\ +3aa{c}^{4}\end{array}zϩ+{a}^{3}{c}^{3}\sqrt{cc-bb}}{{c}^{4}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{2a{b}^{3}c{ϩ}^{3}-2ab{c}^{3}{ϩ}^{3}\begin{array}{c}+4{b}^{3}\\ -2bcc\end{array}z{ϩ}^{3}\sqrt{cc-bb}\begin{array}{c}+2bcc\\ -4{b}^{3}\end{array}{z}^{3}ϩ\sqrt{cc-bb}+2aabbcczϩ\sqrt{cc-bb}+4ab{c}^{3}zzϩ-6a{b}^{3}czzϩ}{{c}^{4}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{{b}^{3}{ϩ}^{3}-bcc{ϩ}^{3}+2bcczzϩ-3{b}^{3}zzϩ+2abcz\sqrt{cc-bb}}{{c}^{3}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\\ \frac{4{b}^{4}z{ϩ}^{3}-5bbccz{ϩ}^{3}+{c}^{4}z{ϩ}^{3}\begin{array}{c}+a{c}^{3}\\ -abbc\end{array}{ϩ}^{3}\sqrt{cc-bb}+3bcczzϩ-4{b}^{3}zzϩ+3abczϩ\sqrt{cc-bb}}{{c}^{4}}& \phantom{\frac{\begin{array}{c}|\\ |\end{array}}{{0}^{0}}}\end{array}\end{array}$

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

Example $dd+dy+xy-yy=0$. Then makeing $d=c$ I write $\frac{bz+ϩ\sqrt{cc-bb}}{c}$, or $\frac{bz+ϩ\sqrt{dd-bb}}{d}$ for x & its square for $xx$ &c. & $\frac{ad-bϩ+z\sqrt{dd-bb}}{d}$ for y , & its square for $yy$. & soe I have this equation, $0=dd+ad-bϩ+z\sqrt{dd-bb}\phantom{\rule{0.5em}{0ex}}\begin{array}{}\frac{+adbz-2bbϩz\phantom{\rule{0.5em}{0ex}}\begin{array}{c}-bϩϩ\\ +bzz\\ +adϩ\\ +2azd\\ +2bϩz\end{array}\sqrt{dd-bb}\phantom{\rule{0.5em}{0ex}}+ϩzdd-aadd+2adbϩ-bbϩϩ\phantom{\rule{0.5em}{0ex}}\begin{array}{c}-ddzz\\ +bbzz\end{array}}{dd}\end{array}$ or by ordering it according to ϩ , $\begin{array}{cc}\begin{array}{ccccccccc}bbϩϩ& +& bϩϩ\sqrt{dd-bb}& +& ddbϩ& -& daϩ\sqrt{dd-bb}& -& {d}^{4}\\ & & & +& 2bbzϩ& -& 2bzϩ\sqrt{dd-bb}& \begin{array}{c}-\\ +\end{array}& \begin{array}{c}abdz\\ aadd\end{array}\\ & & & -& ddz\hfill & & & +& ddzz\\ & & & -& 2abd\hfill & & & -& bbzz\end{array}& \begin{array}{c}+2adz\phantom{\sqrt{0}}\\ -bzz\\ -ddz\end{array}\sqrt{dd-bb}\end{array}\phantom{\rule{0.5em}{0ex}}\begin{array}{cc}=& 0\end{array}$ Then by makeing those quantitys in ye last terme save one {illeg} {illeg} $=0$ . I have this Equation $2bbz-ddz-2bz\sqrt{dd-bb}+ddb-2abd-da\sqrt{dd-bb}=0$ . {illeg}|W|hich I divide into 2 pts makeing those termes $=0$ in wch z is not, & those $=0$ in wch z is of one dimension. & then I have these 2 equation{s} $2bb-dd-2b\sqrt{dd-bb}=0$ . & $db-2ab-a\sqrt{dd-bb}=0$ . by ye f{illeg}|irs|t $4{b}^{4}-4bbdd+{d}^{4}=4bbdd-4{b}^{4}$ . Or $8{b}^{4}-8bbdd+{d}^{4}=0$. That is $bb=\frac{dd}{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{{d}^{4}}{8}}=\frac{dd}{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\frac{dd}{2\sqrt{2}}=\frac{dd\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}dd}{2\sqrt{2}}$. by ye 2d Equation I find $ddbb-4adbb+4aabb=aadd-aabb$. or $\begin{array}{cccccc}& 5aabb& -& aadd& =& 0\\ -& 4adbb\\ +& ddbb\end{array}$. & by writing ye valor of $dd$ wch was found before I have $\frac{5aadd\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}5aadd-4a{d}^{3}\sqrt{2}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}4a{d}^{3}+{d}^{3}\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}{d}^{4}-2aadd\sqrt{2}}{2\sqrt{2}}=0$. Or $\begin{array}{cccccccc}& 3aa\sqrt{2}& -& 4ad\sqrt{2}& +& dd\sqrt{2}& =& 0\\ \phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}& 5aa\hfill & \phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}& 4ad\hfill & \phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}& dd\hfill \end{array}$. Or $aa=\frac{4ad\sqrt{2}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}4ad\phantom{\rule{1em}{0ex}}-\phantom{\rule{1em}{0ex}}dd\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}dd}{3\sqrt{2}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}5}$ . & $a=\frac{d\sqrt{8}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}2d}{18\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}5}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\sqrt{\frac{\begin{array}{c}8dd\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}4dd\sqrt{8}\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}4dd\phantom{\rule{1em}{0ex}}-\phantom{\rule{1em}{0ex}}5dd\\ -dd\sqrt{36}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}5dd\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}dd\sqrt{18}\end{array}}{18\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}5\sqrt{18}\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}25}}$ $a=\frac{d\sqrt{8}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}2d}{\sqrt{18}\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}5}\begin{array}{c}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\\ +\end{array}\sqrt{\frac{dd}{18\phantom{\rule{0.5em}{0ex}}\underset{\cap }{O}\phantom{\rule{0.5em}{0ex}}5\sqrt{18}\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}25}}$ or $\frac{2d\sqrt{2}-2d}{3\sqrt{2}-5}\begin{array}{c}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}\\ +\end{array}\sqrt{\frac{dd}{43-15\sqrt{2}}}=a$. & $\frac{dd\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}dd}{2\sqrt{2}}=bb$. $a=\frac{2d\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}2d\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}d}{3\sqrt{2}\phantom{\rule{0.5em}{0ex}}\stackrel{\cup }{O}\phantom{\rule{0.5em}{0ex}}5}$

<9v>

 $\frac{dd{\varrho }^{2}+2ds\varrho ϩ+ssϩϩ}{ee}=xx$. ${d}^{3}{\varrho }^{3}+3dd\varrho \varrho sϩ+3d\varrho \varrho sϩϩ+{s}^{3}{ϩ}^{3}={e}^{3}{x}^{3}$ . |{d }. $a+y$| ${d}^{4}{\varrho }^{4}+4{d}^{3}{\varrho }^{3}sϩ+6dd\varrho \varrho ssϩϩ+4d\varrho {s}^{3}{ϩ}^{3}+{s}^{4}{ϩ}^{4}={e}^{4}{x}^{4}$. ${d}^{5}{\varrho }^{5}+5{d}^{4}{\varrho }^{4}sϩ+10{d}^{3}{\varrho }^{3}ssϩϩ+10dd{s}^{3}\varrho \varrho {ϩ}^{3}+5d\varrho {s}^{4}{ϩ}^{4}+{s}^{5}{ϩ}^{4}={e}^{4}{x}^{5}$ $\begin{array}{ccccccc}tt\varrho \varrho & -& 2t\varrho vϩ& +& vvϩϩ& =& yy\\ & +& 2t\varrho ce& -& 2cevϩ\\ & & & +& ccee\end{array}$. $\begin{array}{llllllllll}& {t}^{3}{\varrho }^{3}& -& 3tt\varrho \varrho vϩ& +& 3t\varrho vvϩϩ& -& {v}^{3}{ϩ}^{3}& =& {e}^{3}{y}^{3}\\ +& 3tt\varrho \varrho ce& -& 6t\varrho ce& +& 3ce\\ +& 3t\varrho ccee& -& 3ccee\\ +& {c}^{3}{e}^{3}\end{array}$. $\begin{array}{llllllllllll}& {t}^{4}{\varrho }^{4}& -& 4{t}^{3}{\varrho }^{3}vϩ& +& 6tt\varrho \varrho vvϩϩ& -& 4t\varrho {v}^{3}{ϩ}^{3}& +& {v}^{4}{ϩ}^{4}& =& {e}^{4}{y}^{4}\\ +& 4{t}^{3}{\varrho }^{3}ce& -& 12tt\varrho \varrho ce& +& 12t\varrho ce& -& 4ce\\ +& 6tt\varrho \varrho ccee& -& 12t\varrho ccee& +& 6ccee\\ +& 4t\varrho {c}^{3}{e}^{3}& -& 4{c}^{3}{e}^{3}\\ +& {c}^{4}{e}^{4}\end{array}$. $\begin{array}{llllllllllllll}& {t}^{5}{\varrho }^{5}& -& 5{t}^{4}{\varrho }^{4}vϩ& +& 10{t}^{3}{\varrho }^{3}vvϩϩ& -& 10tt\varrho \varrho {v}^{3}{ϩ}^{3}& +& 5t\varrho {v}^{4}{ϩ}^{4}& -& {v}^{5}{ϩ}^{5}& =& {e}^{5}{y}^{5}\\ +& 5{t}^{4}{\varrho }^{4}ce& -& 20{t}^{3}{\varrho }^{3}ce& +& 30tt\varrho \varrho ce& -& 20t\varrho ce& +& 5ce\\ +& 10{t}^{3}{\varrho }^{3}ccee& -& 30tt\varrho \varrho ccee& +& 30t\varrho ccee\\ +& 10tt\varrho \varrho {c}^{3}{e}^{3}& -& 20t\varrho {c}^{3}{e}^{3}& +& 10{c}^{3}{e}^{3}\\ +& 5t\varrho {c}^{4}{e}^{4}-& 5{c}^{4}{e}^{4}\\ +& {c}^{5}{e}^{5}\end{array}$

 $bc=x$. $dc=y$. $df=z$. $bh=c$. $hk:ke\colon\colon d:e$. $ke=\frac{ex}{d}$. $df:de\colon\colon d:f$. $\frac{fz}{d}=de$. $\frac{cd+ex-fz}{d}=y$. $fm=\varrho$. $mh=\frac{d\varrho }{e}$. $mk=\frac{xe-d\varrho }{e}$. $pe=\frac{xe-d\varrho }{d}=$ $p=n$. $fp=\sqrt{{ϩ}^{2}-nn}$. $ep=\frac{e\sqrt{{ϩ}^{2}-nn}+dn}{d}=\frac{ex}{d}$ $eeϩϩ=eenn+ddnn-2dnex+eexx$. $fd:fp\colon\colon${illeg}$f:g$. $\frac{ϩg}{f}=fp$ . $\frac{f\varrho +ϩg}{f}=x$. $fp:pe\colon\colon d:e$. $\frac{eϩg}{df}=${illeg}. $pd=\frac{ϩ\sqrt{ff-gg}}{f}$ $fm=\frac{e\varrho }{d}$. $hm=\varrho$. $\frac{e\varrho +cd}{d}-ϩ\frac{\sqrt{ff-gg}}{f}=y$. $f=d$. &c $\frac{d\varrho +gϩ}{d}=x$. $\frac{e\varrho +cd-ϩ\sqrt{ff-gg}}{d}=y$.

 If any crooked line be revolved about its \owne/ {illeg}|a|xis it generates a Sollid intersected by any plaine \not perpendicular to ye axis/ produceth another line $\left\{\begin{array}{l}\text{not more compound}\phantom{\rule{0.5em}{0ex}}{\mathrm{y}}^{\mathrm{n}}\\ \text{of the same kind with}\end{array}$ ye forme{illeg}|r|. But if it bee revolved by any other line it generates a Sollid which intersected by any plaine \not perpendicular to ye axis/ produceth another whose composition is {n}ot $\left\{\begin{array}{l}\text{lesse}\phantom{\rule{0.5em}{0ex}}{\mathrm{y}}^{\mathrm{n}}\phantom{\rule{0.5em}{0ex}}\text{equall}\\ \text{more}\phantom{\rule{0.5em}{0ex}}{\mathrm{y}}^{\mathrm{n}}\phantom{\rule{0.5em}{0ex}}\text{double}\end{array}\right\}$ to ye formere In the △ adb if $ab=a$, & $db=b$ are definite, but $ad=v$, & $bc=x$ indefined. Then ye Equation is $bb-vv+aa-2ax=0$. But in this case ye maximu or minimum of either v or x cannot bee found according {illeg}|t|o Cartes or Hudde{nius} method, by reason yt $\left\{\begin{array}{c}v\\ x\end{array}\right\}$ hath not 2 divers valors when $\left\{\begin{array}{c}x\\ v\end{array}\right\}$ is determined, wch become equall when $\left\{\begin{array}{c}x\\ v\end{array}\right\}$ is ye least or greatest yt 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 (wch shall pass through a point them all) by an instrument whose angle $edf=\angle abc$. And soe ye sides ed & df being moved close by ye points a & c, ye point $\left(d\right)$ shall describe ye 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 ye fassion {illeg} {illeg}|r|epresented by ye 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 ye 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 wth but sma{illeg}|ll| vibrations. Soe shall the 3 {illeg}|f|iduciall sides of ye 3 plates bee {ground} exactly plane.

<10r>

 Probleme. Of Usury. $ab=a=$ the princi{illeg}|p|le. bc{illeg}={illeg}$x=$ the time of the m{illeg}|o|ney lent $ce=\frac{dx}{e}=$ the use due for ye principle {illeg}|fo|r ye time bc. $de=$ the use upon {illeg}|u|se. {illeg} cd ye $cd=y=$ ye use{illeg} for ye principle & use during ye time bc. $ax:\frac{dx}{e}\colon\colon a+y\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}fc:y$. ab ye sume of princi df a t{illeg}|a|ngent to ye line bd. dg a perpendicular. $cg=v$. Then as ye sume $\left(ab\right)$ or principle drawne into ye time bc, is to $\left(ce\right)$ ye use for it in yt time  So is $ab+dc$ the sume $ab+dc$ drawne into ye time fc, to ye use $\left(dc\right)$ for it in ye time fc. {illeg} therefore $ax:\frac{dx}{e}\colon\colon a+y\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\left\{\begin{array}{c}h\\ fc\end{array}:y$ {illeg}. $axy=\frac{ahdx+yhdx}{e}$. $\frac{eay}{ad+yd}=h:y${illeg}$\colon\colon y:v$. $eav=ady+dyy$. $v=\frac{ady+dyy}{ea}$. $d=e$. $\frac{ay+yy}{a}=v$

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

##  Of Reflection. Suppose ye 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. $a=$ ye body a; $b=$ ye 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. ⊙ ye point of theire occursion.

## Axiome 1st

 Two bodys \$\left(b,c\right)$/ being alike swift {illeg} ye motion of $b:$mot: $c\colon\colon b:c$. for equall pts have equall motion. Therefore $b:c\colon\colon$all ye parts of $b:$all such pts of $c\colon\colon$mot: $b:$the motion of c.

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

Prop: 2d. 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 ym towards k.

Prop 3 suppose i ye center of gravity in d, y in a. z & f ye in wch ye bodys a & d touch in theire meeting. ⊙ ye point of theire meeting. $\left(a\right)$ ye magnitude of \ye body/ $\left(a\right)$, $\left(d\right)$ ye magnitude of d. $m=$ motion of a before meeting, $n=$ mot: {illeg}|d| before me{illeg}|et|ing. ye time in wch a or d moves to $\odot =$ time in wch they both move to y. $p=$ motion of a, {illeg} $q=$ moti{illeg}|o|n of d after occursion{illeg} $m+n=p+q$. $a:d\colon\colon p:q$. or $a+d:d\colon\colon m+n\left(=p+q\right):q=$$qm+qn$ $\frac{dm+dn}{a+d}$. $\frac{am+an}{a+d}=p$. $m:p\colon\colon z\odot :zy$. $\frac{am×z\odot +an×z\odot }{am+dm}=\odot y$. $a=$ magnitude of ye body a, $d=$ mag: of ye body d. {illeg} $o=$ {illeg} ye point of concourse: $z,f=$ ye points of contac{illeg}|t|, at o. $zo=${illeg}|b|. $fo=c$. $op=e$. $t=$ time in wch ye bodys move from z & f to o. $v=$ time in wch they move from o to p. $m=$ motion of a before occursion $n=$ motion of d after occursion $dm${illeg}{illeg}$=\frac{dn+an}{a}$ or $cdm+bam=bdn+ban$. {illeg}|&| $\frac{cdm+bam}{bd+ba}=n${illeg}$\colon\colon m:n$. $tm:vn\colon\colon${illeg}$b:e\colon\colon tbd+tba:vcd+vba$. {illeg}$cd${illeg} $v=\frac{det+aet}{cd+ab}$. . {illeg}{illeg}{&} d {illeg} meete {illeg} {illeg} yt c must be negative yt is $\frac{abv-\phantom{0}}{nt}${illeg}e be negative ye point {illeg} must be t{illeg}on ye same side {illeg}

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## Definitions.

 {illeg} \1st/ When a body \{illeg}|Q|uan{illeg}|ti|ty/ $\left\{\begin{array}{l}\text{is translated}\\ \text{passeth}\end{array}\right\}$ from one pte of Extension to another it is saide to mo{ve.}

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

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

3 One Quantity hath so much more motion yn another, as ye distance through wch it moveth drawne into {illeg}|it|s quantity, is to ye distance {illeg}|t|rough wch ye other moveth in ye same time d{illeg}|raw|ne into its quantity. As if ye line ab move ye length of bc & ef ye length of eh in ye same time, ye motion of ab is to ye motion of cd, as $ab×bc=abcd$, to $ef×eh=iehk$. Alsoe if \ye cube/ $lmqyz=8$ move ye length of $op=5$; & the piramis $tvwx=7$ move ye length of $rs=3$ in ye same tim{illeg}|e|; y{illeg}|n|, as, $op×lmqyz:rs×tvwx\colon\colon 40:21\colon\colon$ ye motion {illeg}|of| lmqyz to ye 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 ye same determination of their motion wch move ye same way, {illeg}|&| those have divers wch move divers ways.

 5 A body /quantity\ is reflected when meeting wth another quantity it looseth ye determination of its motion by rebounding from i{illeg}|t|. As if ye bodys a, b meete one A quantity is said to bee refracted another in ye point c they are parted either by some springing motion in y selves or of \in/ ye matter {crouded} bet{illeg}|w|ixt ym. & as {illeg} ye spring is more dull or {illeg}ety \{illeg}|vi|gor{illeg}|ous|/ /quick\ s{illeg}|o|e ye bodys {illeg}|w|ill bee reflected wth wth more \or lesse/ force; a{illeg}|s| if it endeavour to get liberty \to inlarge it selfe/ wth as greate strength & vigor as ye bodys $a,b$ , pressed it together, ye ye {sic} motion of ye body{illeg} a from b will bee as greate after as before y{illeg}|e| reflection. but if ye spring have but halfe yt vigor, yn ye distance twixt a & b,|at| {illeg}|t|he minute after ye reflection shall bee halfe as much as it was befor at ye minute beef{ore} ye reflecti{illeg}on. 7 Refraction is when {illeg} ye body \{illeg}/ c passing \obliquely/ through ye surface ed at ye point b meets wth more or lesse oppo{illeg}|s|i{illeg}|t|ion on one side of ye surface yn on ye other & soe looseth its determinacon; as if it turne towards a.

 9 Force ii|s| ye pressure or erouding of one body {illeg} upon an{illeg}|o|ther,

10 The center of Gravity \{ Motion }/ \in ye same body/ is such a point wthin a quantity \{illeg}wch rests when a body is moved wth \any/lar but noe progres\sive motion//; yt if it \be considered to as at a/ rest & ye quantity & some quantity line as $\left(mn\right)$ be drawne through it: about wch (as \about/ an axis) ye f{illeg} quantity $\left(dklp\right)$ revolving. {illeg}|t|here shall bee ye same quantity of motion on both sides any {illeg}|p|laine {illeg}|w|th wch $\left(mn\right)$ is coincident; also ye 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 yt (if it bee conceived to rest {&}) if {illeg} the bodys bee moved about it {illeg} wth circular motion they shall both have an equall quantity {o}f motion, the line about wch they move is ye 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\

< insertion from the left margin >

13 Bodys are more or lesse distant as ye distances of {illeg} their pts \centres of motion/ are more or lesse. or as their distances might bee acquired wth more or less motion

< text from f 10v resumes >

## 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 ye same streight line (not changing ye 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.

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

5 If {illeg}{b}ee moved by unequall forces, as ye force moveing $\left(b\right)$ is to ye force {illeg} motion {illeg}|o|f b. to ye m{illeg} of c, so is ye swiftnes of b, to yt of c.

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} ye motion of $\left(b\right)$ shall {illeg}{illeg}

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 8 If two quantitys (a & b) move d{illeg} towards one another & meete in {illeg} o, Then ye 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) therefore one must loose noe more motion yn ye other doth; soe yt ye difference of their motions cannot be destroyed.

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

 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 yr reflection as they did to one another before i{illeg}|t|. For {illeg}ng For firs{illeg}|t| suppose ye {illeg}|sphæricall| bodys $e,f$ to have a springing or elastic{illeg}|k| for{illeg}|c|e soe yt {illeg}|m|eeting one another they will relent & be pressed into a sphæri|o|idicall figure, {illeg}|{&}| in yt moment in wch 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 ye 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 (yt is at ye moment of theire meeting) theire pression is at ye greatest or rather tis \done with/ the whole for{illeg}|c|e by wch theire motion is stopt, \for theire whole motion was stoped b{illeg}|y| ye force of theire pressure upon one another in ysone mome\nt// for /&\ there cannot $\begin{array}{c}\text{bee}\\ \text{succeede}\end{array}$ 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 ye same forcible pressure towards one another: that is soe much force as deprived ye 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) 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, yt they should sepera{illeg}|t|e from{illeg} one another wth as much m{illeg}|{ot}|ion as they they came together.

  11. If a line \$\left(df\right)$/ bee moved not wth a Circular \Progressive/ but onely a Circular motion i{illeg}|t|s middle point \$\left(n\right)$/ shall rest. For if it move let it move towards r soe yt, when ye point $\left(d\right)$ is moved in $\left(p\right)$ & f in $\left(q\right)$, then $\left(n\right)$ shall be moved to $\left(s\right)$ : 11 If {illeg}|a| line $\left(ce\right)$ be bisected in $\left(a\right)$ about wch ye line $\left(ce\right)$ doth circulate {illeg}|&| yt point bee fixed. yn ye whole line hath noe progressive motion. For makeing $ab=ad$, bf, ag, & dh bee parallel, & perpendic to fh, yn is $vb=dp$. & $vf+ph=bf+dh=2ag$. Wherefore ye point c moveing to{illeg}|w|ards n ye point {illeg}|d| shall move soe much towards ye line fh as ye point b doth from it, & all ye points in {illeg} $\left(ac\right)$ or ye line ac move as mue|c|h from to ye line fh as \all/ ye points in $\left(ae\right)$ or ye line $\left(ae\right)$ moves from it soe yt ye whole line ce stays in equilibrio neither moveing to nor from fh, by ye 12th Defin{illeg}|.|

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

13. When $\left(ce\right)$ moves circularly but maketh noe progression it{illeg}|s| midle {sic} point shall rest & is therefore ye center of its motion, for if ye middle point move ye whole line let it bee to r from a soe yt ye line ec bee moved into ye place {$\left(wt\right)$} yn let ye {$\left(wt\right)$} move about ye fixed center r into ye place xs, yn {xs} & {wt} are equally distant from fh (by def: 13 & ax: 11) & alsoe $\left(ln\right)$ & $\left(ce\right)$ are equally distant from ye {illeg}{am} f{illeg} but xs is further from $\left(fh\right)$ yn $\left(ln\right)$ 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 ye line {illeg} pro{gre}ssive motion when it {pressed} into {wt}

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

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

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

 17 If a body move \streight/ forward & circularly its cente{illeg}|r| of motion shall have ye same determinacon & velocity yt {illeg}have had did ye body move Pa∥ll to it selfe \the body hath/. For {suppose} ac wh to be moved into ye place gh & its center of motion b into ye place d, yn let it turne about ye center d into ye place ef ∥ to ac s{illeg}|o|e yt ye point wch was in $\left(a\right)$ bee now in $\left(e\right)$. N{illeg}|o|w since {$\left(gf\right)$} by moveing into ye place $\left(fe\right)$ makes noe progressiv motion (def: 10) it follows yt {illeg}parallell to it selfe into ye place ef it would {illeg} ye same determinon {sic} & quantity of motion ye same quantity \(Or since $gh=ef=ac$) ye same velocity/ & \(axiom 10)/ determinacon of motion \in ye same time/ would translate ac parallell to it selfe into ye place ef yt w{illeg}ld {sic} translate it into ye pla{illeg}|ce| gh , had it bo{illeg}|t|h progressive & circular motion. But ye point d {illeg} hath ye f|s|ame velocity & determinacon wch ye line ef hath when moved ∥ to its selfe (x) therefore ye point d hath ye same determinacon & velocity wch ye line gh hath when moved wth 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 ye same determinacon & velocity wch ye body {hath} for in {illeg} {illeg} any point of ye crooked line its determinaco {sic} is in ye 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 ye croked {sic} line ye motion may bee conceived to be on in ye tangent.

 19 {sic} 2 bodys circu make ye same number of circulations wth ye same dista{nce} from ye center c : yn as ye Radij of ye circles wch they|ir| \centers of motion/ {illeg}|de|scribe are to one another soe are ye 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 ye Radij of those circle {sic} (wch theire centers of motion describe) are to one another (def 3). As: $ec=eo\colon\colon$ velocity of $eb:$ velocity of ac. & $eb×ec:ao×co\colon\colon$ mot $eb:$ mot of ao.

 20 If a s{illeg}|ph|ære ⊙c move wth in \be compelled by/ /move wth in\ {illeg}|y|e concave sphæcall {sic} or cilindricall surface of ye body edf to move circularly abou{illeg}|t| ye center m it shall press upon ye body def for when it is in c (suposeing ye $\mathrm{\odot }$ bhc to be described by it {sic} center of motion {illeg} & ye line cg a tangent to yt $\mathrm{\odot }$ at ⊙) it moves it moves towards g or ye determination of its motion is towards c      therefore if at yt moment ye body edf should cease to check it it would continually move in ye {illeg}|l|ine cg (ax 1. 2.) obliq{illeg}|l|y from ye center m , but if ye 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 ye tangent line in every point of ye ◯ chb , but ye body edf cannot check & curbe ye determinacon of ye body c⊙ unless they continually presse upon one another. |The same may {illeg}|b|e understood if ye body adb bee restrained into ◯lar motion by ye thred $\left(om\right)$ |

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

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

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

 24 If two bodys ( cb & de ) move about a center $\left(a\right)$ yn as ye motion o makeing $bc=a$, $de=b$, $ac=c$, $ae=d$, ye time in wch bc makes halfe a revolution call e, yt in wch de doth ye sam{illeg}|e| call f , ye pressure of cb from ye center a in halfe a revolution call q , {illeg}|&| yt of de call h ; ye motion of cb in halfe a revoluco k & yt of de call l . yn $k:l\colon\colon$

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

 {illeg}|2|6 If y{illeg}|e| body a move through ye space ab \$=b$/ in ye time d \$=be$/, & ye body c {illeg}|thr|ough ye space cd |$=e$| in ye time f . then ye velocity of a is to ye velocity of {illeg} c as $\left(ab×dc\right)$ to $cd×$ {illeg} be . for supposeing $cb=gp$ then ye velocity of a is to ye velocity of c as $\left(ab\right)$ to $\left(cp\right)$ (def 2d) or as $ab×gp:cp×eb\colon\colon ab×cd:ad×eb$ . for since \Then is/ $cp×fd=cd×gp=cd×eb$ . Or {illeg} $cp×fd×ab×\left\{\begin{array}{c}eb\\ gp\end{array}=cd×eb×ab×gp$ yt is $ab×gp:cp×eb\colon\colon ab×fd:cd×eb\colon\colon ab:cp\colon\colon$ velocity of $a:$ velocity of c velocity of $a:$ to ye velocity of $c\colon\colon ab:cp${illeg} For supposeing yt $gp=eb$. yn is $cp=\frac{eb×cd}{fd}$ And (by def: 2) ye velocity of $a:$ is to ye velocity of $c\colon\colon ab:cp\colon\colon ab:\frac{eb×cd}{fd}\colon\colon ab×fd:eb×cd$

 Alsoe ye motion of a is to ye motion of c (by def 3d) $\colon\colon a×ab:c×cp\colon\colon$$a×ab:$\$c×c$/ $a×ab:c×cp\colon\colon a×ab:\frac{c×eb×cd}{fd}\colon\colon a×ab×fd:c×eb×cd\colon\colon a×\text{its velocity}:c×\text{its velocity}$.

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

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

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

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

Hence in equivelox bodys ye powers of persevering in their states are proportionall to their quantitys.

101 Hence may bee perceived what is meant. Supposeing ye bodys aobc & cbqp to be equall & equivelox: Then {illeg}|t|hat cause \hindrence, impedimnt resist{ance}/ or opposi{illeg}|ti|on wch 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} yt cau{illeg}|s| hath ye same {illeg}{illeg} over both ye bodys. Now if {illeg} add ye opposition $\left(a\right)$ wch can \{being} {illeg}ive of its {illeg}/ {illeg}|r|educe cbpq to {illeg}{illeg}ion $\left(b\right)$ wch can reduce aob {illeg} \{illeg}/ ye whole opposition ( $a+b=2a=2b$ ) {illeg} \{illeg}/ both {illeg} bodys $aobc+bcpq=aopq$ ) {illeg} \{illeg}/ motion when they are joyned into one $\left(aopq\right)$ for $a:cbpq\colon\colon b:aobc\colon\colon a+b:cbpq+aobc\colon\colon 2a:aopq$ Also neither {illeg} a or b {illeg} aopq of {illeg} motion for {illeg} <12v> pte ( a or b ) would be equall to ye whole ($a+b=2a=2b$ ). By ye same r{illeg}|e|ason | aopq {illeg}|&| cbqp loosing equall velocity ye resistance /impediment\ of aopq must be double to ye opposition of cbpq .|

 102 By they {sic} same reason yt \Since/ beacuse aopq {illeg} is double to cbpq & both of ym equivelo{x} therfore ye opposition wch can deprive aopq of its motion must be double to yt wch can deprive cbpq of its motion; by ye same reason it will follow yt in equivelox bodys as one body{illeg} \ $\left(a\right)$ / {illeg}|i|s to another \ $\left(b\right)$ / {illeg}|s|oe must ye resistance wch can deprive yt body $\left(a\right)$ of its $\left\{\begin{array}{l}\text{velocity}\\ \text{motion}\end{array}$ bee to ye resistance wch can deprive $\left(b\right)$ of its whole $\left\{\begin{array}{l}\text{velocity}\\ \text{motion}\end{array}$ so is ye resistance wch can deprive $\left(a\right)$ of some pte of its velocity, to ye restance {sic} wch can deprive $\left(b\right)$ of ye same quantity of velocity, soe yt 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 ym of theire whole velocity or of some pt{e} of it wch {illeg}p

103 By ye same reason alsoe If two bodys rest or bee {illeg}|e|quivelox: yn as ye body $\left(a\right)$ is to ye body $\left(b\right)$ soe must ye power orf efficacy \vigor strength/ or virtue of ye cause wch begets new velocity in $\left(a\right)$ {illeg}|b|ee to ye power virtue or efficacy of ye cause wch begets ye = same quantity of velocity in b , soe yt 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 ye 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/ ye {illeg}g perseverance of bodys in theire state, it is said to Indeavour {illeg}|t|o change their {illeg} perseverance.

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

 106 Now if ye {illeg} bodys a & b meete one another ye cause wch hindereth ye progression of a is ye power wch b hath to persever in its state velocity \or state/ & is usually called ye force of ye body b & {illeg}|i|s this power or force are said to |soe yt a body is {so} to be moved wth more or lesse force wch meeting wth another body can cause a greater or lesse mutation in its state, or wch 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 ye progression of a it is said to {illeg} Indeavour ty|o|e hinder ye {illeg} progression of a \wch indeavour in body is performed by pressure/ & by ye same reason ye body b may {illeg}|b|ee said to endeavor to helpe ye motion of a if it should apply its force to move it forward: soe yt it is e{illeg}|v|ident wt ye Force & indeavor of \i{illeg}|n|/ bodys are.

 107 If ye bodys b & c be equiv{illeg}|e|lox yn as {illeg} $b:c\colon\colon$ ye force of \wth wch/ b \is moved/ (or ye 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 ye force of c . For let there be 2 other bodys a & d equivelox to th{illeg}|e|m soe yt a meeting b , & d meeting c they would eqaully hinder one others progression yn is $a=b$ , & $c=d$ (ax 105) & $a\left(=b\right):d\left(=c\right)\colon\colon$ yn force wch can stop $a$ (= to ye force of b ) to ye force wch can stop d (= to ye force of c ). |(vide ax: 102.|

 108 Tis knowne by ye light of nature yt equall forces shall effect an {illeg}|e|quall ch{illeg}|a|nge in equall bodys. Therefore if ye forces g, h, k, m, be equall, & ye bodys a, b, equall {illeg}|&| rest, then let $\left(a\right)$ bee moved by ye force g ; & b by h , a & b shall be equiv{illeg}|e|lox: Also (since tis noe greater change for $\left(a\right)$ 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 $\left(a\right)$ bee againe moved forward by ye force k , its velocity shall be double to ye velocity of b , & if it bee againe moved forward by ye force m its velocity shall be triple to yt of b . &c. Whence as ye force moving $\left(a\right)$ is to ye force moving $\left(b\right)$ soe is |ye| velocity acquired in $\left(a\right)$ to ye velocity acquired in $\left(b\right)$ \{{illeg}|b|y that force}/

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

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

The force wch ye body $\left(a\right)$ hath to preserve it selfe in its state shall bee equall to ye force wc{h}{illeg}{illeg} yt state; not greater for ye effect cannot exceede the {illeg} for {illeg}{illeg}{illeg} ye {illeg} wch was not in ye cause \nor lesse for/ ye cause only {illeg}to its effect{illeg}no reason why its {illeg}{illeg}

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 \112/ A body is saide to have more or lesse motion as {illeg} it is moved wth more or lesse force, yt is as ther{illeg}|e| is more or lesse force required to generate or destroy its \whole/ motion.

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

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

111 By ye same reason as ye celerity of ye body $a\left(=b\right)$ is to ye celerity of b so is ye for{illeg}|c|e wherewth a moveth to ye force wherewth b moveth.

 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| ye same {illeg}|q|uanty {sic} of motion a body suffers ye same quantity of mutacon in its state, & in ye same body equall forces will effect a equall change

 116 If ye bodys $a=3b$ , & a & b {illeg} are moved wth ye same force $\left(d\right)$ yn ye celerity of b {illeg} is tri{illeg}|p|le to ye celerity of {illeg} a . for if a be moved by suppose $b:a\colon\colon d:\frac{ad}{b}$ & let $\left(a\right)$ bee moved by $\left(\frac{ad}{b}\right)$ & $\left(b\right)$ by {illeg} for if $a\left(=3b\right)$ be moved by $3d$ , & $\left(b\right)$ be {illeg} moved by $\left(d\right)$ , ( a & b ) shall {illeg} }{illeg} \moved by/{ for $3b$ moved by $3d$ is equivelox to b moved by d , but since $3b=a$ , therfore a moved {illeg}|b|y $3d$ is equivelox to b moved by d . And (ax 108) as ye ce{illeg}|l|erity of a moved by {illeg}| d | is to ye celerity of a moved by $3d$ , soe is 1 to 3 , soe is ye celerity of a moved by d to ye celerity of b moved by d .

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

 118 If ye body p , be moved by ye forc{illeg}|e| q , & r by ye force s , \to find {illeg} $\left(v\right)$ ye celerity of p & $\left(w\right)$ yt of r ,/ I add $\left(t\right)$ to p , soe yt $p+t=r$ , & yt $\left(p\right)$ , & $\left(p+t\right)$ are moved wth equall force, yn $p+t\left(=r\right):p\colon\colon v:\frac{pv}{r}$ ye celerity of $p+t$, (ax 117) alsoe, (ax 108) $s:q\colon\colon w:\frac{qw}{s}=\frac{pv}{r}$. Or $qrw=pvs$ . that is ye celerity of p is to ye celerity of r as $qr$ is to $ps$. And by ax 113 ye motion of p is to ye motion of r as ye force of p to ye force of r . And by ye same reason if ye motion of p & r bee hindered by ye force q & s , {illeg}ye motion lost in p is to ye motion lost in r , as q is to s . or if ye motion of p be increased by ye force q , by|u|t ye motion of r hindered by ye force s ; the {illeg} as q , to s ∷ so is ye increase of motion in p , to ye decrease of it in r (ax 111

 1{1}9 \121/ If 2 bodys p & r p {illeg}{nest} ye one ye other, ye resistance {illeg}|i|n both is ye 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} wthout {illeg}

120 A body must move yt way wch it is pressed.

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

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

If ye body b move through ye places d , f , r , & ye body c through ye places {illeg}| e | g h & their center of o , p , q , r ye line opqr shall be a {illeg}|s|treight line For nameing ye lines $bc=g$ . $cr=h$ . $br=k$ . $bd=x$. $d:e\colon\colon x:ce$ . Then Then $ce=\frac{ex}{d}$ . $k:g\colon\colon x:\frac{gx}{k}=cv$ & supposeing $bc\parallel dv\parallel ps$ . Therefore {illeg} $ve=\frac{ex}{d}=\frac{gx}{k}$ & then $k:g\colon\colon k-x:\frac{gk-gx}{k}=dv$ . $b+c:c\colon\colon de:pe\colon\colon dv:ps\colon\colon ve:es$ also $k:h\colon\colon x:\frac{hx}{k}=cv$, & $\frac{ex}{d}-\frac{hx}{k}=ve$ therefore $ps=\frac{cgk-cgx}{bk+ck}$. & $es=\frac{cekx-cdhx}{bdk+cdk}$. $cr-ce=er=h-\frac{ex}{d}$ . $er+es=sr=\frac{cekx-cdhx}{bdk+dck}+\frac{dh-ex}{d}$ . Or $rs=\frac{bdhk+cdhk-bkex-cdhx}{bdk+dck}$. {illeg} $b+c:c\colon\colon bc:co\colon\colon g:\frac{cg}{b+c}=co$ . $cr:co\colon\colon h:\frac{cg}{b+c}\colon\colon bh+ch:cg\colon\colon rs:sp\colon\colon bdhk+cdhk-bkex-dhcx:cdgk-cdgx$ . Whence $cdgk\stackrel{.}{bh}-cdgxbh+cdgk\stackrel{.}{ch}-cdgx\stackrel{.}{ch}=bdhk\stackrel{.}{cg}+cdhk\stackrel{.}{cg}+kexbcg-dhcx\stackrel{.}{cg}$

 28. To fin If two bodys b & c mo{illeg}|v| {illeg} through in ye lines br & cr . The body c moveing through ye space cg in ye ti{illeg}|m|e vs , & though g h in ye time nv , & through kr in ye time nr . & ye velocity of ye body b is to ye velocity of c as d , t{illeg}|o| e , & a{illeg}|s| ye line cg to ye line be , or as ck to br , then {illeg} when ye body c is in ye place g , b will bee in e , & when c is in k , b will be in r . to find ye line wch the center of their motion describes, viz dfo . {illeg}|T|hen nameing ye quantitys $br=a$, $cr=f$. $bc=g$. {illeg} $e:d\colon\colon a:\frac{da}{e}=ck$ . $kr=\frac{ef-da}{e}$ . If o be ye center of gravity motion of ye bodys at k {illeg}|&| r ; yn, $b+c:b\colon\colon \frac{ef-da}{e}:\frac{bef-abd}{eb+ec}=or$ . And ye line df must passe through o . againe making $gk=v$ . yn $d:e\colon\colon v:\frac{ev}{d}=er$ . & if $bc\parallel fi\parallel em$ , yn $a:g\colon\colon \frac{ev}{d}=er:\frac{gev}{ad}=em$ . also since f is ye &, $a:f\colon\colon \frac{ev}{d}=er:\frac{fev}{ad}=mr$ $gr=gk+kr=v+\frac{ef-ad}{e}$ . $gm=gr-mr=\frac{ev+ef-ad}{e}-\frac{fev}{ad}=\frac{adev+adef-aadd-feev}{ade}$. since f is ye center of motion in ye bodys at g & e tis, $b+c:c\colon\colon$ {illeg} $ge:fg$ . {illeg} $b+c:c\colon\colon eg:fg\colon\colon em=\frac{gev}{ad}:\frac{cgev}{abd+acd}=fi\colon\colon gm=\text{&c}:gi=\frac{cadev+cadef-caadd-cfeev}{bade+cade}$. $gr=gk+kr=\frac{ev+ef-ad}{e}$. $go=gr-or=\frac{ev+ef-ad}{e}+\frac{abd-bef}{eb+ec}=\frac{bev+cev+cef-cad}{eb+ec}$. $go-gi=io=\frac{adbev+cfeev}{bade+cade}=\frac{abdv+cefv}{abd+acd}$. $co=cr-or=\frac{fec+abd}{eb+ec}$. $b+c:c\colon\colon g:\frac{cg}{b+c}=cd$ .  {illeg}|N|ow if the lines $oi:if\colon\colon oc:cd$ . Then ye line od must be a streight line. but $oi:if\colon\colon adb+cfe:cge\colon\colon oc:cd\colon\colon \frac{fec+abd}{e}:cg\colon\colon fec+abd:ecg$ . therefore ye line do is a streight line, wth wch may bee found by ye two points d & o . |The demonstracon is ye same if ye body b moved from a to b | 29 If two bodys q & c be moved in divers plines {sic}, then find ye shortest {illeg} line $\left(pr\right)$ wch can bee drawne frome one line $\left(cr\right)$ {illeg}|to| ye other line $\left(qp\right)$ in wch those bodys are moved, & yt line pr shall bee perpendicular to both ye lines cr & qp , viz $\angle qpr=$ $\angle rps=prc=\text{recto}$. then draw qb equall & $\parallel pr$ & draw $br=qp$ . Then shall ye plaine qbrp be perpendicular to ye plaine bcr . Suppose also ye body c moves over ye space $\begin{array}{c}cg\\ gk\\ kr\end{array}}\text{in the time}\left\{\begin{array}{c}vw\\ wt\\ tr\end{array}\right\$ {illeg}|&| yt ye body q moves over ye space qa in ye time vw , & ap in ye time wt . Also suppose another body $b\left(=q\right)$ & equivelox to { $\left(q\right)$ } yt is to move over ye space $be=qa$ in ye time vw {illeg}

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Soe yt when $\left(c\right)$ is in $\left(g\right)$ or $\left(k\right)$ {illeg}|,| $\left(b\right)$ will bee in $\left(e\right)$ or $\left(r\right)$ & $\left(q\right)$ in $\left(a\right)$ or $\left(p\right)$ . {illeg} Then \Then drawing {the} streight lines qc , ag , bk / if $b+c:c\colon\colon bc:cd\colon\colon eg:fg\colon\colon rk:ok$ ;|,| the points d , f , & o , shall be ye centers of gravity motion of ye bodys b & c , when they are in ye places b & c , e & g , r & k . & (prop 28) therefore ye line ( dfo {illeg}|)| in a streight line. Likewise if it bee {illeg} $q+c:c\colon\colon qc:lc\colon\colon ag:mg\colon\colon pk:nk$, then ye points l , m , n are centers of motion to ye bodys ( q & c ) being in ye places $\left(q\right)$ & $\left(c\right)$ , a & g , p & k . Then drawing ye lines ld , mf , no , f{illeg} ye (twixt ye neighbouring centers of motion) since $b+c:c\colon\colon$ {illeg} $q+c:c\colon\colon bc:cd\colon\colon q:$ {bc }. therefore $\angle qbc=\angle ldc$ & by ye same reason $\angle gfm=\angle gea$ {illeg}|&| $\angle krp=\angle kon$. Wherefore |all the lines qb , ae , pr , ld , ml , no are parallell t{illeg}|o| one another. And| $b+\phantom{0}:c\colon\colon bc:dc\colon\colon qb:ld\colon\colon eg:fg\colon\colon ea\left(=qb\right):mf\colon\colon kr:ko\colon\colon pr\left(=bq\right):no$ , soe yt $ld=mf=no$ . & since these line line {sic} ld , mf , no , are parallell, equall, in ye same plaine ldon , & stand upon ye 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 ye same streight line lmn , wch line ye line \ $\left(lmn\right)$ / in wch their other ends l , m , n , are are terminated (i.e. in wch /are\ all ye centers of gravity motion of ye bodys ( c & q )) {illeg} must bee a streight line.

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

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

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

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

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

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

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

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

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

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

By ye same r{illeg}|e|as{illeg}|o|n if g reflect {nor} twixt ye bodys, yn $ad:dn\colon\colon$ pression of $\left(n\right)$ tow{illeg}|a|rds $s:$ to ye pression of a towards $r\colon\colon$ motion of n to { $a:$ } motion of a to {illeg}| r |.

Note yt {illeg} & n are taken for ye centers of motion in a & n . & adn for {illeg}

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

 3{sic}6 If ye $\left\{\begin{array}{c}\text{bodys}\\ \text{Globes}\end{array}\right\}$ a & n doe rest, but soe yt ye body $\left(g\right)$ moveing \perpendicularly/ \to qp / to them & refleting on ye line qp (wch is supported by \but not fastened to/ ye bodys a & n \& ought \now/ to be conceived a line onely./) doth move them by communicating its whole motion to ym wch it selfe looseth, Tis required what motion a & n shall have receive from g . Suppose yt in ye same \so much/ time \as/ g moveth to $\left(d\right)$ before reflection in so much time it moveth from d {illeg} to e after reflection & in so much time ye bodys a & n move ye one to r ye other to s ; & yt yn ye point of reflection $\left(d\right)$ is moved to $\left(t\right)$ . Then (by ax 9th) $et=gd$ . & since bq re
flection noe motion is lost or gotten $g×gd=a×ar=n×ns=g×de$ . And (by prop 113) as ye velocit And (by prop 34) ye motion of n is to { v}
Then na{illeg}|m|e|i|ng ye given quantitys $gd=b$. $qd=c$. $qp=e$. $dp=e-c=f$. $qr=z$. ye whole motion of $\left(a\right)$ & $\left(n\right)$ to $\left(r\right)$ & $\left(s\right)$ call $xx$ , yt is $a×qr+n×ps=xx$ , or if $a:n\colon\colon pw:wq$ , (yn w is ye center of ye bodys motion {illeg}|)|     ) & if $wl\parallel dt\parallel qr\parallel ps$ , then $\left(lw\right)$ is a line described by their center of motion & consequently \Then is/ {illeg}|(|     ) $xx=\stackrel{—}{a+n}×\stackrel{—}{lw}$. Now, as ye motion of a to ye motion of n (prop 34) so is $\left(dp\right)$ to $\left(dq\right)$ ; Therefore (ax 113) as ye velocity of $\left(a\right)$ to ye velocity of $n\colon\colon$ wf $n×pd:a×qd\colon\colon qr:ps$ . yt is $fn:ca\colon\colon z:\frac{caz}{fn}=ps$ . but $\frac{xx-az}{\left\{illeg\right\}}$ $az+n×ps=xx$;yt is $az+\frac{caz}{f}=xx$ ; or $\frac{fxx}{af+ac}=z=\frac{fxx}{ae}=qr$ . & $ps=\frac{caz}{fn}=\frac{cxx}{en}$ . $is=\frac{fxx}{ae}-\frac{cxx}{ne}=\frac{fxxx-caxx}{ane}$ . $ri:is\colon\colon ri:tv$ . therefore, $tv=\frac{cfnxx-ccaxx}{anee}$ . & $tdqr-tv=\frac{efnxx-cfnxx+ccaxx}{anee}=$ $\frac{ffnxx+ccaxx}{anee}=td$ . Also (by ax: 9th) $gd=et=b$ . Therefore $ed=\frac{ffnxx+ccaxx-banee}{anee}$ . Now since g hath soe much motion before reflection as all three bodys $\left(g+a+n\right)$ have after=ward {sic}, therefore (ax 113) $gb=xx+g×de$ ; Or, $ed=\frac{gb-xx}{g}=\frac{ffnxx+ccaxx-banee}{anee}$. That is $2abeegn=aeenxx+gffnxx+ccgaxx$. Or, $xx=\frac{2abeegn}{aeen+gffn+ccga}=\stackrel{—}{a+n}×\stackrel{—}{lw}$. Or, calling $a+n=d$ ; then $\frac{2abeegn}{adeen+gdffn+ccgad}=wl$ . Soe yt by this Equation ye point l twixt the bodys r & s being then their center of motion may bee always found. Note y{illeg}|t| ye lines qr & ps must be described by ye \centers of motion of/ 2 bodys on divers sides of ye point d yt is ar by center of motion of {illeg}|ye| body $\left(a\right)$ or $\left(ad\right)$ , & ps by y{illeg}|t| {sic} center of $\left(n\right)$ or $\left(dn\right)$

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

 {Se}{p}t 1664.                          {illeg} $2x+\frac{1}{3}xx={gd}^{2}$. $xx+4x+1={ag}^{2}$.

 64800 $\begin{array}{l}\underset{_}{36}\hfill \\ 2025\\ \underset{_}{162\phantom{0}}\\ \phantom{0}405\end{array}$ $\begin{array}{l}\underset{_}{29376}\\ \underset{_}{162}\\ \underset{_}{29376}\\ \phantom{0}\underset{_}{7776}\\ \phantom{0}\underset{_}{729\phantom{0}}\\ \phantom{00}\underset{_}{486}\end{array}$ $\underset{\text{'}}{a}____________\underset{\text{'}}{b}_____\underset{\text{'}}{c}$ $\begin{array}{l}\underset{_}{32076×25dd}=801900dd\\ 160380\\ \underset{_}{64152}\end{array}$ $\begin{array}{l}729\\ \phantom{0}7290\end{array}$ $\begin{array}{l}134\\ \phantom{0}537\\ \phantom{0}\underset{_}{486}\\ \phantom{00}5184\\ \phantom{00}486\\ \phantom{000}324\end{array}$ $\begin{array}{l}\phantom{0}1296\\ \phantom{0}648\\ 962\end{array}\right\}=\begin{array}{c}\underset{_}{104976×9}\\ 944784\end{array}$ $\begin{array}{c}\phantom{0}900\\ \phantom{0}36\\ 1800\end{array}$ $\begin{array}{l}\phantom{0}405\\ \underset{_}{162\phantom{0}}\\ \underset{_}{2025×25dd}=50625\\ 10125\\ 4050\end{array}$ $\begin{array}{l}Q:\underset{_}{324}=104976\\ \phantom{0}1296\\ \phantom{0}648\\ 972\end{array}$ $\begin{array}{l}\phantom{0}216\\ \underset{_}{106}\\ \phantom{000}6\end{array}$ 1800 $\begin{array}{cc}36& \\ & \begin{array}{r}\phantom{0}216\\ \underset{_}{106}\phantom{0}\end{array}\\ & 1276\\ & 11484\end{array}$ $\begin{array}{rr}9×25×& 36\\ 3& 24\\ 16& 20\\ \underset{_}{64}& \underset{_}{8}\hfill \\ 81& 00\\ \underset{_}{18}& \underset{_}{00}\hfill \\ 99\end{array}$ $\begin{array}{r}36×324\\ \phantom{0}1944\\ \underset{_}{\phantom{0}972\phantom{0}}\\ 11664\end{array}$ $x+\frac{r}{q}xx=9$. $\frac{r}{2}+\frac{rxq}{q}=2$ {illeg}$x=2x-\frac{rx}{2}$ . $\frac{rx}{2}+2x=9$. $x=3$. $rx=6$ $r=2$ $rx+4x=18$ {illeg} $-\frac{qr}{2}+2q=rx$ $2q-\frac{qr}{2}+4x=18$ $\frac{qr}{2}+\frac{72}{r+4}-18=0$

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

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

 {illeg} ${x}^{3}+{y}^{3}$