Newton's Waste Book (Part 3)

Author: Isaac Newton

Source: MS Add. 4004, ff. 50v-198v, Cambridge University Library, Cambridge, UK

Published online: June 2013

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- This document is part of MS Add. 4004
  The previous part of this document is Newton's Waste Book (Part 2) [MS Add. 4004, ff. 15v-50r]

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<50v>

How to Draw Tangents to Mechanicall Lines

Lemma.

\1/^{[Editorial Note 1]} ^[1] If one body move from a to b in y^e same time in w^ch another moves from a to c & a 3^d body move from a w^th motion compoundee|d| of those two it shall (completeing y^e parallelogram abcd ) move to d in y^e same time. For those motion would severally {carry} it y^e one from a to c y^e other from c to d &c

^[2] \2/ In y^e description of any Mechanicall line what ever, {illeg} there may bee found s|t|wo such motions w^ch compound or make up y^e motion of y^e point describeing it, whose ∼ motion being by them found \by y^e Lemma/, its determinacon shall bee in a tangent to y^e mechanicall line.

^[3] Example y^e 1^st. If abe is an helix, {illeg} w^ch {illeg} described by y^e point b move {sic}ing uniformely in y^e line abc from y^e point center a about w^ch y^e line abc circulates uniformely. {illeg} |the line ab increasing uniformely whilest it also circulates uniformely about y^e center a| Let y^e radius of y^e circle dmbd bee ab. & let {dm $b$ {sic}} measure y^e quantity of the giration of ab (viz ad touching y^e helix at y^e center) let bf be a tangent to y^e circle dmbd . yⁿ is y^e motion of y^e line point b towards c to its motion towards f , as ab , to dmb {illeg}. therefore make $bc = fg ∶ bf = cg ∷ ab ∶ dmb$ . & (by y^e Lemma) y^e diagonall bg shall touch y^e helix in b. Or make $bc = fg = ab$ . & $bf = cg = dmb$ . the diagonall bg shall touch y^e helix. (y^e length of bf may be thus found viz; $ae ∶ bc = ad = ab ∷ dmbd ∶ dmb = bf$ .)

^[4] Example y^e 2^d. If y^e center $(a)$ of a globe \ $(bae)$ / moves uniformely in a streight line parallel to eh, whilest y^e Globe uniformely girates. Each point $(b)$ in y^e Globe will describe a Trochoides: to w^ch a y^e point b I thus draw a tangent. Draw y^e radi {sic} ab & bc perpendicular to it yⁿ is y^e circular motion of {illeg}|th|e point b determined in y^e line bc , & its progesive {sic} in bf . If therefore I make $bc = fg$ to $bf = cg$ as y^e circular motion of y^e point b to its progressive y^e Diagonall (by y^e Lemma) bg shall {illeg}g touch y^e Trochoides in b . As if y^e Globe roule upon y^e pl{illeg}|ain|e eh , & I make $bc = fg = ab$ . & $bf = cg = ae$ . yⁿ do{illeg}|th| y^e Diagonall bg touch y^e Trochoides. (Or be , {illeg} passing through y^e point in w^ch y^e globe & plaine touch, is a perpendicular to y^e {Tro}.

^[5] Example y^e 3^d. If y^e line $bh ⊥ ak$ moves uniformely y^e length of ah whilest { ab girate} uniformely from ak to am about y^e center a , y^e point of their intersection $(b)$ will descibe {sic} y^e Quadratrix kbn . Draw $bc ⊥ abp ⊥ pq$ . & $bf ⊥ am$ yⁿ \ $an ∶ am = ak ∷$ / $ka ∶ kpm ∷$ motion of $(b)$ to f ∶ motion of $(p)$ to q , {(sup)}. And motion of $(p)$ to q ∶ motion of $(b)$ to c ∷ \ $am =$ / $ap ∶ ab ∷ kpm ∶ sbt$ . Therefore motion of $(b)$ to f ∶ motion of $(b)$ to c $∷ ka ∶ sbt ∷ an ∶ ab$ . Therefore makeing $ak ∶ sbt ∷$ $an ∶ ab ∷ bf = cg ∶ bc = fg$ . (Or w^ch is makeing $bf = an = cg$ , & $bc = ba = fg$ ) y^e Diagonall bg shall touch y^e Quadratrix at b .

^[6] [Scholium. The tangents of Geometricall \lines/ may be found by their descriptions {after} y^e same manner. As the Ellipsis (whose foci are a & f ) being described by y^e thred abf y^e thred ab lengthens so much as y^e thred bf shortens, or the point b moves equally from a & to f. Therefore I take $bc = bf$ . $cg = fg$ \& $fg ⊥ bf$ ./ & y^e diagonall bg will touch the Ellipsis in b ]. (This should follow y^e 3^d Example's substitute) See fol 57.

^[7] Although y^e nature of a Mechanicall line is not knowne from its description but from some other principle yet may a tang^nt be drawne to it by y^e same method.

As if be is an Hyperbola. tad its asymptote & $cf ∥ tad$ . & $gp ∶ ph ∷ cadf ∶ adeb$ . {illeg} to draw a tangent to y^e tangent {illeg} line gh {illeg} ghm , I consider y^t, $df ∶ de ∷$ increasing of acdf ∶ increasing of abde ∷ increase of gp ∶ increase of ph ∷ motion of y^e point h towards k ∶ motion of h towards r , if $hk ∥ gp$ . Therefore I make { $rh = sk ∶ rs = hk ∷$ } $∷ de ∶ df ∷ hp ∶ pw$ . & y^e diagonall hs or wh shall touch y^e line ghm . Or if $vg = ta = ab = xy$ . $gp = ad$ . $ph = vxy ∥ phr$ . \ $vp ∥ yh$ /. yⁿ doth xhs touch y^e line ghm at h .

Tangents to mechanichanicall lines may sometimes bee found by finding such a point w^ch is immoveable in respect of y^e line described & also doth {{illeg}vary} in distance from y^e describing point. Then in y^e for y^e {Sicunf{illeg}} through y^t point. Thus in y^e Trochoides when y^e point $(e)$ toucheth y^e plaine eh tis immoveable, & tis ever equidistant from y^e describing point b ({illeg} both of y^m fin{illeg} points in y^e Globe). Therefore y^e line {illeg} drawne {fro=} y^e describeing point to y^e touch point of y^e Globe & plaine $(eh)$ is perpendicular to {y^e} trochoides. But in y^e spirall though y^e point {illeg} is {illeg} from y^t {illeg}.

^[8] Instead {in y^e} third example {illeg} {illeg} {illeg}. <51r> Therefore $af ∶ ab ∷ ak ∶ gbl ∷$ {illeg}|abs|olute & whole motion of b towards c (or acf ) ∶ whole motion of b towards d (or ed ). Soe y^t makeing $bc ∶ bd ∷ af ∶ ab$ . { $ab ∥ ed ⊥ ed$ } & $hb ∥ ce ⊥ cb$ . The point b will be moved {illeg} to y^e line {illeg} ce & ed in same times w^ch cannot bee unlesse it move to e (their common intersection). The point b therefore move in y^e line be w^ch doth therefore touch y^e Quadratrix at b : (The same is done by makeing $bd =$ $bl = bd ⊥ de ∥ ab$ . & drawing \y^e tangent/ be through y^e common intersection of ed & aem .)

To resolve these and such like Problems these following propositions may bee very usefull.

May 14. 1666.

^[9] Prop 1. If y^e body a being in a circumference of y^e circle \or sphære/ adce doth move towards its center b its acceleration \motion or velocity/ to|wards| each point \ d , c , e / of y^e circ{illeg}umference is yⁿ as y^e cordes ad , ac , ae , drawn from y^t body to those points are. This may be Demonstrated by Te|h|eorem R pag 57.

^[10] Prop 2^d. If \ ∠ {sic} $adc$ {illeg} is a parallelogram \ sim $△ aec$ although they bee not in y^esame plane &// three bodys move \uniformely from a /. y^e first from a to b d y^e 2^d from a to e , y^e 3^d from a to c their motions being \each to other/ as y^e {illeg} directing lines ad , ae , ac , are in y^e same time, & adce is a parallelogram then is y^e motion of y^e third body compounded of y^e other two. Demonstration. For \makeing $df ⊥ ac$ f & $eb ⊥$ {adba}/ |{illeg} makeing $df ∥ eb ⊥ ac$ | y^e motion of y^e first body towards d is to its motion towards f as ad = {illeg} is to af (prop 1); \{illeg}/ & y^e \{illeg}/ motion of y^e second body towards be is to its motion towards db as ae $= dc$ is to b ba (prop 1). Therefore But $af + ab = ac$ . Therefore &c

Prop 3^d. If a moveing line keepe parallel to it selfe all its pts have equall motion.

Prop 4^th. If a line move in plano, so y^t all its points keepe equidistant from some common point center the motions of those points are as their distances from y^t center.

Prop 5^t. If y^e motion of a line in plano bee mixed of parallell & circular motion, y^e motion of all its points are compound (see prop 2) of that motion which they would have, had y^e line onely its centrall parallel motion, & of y^t w^ch they would have, had y^e line onely its circular motion.

Schol: All motion in plano is reducible to one of these three cases, & in y^e 3^d case any point in y^t plaine may bee taken for a center to y^e circular motion.

^[11] Prop 6^t. If y^e \streight/ line ea doth rest & da doth move: soe y^t y^e point a fixed in y^e line da moveth towards b : Then from y^e moveing line da drawing $de ∥ ab$ , & y^e same way w^ch y^e point a moveth; These motions, viz of y^e \fixed/ point a towards b , of y^e intersection point a in y^e line ad towards d , & of y^e intersection point a towar in y^e line ae towards e , shall bee one to another, as their correspondent lines de , ad , & ae are.

^[12] Prop 7^th. If y^e \streight/ lines adm , ane , doe move, soe y^t y^e point a fixed in y^e line amd moveth towards b , & y^e point a fixed in y^e line ae moveth in y^e towards c : Then from y^e line each line to y^e other draw two lines de , nm parallell to the mo the line amd , draw ∼ $de ∥ ab$ & y^e same way: & from y^e line ae draw $nm ∥ ac$ , & y^e contrary way, to make up y^e Trapezium denm . And if any two of these foure lines de , {illeg} mn , md , ne , bee to any ∼ correspondent two of these foure motions, viz: of y^e point a (fixed in y^e line dma ) towards b , of y^e point a (fixed in y^e line ane ) towards c , of y^e intersection point a moveing in y^e line dma according to y^e order of y^e letters {illeg} m , d & of y^e intersection point a in y^e line ane according to y^e order of y^e letters {illeg} n , e : Also all y^e foure lines shall be one to another as those foure motions are.

Note y^t \in y^e two last propositions/ if y^e moveing lines \may/ bee crooked {illeg} \so y^t/ amd , ane , bee tangents to them in y^e point a .

Note also y^t by y^e place of a body is meant its center of gravity.

To resolve Problems by motion y^e 6 following prop: are necessary & suffcient.

May 16. 1666.

^[13] If y^e body a in y^e perimeter of y^e circle or sphære adce moveth towards its center b . its velocity to each point \ $d. c. e$ / of y^t circumference is as y^e cordes ad , ac , ae , drawne from y^t body to those points are.

Prop 2. If y^e △s adc , aec are alike though in diverse planes; & 3 bodys move from y^e point a uniformely & in equall times, y^e first to d , y^e 2^d to e , y^e 3^d to c : yⁿ is y^e 3^d's {sic} motion compounded of y^e motion of y^e 1^st & {2^d.}

Note y^t by a body is meant its center of gravity.

Prop. 3. All y^e points of a body keeping parallel to it selfe are in equall motion.

Prop. 4. If a body onely move circularly about some axis, y^e motion of its points are as their distances from {illeg}|th|at axis. Call these 2 simple motions

Prop. 5. If y^e motion of a body is considered as comp mixed of simple motions: y^e motions of all its points are compounded of their simple motions, so as y^e motion towards c (in prop 2^d) is compounded of y^e motion towards d & e .

Note y^t all motion is reducible to one of these 3 cases: & in y^e 3^d case any line may bee taken for {the} axis (or if a line or superficies {move} in plano any point of y^t plaine may bee taken for y^e center) of motion.

^[14] Prop. 6. If y^e lines {illeg} ah being moved doe continually intersect; I describe y^e Trapezium abcd {illeg} its diagonall ac : & say y^t y^e proportion & position of these five lines ab , ad , ac , cb , cd being determined by {requisite data} they shall designe y^e proportion & position of these 5 motions: {illeg} of y^e point a fixed in y^e {illeg} moveing towards b ; of y^e point a fixed in y^e line ah {illeg} moveing towards d ; of y^e intersection point a moveing in y^e plaine abcd towards c (for those 5 lines are {illeg} in y^e same plaine though {illeg} & ah may only touch y^e plaine in their intersection point): of y^e intersection point a moveing in y^e line ae parallely to {illeg} cb & according to y^e order of y^e letters c , b : & of y^e {inter}section point a moveing in y^e line ah parallelly to cd & according to y^e order of those {illeg}

^[15] Note y^t a streight line is said to designe y^e position of curved motion in any point {illeg} {illeg} if toucheth y^e line described by y^e motion in y^t point, or when tis (as ab , ad , ac ), or {illeg} tis parallell to such a {illeg} (as ad , {illeg}). Note also y^t one line ah resting (as in Fig 3 & 4) y^e points d & a are coincident & y^e point c shall bee in y^e line ah if {illeg} bee streight {illeg} (fig 3), otherwise in its tangent ac (fig 4) {illeg}. Haveing an equation expressing y^e relation of two lines x & y described by two bodys A & B whose motions {illeg} q ; Translate {illeg} y^e termes to one side & multiply y^m, being ordered according to x {illeg} {illeg} progression { $\frac{3 p}{x} . \frac{2 p}{x} . \frac{p}{x} . 0 . \frac{- p}{x} . \frac{- p p}{x} .$ } &c: & being ordered by y^e dimensions of y multiply those by {illeg} {illeg} &c. y^e summa of those products {illeg} equation expressing y^e relation {illeg} {illeg} motions p & q .

<51v>

To draw a tangent to y^e Ellipsis

^[16] Suppose y^e Ellipsis to be described by y^e thred acb , & y^t ce is its tangent. Since y^e thred ac is diminished with y^e same proportion velocity y^t be increased|t|h, y^t is, y^t y^e point c hath y^e same motion towards a & d , y^e angles dce , ace , must bee equall, by prop 1. I And, so of y^e othe {sic} {conicks}.

To draw a Tangent to y^e Concha.

^[17] Suppose y^t gae , glc , alf are y^e rulers by w^ch y^e concha is usually described, & y^t $gt ∥ af$ $af ⊥ cb = ∥ mn$ , & $ng = cl ⊥ tn ∥ rl$ . And (since equality is more simple yⁿ proportionality) suppose y^t $cb = nm$ is y^e velocity of y^e point c towards b , or of n towards m . Then is nt y^e mot circular motion of y^e point n about g \(prop 1)/; & lr y^e circular motion of y^e point l fixed in y^e ruler ng , ({illeg} prop 4). And lg is y^e motion of y^e intersection point l (y^t is, y^e velocity of y^e point c ) moveing in y^e line glnc from g (prop 6). Now since a two fold velocity of y^e point c is {illeg} known nemely cb toward b & lg towards d , make $fd ⊥ dc = lg$ ; & y^e motion of y^e point c shall bee in y^e line fc y^e diameter of y^e circle passing through y^e points bcdf (prop {illeg} 1) & therefore tang^nt to y^e Concha.

To find y^e point c w^ch distinguisheth twixt y^e concave & convex portion of y^e Concha.

^[18] Those things in y^e former prop: being supposed, make △ gfh like gnt or lbc : & $df ⊥ fr$ ∼ $fr ∥ = hk = 2 gl ⊥ kp$ , & draw kf . Now had y^e line fd onely parallel motion directed by gd or rf , (since $dc = lg$ ) y^e motion of all its points would bee fr , (prop 3): & if it had onely circular motion about g , y^e motion of y^e point f fixed in y^t line df would bee fh (prop 4): But y^e motion of y^e point f is compounded of those two {illeg} simple motions, & is therefore fk (prop 5 & 2); & y^e motion of y^e intersection point f made by y^e lines af df , & moveing in af , shall bee fp , (prop 6). Now if y^e line cf touch y^e concha in y^e required point, tis easily conceived y^t y^e motion of of y^e intersection point f is infinitely little; & therefore y^t y^e points p & f are coincident, df & fk being one streight line, & y^e triãgles gdf , fkh being alike.

Which may bee thus calculated. Make $cb = c$ . $ag = b$ . $cb = y$ . yⁿ is $bl = \sqrt{c c - y y}$ . $2 gl = \frac{2 b c}{y} = hk$ . $bl ∶ bc ∷ ld = \frac{b c}{y} + c ∶ fd = \frac{c b + c y}{\sqrt{c c - y y}}$ . & $\sqrt{c c - y y} ∶ y ∷ bl ∶ bc ∷ gf ∶ fh ∷ df ∶ kh ∷ \frac{c b + c y}{\sqrt{c c - y y}} ∶ \frac{2 b c}{y} ∷ b y + y y ∶ 2 b \sqrt{c c - y y}$ . Therefore $2 b c c - 2 b y y = b y y + y^{3}$ . Or $y^{3} + 3 b y y * - 2 b c c = 0$ .

In stead of y^e ordinary method de Maximis et minimis, it will |be| as convenient (& perhaps more naturall) to use ∼ This; Namely To find y^e motion of y^t line or quantity ∼ ∼ ∼ & suppose it equall to nothing, or infinitely small. But yⁿ y^e motion to w^ch tis compared must bee finite. That is, y^e unknowne quantitys ought not to bee at their greates or least, both at once.

^[19] Example, In y^e triangle bcd , y^e side bc being given & fixed. y^e side dc being given & circulateing about y^e center c , I would know when bd is y^e shortest it may bee. I call $bd = x$ . $da = y$ . \ $bc = a$ . $dc = b$ ./ yⁿ is y^e {illeg} $\sqrt{x x - y y} = ab$ . & $\sqrt{b b - y y} = ac$ . & $\sqrt{x x - y y} + \sqrt{b b - y y} = a$ . or $x x - a a - b b = - 2 a \sqrt{b b - y y}$ . & $\begin{array}{l} x^{4} & - & 2 a a & x x & + & a^{4} + b^{4} & = & 0 \\ - & 2 b b & - & 2 a a b b \\ + & 4 a a y y \end{array}$ . $\begin{array}{l} 4 p x^{3} & - & 4 a a & p x & + & 8 a a q y & = & 0 \\ - & 4 b b \end{array}$ (prop 7). And makeing $p = 0$ , tis $8 a a q y = 0$ . Or $\frac{0}{8 a a q} = y = 0$ . (For q signifieing y^e motion of d towards bc may bee finite though, p , its motion towards b doth perish). Wherefore $x x = a a \overset{\cup}{O} 2 b + b b$ . or $x = a - b$ . $x = b - a$ . $x = b + a$ . $x = - b - a$ . are y^e greatest & y^e least valors of y^e line bd .

Should I have taken $ba = y$ , instead of $da = y$ . {illeg}|Th|e effect would not have followed because both y^e motions p & q would have vanished at once in y^e point {e}. But I might have taken y^e tangent dm for y , or any other line w^ch wou{ld} {illeg} coincidere w^th bc at its being greatest or least.

^[20] Example 2^d. If {ne} is y^e Conchoid ( $ga = b$ . $ae = c = nl$ . $nb = y$ . $ab = x$ .) fo{illeg} em parallell to it. Then is $\sqrt{c c - y y} ∶ y ∷ bl ∶ bn ∷ cg ∶ fl (supra) = \frac{c b y + c y y}{\sqrt{c c - y y}}$ . Then is $\sqrt{c c - y y} = bl ∶ c = cl ∷ dl = cg = c + \frac{b c}{y} (vide supra) ∶ fl = \frac{c c y + b c c}{y \sqrt{c c - y y}}$ . & $fb = \frac{b c c + y^{3}}{y \sqrt{c c - y y}}$ . & $y = cb ∶ fb ∷ ea = c ∶ am = \frac{b c^{3} + c y^{3}}{y y \sqrt{c c - y y}} = z$ . et $b b c^{6} + 2 b c^{4} y^{3} + c c y^{6} - c c y + z z - y^{6} z z = 0$ ponatur p esse motus puncti m & q esse {illeg} motus puncti c versus b . Erit (prop {6}{)} $6 b c^{4} y y q + 6 c c y^{5} q - 4 c c y^{3} z z q -$ {illeg}{illeg} $q z z - 2 c c p y^{4} z - 2 p y^{6} z = 0$ . supposeing $q = 0$ (For when am is y^e least y^t {illeg}{illeg} bee y^e point c is y^t w^ch distinguisheth twixt y^e concave & convex porti{on} of y^e Conchoid, & yⁿ y^e motion q vanisheth.) it will bee $\frac{3 b c^{4} + 3 c c y^{3}}{2 c c y - 3 y^{3}} = z z = \frac{b c^{3} + c y^{3}}{y y \sqrt{c c - y y}}} 2$ . $\frac{3 c}{2 c c - 3 y y} = \frac{b c^{3} + c y^{3}}{c c y^{3} - y^{5}}$ . & {illeg} $c c y^{3}$ {illeg} $= 2 b c^{4} - 3 b c c y y$ . Or $y^{3} = - 3 b y y + 2 b c c$ .

<55r>

Concerning Equations when their rootes relation twixt \ratio of/ their rootes is considered.

^[21] If to {sic} of y^e rootes of an Equation are in proportion y^e one to y^e other as a to b Then multiplying y^e termes of the{illeg} Equation by this progression
|:&c| $\frac{a^{4}}{b^{3}} + \frac{a^{3}}{b b} + \frac{a a}{b} . \frac{a^{3}}{b b} + \frac{a a}{b} . \frac{a a}{b} . 0 . - a . - a - b . - a - b - \frac{b b}{a} . - a - b - \frac{b b}{a} - \frac{b^{3}}{a a}$ . &c. (And that root /or by y^e same progression\ augmented or diminished by any quantity, as if it bee augmented by a it will bee $\frac{a^{4}}{b^{3}} + \frac{a^{3}}{b b} + \frac{a a}{b} + a . \frac{a^{3}}{b b} + \frac{a a}{b} + a . \frac{a a}{b} + a . a . 0 . - b . - b - \frac{b b}{a} . - b - \frac{b b}{a} - \frac{b^{3}}{a a} . - b - \frac{b b}{a} - \frac{b^{3}}{a a} - \frac{b^{4}}{a^{3}}$ . &c. Or were it augmented by $a + b$ {illeg} it would be $\frac{a^{3}}{b b} + \frac{a a}{b} + a + b . \frac{a a}{b} + a + b . a + b . b . 0 . - \frac{b b}{a} . - \frac{b b}{a} - \frac{b^{3}}{a a} . - \frac{b b}{a} - \frac{b^{3}}{a a} - \frac{b^{4}}{a^{3}}$ ). Then shall y^e roote w^ch is correspondent to $(b)$ be a roote of y^e resulting equation: but inverting y^e order of y^e progresion {sic}, y^t roote w^ch is correspondent to $(a)$ shall bee a roote of y^e equation resulting from such multiplication.

As for example did I know y^t two of y^e rootes of y^e Equation $x^{3} - 8 x x + 9 x + 18 = 0$ were in proportion as 1 to 2 & would I have y^e lesser roote (viz y^t w^ch is correspondent to 1) I make $b = 1$ . $a = 2$ . And soe the progression will bee $28.28.12.4.0 . - 2 . - 3 . - \frac{7}{2}$ . &c Or $30.14.6.2.0 . - 1 . \frac{- 3}{2} . \frac{- 7}{4}$ . &c by {illeg} adding 2 . Or by adding one more it will bee $31.15.7.3.1.0 . \frac{1}{2} . \frac{3}{4} . \frac{7}{8}$ . &c. By any of w^ch progressions y^e Equation may bee multiplyed, as by y^e 1^st, $\begin{array}{l} x^{3} & - & 8 x x & + & 9 x & + & 18 & = & 0 \\ 28 & . & 12 & . & 4 & . & 0 & . \end{array}$ . W^ch produceth {illeg} $7 x x - 24 x + 9 = 0$ . Or by y^e 3^d $\begin{array}{l} x^{3} & - & 8 x x & + & 9 x & + & 18 & = & 0 \\ 7 & . & 3 & . & 1 & . & 0 & . \end{array}$ . W^ch produceth {illeg} $7 x x - 24 x + 9 = 0$ . Or by the first Otherwise by destroying y^e 1^st terme. $\begin{array}{l} x^{3} & - & 8 x x & + & 9 x & + & 18 & = & 0 \\ 0 & . & - 2 & . & - 3 & . & - \frac{7}{2} & . \end{array}$ . W^ch produceth {illeg} $16 x x - 27 x - 63 = 0$ . &c the rootes of w^ch products are, viz: of y^e first $x = \frac{3}{7}$ , & $x = 3$ . Of y^e last $x = - \frac{21}{16}$ , & $x = 3$ . There I conclude 3 to be y^e lesse, & consequently 6 the greater of those rootes w^ch are in proportion as one to of y^e Equation $x^{3} - 8 x x + 9 x + 18 = 0$ . w^ch are in double proportion But was y^e greater of those rootes desired yⁿ {illeg} inverting y^e progression it would bee $\begin{array}{l} x^{3} & - & 8 x x & + & 9 x & + & 18 & = & 0 \\ 0 & 1 & 3 & 7 \end{array}$ . Or $\begin{array}{l} x^{3} & - & 8 x x & + & 9 x & + & 18 & = & 0 \\ - \frac{7}{2} & - 3 & - 2 & 0 \end{array}$ . The first producing $8 x x - 27 x - 126 = 0$ whose rootes are $x = - \frac{21}{8}$ , $x = 6$ . {illeg} The 2^d produceth $7 x^{2} - 48 x + 36 = 0$ whose rootes are $x = \frac{6}{7}$ , $x = 6$ . And consequently 6 is the greater & 3 the lesse of y^e rootes in duplicate propor{tion.}

Would I draw

^[22] If in y^e circle adef af is y^e diameter, ah a perpendicular to y^e end of it from w^ch I would draw $he ∥ af$ , w^ch should intersect perpendicular circle in y^e points d & e soe y^t $(de)$ bee triple to $(hd)$ , y^t is he quadruple to hd . Then calling $af = g$ . $ah = y$ . $\begin{matrix} hd \\ he \end{matrix}} = x$ . The equation expresing {sic} y^e relation twixt x & y is {illeg} $x x - g x + y y = 0$ . y^e rootes of w^ch equation must be quadruple y^e one to y^e other: Therefore would I find {illeg} $(hd)$ y^e lesse roote I make $b = 1$ . $a = 4$ . And y^e progression will bee, $21.5.1.0 . - \frac{1}{4} . - \frac{5}{16} . - \frac{21}{32}$ . &c. by w^ch y^e Equation being multiplyed y^e product is $\begin{matrix} x x & - & g x & + & y y & = & 0 \\ 5 & 1 & 0 \end{matrix}$ . Or $x = \frac{1}{5} g$ . Therefore drawing $ab = \frac{1}{5} g$ , Or $ac = \frac{4}{5} g$ . from w^ch /the\ point b , or c raise y^e perpendiculars db , or {illeg} ce. & soe {illeg} draw hde.

Would I have dbec to be a square y^t is $db = de = y$ . Then to find hd I call it x & $dh = x + y = hd + de$ . Soe y^t y^e lesse roote is to y^e greater as x to $x + y$ . Making therefore $b = x$ , $a = x + y$ , The progression will be $\frac{x x + 2 x y}{x}$ $\frac{y y}{x} + 3 x + 3 y . 2 x + y . x . 0 . - \frac{x x}{x + y} . - \frac{x x}{x + y} - \frac{x^{3}}{x x + 2 x y + y y}$ . Or, $\frac{y y}{x} + 2 x + 3 y . x + y . 0 . - x . - x - \frac{x x}{x + y}$ . by w^ch y^e Equation being multiplyed produceth $\begin{matrix} x x & - & g x & + & y y & = & 0 \\ x + y & 0 & - x \end{matrix}$ , Or $x^{3} + y x x - y y x = 0$ , Or $y = \frac{x}{2} \overset{\cup}{O} \sqrt{\frac{5 x x}{4}} = \frac{x \overset{\cup}{O} x \sqrt{5}}{2}$ . And consequently $- x x + g x = y y = \frac{x x \overset{\cup}{O} 2 x x \sqrt{5} + 5 x x}{4}$ . Or $4 g = 10 x \overset{\cup}{O} 2 x \sqrt{5}$ . Or $\frac{2 g}{5 \overset{\cup}{O} \sqrt{5}} = x$ . By w^ch y^e Equation $x x - g x + y y = 0$ , must be multiplyed. & it produceth $\begin{matrix} x x & - & g x & + & y y & = & 0 \\ 2 x + y & . & x & . & 0 \end{matrix}$ . Or $2 x^{3} + y x x - g x x = 0$ Or $- 2 x + g = y$ . & congsequently $- x x + g x = y y = 4 x x - 4 g x + g g$ . y^t is $5 x x = 5 g x - g g$ . And $x = \frac{1}{2} g \overset{\cup}{O} \sqrt{\frac{1}{20} g g}$ . Or $x = \frac{g \sqrt{5} \overset{\cup}{O} g}{2 \sqrt{5}} = \frac{1}{2} g \overset{\cup}{O} \frac{g}{2 \sqrt{5}}$ . And consequently $y = \frac{1}{\sqrt{5}} g$ . Or it might have beene done thus. x substracted from y^e precedent progression it will be, {illeg} $x + y . 0 . - x . - x + \frac{x}{x + y}$ . &c by w^ch y^e Equation being multiplyed produceth $\begin{matrix} x x & - & g x & + & y y & = & 0 \\ x + y & 0 & - x \end{matrix}$ . Or $x x + y x = y y$ . And by extracting the roote, $y = \frac{1}{2} x \overset{\cup}{O} \sqrt{\frac{5}{4} x x}$ . And therefore $g x - x x = y y = \frac{1}{4} x x \overset{\cup}{O} x \sqrt{\frac{5}{4} x x} +$ { $\frac{5}{4} x x$ } Or $4 g = 10 x \overset{\cup}{O} 2 x \sqrt{5}$ . Or $\frac{2 g}{5 \overset{\cup}{O} \sqrt{5}} = x = \frac{g \sqrt{5} \overset{\cup}{O} g}{2 \sqrt{5}}$ . That is $\frac{2 g}{5 + \sqrt{5}} = \frac{g \sqrt{5} - g}{2 \sqrt{5}} = ab$ . And $\frac{2 g}{5 - \sqrt{5}} = \frac{g \sqrt{5} + g}{2 \sqrt{5}} = ac$ . And therefore { $\frac{2 g}{5 - \sqrt{5}} \frac{- 2 g}{5 + \sqrt{5}} = af - ab = \frac{10 g + 2 g \sqrt{5} - 10 g + 2 g \sqrt{5}}{25 + 5 \sqrt{5} - 5 \sqrt{5} - 0} = \frac{g}{\sqrt{5}} = ah = de = db = bc$ }.

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Reductions of Equations may bee very often & readily \perhaps/ performed by this method

^[23] As {illeg} in that problem recited by D: Cartes pag 83, viz: The square ad & y^e right line s being given , to produce ac to e , soe y^t ef drawn towards y^e point b may bee equall to y^e given line {illeg} s . Putting $df = x$ for y^e unknowne quantity. {illeg} $ef = c$ , & $bd = cd = a$ . The Equation will bee $x^{4} - 2 a x^{3} \begin{array}{l} + 2 a a x x \\ - c c \end{array} - 2 a^{3} + a^{4} = 0$ . w^ch having 4 rootes the Equation must have 4 divers resolutions; that is y^e lines ac , cd , produced both ways indefinitely, there may bee 4 divers lines drawne through the point b , whose parts intercepted twixt y^e crosse lines lace , mdch , are equall to the given line s : And they are bih lbk , nbm . And therefore the rootes of this equation are (two affirmative) df , dh , (& two negative) dh , dm . Because $bd = ab$ , $hi = fe$ , Therefore $ae = dh$ , $ai = fd$ , $an = dk$ , $al = dm$ . Soe y^t $fd ∶ bd ∷ ab ∶ ae = dh = \frac{a a}{x}$ . That is one roote $(df)$ of this equation is to another $(dh)$ as x to $\frac{a a}{x}$ . Therefore I may multiply this Equation by this progression $\frac{a^{4}}{x^{3}} + \frac{a a}{x}, \frac{a a}{x}, 0, - x, - \frac{x^{3}}{a a}, - \frac{x^{5}}{a^{3}}$ . And there resulteth $\begin{matrix} x^{4} & - & 2 a x^{3} & \begin{array}{l} + 2 a a x x \\ - c c \end{array} & - & 2 a^{3} x & + & a^{4} & = & 0 \\ + \frac{a a}{x} & . & 0 & . & - x & . & - x - \frac{x^{3}}{a^{2}} & . & - x - \frac{x^{3}}{a^{2}} - \frac{x^{5}}{a^{4}} \end{matrix}$ . That is $a a x^{3} + c c x^{3} - 2 a a x^{3} + 2 a^{3} x x + 2 a x^{4} - a^{4} x - a a x^{3} - x^{5} = 0$ . Or, $x^{4} - 2 a x^{3} \begin{array}{l} + 2 a a x x \\ - c c \end{array} - 2 a^{3} x + a^{4} = 0$ . Which result is y^e {illeg} same w^th y^e first Equation the reason of w^ch is, y^t if I make $df = x$ then is $dh = \frac{a a}{x}$ . Or if $dh = x$ , yⁿ is $df = \frac{a a}{x}$ . Or if $dk = x$ , yⁿ is $dm = \frac{a a}{x}$ . Or if $dm = x$ then is $dk = \frac{a a}{x}$ . Soe y^t y^e relation twixt \all/ y^e rotes {sic} being{illeg} the same & reciprocally the same & {illeg} not distinguishing one roote from another, tis noe wonder if they bee all indifferently expressed in y^e resulting Equation. Otherwise y^e reduction must have succeeded.

Suppose 3 rootes of an Equation are in proportion to each other as a , b , c . Then if that roote w^ch is correspondent to a be required, multiply y^e termes of y^e Equation by this progression any of these Progressions
|1| $\frac{c^{3} + b c c + b b c + b^{3}}{a a} + \frac{b b + b c + c c}{a} + b + c + a . \frac{b b + b c + c c}{a} + b + c + a . + b + c + a . + a . 0.0 . \frac{a^{3}}{b c} . \frac{a^{3}}{b c} + \frac{a^{4} b + a^{4} c}{b b c c} . \frac{a^{3}}{b c} + \frac{a^{4} b + a^{4} c}{b b c c} . \frac{a^{3}}{b c} + \frac{a^{4} b + a^{4} c}{b b c c} + \frac{a^{5} b b + a^{5} b c + a^{5} c c}{b^{3} c^{3}}$ . |2| $\frac{+ b b c + b c c}{a a} + \frac{b b + b c + c c}{a} + b + c . \frac{b c}{a} + b + c . 0 . - a . 0 . \frac{+ a^{3} + a a b + a a c}{b c} . \frac{a^{3} + a a b + a a c}{b c} + \frac{a^{3} b b + a^{3} b c + a^{3} b b + a^{4} b + a^{4} c}{b b c c}$ . |3| {illeg} $\frac{a a b b + a a c c + b b c c}{a^{3} + a a b + a a b} + \frac{b c}{a} . 0 . \frac{- a b - a c - b c}{a + b + c} . - a . 0 . \frac{a^{3}}{b c} + \frac{+ b b + b c + c c}{b a c + b b c + b c c}$ . &c.

As if \3 of/ y^e rootes of this Equation $x^{4} * - 35 x x + 90 x - 56 = 0$ . $x^{4} * - 35 g g x x + 90 g^{3} x - 56 g^{4} = 0$ were to one another as $1.2.4$ . And g would find y^e roote w^ch is correspondent to 1 . Then I make $a = 1$ , $b = 2$ , $c = 4$ , & soe I may have have {sic} by y^e first progression this{illeg} $+ 155 . + 35 . + 7 . + 1 . 0.0 . \frac{+ 1}{7} . \frac{+ 7}{32} . \frac{+ 70}{128}$ . By y^e 2^d; $+ 92 . + 14 . 0 . - 1 . 0 . + \frac{7}{8} . + \frac{45}{32}$ . &c. By y^e first of w^ch y^e Equation being multiplyed produceth $\begin{matrix} x^{4} & * & - & 35 g g x x & + & 90 g^{3} x & - & 56 g^{4} & = & 0 \\ 35 . & 7 . & 1 . & 0 . & 0 . \end{matrix}$ . That is $35 x^{4} - 35 g g x x = 0$ . Or $x x = g g$ . & $x = g$ . Or were{illeg} it multiplyed by y^e 2^d progression thus $\begin{matrix} x^{4} & * & - & 35 g g x x & + & 90 g^{3} x & - & 56 g^{4} & = & 0 \\ 0 . & - 1 . & 0 . & \frac{7}{8} . & \frac{45}{32} . \end{matrix}$ . It would produce $\frac{630}{8} g^{3} x - \frac{2520}{32} g^{4} = 0$ . Or $x = g$ . Soe y^t g being y^e least roote, y^e other two rootes must be $2 g$ & $4 g$ .

If it be desired to know the length of y \& z / when the rootes of this equation are in p $x^{3} - a a x + y y x - a a z = 0$ , when y^e rootes of it are in proportion a {sic} $1.2.4 .$ I multiply it by one of y^e precedent progressions & it is $\begin{matrix} x^{3} & - & a x x & + & y y x & - & \frac{8 a^{3}}{7} & = & 0 \\ 7 & 1 & 0 & 0 \end{matrix}$ . Or $x = \frac{a}{7}$ . w^ch valor of x inserted into its places in y^e Equation there results $\frac{a^{3}}{343} - \frac{a^{3}}{49} + \frac{a}{7} y y - \frac{8 a^{3}}{7} = 0$ . Or $y = \frac{a \sqrt{398}}{7}$ . Or thus $\begin{matrix} x^{3} & - & a x x & + & y y x & - & \frac{8 a^{3}}{7} \\ 1 & 0 & 0 \end{matrix}$

If it be desired to know y^e length of y & z in this equation $x^{3} - y x x + a^{2} x - a a z$ when y^e rootes are in proportion as $1.2.4 .$ I multiply it by y^e precedent progression & y^e results are $\begin{matrix} x^{3} & - & y x x & + & a a x & - & a a z & = & 0 \\ 7 & 1 & 0 & 0 \end{matrix}$ . Or $7 x = y$ . And $\begin{matrix} x^{3} & - & y x x & + & a a x & - & a a z & = & 0 \\ 0 & 0 & \frac{1}{8} & \frac{7}{32} \end{matrix}$ . Or $4 x = 7 z$ . $\begin{matrix} x^{3} & - & y x x & + & a a x & - & a a z & = & 0 \\ 14 & 0 & - 1 & 0 \end{matrix}$ . Or $14 x x = a a$ . And consequently $7 = x = \frac{7 a}{\sqrt{14}} = y = a \sqrt{\frac{7}{2}}$ $\frac{4 x}{7} = \frac{4 a}{7 \sqrt{14}} = z$ .

Likewise were the proportion of 4 or 5 or more rog\o/tes given I might set down progressions to find them but it will bee better to set downe y^e method of finding {those}{these} progressions, And it {illeg} is this. Suppose {illeg} two of y^e rootes of an Equation {illeg} That Equation will bee of this forme $x x \begin{matrix} - a \\ - b \end{matrix} x + a b = 0$ , or of some forme {illeg} of it; And if a corresponds to y^e \desired/ roote of y^e Equation desired this equation $x x$ {illeg} will bee of this forme $a a \begin{matrix} - a \\ - b \end{matrix} a + a b = 0$ . Then assuming two termes ({illeg} { $a . 0$ }) {illeg} third {illeg} progression by w^ch I multiply this equation {illeg} { $a . 0$ } by which {illeg} {illeg} multiplyed produceth $a a \begin{matrix} - a \\ - b \end{matrix} a + a b = 0$ Or { $x a a - a^{3} - a a b = 0$ }. And {illeg} {illeg} termes of y^e {illeg} {illeg}

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Soe y^t I have thus much of y^e progression $a + b + \frac{b b}{a} . a + b . a . 0$ . And by y^e same proceding might continue it or get termes on y^e other side of y^e cipher. As if I multiply y^e Equation by this progression $a . 0 . z$ there is produced $\begin{matrix} a a & \begin{matrix} - a \\ - b \end{matrix} a & + & a b & = & 0 \\ \times & \times & \times \\ a & 0 & z \end{matrix}$ . Or $a^{3} + a b z = 0$ . And $z = - \frac{a a}{b}$ . Againe multiplying y^e Equation by $0 . x$ {illeg} $0 . - \frac{a a}{b} . z$ . It is $\begin{matrix} a a & \begin{matrix} - a \\ - b \end{matrix} a & + & a b & = & 0 \\ \times & \times & \times \\ 0 & . & - \frac{a a}{b} & . & z \end{matrix}$ . Or $\frac{a^{4} + a^{3} b}{b} + a b z = 0$ , And $- \frac{a^{3}}{b b} - \frac{a a}{b} = z$ . Soe that I have thus much of y^e progression viz: $a + b + \frac{b b}{a} . a + b . a . 0 . - \frac{a a}{b} . - \frac{a a}{b} - \frac{a^{3}}{b b}$ .

The proceeding is same when y^e proportion of 3 rootes to one another are given, but there may bee some difficulty /difference\ wⁿ y^e ciphers are {illeg} far distant, as they|r|e bee three {illeg} termes betwixt them, then y^e opr|e|ration may be done thus. Let y^e quanti\t/ys, w^ch beare such proportion to one another as y^e rootes doe bee, a, b, c . let a correspond to y^e roote w^ch w^ch must be knowne And yⁿ y^t Equation will bee of this forme, $a^{3} \begin{matrix} - a \\ - b \\ - c \end{matrix} a a \begin{matrix} - a b \\ - a c \\ - b c \end{matrix} a - a b c = 0$ . or else compounded of it. Then assuming some quantity (as a ) for one of y^e termes of y^e {illeg} progression & placing it conveniently, (as {illeg} \it {illeg} {no{illeg}}/ equidistant from y^e ciphers) feigne two other quantitys as z , y , for y^e deficient termes and there will r progression will bee $0 . z . a . y . 0$ . By w^ch I multiply the Equation $\begin{matrix} a^{3} & \begin{matrix} - a \\ - b \\ - c \end{matrix} a a & \begin{matrix} - a b \\ - a c \\ - b c \end{matrix} a & - & a b c & = & 0 \\ \times & \times & \times & \times \\ 0 & . & z & . & a & . & y \end{matrix}$ . Or $\frac{- a a z - a b z - a c z}{b c} = y$ . Soe have I y^e progression $0 . z . a . \frac{- a a z - a b z - a c z}{b c} . 0 .$ by w^ch I againe multiply y^e Equation & there is produced a correspond to y^e roote w^ch w^ch must be knowne And yⁿ y^t Equation will bee of this forme, $a^{3} \begin{matrix} - a \\ - b \\ - c \end{matrix} a a$ Or $\frac{- a a z - a b z - a c z + a a b + a a c + a b c}{b c} = y$ . Soe that I have the progression, $0 . z . a . \frac{+ a a b + a a c + a b c - a a z - a b z - a c z}{b c} . 0 .$ by w^ch I againe multiply y^e Equation & there results $\begin{matrix} a^{3} & \begin{matrix} - a \\ - b \\ - c \end{matrix} a a & \begin{array}{l} - a b \\ - a c \\ - b c \end{array} a ............... & - & a b c & = & 0 \\ \times & \times & \times & \times \\ z & . & a & . & \frac{+ a a b + a a c + a b c - a a z - a b z - a c z}{b c} & . & 0 \end{matrix}$ . Or $z = \frac{a a b b + a a b c + a a c c + a b b c + a b c c + b b c c}{a a b + a a c + b b a + a c c + 2 a b c + b b c + b c c}$ . w^ch valor of z substituted into its place in y^e valor y There will bee thus much of y^e progre\ssion/ $0 . \frac{a a b b + a a b c + a a c c + a b b c + a b c c + b b c c}{a a b + a a c + b b a + a c c + 2 a b c + b b c + b c c} . a . \{\begin{matrix} a + \frac{a a b + a a c}{b c} - \frac{a^{4} - 2 a^{3} b - 2 a^{3} c - a a b b - 3 a a b c - a a c c - a b c c}{a a b + a a c + b b a + a c c + 2 a b c + b b c + b c c} \\ \frac{- a^{4} b b - a^{3} b^{3} - a^{3} b b c - a^{4} c c - a^{3} b c c - a^{3} c^{3}}{a a b b c + a a b c c + a b^{3} c + a b c^{3} + 2 a b b c c + b^{3} c c + b b c^{3}} \end{matrix}\} . 0$ .

The same done otherwise.

Did I know y^t 2 of y^e rootes of this Equation $x^{3} * - 117 x - 324 = 0$ , were in proportion as 3, $- 4$ . Then I suppose one roote to be $3 a$ , y^e other $- 3 a$ That is $x - 3 a = 0$ . $x + 4 a = 0$ . By \one of/ w^ch I divide y^e Equation as first by $x + 4 a = 0$ And y^e operation is; $\begin{matrix} x + 4 a) & \underline{x^{3} * - 117 x - 324} & (x x - 4 a x + 16 a a - 117 = 0 \\ \underline{0 - 4 a x x - 117 x - 324} \\ \underline{\begin{matrix} 0 + 16 a a x - 324 \\ - 117 x \end{matrix}} \\ \underline{\begin{matrix} 0 - 64 a^{3} \\ + 468 a^{} \end{matrix}} \\ \underline{- 64 a^{3}} & \underline{+ 468 a - 324 = 0} \\ 0 \end{matrix}$ . Againe I divide y^e Quotient by y^{e} other roote $x - 3 a = 0$ . Thus
$\begin{matrix} x - 3 a) & \underline{x x - 4 a x + 16 a a - 117} = 0 & (x - a = 0 \\ \underline{0 - a x + 16 a a - 117} \\ 0 \underline{+ 13 a a - 117} = 0 \\ 0 \end{matrix}$ . By y^e last division I have this equation $13 a a - 117 = 0$ . Or $a a - 9 = 0$ . And $a = 3$ . Therefore y^e rootes of y^e Equation $x^{3} - 117 x - 324 = 0$ are, $3 a = 9 = x$ . $- 4 a = - 12 = x$ .

If I would have y & z of such a length y^t y^e rootes of this equation $x^{3} - 3 y x + g z = 0$ . be in proportion as $+ 1$ , $+ 2$ , $+ 3$ . I suppose $x - a = 0$ , $x + 2 a = 0$ . $x + 3 a = 0$ . And soe first divide y^e Equation by $\begin{matrix} x - a) & \underline{x^{3} + * - 3 y x + g z} & (x x + a x + a a - 3 y = 0 \\ \underline{0 + a x x - 3 y x + g z} \\ 0 \begin{array}{r} + & a a \\ - & 3 y \end{array} x + g z \\ \overline{0 + a^{3}} & \overline{+ 3 a y + g z^{} = 0} \end{matrix}$ . Againe I divide this product
by $\begin{matrix} x - 2 a) & \underline{x x + a x + a a - 3 y} & (x + 3 a = 0 \\ 0 + 3 a x + a a - 3 y \\ \overline{0 + 5 a^{3} - 3 y} & \overline{= 0} \end{matrix}$ . Lastly were it necessary I should have again divided this quote $x + 3 a = 0$ by y^e 3^d supposed roote of y^e Equation (viz {illeg} $= 0$ ). By y^e 2^d {operation} $7 a^{2} - 3 y = 0$ . Or $\frac{7 a^{2}}{3} = y$ . And by y^e first $a^{3} - 3 a y + g z = 0$ . Or {illeg} /{ $18 a y = 7 g z$ }\ Soe y^t If I make {illeg} the rootes of this Equation $x^{3} - 3 y x + g z = 0$ should bee {illeg}

<57r>

^[24] ^[25] .R. An Equation being given, expressing y^e Relation of two or more lines x, y, z, & described in y^e same line by two or more moveing bodys A, B, C &c to find y^e relation of their velocitys p, q, r &c:

Resolution.

Sett all y^e termes on one side of y^e Equation y^t they become equall to nothing. And first Multiply each terme by soe many times $\frac{p}{x}$ as x hath dimensions in y^t terme. Seacondly multiply each terme by soe many times $\frac{q}{y}$ as y hath dimensions in {illeg} --> it. Thirdly multiply each terme by soe many times $\frac{r}{z}$ as z hath dimensions in it &c. The sume of all these products shall be equall to nothing. Which Equation gives y^e relation of p, q, r &c.

^[26] Or more generally thus. Order y^e Equation acording {sic} to y^e dimensions of x , & (putting a & b for any two numbers whither rationall or not) multiply y^e termes \of it/ by any pte of this progression viz : $ↄ& . \frac{a p - 3 b p}{x} . \frac{a p - 2 b p}{x} . \frac{a p - b p}{x} . \frac{a p}{x} . \frac{a p + b p}{x} . \frac{a p + 2 b p}{x} . \frac{a p + 3 b p}{x} . &c$ : Also order y^e Equation according to y & multiply y^e{illeg} {illeg}termes of it by this progression: $ↄ& . \frac{a q - 2 b q}{y} . \frac{a q - b q}{y} . \frac{a q}{y} . \frac{a q + b q}{y} . \frac{a q + 2 b q}{y} . \frac{a q + 3 b q}{y} &c$ . Also order it according to y^e dimentions {sic} of z & multiply its termes by this progression viz $ↄ& . \frac{a r - 3 b r}{z} . \frac{a r - 2 b r}{z} . \frac{a r - b r}{z} . \frac{a r}{z} . \frac{a r + b r}{z} . \frac{a r + 2 b r}{z} . \frac{a r + 3 b r}{z} &c$ . The sume of all these products shall bee equall to nothing. Which Equation gives y^e relation of p, q,r &c.

Example 1^st. If y^e propounded Equation bee $x^{3} - 2 x x y + 4 x x + 7 x y y - y^{3} - 103 = 0$ . By y^e precedent rule y^e first operation will produce $3 x x p - 4 x y p + 8 x p + 7 y y p$ . The seacond produceth $- 2 x x q + 14 x y q - 3 y y q$ . Which two added together make $3 x x p - 4 x y p + 8 x p + 7 y y p - 2 x x q + 14 x y q - 3 y y q = 0$ . (Now suppose a yarde to bee an unit & y^t a|A| hath moved 3 yardes, yⁿ (by y^e 1^st equation) B hath moved two; i,e, {sic} $x = 3$ , $y = 2$ . And \at that time/ by y^e last Equation $55 p + 54 q = 0$ . Or $55 ∶ - 54 ∷ p ∶ q ∷$ velocity of A ∶ velocity of B . Onely if x increaseth yⁿ y decreaseth, y^t is, A & B move contrary ways because p & q are affected w^th divers signes).

Example y^e 2^d. If y^e Equation bee $x^{3} - 2 a^{2} y + z^{2} x - y^{2} x + z y^{2} - x^{3} = 0$ . The first operation will produce $\begin{matrix} \frac{3 p}{x} & \frac{p}{x} & 0 & 0 & 0 \\ x^{3} & * & + z z x - y y x & - 2 a a y & + z^{3} y & - z^{3} \end{matrix}$ . Or $3 p x x + p z z - p y y$ . The second produceth $- 2 a a q - 2 y x q + 2 z y q$ . The third $+ 2 z x r + y y r - 3 z z r$ . The summe of w^ch is $3 p x x + p z z -$ $- p y y - 2 a a q - 2 y x q + 2 z y q + 2 z x r + y y r - 3 z z r = 0$ . (Note y^t in this Example there being three unknowne quantitys x , y , z , There must be two of them & their \two/ velocitys supposed thereby to find y^e 3^d quantity & y^e third velocity. Or else there must be some other equation expressing y^e relation of y^e th two of these x , y , z . (as in y^e first example) whereby one quantity & one velocity being supposed y^e other quantity & velocity may be found & yⁿ by this 2^d Example y^e 3^d quantity & y^e 3^d velocity may bee found)

Example 3^d, Of y^e more generall rule. If y^e Equation bee $x^{4} - 3 g y x x + y y x x - g g y y - 2 y^{4} = 0$ . y^e first operation gives $\begin{matrix} \frac{a p}{x} & . * . & \frac{a p + 2 b p}{x} & . * . & \frac{a p + 4 b p}{x} & . \\ x^{4} & * & \begin{matrix} + & y y \\ - & 3 g y \end{matrix} x x & * & \begin{matrix} - & g g y y \\ - & 2 y^{4} \end{matrix} & . \end{matrix}$ Or $\begin{array}{l} a p x^{3} & + & a p y y x & - & 3 a p g y x & - & \frac{a p g g y y}{x} & - & \frac{2 a p y^{4}}{x} \\ + & 2 b p y y x & - & 6 b p g y x & - & \frac{4 b p g g y y}{x} & - & \frac{8 b p y^{4}}{x} \end{array}$ . the 2^d gives $\overline{a q - 2 b q} \times \overline{- 2 y^{3}} \overline{+ a q} \times \overline{x x y - g g y} \overline{+ a q + b q} \times \overline{- 3 g g x} \overline{+ a q - 2 b q} \times \frac{x^{4}}{y}$ . \The summe of/ w^ch two products is equall to nothing. &c.

Demonstration.

^[27] Lemma. If two bodys $\begin{matrix} A \\ B \end{matrix}$ move uniformely y^e $\begin{matrix} one \\ other \end{matrix}$ from $\begin{matrix} a \\ b \end{matrix}$ to $\begin{matrix} c & , & e & , & g & , \\ d & , & f & , & h & , \end{matrix} &c$ in y^e same line yⁿ are y^e lines $\begin{matrix} ac \\ bd \end{matrix}$ & $\begin{matrix} ce \\ df \end{matrix}$ & $\begin{matrix} eg \\ fh \end{matrix}$ &c as their velocitys $\begin{matrix} p \\ q \end{matrix}$ {illeg}.And though they move not uniformely yet are y^e infinitely little spac lines w^ch each moment they describe {illeg} as their velocitys are w^ch they have while they describe them. As if y^e body A \w^th y^e velocity p / describe y^e infinitely little line o in one moment. Then \In y^e moment/ y^e body B w^th y^e velocity q will describe y^e line $\frac{o q}{p}$ . For $p ∶ q ∷ o ∶ \frac{o q}{p}$ . Soe y^t if y^e described \{lines}/ be x & y in one moment, they will bee $x + o$ & $y + \frac{o q}{p}$ , in y^e next. [or better $p ∶ q ∷ p o ∶ q o$ . &c]

Now if y^e Equation expressing y^e relation of y^e lines x & y be $r x + x x - y y = 0$ . I may substitute $x + o$ & $y + \frac{q o}{p}$ into y^e place of x & y because (by y^e lemma) they as well as x & y doe signifie y^e lines described by y^e bodys A & B . By doeing so there results $r x + r o +$ $+ x x + 2 o x + o o - y y - \frac{2 q o y}{p} - \frac{q q o o}{p p} = 0$ . But $r x + x x - y y = 0$ by supposition: there remaines therefore $r o + 2 o x + o o - \frac{2 q o y}{p} - \frac{q q o o}{p p} = 0$ . On divideing it by o tis $r + 2 x + o - \frac{2 q y}{p} - \frac{o q q}{p p} = 0$ . Also those termes in w^ch o is are {sic} infinitely lesse yⁿ those in w^ch o is not therefore I blotting y^m out there {rests} $r + 2 x - \frac{2 q y}{p} = 0$ . Or $p r + 2 p x = 2 q y$ .

Hence may bee observed: First, y^t those {illeg} termes \ever/ vanish in w^ch o is not because they are y^e propounded Equation. Secondly y^e remaining Equation being divided by o those termes also vanish in w^ch o still remaines because they are infinitely little. Thirdly y^t y^e \still/ remaining termes consist of y {illeg} \will ever/ have y^t forme w^ch by y^e \first/ {root}{rule} they should have. [{illeg} partly appeare by Oughtreds Analyticall table].

The {rule} may bee demonstrated after y^e same manner if there 3 \or more/ unknowne quantitys x, y, z {&c.}

<57v>

By helpe of y^e preceding probleme divers others may bee readily resolved.

^[28] 1. To draw tangents to crooked lines (however they bee related to sreight {sic} ones).

Resolution

^[29] Find (by y^e preceding rule) in w^t proportion those two lines to w^ch y^e crooked line {cheifly {sic}} related doe increase or decrease: produce y^m in y^t proportion {illeg} from y^e given point in y^e crooked line {{illeg}|at|} those ends draw perpendiculars to y^m \lines in which those ends are {enclosed} to move/ through whose intersection y^e tangent shall passe.

Example 1^st. If $ab = id = x$ . $ai = bd = y$ . & $r x - \frac{r x x}{s} = y y$ . Then is $p ∶ q ∷ 2 y ∶ r - \frac{2 r x}{s}$ . (by y^e former rule) Therefore I draw $gd ∶ ge$ {illeg} $de ∶ dg ∷ p ∶ q ∷ 2 y ∶ r - \frac{2 r x}{s}$ . The point g is inclined to move in a parallel to abc & y^e point e in a parallel to aik (for bg \& ie / (by supposition) moves parallel to y^m selves y^e $\begin{matrix} one \\ other \end{matrix}$ upon $\begin{matrix} abc \\ aik \end{matrix}$ ) Therefore I draw $gf ∥ abc$ & $ef ∥ aik$ . & through \y^e intersection/ $(f)$ I draw hdf touching y^e crooked line at d . Soe y^t $gd ∶ de ∷ db ∶ bh ∷ q ∶ p ∷ r - \frac{2 r x}{s} ∶ 2 y$ .

Hence may bee pronounced those theorems in pag 47 Fol 47

^[30] Example y^e 2^d. If $ac = x$ . $bc = y$ . (w^ch move about y^e centers a & b as in y^e Hyperbola or Ellipsis by a thred) And y^e equation bee $- a + x + y = 0$ . yⁿ is $p + q = 0$ . or $p = - q$ . therefore I make $cd ∶ cb ∷ p ∶ q ∷ 1 ∶ - 1$ . (note y^t I draw cd & cB y^e one forward y^e other {backwd} because p & q have contrary signes) y^e points d & B are inclined to move y^e one in a perpendicular to acd y^e other to bBc (for y^e circle they {illeg} move in circles whose centers are a & b ) therefore I draw $de ⊥ acd$ & $Be ⊥ bBc$ & the tangent ce through y^e point e .

^[31] 2. Hitherto may bee reduced y^e manner of drawing tangents in mechanicall lines. see Fol 50.

^[32] 3. To find y^e quantity of crookednes in Geometricall lines.

{illeg} Resolution

^[33] Find y^t point of y^e perpendicular \to y^e crooked line/ w^ch is in least motion, let y^t bee y^e center of a circle w^ch passing through y^e given point shall bee of equall crookednesse w^th y^e line at y^t point. This point of least motion may bee found divers ways, as First. {illeg} From any two {illeg} points in y^e perpendicular \to y^e crooked line/ draw 2 \parallel/ lines in such proportion as y^e perpendicular moves over y^m: through their ends draw another line w^ch shall intersect y^e perpendicular in y^e point required.

Example. Suppose $ab = x$ . $bc = y$ . $x ⊥ y$ . {illeg} $a x - \frac{a x x}{b} = y y$ . $be = a = \frac{a}{2} - \frac{a x}{b}$ . $p =$ motion of b from a {illeg} $y = \frac{- a p}{b} =$ motion of e from b. & $\frac{b p - a p}{b} =$ motion of {c}{e} from a $bk = cn = \frac{y y}{v} = \frac{2 b x - 2 x x}{b - 2 x}$ . $nm = y$ . $nd = v = \frac{a b - 2 a x}{2 b} = \frac{y y}{x} - \frac{a}{2}$ . $cd = ke$

As if $ab = x$ . $bc = y$ . $ce ⊥ ck =$ y^e tangent of y^e crooked line. {illeg} $be = cd ∥ abef$ . & as y^e motion of b from a to y^e motion of e from a so kb to ef . Then, drawing dfg through y^e points d {illeg} & f , cg in y^e radius of a circle as crooked as line acl at c .

Example. Suppose $a x - \frac{a x x}{b} - y y = 0$ . yⁿ is $be = v = \frac{a}{2} - \frac{a x}{b}$ . $kb = \frac{y y}{v} = \frac{2 b y y}{a b - 2 a x}$ . And $cd = ke = \frac{a a b b - 4 a a b x + 4 a a x x + 4 b b y y}{2 a b b - 4 a b x}$ . $bk = p =$ motion \velocity/ of b from a , $bc = y = q =$ velocity of y 's increase $r =$ velocity of v 's increase. $2 b v - b a + 2 a x = 0$ therefore $2 b r + 2 a p = 0$ , or (since $p = \frac{2 b y y}{a b - 2 a x}$ ) tis $r = \frac{2 y y}{- b + 2 x}$ . & $ef = \frac{2 a y y + 2 b y y}{a b - 2 a x}$ \{ $= r + bk =$ {illeg} }/ Lastly $cd - ef ∶ ce ∷ cd ∶ cg$ (or $v - r ∶ y ∷ v + p ∶ ck$ . if { $ce ⊥$ {illeg} }) y^t is $\frac{- 4 b y y + 4 a b x - a b b - 4 a x x}{- 2 b b + 4 b x} ∶ y ∷ \frac{a a b b - 4 a a b x + 4 a a x x + 4 b b y y}{2 a b b - 4 a b x} ∶ ch$ . $\frac{a a b y + 4 b y^{3} - 4 a y^{3}}{a a b} = ch = \frac{4 b y^{3} - 4 a y^{3} + a a b y}{a a b}$ . & $bh = \frac{4 y^{3}}{a a} - \frac{4 y^{3}}{a b}$ .

Hence may bee pronounced those theorems in Fol 49.

<62v>

Addition connects affirmative qua numbers into an affirmation sume, & negative ones into a negative
one. as $\begin{array}{l} \begin{matrix} + 1352 \\ + 7460 \end{matrix}} \\ + 8812 \end{array}$ $\begin{array}{l} \begin{matrix} - 137905 \\ - 68432 \end{matrix}} \\ - 206337 \end{array}$

Substractions takes y^e greater \lesse/ number from y^e lesse \greater/, the difference having y^e same signe prefixed w^ch y^e greater quantity \number/ {hats} as $\begin{array}{l} \begin{matrix} + & 6 & 3 & 5 & 7 & 9 & 2 \\ - & 1 & 4 & 7 & 0 & 3 & 1 \end{matrix}} \\ \begin{matrix} + & 4 & 8 & 3 & 7 & 6 \end{matrix} \end{array}$ . $\begin{array}{r} + 44503 \\ - 26794 \end{array}$ $\begin{array}{l} \begin{matrix} - 26791 \\ + 4503 \end{matrix}} \\ - 22288 \end{array}$ .

Multiplicacon adds one factor soe often {illeg} to it selfe as there are units in y^e other, & if y^e signes of y^e factors bee y^e same y^e product is affirmative, if divers tis negative. As to multiply $+ 735$ by $+ 47$ doe thus $735 in 7$ / $\begin{array}{l} \begin{array}{r} 5145 = 735 in 7 \\ \underline{29400 = 735 in 40} \end{array}} \\ \begin{array}{r} + 34545 = 735 in 47 \end{array} \end{array}$ \ Or thus $\begin{array}{r} 735 \\ 47 \\ 5145 \\ 29400 \\ + & 34545 \end{array}$ . Or thus $\begin{matrix} \begin{array}{r} 735 \\ 5145 \\ 29400 \\ + & 34545 \end{array} & \begin{array}{l} \times \\ 7 \\ 40 \end{array} \end{matrix}$ . Thus to multiply $- 3241$ by / $- 3175$ the operation will\ bee $\begin{matrix} \begin{array}{r} 3241 \\ 16205 \\ 22687 \\ 3241 \\ + & 567175 \end{array} & \begin{array}{l} 5 \\ 7 \\ 1 \end{array} \end{matrix}$ Alsoe 465 multiplyed by $- 32$ /will produce\ $\begin{matrix} \begin{array}{r} 465 \\ 930 \\ 1395 \\ - & 14880 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} \end{matrix}$

Division takes y^e number w^ch signifies how often y^e divisor $\{\begin{array}{l} is contained in & \\ may bee substracted from \end{array}\}$ y^e divisor, \y^e sign of/ w^ch number or Quote {illeg} is affirmative if y^e dividend & divisor have not divers signes, but negative{illeg} if they have. For if $x = \frac{a}{b}$ . then $b x = a$ , Or $a - b x = 0$ . Suppose 34545 to be divided by 47 . First get a {illeg} Table of y^e Divisor drawn into y^e 9 first units {illeg} as defg . {illeg} $\begin{matrix} d & f \\ \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ - 47 & . & - 94 & . & - 141 & . & - 188 & . & - 235 & . & - 282 & . & - 329 & . & - 376 & . & - 423 & . \end{matrix} \\ e & g \\ \begin{matrix} \begin{array}{r} 345 4 5 \\ 016.4 5 \\ 02 3.5 \\ 0 0 0 . \end{array} & \begin{array}{l} + & 735 \\ - & 000 \\ 735 \end{array} \end{matrix} \end{matrix}$ cut at y^e bottome $(eg)$ close to the figures. Then looke w^ch of those 9 quantitys are most like y^e dividend. As in this case y^e 7^th 329 is therefore substract it from y^e {illeg} dividend 34545 , & there will remaine $16.45$ , & then set downe its caracteristick 7 in y^e quote. I make a prick twixt those figures $(16)$ w^ch have or might have beene altered & those $(45)$ w^ch could not bee altered by the subtraction, & the places of y^e pricks will will {sic} skew the places of y^e figures in y^e quotient. Againe I substract 141 from $16.4$ &c: & set 3 in y^e quote &c.

If 19489012 was to be divided by {illeg} 732 $\begin{matrix} \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 732 & . & 1464 & . & 2196 & . & 2928 & . & 3660 & . & 4392 & . & 5124 & . & 5856 & . & 6588 \end{matrix} \end{matrix}$ $\begin{array}{l} \begin{matrix} \begin{array}{r} + 3 . & + 1948 9 012 \\ - 2 . & - 147.0 988 \\ - 9 . & - 000 6.988 \end{array} & \begin{array}{r} + & 30000 00355 \\ - & 02009, 55000 \\ - & 27990, 45355 \end{array} \end{matrix} \\ \begin{array}{r} - 5 . & - 0 400 & . 0 \\ - 5 . & - 034 & 0.0 \\ + 3 . & + 02 & 6 0.0 \\ + 5 . & + 0 & 4 0 4.0 \\ + 5 . & + & 0 3 8 0 . \end{array} \end{array}$

<64r>

The resolution of y^e affected Equation $x^{3} + p x x + q x + r = 0$ . Or $x^{3} + 10 x^{2} - 7 x = 44$ First having found two or 3 of y^e first figures of y^e desired roote viz $\begin{matrix} \begin{matrix} 2 \end{matrix} & \begin{matrix} \underline{2} \end{matrix} \end{matrix}$ (w^ch may bee done either by rationall of Logarithmicall tryalls as M^e Oughtred hath thought, or \Geometrically/ by Geometricall descriptions of lines, or by an instrument consisting of 4 or 5 or more lines of numbers made to slide by one another w^ch may be oblong but better circular.) this knowne pte of y^e root I call g, y^e other unknowne pte I call y then is $g + y = x$ . Then prosecute y^e Reduction /Resolution\ after this manner (making $x + p in x = a$ . $a + q in x = b$ . $b + r in x = c$ &c.) $\begin{array}{l} \times & 12 = x + p \\ x = 2, & 24 = a \end{array}$ . {illeg} $\begin{array}{l} a + q = 17 & \times \\ b = 34 & 2 = x \end{array}$ . $r - b = 10 = h$ . by supposing $x = 2$ . Againge supposing $x = \begin{matrix} \begin{matrix} 2 \end{matrix} & \begin{matrix} \underline{2} \end{matrix} \end{matrix}$ Then $\begin{array}{r} x + p = 12,2 \\ 244 & 2 \\ 244 & 2 \\ 26,84 & = a \end{array}$ . $\begin{array}{r} a + q = 19,84 \\ 3968 & 2 \\ 3968 & 2 \\ 43,648 & = b \end{array}$ . $r - b = 0,352 = k$ . $h - k = 9,648$ . That is y^e $\{\begin{cases} latter r - b substracted \\ difference twixt \end{cases}$ $\begin{array}{l} from the former r - b there remaines \\ this & y^{e} former valor of r - b is \end{array}\} 9,648$ . & y^e difference twixt this & y^e former valor of x is 0,2 . Therefore make $9,648 ∶ 0,2 ∷ 0,352 ∶ y$ . Then is $y = \frac{0,0704}{9,648} = 0,00728$ &c. the first figure of w^ch being added to y^e last valor of x makes $2,207 = x$ . Then w^th this valor of x presecuting y^e operacon as before tis $\begin{array}{r} x + p = 12,207 \\ 24414 \\ 24414 & 7 \end{array}$ $\begin{array}{r} x + p = 12,207 \\ 65449 & 7 \\ 244140 & 02 \\ 244140 & 2 \\ 26,940849 & = a \end{array}$ $\begin{array}{r} a + q = 19,94084 \\ 13958588 & 7 \\ 39881680 & 02 \\ 3988168 & 2 \\ 44,00043388 & = b \end{array}$ . $r - b = - 0,00943388$ . w^ch valor of $r - b$ substracted from y^e precedent valor of $r - b$ y^e {diff} is $+ 0,36143388$ . Also y^e {diff} twixt{illeg} tis & y^e precedent valor of x is 0,007 . Therefore I make { $36143388 ∶ 0,007 ∷ - 0,0094388 ∶ y$ }. That is $y =$ {illeg}

<67v>

^[34]

Of the construction of Problems.

^[35] If y^e equation to be resolved bee $y y \overset{\cup}{O} a y - bb = 0$ . Or yy−ay+bb=0 in w^ch y^e roote of y^e last terme (viz b) is knowne, they may bee resolved {illeg} conveniently resolved by D. Cartes his rules. Otherwise y^e rootes of y^t ter{illeg}|me| must bee first extracted as in this yy−py+q=0. Where I take $ln = \frac{1}{2} q . lg = \frac{q - 1}{2} q . gs = \frac{q + 1}{2} q$ . & soe describing y^e circle smf erect lm⊥ln & from m y^e point of intersection draw mr∥ln. y^e rootes of y^e Equation shall bee mq & mr. ln being y^e radius & n the center of the circle

Or it may bee done thus. Let{illeg} the Equation bee $y y \overset{\cup}{O} p y \overset{\cup}{O} q = 0$ . {illeg}Then in the indefinite line af take $ab = \frac{\underset{\cap}{O} p}{2}$ . erect y^e perpendicular { $bc = \frac{pp}{8 c} \frac{\underset{\cap}{O} q}{2 c} - \frac{1}{2} c$ . $(bc = \frac{pp \overset{\cup}{O} 4 q - 4 cc}{8 c})$ } & w^ch y^e Radius cd erect the perpendicular db=c. And from y^e point & towards b draw $dc = \frac{pp \underset{\cap}{O} 4 q + 4 cc}{8 c} (= \frac{pp}{8 c} \underset{\cap}{O} \frac{q}{2 c} + \frac{c}{2})$ w^th w^ch radius describe y^e circle edf & ae, af shall {illeg}bee y^e rootes of y^e Equation. When note that any quantity may be taken for c, soe y^t if it may bee y^e val{illeg}or of dc may bee noe fraction, & that db & dc bee as {illeg}|equall| little differing as may bee. Soe y^t y^e operacon {illeg}may thereby be made convenient, & {illeg} to y^t purpose the difference twixt db & dc must bee as little as may bee,|(|that is twixt $pp \underset{\cap}{O} 4 q$ & 4cc.|)| soe y^t y^e circle intersect not (ef) {illeg} over obliquely nor y^e circle be over greate.

^[36] As if I had this Equation yy+6y−9=0 Or yy=−6y+9. Then must I make $dc = \frac{9}{c} + \frac{1}{2} c$ . Then if I make c=6 it will bee $d = \frac{3}{2} + 3 = \frac{9}{2} = 4 \frac{1}{2}$ . Therefore I take $ab = - \frac{p}{2} = bd =$ $ab = - \frac{1}{2} p = -3$ . $bd = c = 6$ . $dc = 4 \frac{1}{2} = \frac{3 c}{4}$ . And soe describing y^e circle efc, I have one affirmative roote af, another negative ae. Or had I taken any other convenient valor for c as 1, or 3. or 4 the li{illeg}ne ae & af would still have bene y^e same.

Had I this equation yy−8=0. or yy=8. Then is $dc = \frac{4}{c} + \frac{1}{2} c$ . soe y^t Or makeing c=2; tis: dc=3. Soe y^t since p is wanting y^e p{illeg} I take ab=0. ad=c=2 dc=3. & describing a circle y^e rootes will bee ea, af.

Note y^t if dc is negative or not greater then $\frac{1}{2} ad$ y^e circle cannot intersect y^e line eaf & therefore y^e rootes of y^e equation are imm̄agina{illeg}|{rie}|.

< insertion from f 67v >

^[37] Or they may bee construed by drawing streight lines onely{.} thus. Let y^e Equation be $2 y y = 2 a y + b$ . or $y = a \overset{\cup}{O} \sqrt{aa + b}$ First I {make} {illeg} {the} {illeg} ab equall divide aa+b into square numbers (as {for} of y^m as may bee) (It may ever bee divided (though not) into (y^e fewest) squares by taking the greatest {square} out of aa+b & y^e greatest out of y^e remainder &c) as if in numbers y^e Equation were yy=2y+4 Or $y = 1 \overset{\cup}{O} \sqrt{5}$ . I take then square 4 out of 5 & there rests 1 w^ch is also a square. Then I draw ab $\sqrt{4}$ . & $bc = \sqrt{1} = 1$ . & make ab⊥bc. soe is $ac = \sqrt{5}$ . to w^ch I add ad=1. & soe is $dc = y = 1 + \sqrt{5}$ .

Were y^e Equation yy=−4y+34. Or $y = -2 \overset{\cup}{O} \sqrt{38}$ . Then is 38−36=2. 2−1=1. & 38=36+1+1 w^ch are square numbers. Therefore I make $ad = \sqrt{36} = 6$ . $ad ⊥ de = \sqrt{1} = 1$ . & draw $ae ⊥ ec = \sqrt{1} - 1$ . & draw $ac = \sqrt{38}$ . from w^ch take ab=2, & there rests $bc = -2 + \sqrt{38} = y$ .

Were y^e Equation { $6 y y = 15 y - 2$ } {illeg} $y = \frac{15}{6}$ $y = 6 + \sqrt{\frac{15}{7}}$ . Or $y = 6 + \frac{\sqrt{15}}{\sqrt{7}}$ . Find $\sqrt{15}$ & $\sqrt{7}$ before, &c:

< text from f 67v resumes > <68r>

^[38] If y^e Probleme be sollid it may bee readily resolved by {illeg} the intersection of y^e Parabola & circle. \as D: Cartes hath shewed/ If it bee of 5 or 6 dimensions it may bee resolved by y^e intersection of y^e line b y³−byy−cdy+bcd+dxy=0. Or y³−byy+bcd+dxy=0 & y^e circle \when pp={illeg} 4q. & q & v affirmative./ as D: C: hath explained. Or it might beee done by y^e intersection of a circle & one of these lines, viz y³+byy−hx=0 when y^e equation is reduced to such a forme y^t pp=4q. Or this y³+byy+gy−hx=0. Or this y³+gy−hx={illeg}|{0}|, s being affirmative & {illeg} p=0. Or this y³+d−fyx=0 when p=0, & q & v affirmative. &c.

But since all Equa{tions}

^[39] But all Equations in Generall may bee resolved by y^e line a²x=y³, after this manner. First (making a=1{.}|)| describe y^e line x=y³ uppon a plate. {illeg} (as cadce. Then in w^ch ab=x. bc=y). Then suppose y^e Equation to bee resolved bee y⁹*+my⁷+ny⁶+py⁵+qy⁴+ry³+syy+tq+v=0. (in w^ch y^e letters m, n, p &c: signifie y^e ^[40]termes knowne quantitys of each terme affected w^th its signe + or −). I describe another line cdce, whose nature (making ab=x, bc=y) is the exprest $\begin{matrix} x^{3} & + & m y x x & + & p y y & + & s y y \\ + & n & + & q y & x & + & t y & = 0 \\ + & r & + & v \end{matrix}$ & letting & let fall perpendicula{{illeg}|r|} from{e} every point where these two lines intersect as, df eg, they shall bee y^e rootes of y^e propounded eqation.

In like manner was y^e Equation to bee resolved y¹⁰*⁺my⁸+my⁷+py⁶+qy⁵+ry⁴+sy³+ty²+ry/+w=0\ the nature of y^e line cdce woluld bee $\begin{matrix} x^{4} & + & m y & + & p y y & + & s y y \\ + & n & x^{3} & + & q y & x x & + & t y & x & + & w y y & = 0 \\ + & r & + & v \end{matrix}$ . Or else it might bee $\begin{matrix} y x^{3} & + & m y y & + & q y y & + & t y y \\ + & n y & x x & + & r y & x & + & v y & = 0 \\ + & p & + & s & w \end{matrix}$ . Or had I this Equation y¹⁰+{illeg}|{l}|y⁹+my⁸+ny⁷+py⁶+qy⁵+ry^3|4|+sy³{illeg}+ty²+vy\+w/=0. The nature of y^e line cdce would bee, $\begin{matrix} x^{4} & + & l y y & + & p y y & + & s y y \\ + & m y & x^{3} & + & q y & x x & + & t y & x & + & w y^{2} & = 0 \\ + & n & + & r & + & v \end{matrix}$ . Or, $\begin{matrix} y & + & m y y & + & q y y & + & t y y \\ + & l & x^{3} & + & n y & x x & + & r y & x & + & v y & = 0 \\ + & p & + & s & + & w \end{matrix}$ . Or {illeg} it might bee, $\begin{matrix} y y & + & n y y & + & r y y & + & v y y \\ + & l y & x^{3} & + & p y & x x & + & s y & x & + & w y & = 0 \\ + & m & + & q & + & t \end{matrix}$ . If y^e resolved Equation have fewer dimensions y^t is if some of y^e \ultimate/ termes {illeg} as, w, v, t &c: (or intermediate termes as m, n &c \be blotted out/: Or if y^e Equation have more yⁿ 10 dimensions{{illeg}|th|}e nature of y^e lines \cdce/ to bee described may \be/ known by y^e same manner observing y^e order of y^e progression

Tis evident alsoe y^t there are 3 divers lines {any of w in} by w^ch {illeg} \any/ Probl: may bee resolved \unless some of them {chanch} to be y^e same/, the easiest whereof is to bee chosen. It appeares also how Equations of 2 & 3 dimensions may be resolved by drawing streight lines; of 4, 5, & 6 by describing some conick section; of 7, 8, 9, by describing a line of 3 dimensions; of 10, 11, 12, by a line of 4 dimensi{illeg}|ons|, {illeg} &c: but yet y is never above 2 dimensions & consequently all these lines may bee described by y^e rule & compasses.

{illeg}Some Those {illeg} Equations of more then 9 dimensions may bee (though seldome soe

Had I this line y⁴=x. described on a plate & this Equation to bee resolved viz: y¹³+ly¹²*+ny¹⁰+py⁹+qy⁸+ry⁷+sy⁶+ty⁵+vy⁴+wy³+ayy+by+c=0. It might bee resolved by describing y^e line whose nature is $\begin{array}{l} + & y x^{3} & + & n y y & + & r y^{3} & + & w y^{3} \\ + & l & + & p y & x x & + & s y y & x & + & a y y & = 0 \\ + & q & + & t y & + & b y \\ + & v & + & c \end{array}$ . A line of y^e 2^d sort. Whereas by y^e preceding rule was required y^t a line of y^e 3^d sort should have ^[41] beene described. And here observe y^t taking y^e square number w^ch is next greater yⁿ y^e number of y^e resolvend equation termes \dimensions/ of the resolvend equation,|.| That Equation may bee resolved by lines, y^e number of whose dimensions is not greater than the roote of y^e {illeg} square number. And the rectangles of th{illeg}|os|e numbers w^ch signifie y^e dimensions how many dimensions th{illeg}|{e}| lin{illeg}|es| have, {must \even/} either equall or greater may always bee greater or equall but never lesse yⁿ y^e number of dimensions of y^e resolvend Equations. For y^e number of points in w^ch two lines may intersect can never bee ^[42] greater yⁿ y^e rectangle of y^e numbers of theire dimensions. And they always intersect in soe many points, excepting those w^ch are im̄aginarie onely. Soe that all Equations $\begin{matrix} \begin{array}{l} which have \\ noe more than \end{array} & \{\begin{matrix} 1 \\ 2 \\ 4 \\ 6 \\ 9 \\ 12 \\ 16 \end{matrix}\} & \begin{matrix} dimensions, may always be \\ resolved by the \\ intersection of \end{matrix} & \{\begin{array}{l} 2 streight lines \\ a streight line & a c o nick section. \\ two conic k sections. \\ a conick section of a line of 3 dimensions. \\ two lines of 3 dimensions. \\ a line of 3. & another of 4 dimensions. \\ two lines of 4 dimension; or by one of 1 & another of 6 dim: &c: \end{array}\} & \begin{array}{l} but not by any simpler \\ lines. (From this conside r \\ may Problems be di st in- \end{array} \end{matrix}$ guished into sorts.) {St} will often bee very intricate to resolve Equations \of many dimensions/ by the simplest line by w^ch they may be resolved & also for y^e most ꝑt will regaine a des{illeg}|cr|iption of two {illeg} lines for every probleme. And then {if maybe often} {illeg} end to use two lines whereas {illeg} compound y^e other more simple {illeg}|&| {illeg} As perhaps an Equation of 16 dimension may bee more speedily resolved by two lines {illeg} of 6 dimensions then by two lines {illeg} 4 dimensions.

<68v>

^[43] But it will not bee {amisse} to shew \more/ particularly how these resolutions may bee performed. A{illeg}|nd| that firs by y^e parabola

T Suppose therefore I had y^e parabola x=yy exactly described & would resolve \{illeg} plaine probleme/ the Equation yy+ky+{illeg}l=0. I take ag={illeg}l. gf=k. fh=1=lateri recto Parab: & so draw y^e line gh & from y^e intersection points d, e, draw db, ec perpendicular to y^e axis gc. w^ch shall bee y^e rootes of y^e Equation w^ch are affirmative when they fall on y^e contrary side to fh, but negative if {illeg}|on| y^e same, as in this case.

^[44] But were I to resolve a sollid problem the Equation being of 4 dimensions, I take away y^e 2^d terme|,| {illeg} makeing it of this forme y⁴*⁺lyy+my+n=0. Then take $ae = \frac{1}{2}$ $ep = \frac{1}{2} l$ . $pq = \frac{1}{2} m$ . Then perpendicular to ap draw af=aq. Also draw fk∥ap, & from y^e point of intersection k draw{illeg} kh={illeg}n. lastly draw kr⊥ap, & w^th the radius wr upon the center q describe y^e circle tsm. {illeg} (or, w^ch is y^e same, take $ar = \frac{1 - 2 l + ll + mm - 4 n}{4}$ . & soe erecting y^e perpendicular rw, w^th y^e Radius rw describe y^e circle tsm) & from y^e points where it intersects y^e Parabola let fall perpendiculars to y^e axis, (tv, nm) they shall bee the rootes of y^e Equation y^e affirmative ones falling on y^e contrary side to pq. when m is affirmative.

If n=0, that is if y³*+ly{illeg}+m=0, then must the circle bee described w^th y^e Radius aq; for then is wr=fa=aq.

^[45] If I would resolve y^e cubick Equation y³+ky²+ly+m=0 (w^ch multiplyed by y−k=0 produceth $\begin{array}{l} y^{4} * & + & l y y & + & m y & - & k m & = 0 \\ - & k k & - & k l \end{array}$ ) I make $ae = \frac{1}{2}$ . $ep = \frac{l - kk}{2}$ . $pq = \frac{m - kl}{2}$ . fk∥ae⊥af={illeg}aq. kd={illeg}km. And w^th y^e radius cg upon y^e center q describe y^e circle wf. Or else doe thus (since k is one of y^e rootes of y^e Equation $\begin{array}{l} y^{4} * & + & l & y y & + & m & y & - & m k & = 0 \\ - & k k & - & l k \end{array}$ ) make k=ab+ar & draw bw∥ae (or make ar=kk, & wr{illeg}⊥ap) & describe a circle w^th y^e radius wq. Then letting fall perpendiculars from y^e intersection points, they (being y^e rootes of y^e Equation $\begin{array}{l} y^{4} * & + & l & y y & + & m & y & - & m k & = 0 \\ - & k k & - & l k \end{array}$ ) shall all, except wr=k, bee y^e rootes of y^e Equation y³+kyy+ly+m.

This operation will bee much shortened when y^e 2^d terme is wanting for {y^t} since k=0. it will bee $ae = \frac{1}{2}$ . $ep = \frac{1}{2} l$ . $pq = \frac{1}{2} m$ & aq y^e radius of y^e circle.

^[46] And if y^e last terme {vanish} that is if I would resolve this equation yy+ky+l=0. by y^e intersection of a circle & parabola. I must take $ae = \frac{1}{2}$ . $eπ = \frac{1}{2} l$ . $πq = \frac{kk}{2}$ . $πq = \frac{kl}{2}$ & soe w^th y^e radius aq upon y^e center q describe a circle, & y^e perpendiculars from y^e intersection points to y^e axis (a{{illeg}}, tv) are y^e rootes excepting one w^ch is equall to k.

<69r>

^[47] If I had y^e crooked \line/ described \fig 1st/^[48] whose nature is x=y³, & would resolve y^e Equation y³*+lyy+m=0. (calling ad=x, dg=y; Or a=-x. ce=-y) I take ab=m. bd=l. df=1. & df⊥bd & draw bf infinitely both ways. From y^e intersection points (as e) letting fall perpendiculars, they shall bee y^e {illeg} rootes of y^e Equation y³*+lyy+m. as ce w^ch in this case is negative because on that side on w^ch y is negative.

Would I resolve this equation yy+ky+l=0. (w^ch multiplyed by y-k produceth $\begin{array}{l} y^{3} * & + & l & y & - & kl = 0 \\ - & kk \end{array}$ ) I take ab=kl, (fig 2^d)^[49] bδ=l, δd=kk. df=1, & soe through y^e points b & f draw the streight line bfλ |(|Or w^ch is y^e same take ab=kl. k=ah⊥ab. & draw hλ∥ab untill it intersect y^e crooked line in λ (i. e. untill hλ=k³ & soe through the points λ & b draw λbfe|)|. Then from y^e intersection points to y^e axis letting fall perpendiculars they (being y^e rootes of y^e Equation $\begin{array}{l} y^{3} * & + & l & y & - & lk = 0 \\ - & kk \end{array}$ .) shall all, except βλ=k, be y^e rootes{illeg} of y^e Equation yy+ky+l=0.

^[50] Would I resolve the Equation {y⁴}+ z⁴{illeg}+az³+bzz+cz+d=0. It may bee done by a circle thus. M{illeg}ultiply it by this Equation zz−az+aa−b=0, & it will produce $\begin{array}{r} z^{6} * * & + & c z^{3} & + & d z z & - & ad z & + & aad & = 0 \\ - & 2 ab & - & ac & + & aac & - & bd \\ + & a^{3} & + & aab & - & bc \\ - & bb \end{array}$ , Of this forme z⁶**+mz³+nzz+pz+q=0. In w^ch (n) ought to be affirmative, & if it bee not, yⁿ augment or diminish y^e rootes of y^e Equation z⁴+az³+bzz+cz+d=0. & then repeate y^e operacon again untill there bee an Equation of this forme z⁶**+mz³+nzz+pz+q=0 in w^ch n is affirmmative. Then (dividing this equation by $\sqrt{4 : n}$ it is $z^{6} * * + \frac{m}{\sqrt{n \sqrt{n}}} z^{3} + z z + \frac{p}{n \sqrt{4 : n}} z + \frac{q}{n \sqrt{n}} = 0$ therefore) take $ab = \frac{m}{2 \sqrt{n \sqrt{n}}}$ . $bc = \frac{p}{2 n \sqrt{4 : n}}$ . {illeg} & w^th y^e radius $cd = \sqrt{\frac{mmn + pp - 4 nq}{4 nn \sqrt{n}}}$ , describe y^e circle dk & y^e perpendiculars (as dh ck) multiplyed by $\sqrt{4 : n}$ shall bee y^e rootes of y^e Equation.

<71r>

Theoremata Optica.

Si radius divergens a puncto \dato/ A vel convergens ad punctum \idem/ A incidit in Sphæram CVD ad punctum D, et refractus ejus convergit ad puncto B vel divergit ab eodem, sit sphæræ centrum C {illeg} in recta AB situm, secet {sic} AB sphæram in v tr sit sph{illeg}|æ|ræ centrum C, & secet AC producta sphæram in V et radium refractum in {illeg} DR in B: a punctis D et C ad AB, AD, BD demitte normales DH, CI, CR; sit sinus incidentiæ ad sinum refractionis seu Ci ad CR ut I ad R; et facto R,AC.I,AV∷CF.VF. {illeg} erit {illeg}F focus, seu locus imaginis puncti A radios quaquaversum emittentis.

2^do A puncto v versus A cape VG ad VA {illeg} ut est R ad I et error radij refracti DR a loco imaginis {seu} in axe AV, seu distantia punctorum B et F erit {illeg} AC, BC, BG, VD AC, FC $\frac{AC, FC, FG, {VD}^{q}}{2 {AV}^{q}, {CV}^{q}}$ , sive {illeg} $\frac{AC, FC, FG, VH}{{AV}^{q}, CV}$ quamproxime. Sed regula prior in f{illeg} radios est.

3. Ubi punctum infinite distat ita ut radius incidens parallelus sit axi, pro AC scrito AV, et pro FG scripto $\frac{R, VA}{G}$ (nam hæ \jam/ sunt æquipollentia) error BF {illeg}|{fist}| et $\frac{\frac{R}{G} FC, {MD}^{q}}{2 {CV}^{q}}$ , vel {illeg} $\frac{\frac{R}{G} FC, VH}{CV}$ , vel $\frac{RR}{II - IR} VH$ .

4. Si radius non refringi{illeg}|tu|r sed reflectitur a superficie sphærica VD, eadem regula obtinet si modo ponatur S.R∷ 1.-1. et perinde capiatur VG ad contrarias partes VA {illeg} {illeg} f{illeg} ipsi VA æqualis. Erit enim{illeg}|{adh}|uc error $FB = \frac{AC, FC, FG}{{AV}^{q}, CV} VH$ . vel AV^q×CV. ACF×FG∷VH.FG.

<71v>

Hujus autem Theorematis inventio totis est.

${AD}^{q} = {AV}^{q} + 2 AC, VH$ . ${DF}^{q} = {VF}^{q} - 2 CF, VH$ . Et extractis radicibus, $AD = AV + \frac{AC}{AV} VH - \frac{{AC}^{q}}{2 {AV}^{cub}} {VH}^{q}$ &c $DF = VF - \frac{CF}{VF} VH - \frac{{CF}^{q}}{2 {VF}^{cub}} {VH}^{q}$ &c fluxio \ipsius/ $AD = \frac{AC}{AV} fl : VH - \frac{{AC}^{q}, VH}{{AV}^{cub}} f l VH$ &c |De|fluxio ipsius $DF = - \frac{CF}{VF} fl VH - \frac{{CF}^{q}, VH}{{VF}^{cub}} fl VH$ &c $\frac{R}{I} fl : AD + defl : DF =- \frac{R, {AC}^{q}}{I, {AV}^{cub}} - {CF}^{q} {VF}^{cub}$ in VH, fl VH &c {illeg} Qu{illeg} si nihil esset {illeg}|ra|dij omnes accuratè refrigerentur ad focu{illeg} F. Tunc enim AD et DF fluerent in data ratione, jux{illeg} ea quæ Cartesins in Optica probavit: {A}{'} Sed qui{a} nihil n{illeg}|on| est, error{illeg} \fluxio {obliquitatis}/ superficiei \{illeg}/ erit \VD/ ut illud $\frac{R}{I} fl AD + defl : DF$ . Et ut error ille sive defluxio a legitima figura \obliquitate/ ita \{illeg}/ error angular radij {illeg} refracti. Jam vero est $\frac{R, {AC}^{}}{I, AV} = \frac{CF}{VF}$ Ergo {fluxio} error{illeg} angularis radij refracti \est/ ut $\frac{AC}{{AV}^{q}} - \frac{CF}{{VF}^{q}}$ in $\frac{CF, VH}{VF}$ , fl VH. Vel etiam ut $\frac{I, CF}{R, AV, VF} + \frac{CF}{{VF}^{q}}$ in $\frac{CF, VH}{VF}$ , fl VH seu ut {illeg}, $VF + \frac{R}{I} AV$ in $\frac{{CF}^{q}, VH}{{VF}^{c}, AV}$ fl VH seu $\frac{FG, {CF}^{q}, VH}{AV, {VF}^{c ub}}$ , fl {illeg} posito fl VH=1 & $VF + \frac{R}{I} AV = FG$ . Dat{{illeg}ur} autem ratio $\frac{CF}{VF}$ ad $\frac{AC}{AV}$ ergo substituto posteriore fiet error ille ut $\frac{FG, CF, AC, VH}{{AV}^{q}, {VF}^{q}}$ . Duc in VF^q et error in axe FB erit ut $\frac{FG, CF, AC, VH}{{AV}^{q}}$ qua{illeg}do circuli radius determinatur. Divide per radium circuli et fiet $\frac{FG, CF, AC, VH}{CV, {AV}^{q}}$ ut error BF in omni casu. Dato igitur errore illo in uno casu datur in omni. At in eo casu ubi est radius incidens axi pall|r|allelus datur error {e}odem cum quantitate {illeg} $\frac{FG, CF, AC}{CV, {AV}^{q}} VH$ ergo semper idem est cum hac quantitate.

<71a(r)>

me.Ce∷d.y. mO.CO∷e.y. $\frac{d, Ce}{me} = \frac{e, CO}{mO}$ . $\frac{d, Cq}{mq} = \frac{e, Cp}{mp}$ . DO.DR∷Dq.Dp d, Cq, Cp−Cp|m|=e, Cp, Cq-Cm. d, Ce, CO−Cm=e, CO, mC+Ce d, Cp $\frac{d, qCp - e, qCp}{d, Cq - e, Cp} = Cm = \frac{d, eCO - e, eCO}{d, Ce + e, CO}$ . d−e=f. | $\frac{{Dn}^{q}}{DR} = D^{q}$ .| $\frac{f, qCp}{d, Cq - e, Cp} = \frac{f, eCO}{d, Ce + e, CO}$ . {Se CD = 0 sit CO.CR∷Cq.Cp.} |=Cm|

$\begin{array}{l} - df, Ce, Cp, Cq - ef, CO, Cp, Cq = 0 . & d, f, Ce, Cp, pO = e, f, CO, Cp, qe = e, f, {Dn}^{q}, qe . \\ \frac{+ df, Ce,}{7) 20 \frac{1}{4} (Dq)} \binom{CO, Cq - ef, CO, Cp, Ce}{} & PO = \frac{e,, {Dn}^{q}, qe}{d, Ce, Cq} = R . \sqrt{} \frac{RR + 4 {Dn}^{q}}{4} :\pm \frac{1}{2} R = \binom{CO.}{CP.} \end{array}$ $\frac{9, 20 \frac{1}{4}, 9 \frac{6 \frac{1}{4}}{7} .}{14, 7, 2 \frac{6 \frac{1}{4}}{7}} = \frac{89}{14} = 6 \frac{1}{3} = PO = R . \sqrt{} 30 \frac{1}{4} \pm 3 \frac{1}{6} = \binom{CO}{CP} = 5 \frac{1}{2} \pm 3 \frac{1}{6} = \binom{8 \frac{2}{3} .}{2 \frac{1}{3} .} CO = 8 \frac{2}{3} . CP = 2.$ $Cm = \frac{5, 60 \frac{2}{5}}{98 + 78} = \frac{303 \frac{1}{5}}{176} \begin{array}{l} 910 & 455 \times 12^{inch} \\ 528 & 264 \end{array} = \begin{matrix} 455 & 20, 6 \\ {xx}_{15} \end{matrix}$ | $Cm = 20 \frac{2}{3} inches$ |.

<71a(v)>

$Z = N + \frac{\frac{far - fz}{a - r + f} x - aNx - ffx}{a - r + f \times a - r - f.} \frac{e}{d} a = f . AV = t = a - r$ $Z = N + \frac{\frac{far - fz}{t + f} x - aN - ff}{tt - ff} x = N + \frac{far - fft - 2 fz - faN - taN}{\bar{t \times f} \times \bar{t + f} \times \bar{t - f}} . \binom{N = CD .}{N = \frac{fr}{t - f} .}$ $t . f ∷ \binom{BG . CF .}{VD . CD} t . a ∷ BG . \frac{d}{e} CF$ Be.BC\BF/∷Bn.BQ.BR. B{illeg}|{e.}|BF∷Bn.{illeg}|BR| $\frac{FBn}{Be} = \frac{BS, FC}{FS} .$ {illeg} t²ft−\2/ffar−tf{z}±tfar−ttff {illeg}

^{[Editorial Note 2]}

$zz = \frac{2 zz \pm rr}{\frac{Id - ee}{ee}} + \frac{Idrr - 2 Idax}{eeaa} .$

<72r>

Probl.

Habita Lente convex plano-convexa, invenire tum convexitatem, tum refractionem vitri.

{Sit} Lens RS, ejus superficies plana RTS, convexa RVS \axis KF/{.} Lentis superficie plana solem respiciente, observentur imaginum sola{ntum} ubi \a/ radijs {tum} trajectis tum reflexis congr{illeg} convergentibus \in charta obversa/ distinctissimè pict{æ}rum loci duo F et F|G|; F locus imaginis trajectæ {illeg}G locus reflexæ: et mensurentur quam accuratissime distantiæ VF, TG, ut et crassities vitri TV. Dein fac [VF{+}\2/GT.GT∷TF\2TV./.KT, et erit 2KV semidiameter circuli RVS. Et sinus incidentiæ ex aere in vitrum erit ad sinum refracionis ut KT ad GT, vel ut \{illeg}/ +KF ad VF. vel] ut T|V|F+2TV ad VF−2GT, {ita} sinus incidentiæ ex aere in vitrum ad sinum refractionis, {ita} KT ad GT et erit 2KV radius circuli RVS

Probl.

Habita Lente quavis convexo-convexa, invenire ve{illeg} etiam convexa-concava cujus concavitas {illeg} convexit{illeg} major |multo minor|, invenire tum refractionem vitri, tum convexit{illeg} Lentis.

Sit Lens RS, superficies magis convexa RVS, minus convexa vel concava RTS, axis KF, vertices V ac T. Lentis hujus superficie minus convexa vel concava RTS solem directe respiciente, observentur quam accuratissim{e} sol{{illeg}|em|} imaginis in charta obversa d{illeg}|is|tinctissimè pictæ ta{tam} trajectæ locus F quam reflexæ locus G, et mensure{illeg} distantiæ VF, TG, et crassities vitri TV. Dein alter{illeg} {illeg}|Le|ntis superficie RVS solem respiciente observetur qu{æ} locus imaginis reflexæ H et mensuretur distantia V{H} qu{illeg}|æ| est imaginis illius a vitro. Biseca TV in X. Et fac $\frac{1}{HX} - \frac{2}{FX} = A$ . Et $\frac{1}{GX} + A = B$ . Et $\frac{1}{B} = gX$ . Et {L}{illeg} plano-convexa \ex consimili vitr{illeg}|{o}| confect{illeg}|{æ}|/ cujus vertices si|u|nt T, V, et convexitas \versus F {sita}/ æqu{illeg} summæ convex{|i|tatum} RTS, RVS et versus F sita{illeg} est \in fig. 1 vel differentiæ convexitatis et concavitatis in fig. 2/, pro{jectis} solis imaginem refractam ad locum \priorem/ F, reflexam \vero/ ad locum quamproximè. Unde \^†^[51]/ si fiat (juxta Problema prius) VF−\2/g{T} gT∷VF\+2TV/.KT erit sinus incidentiæ ex aere in vitrum ad sinum refractionis ut KT ad gT, vel ut KF ad VF. {Sec} ista ratio{.} S|ec| ad R, et erit {illeg} $\frac{4 S - 4 R}{R \times A}$ radius circuli {RVS} sit ista {C}{illeg}{−}{illeg}{−}{illeg}{D}{illeg}{−}

<72v>

et {poni} debet {illeg} $\frac{1}{2 gV} + \frac{1}{C} = D$ .

Exempli gratia. In Telescopij cujusdam vitro objectivo observabam VF=13^ped.11^digit. {TH=} $VH = 6^{ped.} {9 \frac{13}{16}}^{digit.}$ $TG = 2^{ped .} {4 \frac{13}{16}}^{dig.}$ . Et $TV = {\frac{2}{9}}^{dig.}$ . Ergo Seu {VF=} VF=167^dig. TG= VH={illeg}1,8125^dig. TG=28,8125^dig. TV=0,2222 &c \dig./. Adeo $XF = 167_{,}^{d} 1111$ . $XH = 8_{,}^{d} 9236$ . $XG = 28_{,}^{d} 9236$ . Unde prodit A= 0,00023{illeg}|{84}|{illeg} dig. B=0,0348{illeg}|{127}|^dig. gx=28,72{illeg}^dig|56^dig.| KT=34,53{2}. dT={28,536 {illeg}T} VF+2TV=167,{illeg}|{444}||4|&c VF−2gT= {109,}{illeg} 109,{illeg}{60}|771|{illeg}. Ergo 167,444. 109,7{60}|71|∷I.R. vel in minoribus numeris 29.19∷I.R aut magis accuratè $\begin{matrix} 61 & . & 40 \\ 90 & . & 59 \end{matrix}$ ∷ I.R. $\frac{4 I - 2 R}{RA} =$ $\frac{}{R} 1243^{dig} = \frac{R}{g} {937 \frac{3}{4}}^{ped} = radio cir culi$ {illeg}=17161{illeg}^dig=1430^ped. {Cir}|Un|nde alterius RVS semidiameter erat quasi 7^ped 4^dig. At hæc ita se habebant in vitro objectivo Telescopij D^ris Babington.

In altero Telescopio quad erat \in archivis/ Academiæ, measura{ve} distantiam imaginis trajectæ a vitro objectivo VF=14^ped3^{gis} $+ \frac{9}{10} dig$ .

<73r>

Telescopij novi delineatio

Vitrum objectivum CD parallelos radios refringat versus O. Imago O per refractionem concavæ superficiei GEH transferatur ad P, et inde per reflexionem superficiei specularis ad Q, et inde per refractionem secundam superficiei GEH ad R ubi a speculo obliquo T detorquetur per vitrum oculare perexiguum \V/ ad oculum.

Sit \imaginis/ translatio \angularis/ ab O ad P et a P ad S tanta quanta {debendis} \corrigendis/ vitri objectivi refractionibus erroneis \ab inæquali refrangibilitate ortis / sufficit et erit refract \angularis/ translatio imaginis a Q ad R tanta quanta est a P ad S, et punctum S invenietur faciendo ut sit BE.EO∷EO.ES.

Sit X centrum circuli specularis JFK et Y centrum circuli refrigentis concavi GEH. Et quoniam imaginis angulares translationes PX, XQ æquales sunt, ut et PS, QR; erunt etiam translationes SX, RX æquales: adeo si fiat {illeg} ES.SX∷EQ|R|.RQ{illeg} \RX,/ vel ES+SX.SX∷EX.Q|R|X, \ex dato puncto X/ habebitur ultimæ imaginis locus {illeg} R, e cujus regione consistet oculus.

Sit insuper Y centrum superficiei concavæ GEH, et quoniam est QP {illeg} EP.EQ∷PX.QX, et I×P{illeg} I×OE.R×OY∷EP.YP. et I×ER.R×YR∷QE.QY: inde derivabitur hæc conclusio. Hac ER.EX∷{illeg}OR.P, et $\frac{2 I}{R} EO . OX ∷ P . XY$ et habebitur circuli GEH centrus Y. Ubi nota quod usurpo $\frac{I}{R}$ pro ratione sinus incidentiæ ex aere in vitrum ad sinum refractionis. Et suppono in super vitri crassitiem EF ad instar nihili esse. ff Fac $\frac{2 I \times EO}{R \times EX} = a . \frac{EO}{ER} = b . b \times RX = c$ . et {illeg} $\frac{OX - c}{a - b + 1} = XY$ . Et habebitur circuli GEH centrum Y. Ubi nota quod usurpo $\frac{I}{R}$ pro ratione sinus incidentiæ ex aere in vitrum ad sinum refractionis: et suppono insuper vitri crassitiem EF ad instar nihili esse.

<77r>

Ghetaldus in his Promotus Archimededs compu{illeg}tes y^e w{h}eights of y^e following equall bodys to bee in y^e proportions following.

$\begin{matrix} Glasse & Oyle. & Wax. & Wine. & Water. & Honey. & Tinn. & Iron. & Brasse. & Silver. & Lead. & Quicksilver. & Gold \\ 1 . & 1 \frac{5}{121} . & 1 \frac{4}{55} . & 1 \frac{1}{11} . & 1 \frac{32}{55} . & 8 \frac{4}{55} . & 8 \frac{8}{11} . & 9 \frac{9}{11} . & 11 \frac{3}{11} . & 12 \frac{6}{11} . & 14 \frac{62}{77} . & 20 \frac{8}{11} . \\ Or & 4235. & 4410. & 4543. & 4620. & 6699. & 34188. & 36960. & 41580. & 4774. & 53130. & 62700. & 87780. \\ 1562 & Or & 55. & \frac{630}{11} . & 59. & 60. & 87. & 444. & 480. & 540. & 620. & 690. & \frac{5700}{7} & 1140. \\ Or & 111. & 120. & 135. & 155 & 172 \frac{1}{2} & \frac{1425}{7} . & 285. \\ Or & 777. & 840 & 945 & 1085 & 1207 \frac{1}{2} & 1425. & 1995. \\ Or & 4 \frac{47}{57} & 5 \frac{5}{209} & 5 \frac{10}{57} . & 5 \frac{5}{19} . & 7 \frac{12}{19} . & 38 \frac{18}{19} . & 42 \frac{2}{19} . & 47 \frac{7}{19} . & 54 \frac{22}{57} & 60 \frac{10}{19} & 71 \frac{3}{7} & 100. \\ 2 \frac{3}{5} . & Or & \frac{11}{12} . & \frac{21}{22} . & \frac{59}{60} . & 1. & \frac{29}{20} . & \frac{37}{5} . & 8. & 9. & \frac{31}{3} . & \frac{23}{2} . & \frac{95}{7} & 19 \end{matrix}$

If Sphæres bee made of y^e following metalls \each of/ whose diameters are each one foote their weights will bee as followeth. {{illeg}} |Note y^t 1^li|=12′. 1′=24″. 1″=24‴

|Or| $\begin{matrix} Tinn. & Iron e . & Brasse. & Silver. & Lead & Gold \\ 304^{lib} . & 328^{li} . 7' . 18 ″ . 19 \frac{17}{37} ‴ & 369^{li} . 8' . 18 ″ . 3 ‴ \frac{33}{37} & 424^{li} . 6' . 1 ″ . 7 ‴ \frac{5}{37} & 472^{li} . 5' . 4 ″ . 12 ‴ \frac{36}{37} & 780^{li} . 6' . 11 ″ . 16 ‴ \frac{8}{37} \\ 304^{lib} . & {\frac{12160}{37}}^{li} . & {\frac{13680}{37}}^{li} . \end{matrix}$

A Sphære of tinn whose diameter is six inches weighs Thirty and Eight pounds. The following line being {6} of Ghetaldus his inches, w^ch is half y^e Roman foot by Villalpandus account from y^e Far{nesian urn}. $\begin{matrix} |-----------------------------------------------------------------------| \end{matrix}$ Soe y^t y^e weight of a circumscribed cilinder is {illeg} 57 \lib/ \& of such a cilinder of water $5 \frac{}{57}$ lb $7 \frac{26}{37}$ lb/. And of a circumscribed cube (viz whose side is 6 ^inches)|is| $\frac{836}{}$ ^lib $\frac{627}{}$ ^lib {Or more} exactly ${\frac{798}{11}}^{} = \overset{li}{72 \frac{6}{11}}$ \li/. Or more exactly ${\frac{25764}{355}}^{lib} = \overset{lib}{72, \frac{204}{355}} = \overset{lib}{72,57465}$ {-.} \that is $72 \frac{4}{7}$ ./ Or more exactly $\frac{228^{lib}}{3,14159265359}$ And such a cube of water $9 \frac{4}{5}$ ^lb or more exactly 9,80752^lib. & a foot cube of water $78 \frac{6}{13}$ ^lb or more exactly 78,46016^lb.

The {P}es Regius Gallorum, the Rhinlandick foot, the old Roman foot, & the English foot are as $12, 11 \frac{1}{2}, 10 \frac{5}{6} & 11$ {illeg} Ga{illeg} But by y^e Farnesian Urn \printed by Villalpand/ the French royal foot is to {illeg} old {illeg} foot as 12 to 11{illeg} \or perhaps 11{illeg} to 10{illeg}/. The urn conteined \a longius of water weighing/ 10 Roman pounds \of 12 {illeg}/ {illeg} Gaffendus by weighing found it contein 7 french pounds of 16 ounces. \of water/ Eight such vessels make a Roman foot cube called a Quadrantal or Amphora Romana weighing 56 French pounds. A cubic French Foot of water by Mersennus trial weighed 74^lb, by the vulgar estimat{illeg} { $70 \frac{1}{2}$ } or 72, suppose 72 & two meane proportional{s} between 7{2} & {56} will be as $24 \frac{1}{3}$ to 23 or 12{ $\frac{1}{2}$ } to {11}{ $\frac{1}{2}$ }. Suppose 74{^lb} two {meane} will be |{as}| 11 to 10 or 12 to $10 \frac{10}{11}$ & this is {y^e} proportion of y^e french foot to y^e Roman. {illeg} Royal French foot {illeg} {illeg} 12. $11 \frac{1}{2}$ . {11} {illeg} $10 \frac{9}{10}$ . {illeg} $\frac{7}{8}$ . $12 \frac{13}{5}$ {illeg} ({illeg} $10 \frac{}{4}$ The Roman {illeg} to y^e {horary} foot as 8 to 9. {illeg} 9 {illeg} $12 \frac{17}{52}$ {illeg} $10 \frac{71}{81}$ .

<77v>

Some Problems of Gravity & levity &c

^[52] Prob 1. To find y^e proportion of y^e weights of two equall bodys y^e one being sollid y^e other liquid. Resp. If y^e Sollid body A bee heavier yⁿ y^e liquid B weigh it in y^e aire & in y^e liquid Body B; & y^e difference of those two weights is y^e weight of soe much water as is equall to y^t sollid body |& so much {wyer} as {deppend} in y^e {water}|. But if it bee lighter yⁿ y^e liquid body B, hang a heavier body C {at} to it, y^t will sinke it; & weigh y^m first y^e body A being in y^e air & C in y^e water{;} 2^dly liquid body, 2^dly both A & C being in y^e liquid body; & y^e difference of these{s} weights is y^e weight of so much water as is equall to A (& also to y^e soe much thred wyer or hayre as was weighed both{illeg} in y^e water & air.)

^[53] Note y^t y^e weight of soe much water as is equall to y^t ꝑte of y^e wyer (to w^ch A & C are fastened) |w^ch| was weighed both in y^e air & water, bee subducted from y^e whole weight of y^e water {illeg}|&| y^e remaining weight shall bee y^e weight of y^e water requi{illeg}|re|d.

__________________________________________________________________________________________________________________________

Or Thus, Make y^e scoale B as light as may bee, yet soe y^t it will sinke it selfe in y^e liquid body CC, & \sinke/ y^e body A also, if A chance to bee lighter yⁿ CC. Suppose y^t f y^e weight of scoale B in y^e water, & g in y^e aire; & y^t e is y^e weight of B+A in y^e water, & d in y^e aire: yⁿ is d−g+f−e y^e weight of soe much water as is equa{illeg}|ll| to A. |Note y^t y^e water must reach {'}at both weighings to y^e same point {illeg} of y^e wyer & y^t y^e wyer bee as small as may bee.|

{P} Cor: Hence y^e proportion of all weights of all bodys sollid or liquid or both may bee gathered. & wee may hereby make hence deduce tables of y^e weights of equall bodys, & of y^e quantitys of bodys equally heavy.

^[54] Prob 2. Two{illeg} bodys {illeg} \D & E/ given to find y^e proportion of their quantity. Weigh y^m in y^e scale B, let their weight \& y^e weight of y^e scale/ in y^e air bee h & k, in y^e water, m & n yⁿ is D{illeg} E∷h−g+f−m:k−g+f−n. For their weight in air is h−g, & k−g; in the water m−f & n−f. &c.

^[55] Prob 3. A compound body cd being \given/ to find y^e weights \& proportions,/ of its two compounding parts c & d. Answer. a & c, b & d, are of y^e same matter. That y^e weights of y^e 5 bodys a, b, cd, c, d, in y^e aire are ef e, f, n, m, n−m; & in y^e water g, h, q, p, q−p. Then is $e : g ∷ m : p = \frac{gm}{e}$ . & $f : h ∷ n - m : q - p = \frac{hn - hm}{f}$ . The{re}fore $q = \frac{gm}{e} + \frac{hn - hm}{f}$ . Or $\frac{feq - hen}{gf - eh} = m = to$ y^e weight of C in y^e aire. Also $\frac{gfn - feq}{gf - eh} = n - m$ . & $\frac{gfq - hgn}{gf - eh} = p$ . & $\frac{hgn - qeh}{gf - eh} = q - p$ . & h|f|eq{+} \−/h|f|gq−f|h|en+f|h|gn:−gf|h|n+gh|f|n+f|h|eq−h|f|eq∷c:d. (Prob 1). feq−hen+hgn−fgq:gfn−efq+ehq{illeg}|−gh|n∷c:d. And, gfn+ehq−ehn−gfq:efq+ghn−ehn−gfq∷c+d:{c}.

^[56] Prob{:} 4. A body f compounded of 3 severall sorts of matter d, e, f−d−e, begin given{;|:|} w^th y^e proportion of y^e weight of y^e two bodys d & e as {1} to r. To find y^e weight of y^e body d. Resp. Suppose y^t y^e bodys a & d, b & e, c & f−d−e are of y^e same matter; & {illeg} & y^t y^e weight of y^e bodys a, b, c, f, d, e, f−d−e, {are} in y^e aire are g, h, k, p, x, rx, p−x−rx{;} & in y^e water l, m, n, v, s, t, v−s−t. Then is g:l∷x:s. & h:m∷rx:t. & k:n∷p−x−rx:v−s−t. Therefore $v - \frac{lx}{g} - \frac{mrx}{h} = v - s - t = \frac{np - nx - nrx}{k}$ . And ghkv−ghnp=hklx+gkmrx−ghnx−ghnrx. Or $\frac{ghkv - ghnp}{hkl + ghmr - ghn - ghnr} = x$ .

If y^e weights g=h=k, (as may bee either by experience or cal{illeg}|cu|lacon (see coroll: Prob 1) Then is $\frac{gv - np}{l + mr - n - nr} = x$ . Now because gold is usually allayed by mixing w^th it brasse & silver of each {illeg} |an| equall weight; suppose y^t a & d are brasse, b & e silver, c & f−d−e gold, & y^t x=rx, or r=1. Then is $\frac{gv - np}{l + m - 2 n} = x$ y^e weight of y^e brasse or siver {sic} in y^e masse f, & $\frac{pl + pm - 2 gv}{l + m - 2 n} = p - 2 x$ y^e weight of y^e gold in it.

<78r>

Descriptio cujusdam {illeg}|ge|neris curvarum
{terty} \secundi/ ordinis.

Concipe lineas PED datum angulum PED continentes ita moveri ut una earum EP perpetuo transeat per polum P positione datum, et altera{illeg} {illeg} ED datæ longitudinis existens perpetuò tang{illeg}|{a}|t rectam AB positione datam . Age PA {illeg}|co|nstituentem angulum PAD æqualem angulo PED sit CD æqualis AP et quodvis punctum C in recta ED. datum describet curvam secundi ordinis.

Age CB constituentem angu{illeg}|lu|m CD{illeg}|BD| æqualem angulo PED, et ad AD demitte normalem C{illeg}|F.| et dictis {illeg}. DC=c. CE=b. AB=x BC=y. et posito 1. E∷BC.BF, erit $\begin{array}{l} y^{3} & + & 2 b & y y & + & b b & y & + & c b b & = \\ + & c & + & 2 b c & - & c x x \\ - & 4 e b & + & 2 b x \\ - & 4 e e c & + & 4 c x \\ - & 2 e x & + & x x \end{array}$

Et nota quod ubi angulus PED rectus est, et recta ED bisecatur in C, curva erit cissoides veterum.

<80r>

A Method Whereby to Square lines Mechanichally.

^[57] Lemma: |Prop 1.| Supposing ab=x⊥bc=y. If y^e valor of y (in y^e Equation expressing y^e relation twixt x & y) consist of simple termes, Multiply {each} terme by x, & divide it by y^e number of y^e dimensions of x in that terme, & y^e quote shall signify y^e area acb.

Example. If ax=yy. Or $\sqrt{a x} = y$ . yⁿ is $\frac{2}{3} \sqrt{a x^{3}} = \frac{2}{3} x \sqrt{a x} = acb$ . Soe if $\frac{x x}{a} = y$ . yⁿ is $\frac{x^{3}}{3 a} = acb$ . Soe if $a + x + \frac{a b b}{x b} + \frac{x^{3}}{a a} = y$ . yⁿ is $a x + \frac{x x}{2} + \frac{a b b}{x} + \frac{x^{4}}{4 a a x} = acb$ Soe if $\frac{a^{4}}{x^{3}} + \sqrt{\frac{a^{5}}{x^{3}}} - \sqrt{\frac{x^{3}}{a}} + \frac{6 x^{6}}{a^{6}} = y$ . yⁿ is $\frac{{-^{a}}^{4}}{2 x x} - 2 \sqrt{\frac{a^{5}}{x}} - \frac{2}{5} \sqrt{\frac{x^{5}}{a}} + \frac{7 b x^{7}}{a^{6}} = acb$ .

Lemma 2^d |Prop 2.| If {illeg}any terme in y^e valor of y bee a compound ter{illeg} Reduce it to simple ones by Division or Extraction of Rootes or by Vie{illeg}|{s}| Method of Resolving Affected Equations, as you would doe in Decimall Numbers; & yⁿ find y^e Area by Prop 2^d 1^st.

Example. If $\frac{a a}{b + x} = y$ . bee divided as in decimall fractions it produce{illeg} $\frac{a a}{b + x} = \frac{a a}{b} - \frac{a a x}{b b} + \frac{a a x x}{b^{3}} - \frac{a a x^{3}}{b^{4}} + \frac{a a x^{4}}{b^{5}} - \frac{a a x^{5}}{b^{6}} + \frac{a a x^{6}}{b^{7}} - \frac{a a x^{7}}{b^{8}}$ &c. & by y^e {2}|1|^{d|s|t} Proposition $\frac{a a x}{b} - \frac{a a x x}{2 b b} + \frac{a a x^{3}}{3 b^{3}} - \frac{a a x^{4}}{4 b^{4}} + \frac{a a x^{5}}{5 b^{5}} - \frac{a a x^{6}}{6 b^{6}} + \frac{a a x^{7}}{7 b^{7}} - \frac{a a x^{8}}{8 b^{8}} + \frac{a a x^{9}}{9 b^{9}} &c = abc$ . y^e Hyperbolas Area.

As if a=1=b\=ab=bc./ & x=0,1=be The Calculation is as followeth,

$\begin{array}{r} \begin{matrix} 0,10033,53477,31075,58063,57265,52060,03894,52633,62869,14595,91358,63 \end{matrix} \\ \frac{a a x}{b} = 0,10000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00000 00 \\ \frac{a a x^{3}}{3 b^{3}} = 33,33333,33333,33333,33333,33333,33333,33333,33333,33333,33333 33 \\ \frac{a a x^{5}}{5 b^{5}} = 20000,00000,00000,00000,00000,00000,00000,00000,00000,00000 00 \\ \frac{a a x^{7}}{7 b^{7}} = 142,85714,28571,42857,14285,71128,57142,85714,28571,42857 14 \\ \frac{a a x^{9}}{9 b^{9}} = 1,11111,11111,11111,11111,11111,11111,11111,11111,11111 11 \\ \frac{a a x^{11}}{11 b^{11}} = 909,09090,90909,09090,90909,09090,90909,09090,90909 09 \\ \frac{a a x^{13}}{13 b^{13}} = 7,69230,76923,07692,30769,23076,92307,69230,76923 07 \\ 6666,66666,66666,66666,66666,66666,66666,66666 66 \\ 58,82352,94117,64705,88235,29411,76470,58823 52 \\ 52631,57894,73684,21052,63157,89473,68421 05 \\ 476,19047,61904,76190,47619,04761,90476 19 \\ 4,34782,60869,56521,73913,04347,82608 69 \\ 4000,00000,00000,00000,00000,00000 00 \\ 37,03703,70370,37037,03703,70370 37 \\ 34482,75862,06896,55172,41379,31034482 \\ 322,58064,51612,90322,58064 51 \\ 3,03030,30303,03030,30303 03 \\ 2857,14285,71428,57142,85 \\ 27,02702,70270,27027,02 \\ 25641,02564,10256,41 \\ 243,90243,90243,90 \\ 2,32558,13953,48837,20930,23255 \\ 2222,22222,22222 \\ 21,27659,57446 \\ 20408,16326 \\ 196,07843 \\ 1,88679 \end{array}$

$\begin{array}{r} 3 10 \\ 000502,51679,26750,72059,17744,28779,27385,30427,57503,83731,49363,62 \\ 502,51666,66666,66666,66666,66666,66666,66666,66666,66666,66666,66 \\ \frac{a a x^{8}}{8 b^{8}} = 12,50000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{10}}{10 b^{10}} = 10000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{12}}{12 b^{12}} = 83,33333,33333,33333,33333,33333,33333,33333,33333,33 \\ \frac{a a x^{14}}{14 b^{14}} = 71428,57142,85714,28571,42857,14285,71428,57142,85 \\ \frac{a a x^{16}}{16 b^{16}} = 625,00000,00000,00000,00000,00000,00000,00000,00 \\ 5,55555,55555,55555,55555,55555,55555,55555,55 \\ 5000,00000,00000,00000,00000,00000,00000,00 \\ 45,45454,54545,45454,54545,45454,54545,45 \\ 41666,66666,66666,66666,66666,66666,66 \\ 384,61538,46153,84615,38461,53846,15 \\ 3,57142,85714,28571,42357,14285,71 \\ 3333,33333,33333,33333,33333,33 \\ 31,25000,00000,00000,00000,00 \\ 29411,76470,58823,52941,17 \\ 277,77777,77777,77777,77 \\ 2,63157,89473,68421,05 \\ 2500,00000,00000,00 \\ 23,80952,38095,23 \\ 22727,27272,72 \\ 217,39130,43 \\ 2,08333,33 \\ 2000,00 \\ 19,23 \\ 18 \end{array}$ |The summe of these two summes is equall to {y^e area} dbfc, supposing ad={0,9}. And their difference {is equall to} y^e area bche, supposing ae=1,1. & ab=1=bc∥df∥he. {that is}|

$\begin{array}{r} bfc & = & 0,10536,05156,57826,30122,75009,80839,31279,83001,20372,98327,40795,43 \\ he & = & 0,09531,01798,04324,86004,39521,23280,76509,22206,05365,30864,41990,83 \end{array}$

<80v>

In the manner If a=b=1=ab=bc. & x=0,2=be. The calculatio is as followeth

$\begin{array}{r} 0,20273,25540,54082,19098,90065,57732,17456,82859,95211,73124,70987,67 = summe \\ \frac{a a x}{b} = 0,20000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{3}}{3 b^{3}} = 266,66666,66666,66666,66666,66666,66666,66666,66666,66666,66666,66 \\ \frac{a a x^{5}}{5 b^{5}} = 6,40000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{7}}{7 b^{7}} = 18285,71428,57142,85714,28571,42857,14285,71428,57142,85714,28 \\ \frac{a a x^{9}}{9 b^{9}} = 568,88888,88888,88888,88888,88888,88888,88888,88888,88888,88 \\ 18,61818,18181,81818,18181,81818,18181,81818,18181,81818,18 \\ 63015,38461,53846,15384,61538,46153,84615,38461,53846,15 \\ 2184,53333,33333,33333,33333,33333,33333,33333,33333,33 \\ 77,10117,64705,88235,29411,76470,58823,52941,17647,05 \\ 2,75941,05263,15789,47368,42105,26315,78947,36842,10 \\ 9986,43809,52380,95238,09523,80052,38095,23809,52 \\ * 364,72208,69565,21739,13043,47826,08695,65217,39 \\ 13,42177,28000,00000,00000,00000,00000,00000,00 \\ 49710,26962,96296,29629,62962,96296,29629,62 \\ 1851,27900,68965,51724,13793,10344,82758,6206896551 \\ 69,27366,60645,16129,03225,80645,16129,03 \\ 2,60301,04824,24242,42424,24242,42424,24 \\ 9817,06810,51428,57142,85714,28571,42 \\ 371,45663,10054,05405,40540,54054,05 \\ 14,09630,29202,05128,20512,82051,28 \\ 53634,71355,00487,80487,80487,80 \\ 2045,60302,84204,65116,27906,976744 \\ 78,18749,35307,37777,77777,77 \\ 2,99441,46458,58042,55319,1489 \\ 11488,77455,96186,12244,897959 \\ 441,52937,52324,01568,6274 \\ 16,99471,55749,83003,77358 \\ 65506,90367,08435,78 \\ 2528,33663,29097,52 \\ 97,70521,22548,17 \\ 3,78007,05069,07 \\ 14640,27307,43 \\ 567,59212,53 \\ 22,02596,302 \\ 85550,117 \\ 3325,609 \\ 129,37 \\ 5,03 \end{array}$

$\begin{array}{r} 0,2041,09972,60127,56477,72885,32577,65993,50886,05873,81676,01148,45 = sum m e \\ 0,2041,06666,66666,66666,66666,66666,66666,66666,66666,66666,66666,66 = \frac{a a x x}{2 b b} + \frac{a a x^{4}}{4 b^{4}} \\ \frac{a a x^{8}}{8 b^{8}} = 3200,00000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{10}}{10 b^{10}} = 102,40000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{12}}{12 b^{12}} = 3,41333,33333,33333,33333,33333,33333,33333,33333,33333,33 \\ \frac{a a x^{14}}{14 b^{14}} = 11702,85714,28571,42857,14285,71428,57142,85714,28571,42 \\ \frac{a a x^{16}}{16 b^{16}} = 409,60000,00000,00000,00000,00000,00000,00000,00000,00 \\ 14,56355,55555,55555,55555,55555,55555,55555,55555,55 \\ 52428,80000,00000,00000,00000,00000,00000,00000,00 \\ 1906,50181,81818,18181,81818,18181,81818,18181,81 \\ 69,90506,66666,66666,66666,66666,66666,66666,66 \\ 2,58111,01538,46153,84615,38461,53846,15384,61 \\ 9586,98057,14285,71428,57142,85714,28571,42 \\ 357,91394,13333,33333,33333,33333,33333,33 \\ 13,42177,28000,00000,00000,00000,00000,00 \\ 50529 02701,17647,05882,35294,11764,70 \\ 1908,87435,37777,77777,77777,77777,77 \\ 72,33629,13010,52631,57894,73684,2105263 \\ 2,74877,90694,40000,00000,00000,00 \\ 10471,53931,21523,80952,38095,23 \\ 399,82241,01003,63636,36363,63 \\ 15,29755,30821,00869,56521,739130434 \\ 58640,62014,80533,33333,33 \\ 2251,79981,36852,48000,00 \\ 86,60768,51517,40307,692 \\ 3,33599,97239,78145,18 \\ 12867,42750,67728,45 \\ 496,94892,43995,03 \\ 19,21535,84101,14 \\ 74382,03255,528 \\ 2882,30376,151 \\ 111,79844,893 \\ 4,34041,03 \\ 16865,59 \\ 655,88 \\ 25,52 \\ 99 \end{array}$ |The summe of these two summes is Equall to y^eArea dbfc, supposing ad=0.8. And their Difference is equall to y^e area bche, supposing ae=1,2. & 1=ab=bc∥df∥he. {that is} $\begin{array}{r} dbfc & = & 0,22314,35513,14209,75576,60603,07701,13885,12006,88042,06974,63440,46 . \\ bche & = & 0,18232,15567,93954,62621,14832,42545,81898,10234,76294,43622,61143,46 . \end{array}$ |

$\begin{array}{r} dbfc & = & 0,22314,35513,14209,75576,62950,90309,83450,33746,01085,54800,72136,12 \\ bche & = & 0,18232,15567,93954,62621,17180,25154,51463,31973,89337,91448,69839,22 \end{array}$

{illeg} such respect to y^e superficies bcfd{,} bche {y^e numbers} {illeg} {(viz as y^e lines ad} {illeg} {illeg} {illeg} {illeg}

<81r>

Soe y^t since $\frac{1,2 \times 1,2}{0,8 \times 0,9} = 2 . \frac{1,2 \times 1,2 \times 1,2}{0,8 \times 0,8 \times 0,9} = 3 = \frac{1,2 \times 2}{0,8}, \frac{1,2 \times 1,2 \times 1,2 \times 1,2}{0,8 \times 0,8 \times 0,8 \times 0,9 \times 0,9} = 5 = \frac{2 \times 2}{0,8} = \frac{4}{0,8} . 2 \times 5 = 10 . 10 \times 10 = 100$ 10×100=1000 &c. 10×1,1=11 &c. The Superficies answere|i|ng to these lines{,} 2. 3. 9. 11. & their products (of one of y^m multiplying another) may bee found. Viz. if ab=1=bc⊥ab{illeg}

|If y^e line ak is| Then, y^e superficies bcgh is

$\begin{array}{r} 2. & 0,69314,71805,59945,30941,72321,21458,17656,80755,00134,36025,52539,99 \\ 3. & 1,09861,22886,68109,69139,52452,36922,52570,46474,90557,82274,94515,33 \\ 10. & 2,30258,50929,94045,68401,79914,54684,36420,76011,01488,62877,29756,09 \\ 100. & 4,60517,01859,88091,36803,59829,09368,72841,52022,02977,25754,59512,18 \\ 1000. & 6,90775,52789,82137,05205,39743,64053,09262,28033,04465,88631,89268,27 \\ 10000. & 9,21034,03719,76182,73607,19658,18737,45683,04044,05954,51509,19024,36 \\ &c— \\ 11. & 2,39789,52727,98370,54406,19435,77965,12929,98217,06853,93741,71747,92 \end{array}$

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Having already found y^e areas correspondent to y^e lines 1,1. 0,9. 1,2. 0,8. tis easy by y^e help of those operations to ding y^e areas correspondent to y^e lines 1,01. 1,001. 1,0001. &c: 0,99. 0,999. 0,9999. &c: 1,02. 1,002 &c: 0,98. 0,998. 0,9998. &c. And since $7 = \sqrt{\frac{100 \times 0,98}{2}} . 17 = \frac{100 \times 1,02}{6} . 13 = \frac{1000 \times 1,001}{7 \times 11} = \frac{1001}{77}$ . &c. Therefore y^e areas correspond{ent} to y^e lines 7. 13. 17. {illeg}|&c|: are easily found, as followeth. Viz: if x=0.02. Then

$\begin{array}{r} 0,02000,26673,06849,58071,70371,83954,64639,04807,62055,62238,59310,49 & = & summe \\ \frac{a a x}{b} = 0,02000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{3}}{3 b^{3}} = 26666,66666,66666,66666,66666,66666,66666,66666,66666,66666,66 \\ \frac{a a x^{5}}{5 b^{5}} = 6,40000,00000,00000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{7}}{7 b^{7}} = 182,85714,28571,42857,14285,71428,57142,85714,28571,42 \\ \frac{a a x^{9}}{9 b^{9}} = 5688,88888,88888,88888,88888,88888,88888,88888,88 \\ \frac{a a x^{11}}{11 b^{11}} = 1,86181,81818,18181,81818,18181,81818,18181,81 \\ \frac{a a x^{13}}{13 b^{13}} = 63,01538,46153,84615,38461,53846,15384,61 \\ 2164,53333,33333,33333,33333,33333,33 \\ 77101,17647,05882,35294,11764,70 \\ 27,59410,52631,57894,73684,21 \\ 998,64380,95238,09523,80 \\ 36472,20869,56521,73 \\ 13,42177,28000,00 \\ 497,10269,62 \\ 185,12,79 \\ 6,92 \end{array}$

$\begin{array}{r} 0,00020,00400,10669,86769,10081,17069,54599,73717,71328,11118,23899,70 & = summe \\ \frac{a a x^{2}}{2 b b} + \frac{a a x^{4}}{4 b^{4}} + \frac{a a x^{6}}{6 b^{6}} = 0,00020,00400,10666,66666,66666,66666,66666,66666,66666,66666,66666,66 \\ \frac{a a x^{8}}{8 b^{8}} + \frac{a a x^{10}}{10 b^{10}} + \frac{a a x^{12}}{12 b^{12}} = 3,20102,43413,33333,33333,33333,33333,33333,33333,33 \\ \frac{a a x^{14}}{14 b^{14}} = 1,17028,57142,85714,28571,42857,14285,71 \\ 40,97456,87984,35555,55555,55555,55 \\ 19,06501,81818,18181,81 \\ 699,05066,66666,66 \\ 25811,10153,84 \\ 9,58698,05 \\ 357,91 \\ 13 \end{array}$

The sume & difference of w^ch two summes give y^e areas bcfd, bche as before. That is

$\begin{array}{l} \begin{matrix} If fd = 0,98 \\ he = 0,98 \end{matrix} & \begin{array}{r} bcfd & = & -0,02020,27073,17519,44840,80453,01024,19238,78525,33383,73356,83210,19 \\ bche & = & +0,01980,26272,96179,71302,60290,66885,10039,31089,90727,51120,35410,79 \end{array} \end{array}$

And since $7 = \sqrt{\frac{100 \times 0,98}{2}} = \sqrt{\frac{98}{2}} = 49$ . & $17 = \frac{100 \times 1,02}{6}$ . |Therefore|

|{If y^e} line {illeg}|{c}|k is| The superficies bcgk is $\begin{array}{r} 7. & 1,94591,01490,55313,30510,53527,43443,17972,96370,84729,58186,11881,00. \\ 17. & 2,83321,33440,56216,08024,95346,17873,12653,55882,03012,58574,47867,65. \end{array}$

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|{illeg}if x=0,001. The operation is as followeth{.}| $\begin{aligned} + \frac{a a x^{5}}{5 b^{5}} &c = 0,00100,0003,33333,53333,34761,90476,19047,61904,76190,47619,04761,90 \\ 111,11120,20202,02020,20202,02020,20 \\ 76923,07692,30769,23 \\ 6666,66666,66 \\ 588,23 \\ 0,00100,00003,33333,53333,34761,90587,30167,82107,55133,82180,04806,22 & = summe \\ 0,00000,05000,00250,00016,66667,91666,76666,67500,00071,42863,39286,26 & = su mm e of + + \\ for befd = 0,00100,05003,33583,53350,01429,82254,06834,49607,55205,25043,44092,48 & If ad = 0.97 \\ bche = 0,00099,95003,33083,53316,68093,98929,53501,14607,55062,99316,65519,96 & If ad = \end{aligned}$ {Where} {illeg} $\frac{1000 \times 1,001}{77} = 13$ . & $\frac{1000 \times 0,999}{27} = \frac{999}{27} = 37$ . |Therefore| |{illeg}ak is {illeg}| $\begin{matrix} 13 & 2,64949527881836330058457441001318604805267549760307115931 \\ 37 \end{matrix}$ {illeg}

<81v>

Which Area may bee otherwise thus found s{illeg}|upp|osing x=db=-0,0016. Viz. $\begin{array}{r} \frac{a a x}{b} + \frac{a a x^{3}}{3 b^{3}} = 0,00160,00013,65333,33333,33333,33333,33333,33333,33333,33333,33333,33 & . \\ \frac{a a x^{5}}{5 b^{5}} = 2,09715,20000,00000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{7}}{7 b^{7}} = 38347,92228,57142,85714,28571,42857,14285,71 \\ \frac{a a x^{9}}{9 b^{9}} = 7635,49741,51111,11111,11111,11111,11 \\ \frac{a a x^{11}}{11 b^{11}} = 1599,28964,04014,54545,45454,54 \\ 346,43074,05669,61230,77 \\ 76 86143,36404,56 \\ 17,36164,14 \\ 3 98 \\ The Su mm e of \frac{a a x}{b} + \frac{a a x^{3}}{3 b^{3}} &c = 0,00160,00013,65335,43048,91681,33197,41816,99469,20181,33677,37988,14 \\ \frac{a a x x}{2 b b} + \frac{a a x^{4}}{4 b^{4}} + \frac{a a x^{6}}{6 b^{6}} = 0,00000,12800,01638,40279,62026,66666,66666,66666,66666,66666,66666,66 \\ \frac{a a x^{8}}{8 b^{8}} = 53,68709,12000,00000,00000,00000,00000,00000,00 \\ \frac{a a x^{10}}{10 b^{10}} = 10,99511,62777,60000,00000,00000,00 \\ 2,34562,48059,22133,33333,33 \\ 51469,71002,70913,82 \\ 11529,21504,60 \\ 2623,53 \\ The su mm e of \frac{a a x x}{2 b b} + \frac{a a x^{4}}{4 b^{4}} &c = 0,00000,12800,01638,40279,62080,35386,78180,64007,26195,71331,95041,94 \\ The su mm e of which two summes is equall to the area bcfd, Which summe is \\ 0,00160,12813,66973,83328,53761,68584,19997,63476,46377,05009,33030,08 & . \end{array}$

As was found before excepting y^t their differenc is e in y^e two last figures is 28. Which agreement could scarce thus happen in more yⁿ 50 figures, were not y^e reas, correspond{illeg}|in|g to y^e lines 2. 3. 5. 7. 13. &c, calculated {aright} {illeg}|in| so many figures.

Octob. 1676.

Memorandum. The letters baccdæ13eff7i3lgn4049 rr458t12vx {illeg}|in| my second epistle to |M.| Leibnitz contein this sentence Data æquatione quotcun fluentes quantitates involvente, fluxiones invenire: et vice versâ.

The other letters in y^e same Epistle, viz: 5accdæ10effhui4l3m 9nboqqr{illeg}|{8}|s11t9v3x: 11ab3cdd10eæg10ill4m7n603p3q6r5511{illeg}|t8|vx, 3acæ 4egh5i4l4m5n80q4r3s6t4v aaddæ5eüjmmnnooprrr5sttuv, express this sentenc{illeg}|e|. Una Methodus consistit in extraction{e} fluentis quantitatis ex æquatione simul involvente fluxionem ejus. Altera tantum {illeg} in assumptionæ {seriei} pro quantitate qualibet incognita ex qua cætera commodè derivari posunt, et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ {seriei}.

<82r>

$\begin{aligned} Series Arithmetica & Series correspondens. \\ A + x . & a + b x + c x x + d x^{3} + e x^{4} + f x^{5} + g x^{6} + h x^{7} + i x^{8} + k x^{9} + l x^{10} \\ \begin{array}{l} A + 6 \\ A + 5 \\ A + 4 \\ A + 3 \\ A + 2 \\ A + 1 \\ A + 0 \\ A - 1 \\ A - 2 \\ &c \end{array} & \begin{array}{r} a & + & 6 b & + & 36 c & + & 216 d & + & 60466176 l & + & 362797056 m \\ a & + & 5 b & + & 25 c & + & 125 d & + & 625 e & + & 3125 f & 15625 g & + & 78125 h & + & 390625 i & + & 1953125 k & + & 9765625 l & + & 48828125 m \\ a & + & 4 b & + & 16 c & + & 64 d & + & 256 e & + & 1024 f & + & 4096 g & + & 16384 h & + & 65536 i & + & 262144 k & + & 1048576 l & + & 4194304 m \\ a & + & 3 b & + & 9 c & + & 27 d & + & 81 e & + & 243 f & + & 729 g & + & 2187 h & + & 65611 i & + & 19683 k & + & 59049 l & + & 177147 m \\ a & + & 2 b & + & 4 c & + & 8 d & + & 16 e & + & 72 f & + & 64 g & + & 128 h & + & 256 i & + & 512 k & + & 1024 l & + & 2048 m \\ a & + & b & + & c & + & d & + & e & + & f & + & g & + & h & + & i & + & k & + & l & + & m \\ a \\ a & - & b & + & c & - & d & + & e & - & f & + & g & - & h & + & i & - & k & + & l & - & m \\ a & - & 2 b & + & 4 c & - & 8 d & + & 16 e & - & 32 f & + & 64 g & - & 128 h & + & 256 i & - & 512 k & + & 1024 l & - & 2048 m \\ &c \end{array} \\ \begin{array}{l} T h e first Dif- \\ ference of these \\ termes. \end{array} & \begin{array}{r} &c \\ b & + & 3 c & + & 7 d & + & 15 e & + & 31 f & + & 63 g & + & 127 h & + & 255 i & + & 511 k & + & 1023 l & + & 2047 m \\ b & + & c & + & d & + & e & + & f & + & g & + & h & + & i & + & k & + & l & + & m \\ b & - & c & + & d & - & e & + & f & - & g & + & h & - & i & + & k & - & l & + & m \\ b & - & 3 c & + & 7 d & - & 15 e & + & 31 f & - & 63 g & + & 127 h & - & 255 i & + & 511 k & - & 1023 l & + & 2047 m \\ &c \end{array} \\ \begin{matrix} The second \\ Difference \end{matrix} & \begin{array}{r} &c \\ 2 c & + & 6 d & + & 14 e & + & 30 f & + & 62 g & + & 126 h & + & 254 i & + & 510 k & + & 1022 l & + & 2046 m \\ 2 c & + & 0 & + & 2 e & + & 0 & + & 2 g & + & 0 & + & 2 i & + & 0 & + & 2 l & + & 0 \\ 2 c & - & 6 d & + & 14 e & - & 30 f & + & 62 g & - & 126 h & + & 25 i & - & 510 k & + & 1022 l & - & 2046 m \\ &c \end{array} \\ \begin{matrix} The 3^{d} \\ diff. \end{matrix} & \begin{array}{r} &c \\ 6 d & + & 36 e & + & 150 f & + & 540 g & + & 1806 h & + & 5796 i & + & 18650 k & + & + & 171006 m \\ 6 d & + & 12 e & + & 30 f & + & 60 g & + & 126 h & + & 252 i & + & 510 k & + & 1020 l & + & 2046 m \\ 6 d & - & 12 e & + & 30 f & - & 60 g & + & 126 h & - & 252 i & + & 510 k & - & 1020 l & + & 2046 m \\ 6 d & - & 36 e & + & 150 f & - & 540 g & + & 1806 h & - & 5796 i & + & 18650 k & + & 171006 m \\ + & 204630 & + & 3669006 \\ + & 1225230 & + & 36774606 \\ + & 228718446 \\ &c \end{array} \\ \begin{matrix} The 4^{th} \\ diff. \end{matrix} & \begin{array}{r} &c \\ 24 e & + & 120 f & + & 480 g & + & 1680 h & + & 5544 i & + & 17640 k & + & + & 168960 m \\ 24 e & + & 0 & + & 120 g & + & 0 & + & 504 i & + & 0 & + & 2040 l & + & 0 \\ 24 e & - & 120 f & + & 480 g & - & 1680 h & + & 5544 i & - & 17640 k & + & - & 168960 m \\ - & 186480 & - & 3498000 \\ - & 1020600 & - & 33105600 \\ 191943840 \\ &c \end{array} \\ \begin{matrix} The 5^{t} \\ diff \end{matrix} & \begin{array}{r} &c \\ 120 f & + & 1080 g & + & 6720 h & + & 35280 i & + & 168840 k & + & + & 3329040 m \\ 120 f & + & 360 g & + & 1680 h & + & 5040 i & + & 17640 k & + & + & 168960 m \\ 120 f & - & 360 g & + & 1680 h & - & 5040 i & + & 17640 k & - & + & 168960 m \\ 120 f & - & 1080 g & + & 6720 h & - & 35280 i & + & 17640 k & - & + & 3329040 \\ + & 29607600 \\ + & 15883824 \\ &c \end{array} \\ \begin{matrix} The 6^{t} \\ Diff \end{matrix} & \begin{array}{r} &c \\ 720 g & + & 5040 h & + & 30240 i & + & 151200 k & + & + & 3160080 m \\ 720 g & + & 0 & + & 10080 i & + & 0 & + & + & 0 \\ 720 g & - & 5040 h & + & 30240 i & - & 151200 k & + & - & 3160080 m \\ - & 26298560 \\ - & 129230640 \\ &c \end{array} \\ \begin{matrix} The 7^{th} \\ diff \end{matrix} & \begin{array}{r} &c \\ + 5040 h & + & + & 514080 k & + & + & 23118480 m \\ + 5040 h & + & 20160 i & + & 151200 k & + & + & 3160080 m \\ + 5040 h & - & 20160 i & + & 151200 k & - & + & 3160080 m \\ + 5040 h & - & + & 514080 k & - & + & 23118480 m \\ + & 10295208 \\ &c \end{array} \\ \begin{matrix} The 8^{th} \\ diff. \end{matrix} & \begin{array}{r} &c \\ 40320 i & + & 514080 & + & + & 19958400 m \\ 40320 i & + & 0 & + & + & 0 \\ 40320 i & - & 362880 & + & - & 19958400 m \\ - & 79833600 m \\ &c \end{array} \\ \begin{matrix} The 9^{th} \\ diff \end{matrix} & \begin{array}{r} 362880 k & + & + & 19958400 m \\ 362880 k & - & + & 19958400 m \\ + & 59875200 m \end{array} \\ \begin{matrix} The 10^{th} Diff. \end{matrix} & \begin{array}{l} + \\ + \end{array} \\ \begin{matrix} The 11^{th} Diff. \end{matrix} & \begin{array}{l} 39916800 m \end{array} \end{aligned}$

The use of these differences is for composing rules to find the differences of {y^e terms of a table w^ch {illeg} to be interpoled by y^e contiuall addition of those differences & also following a geo {illeg}}

<82v>

Prob.

Recta aliqua AA9 in æquales quotcun partes AA2 A2A3, A3A4, A4A5, A5A6, A6A7, A7A8, A8A9 &c divisa et ad puncta divisorum erectis parallelis AB, A2B2, A3B3 &c: invenire curvam geometricam quæ per omnium erectarum extremitates B, B{2} B, B2, B3 &c transibit.

Erectarum {illeg}|AB|, A2B2. A3B3 &c quære differentias primas b, b2, b3, &c secundas c, c2, c3 &c tertias d, d2, d3 &c & sic deincep{s} us dum veneris ad ultimam differentiam \i/. Tunc incipiendo ab ul{illeg}|ti|ma differentia excerpe medias differentias in alterna|i|s $\{\begin{cases} columnis \\ seriebus \\ ordinibus \end{cases}$ \differentiarum/ et arithmetica media inter duas medias inte reliquorum ordinum.♀ < insertion from the left margin of f 82v > ♀pergendo ad us seriem \primorum/ terminorum A, A2, A3, &c. Sint hæc k, l, - - - s & Nempe k ultim{æ} di{illeg}|ff|. l medium arithmeticum inter duas penultimas, m antepe media \trium/ antepenultimarum differentiarum &c

Cas 1

{illeg} \{Igitur} Si numerus \prignorū./ terminorum A, A2, A3, &c sit impar ultim medius {servimus} eorum erit ultimus terminus series ejus k, l, m, &c. Et tunc sic pergendum erit. Sit numerus primerum terminorum 9 et erit k=i, $l = \frac{h + h2}{2}$ {&c}/ < text from f 82v resumes > [{illeg}sint {la} k, l, m, n, o, p, q, r, s, {illeg}, nempe k ultim{æ}=i differentiæ ultimæ, l= $\frac{h + h2}{2}$ , m=g2, $n = \frac{f2 + f3}{2}$ , o=e³, $p = \frac{d3 + d4}{2}$ , {illeg} q=c4, $r = \frac{b4 + b5}{2}$ s=A5; nam series terminorum A, A2, A3, hic est |{in}|star serieri differentiarum, adeo ut medius {illeg}|ej|us terminus A5 {illeg} ultimo termino seriei k, l m, &c existente vel medio termino seriei hujus A, A2, A3 si constat impare|i| numero terminorum ut in hoc casi, vel arithmetico medi{illeg}|{æ}| Erige {s}{illeg}g|or|dinatim applicatam PQ et bis{e}tp|a| AA9 in A5 die A5P=x, PQ ={y} pergendo us ad seriem primorum terminorum A|{illeg}|B||, A2|B2|, A3|B3|, &c. Sint h{illeg}|æ|c k, l, m, n, o, p, q, r, s &c quorum ultimu{m} significet ultimam differentiam, penultimum medium arithmeticum inter duas pe{illeg}|nu|ltimas differentias, antepenultimu{m} mediam trium antepenultimarum differentarum, & \sic/ deinceps us ad primum quod erit vel medius terminorum A, A2, A3, vel arithmeticum medium int{illeg}|er| {illeg}|du|os medios. Prius accidit ubi numerus terminorum A, A2, A3 & est impar, posterius ubi par.

Cas. 1.

In casu priori sit A5|B5| iste medius terminus, {illeg} \hoc est/ A5|B5|=k, $\frac{b4 + b5}{2} = l$ , c4=m, $\frac{d3 + d4}{2} = n$ , e3=0, $\frac{f2 + f3}{2} = p$ , g2=q, $\frac{h + h2}{2} = r$ {illeg}i=s. Et erecta ordinatim applicata PQ die A5P=x, due terminus hujus pr{illeg}|ogr|essionis $1 \times \frac{x^{2}}{2} \times \frac{x x - 1}{3, 4} \times \frac{x x - 4}{5, 6} \times \frac{x x - 9}{7, 8} \times \frac{x x - 16}{9, 10} \times \frac{x x - 25}{11, 12} \times \frac{x x - 36}{13, 14} \times$ &c in se continuò {illeg}|&| orientur coefficientes termini $1 . \frac{x x}{2} . \frac{x^{4} - x x}{24} . \frac{x^{6} - 5 x^{4} + 4 x x}{720} .$ $\frac{x^{8} - 14 x^{6} + 49 x^{4} + 36 x x}{40320}$ &c $1 \times \frac{x}{1} \times \frac{x}{2} \times \frac{x x - 1}{3 x} \times \frac{x}{4} \times \frac{x x - 4}{5 x} \times \frac{x}{6} \times \frac{x x - 9}{7 x}$ $\times \frac{x}{8} \times \frac{x x - 16}{9 x} \times \frac{x}{10} \times \frac{x x - 25}{11 x} \times \frac{x}{12} \times \frac{x x - 36}{13 x}$ &c. \in se continuo/ Et orientur termini 1. x. $\frac{x x}{2} . \frac{x^{3} - x}{6} . \frac{x^{4} - x x}{24} . \frac{x^{5} - 5 x^{3} + 4 x}{120} . \frac{x^{6} - 5 x^{4} + 4 x x}{720} . \frac{x^{7} - 14 x^{5}}{5040} +$ $\frac{+ 49 x^{3} - 36 x}{5040}$ &c per quos si termini seriei k, l, m, n, o, p &c resspectivè multiplicentur, prodibunt aggregatum factorum $k + x l + \frac{x x}{2} m$ $+ \frac{x^{3} - x}{6} n + \frac{x^{4} - x x}{24} o + \frac{x^{5} - 5 x^{3} + 4 x}{120} p$ &c erit longitudo ordinatim applicatǽ PQ

Cas 2.

In casu posteriori sint A{illeg}, A5 \A4B4, A5B5/ duo medij termini, hoc est $\frac{A4 B4 + A5 B5}{2}$ =k, b4=l, $\frac{c3 + c4}{2} = m$ , d3=o|n|, e2+e3={illeg}|{o}|, f2=p, $\frac{g + g2}{2} = q$ & h=r{,|.|} Et erecta ordinatim applicata PQ, biseca A4|5|A5 in O {et} d{illeg} OP=x due terminos hujus progressionis $1 \times \frac{x}{1} \times \frac{x x - \frac{1}{4}}{2 x} \times \frac{x}{3} \times \frac{x x - \frac{9}{4}}{4} \times \frac{x}{5}$ $\times \frac{x x - \frac{25}{4}}{6} \times \frac{x}{7} \times \frac{x x - \frac{49}{4}}{8}$ &c in se continuo. Et orientur termini $1 . x . \frac{4 x x - 1}{8}$ $\frac{4 x^{3} - x}{24} . \frac{16 x^{4} - 40 x x + 9}{384}$ &c per quos si termini seriei k, l, m, n, o, p &c respectivè multiplicentur, aggregatum factorum $k + x l + \frac{4 x x - 1}{8} m$ $+ \frac{4 x^{3} - x}{24} n + \frac{16 x^{4} - 40 x x + 9}{384} o$ &c erit longitudo ordinatim, applicat{illeg} PQ.

Sed hic notandum est j. Quod intervalla A,A2, A2A3, A3A4, &c hic supponuntur esse unitates, Et quod differentiæ colligi debent {illeg}ferendo inferiores quantitates de superioribus A2B\2/de AB, A3B\3/de A2B\2/, b2 de b &c, faciendo AB−A2B\2/=b{,} A2B\2/−A3B\3/=b2{,} b−b2=c {illeg}|&|c adeo quando differentia illa {hoc modo} {illeg} negativæ sig{m}a{-}earum {ubi mutand{illeg}em|p| sunt}.

<83r>

For taking of unknowne quantitys out of intricat Equations it may be convenient to have severall formes. Now suppose x, was to be taken out of y^e Equations ax³+bxx+cx+d=0 & fx³+gxx+hx+k=0.

I feighe y^e 3 valors of x in y^e first Equation to bee -r, -s, & -t. [{illeg} is {illeg}|[| $r + s + t = \frac{b}{a}$ . $r r + s s + t t = \frac{b b}{a a} - \frac{2 c}{a}$ . $r^{3} + s^{3} + t^{3} = \frac{b^{3} - 3 a b c + 3 a d d}{a^{3}}$ . $r s + r t + s t = \frac{c}{a}$ . & that is] {illeg} y^e summe of y^e rootes is $\frac{b}{a}$ {;} of their squares is $\frac{b b - 2 a c}{a a}$ of their cubes is {illeg} $\frac{b^{3} - 3 a b c + 3 a a d}{a^{3}}$ ; of their rectangles is $\frac{c}{a}$ &c] that is,] supposing a=1, {illeg}|ev|ery r is b every rr=bb−2c. every r³=b³−3{illeg}bc+3d{illeg}, rs=c, rrs=bc-3d, r³s=bbc−2cc−bd. rrss=cc−2bd. r³ss=bcc−2bbd−cd. r³s³=c³−3bcd+3dd. rst=d. rrst=bd. r³st=bbd−2cd. rrsst=cd. r³sst=bcd−3dd. {rrsstt}r³s³t=ccd−2bdd. rrsstt=dd. r³sstt=bdd.r³s³tt=cdd. r³s³t³=d³

— — — — — — — — — — — — — — — — —

Or thus, every r⁴=b. rs⁴=c. {illeg} rr⁴=bb−2c{illeg}. rst⁴=d. rrs⁴=bc−3d. r^3⁴=b³−3bc+3d. rrst⁴=bd. rrss=cc-2bd. r³s⁴=bbc-2cc-bd. r^4⁴=b⁴−4bbc+4bd+2cc. rrsst⁴=cd. r³st⁴=bbd−2cd. rrrss⁴=bcc−{illeg}dc−2b{{illeg}b}d^{illeg}. rrsstt⁴=dd. r³sst⁴=bcd−3dd. ^*r³sstt⁴=bdd. r³s³t⁴=ccd−2bdd. r³s³tt⁴=cdd. r³s³t^3⁴=d³. ^*r³s^3⁴=c³−3bcd+3dd.

Now supposing k (or any other quantity of y^e second Equation) to bee {illeg} i|a|n unknow{ne} quantity, it must have 3 severall valors by reason of y^e 3 valors of x in y^e first Equation, & therefore x being taken away, \h will bee of three dimensions in/ y^e resulting equation.

The 3 valors of {illeg}h are −grr+hr +fr³{illeg}−grr+hr=k. & fs³−gss+hs=k & {fts} ft³−gtt+ht={k.} Which I multiply into one another that they may produce an equation expressing y^e 3 fold valor of k: out of w^ch equation I take {illeg} out r, s, t by writing b for their summe c for y^e sum̄e of their rectangles rs+rt+st. bc−3d for y^e summe of all theire rectangles of this for{me {illeg}|rrs|} (viz: for rrs+rrt+{illeg}|rss|+rtt+rtt+sst+stt) &c as in y^e Table. W^ch substitution may bee most briefly done in y^e said multiplication, thus; writing a to make up six dimensions $\begin{matrix} \begin{array}{l} k - h r + g r r - f r^{3} \\ k - h s + g s s - f s^{3} \\ k - h t + g t t - f t^{3} \end{array} & \begin{matrix} Produceth, \end{matrix} & \begin{matrix} \begin{array}{l} a^{3} h^{3} - a a b h k k + a b b g k k - 2 a a c g k k - b^{3} f k k + 3 a b c f k k \\ - 3 a a d f k k + a a c h h k - a b c g h k + 3 a a d g h k + b b c f h k \\ - 2 a c c f h k - a b d f h k + a c c g g k - 2 a b d g g k - b c c f g k \\ + 2 b b d f g k + a c d f g k + c^{3} f f k - 3 b c d f f k + 3 a d d f f k \\ - a a d h^{3} + a b d g h h - b b d f h h + 2 a c d f h h - a c d g g h \\ + b c d f g h - 3 a d d f g h - c c d f f h + 2 b d d f f h + a d d g^{3} \end{array}\} =0 \end{matrix} \end{matrix}$

For solving this Problem generally Datis quotcun punctis Curvam describere quǽ per omnia transibit: Note these differences

$\begin{matrix} \begin{array}{r} A \\ A + x \end{array} & \begin{array}{r} a \\ a & + & b x & + & c x x & + & d x^{3} + & e x^{4} & + & f x^{5} & &c \end{array} \\ \begin{array}{r} A + p \\ A + q \\ A + r \\ A + s \\ A + t \end{array} & \begin{array}{r} a & + & b p & + & c p p & + & d p^{3} + & e p^{4} & + & f p^{5} & = & α \\ a & + & b q & + & c q q & + & d q^{3} + & e q^{4} & + & f q^{5} & = & β \\ a & + & b r & + & c r r & + & d r^{3} + & e r^{4} & + & f r^{5} & = & γ \\ a & + & b s & + & c s s & + & d s^{3} + & e s^{4} & + & f s^{5} & = & δ \\ a & + & b t & + & c t t & + & d t^{3} + & e t^{4} & + & f t^{5} & = & ε \end{array} \\ \begin{array}{r} p - q & ) & α - β & = \\ q - r & ) & β - γ & = \\ r - s & ) & γ - δ & = \\ s - t & ) & δ - ε & = \end{array} & \begin{array}{r} b & + & c \times \bar{p + q} & + & d \times \bar{p p + p q + q q} & + & e \times \bar{p^{3} + p p q + p q q + q^{3}} & + & f \times p^{4} & &c & = & ζ \\ b & + & c \times \bar{q + r} & + & d \times \bar{q q + q r + r r} & + & e \times \bar{q^{3} + q q r + q r r + r^{3}} & + & f \times q^{4} & &c & = & η \\ b & + & c \times \bar{r + s} & + & d \times \bar{r r + r s + s s} & + & e \times \bar{r^{3} + r r s + r s s + s^{3}} & + & f \times r^{4} & &c & = & θ \\ b & + & c \times \bar{s + t} & + & d \times \bar{s s + s t + t t} & + & e \times \bar{s^{3} + s s t + s t t + t^{3}} & + & f \times s^{4} & &c & = & ι \end{array} \\ \begin{array}{r} p - r & ) & ζ - η & = \\ q - s & ) & η - θ & = \\ r - t & ) & θ - ι & = \end{array} & \begin{array}{r} c & + & d \times \bar{p + q + r} & + & e \times \bar{p p + p q + q q + p r + q r + r r} & + & f p^{3} & &c & = & λ \\ c & + & d \times \bar{q + r + s} & + & e \times \bar{q q + q r + q s + r r + r s + s s} & + & f q^{3} & &c & = & μ \\ c & + & d \times \bar{r + s + t} & + & e \times \bar{r r + r s + r t + s s + s t + t t} & + & f r^{3} & &c & = & ν \end{array} \\ \begin{array}{r} p - s & ) & λ - μ & = \\ q - t & ) & μ - ν & = \end{array} & \begin{array}{r} d & + & e \times \bar{p + q + r + s} & + & f \times \bar{p p + p q + p r + p s + q q + q r + q s + r r + r s + s s} & = & ξ \\ d & + & e \times \bar{q + r + s + t} & + & f \times \bar{q q + q r + q s + q t + r r + r s + r t + s s + s t + t t} & = & π \end{array} \\ \begin{array}{r} p - t & ) & ξ - π & = \end{array} & \begin{array}{r} e & + & f \times \bar{p + q + r + s + t} & = & σ \end{array} \end{matrix}$

<83v>

Prob
Curvam Geometricam describere quæ per
data quotcun puncta transibit.

Sint ista puncta B, B2, B3, B4, B5, B6 B7 {illeg}|&|c. Et ad rectam quamvis {demitis} AA7 demitte perpendicula BA, B2A2, B3A3 &c Et fac $\frac{A B - A2 B2}{A A2} = b$ , $\frac{A2 B2 - A3 B3}{A2 A3} = b2$ , $\frac{A3 B3 - A4 C4}{A3 A4} = b3$ , $\frac{A4 C4 - A5 C5}{A4 A5} = b4$ , $\frac{A5 C5 - A6 C6}{A5 A6} = b5$ {,} $\frac{A6 C6 + A7 C7}{A6 A7} = b6$ . $\frac{- A7 C7 - A8 B8}{A7 B8} = b7$ . Deinde $\frac{b - b2}{A A2} = c$ . $\frac{b2 - b3}{A2 A4} = c2$ . $\frac{b3 - b4}{A3 A5} = c3$ &c. Tunc $\frac{c - c2}{A A4} = d$ . $\frac{c2 - c3}{A2 A5} = d2$ . $\frac{c3 - c4}{A3 A6} = d3$ &c. Tunc $\frac{d - d2}{A A5} = e$ . $\frac{d2 - d3}{A2 A6} = e2$ . $\frac{d3 - d4}{A3 A7} = e3$ &c. Differe Sic pergendum est ad ultimam differentiam. Differentijs sic collectis & divisis per intervalla ordinati{illeg} applicatorum: in alter{eris} \earum/ columnis sive {illeg} \seriem vel/ ordinibus {illeg} excerpe{illeg} medias inc{illeg}|ip|iendo ab ultima et in reliquis columnis excerpe{illeg} du{a} media arithmetica inter duas medias, pergendo us ad seriem primorum terminorum AB, A2B2, &c. Sunto hæc k, l, m, n, o, p, q, r &c non quorum ultimus \terminus/ significet ultimam diff. penultimus mediam arithmeticum inter duas penultimas, antepenultimus mediam trium antepenultima{illeg}|rum| &c. Et primus {illeg} erit medias {duarum} ordinatim applicata si numerus {illeg}p datorum punctorum est impar, vel medium a{illeg}ritthmeticum inter duas medias si numerus earum est {illeg}par.

Cas 1

In casu priori sit A4B4 ista media ordinatim applicata, hoc est A4B4=k, $\frac{b3 + b4}{2} = l$ . c3=m, $\frac{d2 + d3}{2} = n$ , {illeg} e2=0. f+f2=p. g=q. Et erecta ordinatim applicata PQ et in basi AA5 sumpto quovis puncto O dic OP=x, et duc \in {se}gradatim/ terminos hujus progessionis {sic} $1 \times \bar{x - O A4} \times \bar{x - \frac{4 O A3 + O A5}{2}} \times \frac{\bar{x - O A3} \times \bar{x - A 05}}{x - \frac{O A3 + O A5}{2}} \times x - \frac{+ O A2 + O A6}{2} \times$ &c \et {illeg} ortam progressionem asserva./ Vel quod perinde est duc terminos hujus progressionis $1 \times \bar{x - O A4} \times \bar{\bar{x - O A3} \times \bar{x - O A5}}$ $\times \bar{\bar{x - O A2} \times \bar{x - O A6}} \times \bar{\bar{x - O A} \times \bar{x - O A7}} \times$ &c in se gradatim et terminos ex ortos duc \respective/ in terminos hujus progressionis $1 . x - \frac{+ O A3 + O A5}{2} . x - \frac{+ O A2 + O A6}{2}$ $x - \frac{+ O A + O A7}{2}$ . &c et orientur termini intermedij: {illeg} tota progressione existente $1 . x - O A4 . x x - \frac{+ O A3 + 2 O A4 + O A5}{2} x + O A3 \times O A4 + O A5 \times O A4$ &c Vel dic OA=α, OA2=β, OA3=γ{,} OA4=δ, OA5=e, OA6=ζ, OA7=η $\frac{O A3 + O A5}{2} = θ$ , $\frac{O A2 + O A6}{2} = μ$ . $\frac{O A + O A7}{2} = λ$ . Et ex progressione $1 \times \bar{x - δ} \times$ $\bar{\bar{x - γ} \times \bar{x - ε}} \times \bar{x - β} \times \bar{x - ζ} \times \bar{x - α} \times \bar{x - η}$ collige ter{i|m|in}os \e/ quibus multiplicatis per 1×. x−θ.× x−μ. x−λ {illeg}|&|c collige alios terminos intermedios tota serie prode{ants} 1. x−δ. xx\−δ//−θ\x+δθ. x³−\−δ//−2θ\xx+\+γε//+δθ\x−{γ.} Per cujus terminos multiplica terminos seriei k. l. m, n. o &c et aggregatum productorum erit k+x−δ×l+xx\−δ//−θ\×m &c erit longitudo ordinatim applicata PQ.

Cas. 2.

In casu posteriori sint A4B4 et A5B5 dua media ordinatim applicata hoc est $\frac{A4 B + A5 B5}{2} = k$ {.} b4=l. $\frac{c3 + c4}{2} = m$ . d3=n. $\frac{e2 + e3}{2} = o$ . f2=p{illeg} Et {illeg} \{illeg} k, m, {illeg}, o, {illeg} &/ coefficientes orientur \en/ multiplicatione terminorum hujus progressiones {illeg} $1 \times \bar{\bar{x - O A4} \times \bar{x - O A5}} \times \bar{\bar{x - O A3} \times \bar{x - O A6}} \times \bar{\bar{x - O A2} \times \bar{x - O A7}}$ $\times \bar{\bar{x - O A} \times \bar{x - O A8}}$ &c et ubiquorum coefficientes en multiplicatione {illeg} {illeg} {c}ujus progressionis $x - \frac{\bar{+ O A4 + O A5}}{2}$ . $x - \frac{\bar{+ O A3 + O A6}}{2}$

<84r>

Of the nature of Equations

Every Equation as x⁸+px⁷+qx⁶+rx⁵+sx⁴+tx³+vxx+yx+z=0. hath so many roots as dimensions, of w^ch y^e summ is −p, the summ of the rectangles if each two +q, of each three −r, of each foure +s: &c: & of all together ♉z.

Also the summe of theire squares, cubes, &c{illeg} is as followeth.

$\begin{array}{lrl} \begin{array}{l} summe of \\ squares \end{array} & p p & - & 2 q \\ cubes & - p^{3} & + & 3 p q & - & 3 r \\ sq: square & p^{4} & - & 4 p p q & + & 4 p r & + & 2 q q & - & 4 s \\ sq: cubes & - p^{5} & + & 5 p^{3} q & - & 5 p q q & - & 5 p p r & + & 5 p s & + & 5 q r & - & 5 t \\ cube . cubes & p^{6} & - & 6 p^{4} q & + & 6 p^{3} r & + & 9 p p q q & - & 6 p p s & - & 12 p q r \\ sq: sq: cubes \\ sq: cu: cubes \end{array}$

Also y^e summe of their square cubes &c is as followeth $\begin{array}{l} squares & + p p & - & 2 q \\ cubes & - p^{3} & + & 3 p q & - & 3 r \\ sq: squares & + p^{4} & - & 4 p p q & + & 4 p r & - & 4 s & + & 2 q q \\ - p^{5} & + & 5 p^{3} q & - & 5 p p r & + & 5 p s & - & 5 t & - & 5 p q q & + & 5 q r \\ + p^{6} & - & 6 p^{4} q & + & 6 p^{3} r & - & 6 p p s & + & 6 p t & - & 6 v & + & 9 p p q q & - & 12 p q r & + & 6 q s & - & 2 q^{3} \\ - p^{7} & + & 7 p^{5} q & - & 7 p^{4} r & + & 7 p^{3} s & - & 7 p p t & + & 7 p v & - & 7 q \\ + p^{8} & - & 8 p^{6} q & + & 8 p^{5} r & - & 8 p^{4} s & + & 8 p^{3} t & - & 8 p p v & + & 8 p q & - & 8 z \end{array}$

Or thus If their sume is {illeg}|−|p=\−/a. Then is y^e sume of their squares ap−2q=\b/ of their cubes −pb+qa−3r=c. square squares pc−qb+ra−4s=d. sq: cubes −pd+qc−rb+sa−5t =−e. cube cubes +pe+/−\qd+rc+\−/st|b|+ta{illeg}−6v &c.

Non of these rootes some are true {illeg} other some false & some imaginary

$x - \frac{+ O A2 + O A7}{2}$ . $x - \frac{+ O A + O A8}{2}$ &c Hoc est eritk+x−OA4+ $k + \bar{x - \frac{+ O A4 + O A5}{2}} \times l + \bar{x x - \bar{O A4 - O A5} \times x + O A4 \times O A5} \times m$ &c=Ordinatim applicatæ $PQ /= \begin{array}{l} k & \begin{matrix} \begin{array}{l} + x \\ - \frac{1}{2} O A4 \\ - \frac{1}{2} O A5 \end{array}× & l \end{matrix} & \begin{matrix} \begin{array}{l} + x \\ - O A4 \end{array}× & \begin{array}{l} x \\ - O A5 \end{array}× & m \end{matrix} & \begin{matrix} \begin{array}{l} + x \\ - O A4 \end{array}× & \begin{array}{l} x \\ - O A5 \end{array}× & \begin{array}{l} x \\ - \frac{1}{2} O A3 \\ - \frac{1}{2} O A6 \end{array}× & n \end{matrix} \end{array}$ $\begin{matrix} \begin{array}{l} + x \\ - O A3 \end{array}× & \begin{array}{l} x \\ - O A4 \end{array}× & \begin{array}{l} x \\ - O A5 \end{array}× & \begin{array}{l} x \\ - O A6 \end{array}× & o \end{matrix}$ . {illeg} Sive dic $x - \frac{+ O A4 + O A5}{2} = π$ . $\bar{x - A4} \times \bar{x - A5}$ =ρ. $ρ \times \bar{x - \frac{+ O A3 + O A6}{2}} = σ$ . $ρ \times \bar{x - O A3} \times \bar{x - O A6} = τ$ . $τ \times x - \frac{+ O A2 + O A7}{2}$ =ν. $τ \times \bar{x - O A2} \times \bar{x - O A7} = φ$ . $φ \times x - \frac{+ O A + O A8}{2} = χ$ . $φ \times \bar{x - O A} \times$ $\bar{x - O A8} = ψ$ Et erit k+πl+ρm+σn+τo+νp+φq χ{illeg}r+ψs=PQ.

<84v>

Prob{l}

<85r>

Of Equations,

Every equation hath soe many roots as dimensions of w^ch some may be true some false & some imaginary or i{illeg}|mp|ossible.

If there bee some imaginary then the true & false rootes may be knowne by y^e signes of y^e Equations termes: Namely there are soe many true rootes as variations of signes & soe many false ones as successions of y^e same signes. When any termes a{illeg}|re| wanting supply their{e} voyd places w^th ±0.

But if any \because imaginary roots are properly neither true nor false/ roots bee imaginary, this r{oote} soe far admitts of exception.{illeg} Thus the signes of{illeg} this Eq: x³−pxx+3{illeg}|pp|x−p³=0. show it to have three true roots, {illeg}wherefore if it bee multiplyed by x+\2/a=0 the resulting equation have {three} true x⁴+px³+{illeg}|pp|xx\+6p³//−q³\x−2{illeg}|pq|³=0 should have thre true rootes & a false one, g|b|ut the signes shew it to have three false & one true,|.| I conclude therefore that the two roots w^ch in y^e one case appeare true, |&| in a|y^e|nother false are neither, but imaginary; & that of y^e other two roots one is true {y^e} other false.

Hence it appeares y^t to know y^e particular constitution of any Equation it is {chi{illeg}fely} necessary to understand\{illeg}/ w^t imaginary roots it hath. And this in some of the simplest Equations is easily discovered, thus in xx±ax+bb=0, {illeg} both roots are imaginary if 4bb<aa, otherwis both reall. And thus in {illeg}|px|³−{illeg}|apx|−q\{illeg}/=0 two roots are imaginary if 4p³<27qq, otherwise all reall. But to give \accurate/ rules for determining the {illeg} \number/ of these roots in all so{illeg}|rt|s of Equations would bee a thing not onely {illeg}|ver|y difficult, but usele{illeg}|sse|:|,| because in Equations of many dimensions y^e rules would bee more in{illeg}|tr|icate & laborious than y^e to put in practise then to solve the Equations either by lines or numbers. Soe y^t y^e accurate determination{s} of those roots is {illeg} for the most part {esilyest} acquired by solving the Equations.

But yet because the discovery of {illeg}|th|ese roots is very usefull I shall lay downe rules whereby they may bee many times discovered at first sight, & almost always w^thout much labour.

First then {illeg} if you see any terme betwixt

First yⁿ if in any three termes together y^e two extreame termes having y^e same signes bee neither of them \as little or lesse/ greater ({illeg}tis as more \little or l{illeg}|es|se/ remote from nothing) then y^e terme betwixt them, conclude there are two imaginary roots. Thus \{in}/{illeg} +aaax−b³{illeg} x³−2|3|xx+2x{\{illeg}/}−4=0 has two roots imaginary because neither 3 nor 4 are lesse then 2. And y^e like of x³−3xx−2x\{illeg}/-4=0. And soe of x³+2x−4=0, or x³+0xx+2x−4=0, because ne{illeg}|it|her 1 nor 2 {illeg} is lesse then 0.

Secondly if uppon sight you discover three \such/ termes together \that/ the two extreames having the same signes their rectangle bee as greate or greater then y^e square of the meane terme, conclude there are two imaginary roots. Thus in x³+\−/pxx+3ppx−q³=0 are two imaginary roots because 1×3pp<−px−p. And soe of x³−pxx+ppx−\2/p³=0 because −p{x} 1×pp{illeg}=-px-p. or −px−\2/p³= pp×pp.

Thirdly if

First then the reall roots of an Equation are not more then the number of \its/ termes.

First yⁿ y^e number of impossible roots is always eaven. If one bee {illeg}|imp|ossible there must bee two, if three there must bee foure &c. And hence Equations of odd dim{en}|sions must have one roote reall at least.|

Secondly the number of reall roots \of any Equation/ are not more then the number of its termes. Symbol (two surmounted by circle) in text If Thus x⁴−2x+3=0 {illeg} have all foure roots reall & therefore must have two imaginary. Symbol (one surmounted by circle) in text Thus x⁵−3x⁴+4=0 can have but thre reall roots & then other two bee \must bee/ imaginary. Hence {illeg}|are| to bee excepted equations w^ch {want} all their odd termes as x⁶−2x⁴+3xx−2=0. And s|i|n \{illeg} like{illeg}/ cases write y for xx. And so many {illeg}|term|e roots as y^e product y³−2yy+3y−2=0 hath \{times}/ soe many reall roots, halfe true halfe false, the other shal & four times soe imaginary ones y^e other shall x⁶−2x⁴+3xx−2=0 shall have.

Thirdly, if under the termes of any Equation you set a progression of {fractions} {illeg} \each having/ y^e dimension of the terme{illeg} above {illeg} it for its {illeg} & number {illeg} {denominating} y^t terme first second third &c for its denominator & \{illeg} if/ {illeg} {illeg} any {illeg} together so y^t the \rectangle of the/ first {illeg} have {illeg} {illeg} multiplyed {viz the fraction} {illeg} first {illeg} square of the {illeg} terme multipl{illeg} by {y^e fraction} {illeg} conclude there are two imaginary roots {illeg} if {illeg} {illeg} one {illeg} <85v> in all throughout y^e equation conclude \also/ there are two imaginary roots at least. If equall in all cases throughout the Equation, conclude th{illeg}|at|{illeg} all the roots of the equation are equall. If it be greater or equal to it in two {illeg} pla\If the said factus be greater or equall to y^e said squ/ces of y^e equation & not in all places betwixt conclude there are foure imaginary roots at least. If it bee greater or equall to it in three places of y^e equation & not in all places betwixt, conclude there are six imaginary roots at least. And soe of the rest.

{Thus if} \Thus if the Equat be x³−3xx+4x−2=0./ /Th{illeg} /en\ y^e progre{illeg}|ss| is $\frac{3}{1} . \frac{2}{2} . \frac{1}{3}$ .\ & because $- 3 x x - 2 \times \frac{2}{2} (6) < 4 \times 4 \times \frac{1}{3} (\frac{16}{3})$ , I conclude there are two roots imaginary at least.

$\begin{matrix} Th u s in & x^{3} & - & 6 x^{2} & + & 6 x & - & 2 & = & 0. & because - 6 x - 2 \times \frac{2}{2} (12) = 6 \times 6 \times \frac{1}{3} (12) but \\ \frac{3}{1} & . & \frac{2}{2} & . & \frac{1}{3} & . & 1 \times 6 × \frac{3}{1} (18) < - 6 \times - 6 \times \frac{2}{2} (36 I conclude \end{matrix}$ there are two imaginary roots.

$\begin{matrix} T hus in & x^{3} & - & 6 x x & + & 12 x & - & 8 & = & 0 & because 1 \times 12 \times 3 (36) = - 6 \times - 6 \times 1 (36) and also \\ 3 & . & 1 & . & \frac{1}{3} & . & - 6 × - 8 \times 1 (48) = 12 \times 12 × \frac{1}{3} (48) I conclude that \end{matrix}$ all the 3 roots are equall.

$\begin{matrix} Bu t in & x^{3} & + & 6 x x & + & 12 x & - & 8 & = & 0, & because 1 \times 12 \times 3 (36) = 6 \times 6 \times 1 but \\ 3 & . & 1 & . & \frac{1}{3} & 6 × - 8 \times 1 (- 48) < 12 \times 12 × \frac{1}{3} (+ 48) I \end{matrix}$ conclude two rootes are imaginary.

$\begin{matrix} In & x^{4} & - & x^{3} & + & 2 x x & - & 2 x & + & 3 & = & 0, & because 1 \times 2 \times \frac{4}{1} (8) < - 1 \times - 1 \times \frac{3}{2} \\ \frac{4}{1} & \frac{3}{2} & \frac{2}{3} & \frac{1}{4} & (\frac{3}{2}), I conclude there are two \end{matrix}$ imaginary roots at least, also by the three last termes $\frac{2 \times 3}{} \times \frac{2}{3} (4) < - 2 \times - 2 \times \frac{1}{4}$ (1) therefore there {illeg} are two more imaginary roots \all 4 roots are imaginary/ unlesse the like happen in the three middle termes. I try therefore & find $- 1 \times - 2 \times \frac{3}{2} (3) < 2 \times 2 \times \frac{2}{3} (\frac{8}{3}$ & soe can conclude but two rootes imaginary.

$\begin{matrix} In & x^{4} & - & x^{3} & + & 3 x x & - & 2 x & + & 3 & = & 0 & because 1 \times 3 \times \frac{4}{1} (12) < - 1 \times - 1 \times \frac{3}{2} (\frac{3}{2}) and also \\ \frac{4}{1} & . & \frac{3}{2} & . & \frac{2}{3} & . & \frac{1}{4} & . & 3 \times 3 \times \frac{2}{3} (6) < - 2 \times - 2 \times \frac{1}{4} (1), but not - 1 \times - 2 \times \frac{3}{2} (3) < \end{matrix}$ $< 3 \times 3 \times \frac{2}{3} (6) .$ Therefore I conclude all four roots imaginary.

$\begin{matrix} In & x^{7} * & + & x^{5} & - & 2 x^{4} & + & 3 x^{3} & - & 3 x^{2} & - & 2 x & - & 1 & = & 0 & because the three first termes \\ \frac{7}{1} & . & \frac{6}{2} & . & \frac{5}{3} & . & \frac{4}{4} & . & \frac{3}{5} & . & \frac{2}{6} & \frac{1}{7} & give 1 \times 1 \times \frac{7}{1} (7) < 0 \times 0 \times \frac{6}{2} (0) there \end{matrix}$ are two img|a|ginary roots{.} Also the 3^d 4^th & 5^t terme give $1 \times 3 \times \frac{5}{3} (5)$ therefore since by the 2^d 3^d & 4^th terme tis $0 * - 2 \times \frac{6}{2} (0) < 1 \times 1 \times \frac{5}{3} (\frac{5}{3})$ I conclude ther are 4 roots imaginary. Also \by y^e 4^th 5^t & 6^t termes/ I find $- 2 \times - 3 \times \frac{4}{4} (6) < 3 \times 3 \times \frac{3}{5} (\frac{17}{5})$ but thence nothing can be concluded because those three termes are of the same condition w^th y^e 3^d 4^th & 5^t termes w^ch immediately precede them. Lastly I find by the three last termes $- 3 \times - 1 \times \frac{2}{6} (1) < - 2 \times - 2 \times \frac{1}{7} (\frac{4}{7})$ ; And by the termes pr{e}ceding them $3 \times - 2 \times \frac{3}{5} (- \frac{18}{5}) < - 3 \times - 3 \times \frac{2}{6} (3)$ . Therefore I conclude there are \two more imaginary/ roots; imaginary \y^t is in all 6/ & but one reall.

Had it been

Thus in litterall Equations, if $\begin{matrix} x^{3} & - & p x x & + & 3 p p q & - & q^{3} & = & 0 \\ \frac{3}{1} & \frac{2}{2} & \frac{1}{3} \end{matrix}$ , because 1×3ppx{illeg} $(9 p p) < - p x - p \times \frac{2}{2} (p p)$ \therefore/ what ever numbers are taken for p and q two roots shall bee im{illeg}|ag|inary. And soe of the rest.

You may set the

This rule may be otherwise thus exprest. Over y^e termes of y^e equation set a series of fractions each having y^e dimensions of the terme under it for its numerator, & the number denominating y^e terme first, second {third} &c for its denominator. Then in every three termes observe{illeg} whither the square of the middle terme \multiplyed by the fraction above/ be greater equall or lesse yⁿ y^e factus of the {terme} before & after it multiplyed by y^e fraction over y^e terme before it. If greater write y^e signe + underneath; if equal or lesse write {illeg} the signe − underneath y^e middle terme {illeg} & lastly set + under y^e first terme of y^e equation. Then observe how may {sic} changes there are from + to − & conclude that there \are/ soe many pa{ires} of imaginary roots{.} |unlesse {illeg}|all| y^e roots bee equall|

Thus $\begin{matrix} \frac{3}{1} & . & \frac{2}{2} & . & \frac{1}{3} & . \\ x^{3} & - & 3 x x & + & 4 x & - & 2 & = & 0 \\ + & . & + & . & - & . \end{matrix}$ hath 2: & $\begin{matrix} \frac{4}{1} & \frac{3}{2} & \frac{2}{3} & \frac{1}{4} \\ x^{4} & - & x^{3} & + & 3 x x & - & 2 x & + & 3 & = & 0 \\ + & . & - & . & + & . & - & . \end{matrix}$ hath 4 {illeg} /many {roots}\

<86r>

$\begin{matrix} 5 & 4 & 3 & 2 & 1 \\ 1 & 2 & 3 & 4 & 5 \end{matrix}$

If you would bee more exact set downe \after their signes/ the differences {illeg} of y^e said squares & rectangles multiplyed. And then if you see three di{illeg}|ff|erences together w^th the same signe soe y^t y^e square of the meane{illeg} diff bee less then y^e rectangle of the other two change the signe of the said meane difference

$\begin{matrix} \frac{4}{1} & . & \frac{3}{2} & . & \frac{2}{3} & . & \frac{1}{4} & . \\ Thus if & x^{4} & - & x^{3} & + & 2 x x & - & 2 x & + & 3 & = & 0. & becaus - \frac{23}{4} \times - 3 (\frac{69}{4}) < - \frac{1}{3} \times - \frac{1}{3} (\frac{1}{9}) I change \\ + \frac{4}{1} & - & \frac{23}{4} . & - & \frac{1}{3} . & - & 3 . & the signe of - \frac{1}{3} & soe the signes +.−.+.− \\ + & . & - & . & + & . & - & . & shew all foure roots imaginary. \end{matrix}$

If you would bee yet more exact, augment y^e roots of the Equatio the more the better, & at least soe much as to make them a{illeg}|ll| true. then set y^e afforesaid differences \w^th their signes/ underneath as before. And under them t{illeg}|he| progression of fractions squares. Then if you see three differences together w^th y^e same signe soe y^t y^e square of the middle difference multiplyed by the fraction under {illeg}|it| bee not greater yⁿ y^e rectangle of the other two differences multiplyed by y^e fraction under the first: change y^e signe of y^e middle difference.

Any Equation being propounded, set down a series of \so many/ fractions as y^e Equation hath dimensions, whose {illeg}|nume|rators are a prog & denominators are a progression of units backward & forward. Divide each fraction by y^t p^eceding it & set the quotes in order overal |y^e| middle termes of the Equation. Then observe whither of every {thee term such} of every middle terme whither it square multiplyed by y^e fraction over it bee greater equall or lesse yⁿ y^e rectangle of y^e two termes on either hand. If greater write + underneath, if equall or lesse write {illeg} −. Lastly set + under y^e first \& last/ terme & soe many {illeg}|chan|ges as there are from + to {illeg} & there shall bee soe many impossible roots as there are changes of signes. |Unlesse it happen y^t all y^e roots are equall, for &c:|

Thus if x³−6|3|xx+4|6|x−2|4|=0. The \series/ fractions will bee $\frac{3}{1} . \frac{2}{2} . \frac{1}{3} .$ & dividing $\frac{2}{2}$ by $\frac{3}{1}$ , & $\frac{1}{3}$ by $\frac{2}{2}$ their quotes will be $\frac{1}{3}$ , $\frac{1}{3}$ , to be set over y^e middle termes of the equation thus $\begin{matrix} \frac{1}{3} . & \frac{1}{3} . \\ x^{6} & - & 3 x x & + & 6 x & - & 4 & = & 0. \\ + & - & - & + \end{matrix}$ . Then I observe in y^e 2^d terme that $- 3 \times - 3 \times \frac{1}{3} (3)$ is lesse then {illeg}4|1×6| & therefore I write − under it. so in y^e 3^d terme I find $6 \times 6 \times \frac{1}{3} (12) = - 3 \times - 4 (12)$ , therfore I write − under it. Lastly seting + under y^e first & last terme I find two changes of sines & so conclude there are

Thus if x⁵−\5/{illeg}x⁴+5x³− xx− x⁵−2|4|x⁴+x³−2xx−4|5|x−6|4|=0. The series of fractions will bee $\frac{5}{1} . \frac{4}{2} . \frac{3}{3} . \frac{2}{4} . \frac{1}{5}$ And dividing $\frac{4}{2}$ by $\frac{5}{1}$ & $\frac{3}{3}$ by $\frac{4}{2}$ &c there results $\frac{2}{5} . \frac{1}{2} . \frac{1}{2} . \frac{2}{5}$ to bee set over y^e middle termes of the equation th{illeg}|{illeg}s| $\begin{matrix} \frac{2}{5} . & \frac{1}{2} . & \frac{1}{2} . & \frac{2}{5} . \\ x^{5} & - & 4 x^{4} & + & 4 x^{3} & - & 2 x x & - & 5 x & - & 4 & = & 0. & Then in the second time I find - 4 \times - 4 \times \frac{2}{5} (\frac{32}{5}) \\ + & + & - & + & + & + & < 1 \times 4 (4) : therefore I set + under it. In \end{matrix}$ the third $4 \times 4 \frac{1}{2} (8) = - 4 \times - 2 (8)$ : therefore I write −. In the 4^th $- 2 \times - 2 \times \frac{1}{2} (2) < 4 \times - 5$ (-20) therfore I write +. In y^e 5^t $- 5 \times - 5 \times \frac{2}{4} (10) < - 2 \times - 4 (8)$ therefore I w{illeg}|rit|e +. lastly under y^e first & last terme I w{illeg}|rit|e +. And soe finding two changes of termes I conclude two roots to bee impossible.

$\begin{matrix} \frac{3}{8} & \frac{4}{8} & \frac{3}{8} \\ T hus in & x^{4} & + & 3 x^{3} & - & 6 x^{2} & - & 3 x & - & 2 & = & 0. & two roots are impossible \\ + & +. & + & - & + \end{matrix}$ $\begin{matrix} \frac{1}{3} & \frac{1}{3} \\ In & x^{3} & + & 0 x x & + & p p x & - & q^{3} & = & 0 & two roots are impossible \\ + & - & + & + \end{matrix}$ $\begin{matrix} \frac{1}{3} & \frac{1}{3} \\ x^{3} & - & 6 x x & + & 12 x & - & 8 & = & 0 & All the roots are equall. \\ + & - & - & + \end{matrix}$

<86v>

Sometimes there may bee impossible not by this meanes discovered, w^ch if you suspect, augment or diminish y^e roots of the Equation a little, not soe much as to make them all affirmative or all negative, or at most not much more. & try the rule againe. And if there bee any impossible roots twill rarely happen y^t they shall not bee discovered at two or three such tryalls. Nor can there bee an Equation whose impossible roots may not bee thus discovered.

Thus if x³−3ppx−3p³=0, because i{f} \in w^ch noe/ impossible appeare I put x−p{illeg} x=y−p & the result is y³−3pyy−p³=0 in w^ch two appeare, Or if I put x=y−2p the result is y³−6pyy+9ppy−5p³=0 in w^ch also two appeare.

Thus |if| $\begin{matrix} x^{5} & + & x^{4} & - & 4 x^{3} & + & 5 x x & - & 2 x & + & 1 & = & 0 \\ + & + & + & + & - & + \end{matrix}$ .|,| I set y^e signes + & − under it as before and find two imaginary roots & to try if it have any more I suppose x=y+1 & y^e results is x^4|5|{illeg}|+|6x⁴+10x³+9xx+5x+0

Now by this rule false roots may bee often discovered at first sight; as if you see a terme {illeg}|wan|ting twixt two others of {illeg}the same signes, or if it bee lesse \greater/ there either of those two or its square {illeg}not greater then their rectangle; conclude there a paire of impossible roots, may soe many pa{illeg}|ir|e as these {caseses} happe{n}. Thus x⁷+\3/x⁵−5x⁴+\4/x³ \set the signe − under that terme/ /conclude\ conclude there is a paire of impossible roots at least & set the signe − under y^t terme \also set y^e signe + on either side the term {wanting}/. As in this $\begin{matrix} x^{7} & + & 0 x^{6} & + & 2 x^{5} & - & 2 x^{4} & + & 3 x^{3} & - & 4 x x & + & 6 x & - & 2 & = & 0 \\ + & - & + & - & - & + \end{matrix}$ . In w^ch it appears there are 4 if not 6 impossible roots

If there bee {illeg}|tw|o or more termes wanting set signes under them successively to{illeg} {the best} advantage {illeg} indicating {im{illeg}|pos|:} roots \begining w^th a negative{illeg} only end w^th an affirmative/ /if the terms on either hand have contrary signes\. As in {illeg} $\begin{matrix} x^{5} & - & a x^{4} & * & * & * & + & a^{5} & = & 0 \\ - & + & - \end{matrix}$ so in $\begin{matrix} x^{5} & + & a x^{4} & * & * & * & - & a^{5} & = & 0. \\ - & + & + \end{matrix}$ The first shows 4, y^e last two roots imaginary: Soe{.} in x⁹ * x⁸ * $\begin{matrix} x^{10} & * & + & x^{8} & * & * & - & 3 x^{5} & * & * & * & * & - & 6 & = & 0. \\ + & - & + & - & + & + & - & + & - & + & + \end{matrix}$ w^ch hath {illeg}|{8}| roots imaginary

<88r>

To reduce sev{{illeg}}erall equations of four \by divisors of {three}/ dimensions.

Get y^e divisors of 6 or 7 \or 8/ such numbers as were described before. Add & sustract them from 29. 8. 1. 0. -1. -8. -27. Make some number in the Take any three numbers out of the \{three}/ middle ranks, r, s. t. Make -s=c. $\frac{r + t}{2} - s = a$ $\frac{r - t}{2} = b$ . Then see if you can find 4a+2b+c in y^e rank peceding {sic} these & 9a+3b+c in y^e rank preceding that also 4a−2b+c in the rank following these & 9a−3b+c in the rank after that{;} if you can try the division by x³−axx+bx−c.

Or take \better/ multiply all y^e numbers in y^e middle rank by \4|5| &/ 9|1|0. Let s, {illeg}|{o}| {illeg}|{v}| signify y^e products, out of y^e two ranks on either side take any two eaven or two odd numbers out of y^e two ranks on either hand. Let those be r & t. {illeg} Then observe if you can{illeg} find \make|i|ng {of}/ $\frac{r + t}{2} = m$ . $\frac{r - t}{2} = n$ observe if you can find m−any s {illeg} 4m+any s {illeg} 4m+2n+any s in y^e rank preceding those or 4m−2n{illeg}|+|any s in y^e rank following those or|&| yⁿ 9m+3n+any v in y^e 2^d rank preceding those & 9m−3n+any v in {illeg}|y|^e 2^d rank following them. If so try y^e division by $x 3 - m x x + \frac{1}{5} s + n x - \frac{1}{5} s$ .

Or yet better. Do not add {illeg}|&| subduct y^e divisors from 9, 4, 1, 0, 1, 4, 9 but try if of those in {illeg}|y|^e first & last rank y^e difference of any two eaven or two odd ones be divisible by 6. Call $\frac{1}{2}$ that difference G & $\frac{1}{2}$ y^e summ of y^e same terms H. Then {illeg}|try| in y^e middle collumn there be any terms w^ch subducted from H, or added to it produces a number divisible by {illeg} 9. Call that \term/ + {illeg} c if it be subducted or − {illeg} c if added. [Then try if in that column \next/ before or after y^e middlemost w^ch has least /fewest\ divisors there be any te{rm} {illeg} w^ch added |to| or subducted from $\frac{1}{3} G$ produces a number divisible by 9] Then making & y^e summ or difference {illeg}K, & putting $\frac{1}{9} K - 3 = a$ , & G+ /−\=b. Try if you can find 8−4a+2b−c {illeg}|on| y^e {illeg} |2^d| rank next above y^e middle one or 84|+|4a+2b+c in y^e \2^d/ rank 2^d next \ra{illeg}/ below {illeg}|it|. If so {illeg}|tr|y y^e division by x³−{illeg}ax²+bx−c. Or thus against those divisors added & subducted from 20|7|. 8. 1. 0. 1. 2|8|. 27. set 9a+ {illeg}b+c. 4a+2b+c. a+b+c. c. a−b+c. 4a−2b+c. 9a−3b+c. then choose |y^e| three ranks \of the fewest terms/ & in them those numbers \one in each/ {b}by w^ch get y^e valor of a, b, & c. & those gotten will for y^e give you numbers to be sought in the other ranks, w^ch if you find there try the division by x³−axx−bx−c. otherwise \c{illeg}|l|ose/ there other numbers out of the first \same 3/ rak|n|ks, & doe so till you have gone through all variety.

Note that if y^e last term of y^e æquation divided by {c} be p y^e last but one q, Then if c & $\frac{q}{c}$ {must {illeg}ust} have a common divisor w^ch divides {illeg}|{n}|ot q, that c is to be rejected. Also if $\frac{1}{4} b \times \frac{p}{c}$ is greater then {+} {×} y^e grea or $\frac{1}{4} a \times \frac{p}{c}$ be greater then {illeg} {illeg} y^e greatest term of y^e æquation y^t b or a is to be rejected.

Note also y^t if y^e æquation to be reduced be of six dii|m|ensions it is not necessary both to ad & substract y^e divisors fro 20|7|. 4|8|. 1. 0. {illeg}1. {illeg} 8. -27

Or thus best. L|I|f t{illeg}|he| Let y^e numbers w^ch arose by substituting 2. 1. 0. -1. -2 for x be G. H. I. K. L. Seek If I end not in 5 or {illeg}|{0}| substitute 10 & -10 for x & let y^e numbers arising be F. {illeg}|& M{illeg}| but if I end in 5 {illeg}|or| 0 increase y^e root of & H or K do not, increase \or decrease/ y^e root of y^e æquation or little by an unit. Do so also if {H} be an eaven number and H or K an o{illeg}|dd| one w^th fewer divisors & then substitute 10 {&} -10 for x{.}

<88v>

How numeral æquations are to be
reduced by divisors of 3 or 4 \or more/ dimensions

Substitute 5, {illeg}|{6}|, 3, 2, 1, 0, -1, -2, -3, -{illeg}, -5 {illeg} & also 4 & -4 if need be, for x, & find \all/ y^e divisors of \suppose/ y^e resulting terms, w^ch set to be F, G, H, I, K, L, M, N, O, P, Q. Find all their divisors set those of F & Q together \by pairs/ whose last figures are equal . Let y^e sum Take any one of these pairs \or differ by 5. Gather the summs & differences of these pairs/. Let Let {sic} their t summ \of any two/ be R, \the tenth parts of/ their {illeg} difference S \if their last figures/, But for finding this {illeg} differences you must subduct y^e divisor of Q from y^e divisor of F not y^t of F from y^t of Q so y^t S will be negative if y^e divisor of Q be y^e greater ✝ < insertion from the left margin > ✝ And if y^e æquatiō be not of more yⁿ {{illeg}}5 dimensions so y^t {{illeg}} it must be divisible (if at all) by a divisor of 2 dimensions set down xx∓Sx $\mp \frac{1}{2} R - 25$ to be tried for such a divisor. Where R & S must have y^e same signes if s {illeg} y^e divisor of F{illeg} was greater then y^e divisor of K otherwise contrary signes. < text from f 88v resumes > Quadruple y^e divisors of L. L {F} Take one & if {illeg}|th|e two last figures of any one be the same w^th y^e two last figures of 2R, take it from 2R. Let y^e residue {be T} divided by 100, be T. Or if y^e two last figures of any one added to y^e two last figures of 2R make 100 add it to 2R & let y^e summ \divided by 100/ be {illeg} T & let y^e number \whose quadruple is/ added to or subducted from 2R be a And [if y^e æequation be not of more then 5 dimensions so that it must be divisible (if at all) by a divisor of \but/ two dimensions, set down xx+{²⁵}+Sx. a to be tried for a divisor. But] if y^e æquation be of 6 or 7 dimensions & no more set down x³+Txx+{S^-25}−+a to be tried Where note that S & T must be negative if they were found so above {illeg}|&| a must be negative if it was added to 2R to make T, or els {illeg}|ff|irmative if it was subducted. |& the same is to be observed of the signes in y^e following operations.|

But if y^e æquation be of more then 7 dimensions then look in y^e columns{illeg} among y^e divisors of K for a number w^ch added to or subducted from {illeg}S+T+a gives a number divisible by 24. (all This number divided by 24, call v & the divisor w^ch gave it call β And set down $\begin{array}{l} x^{4} & + & v x^{3} & + & T x x & + & S x^{} & + & a \\ - & 1 & - & 25 & - & 25 v \\ - & 25 \end{array}$ to be tried for a divisor \if the æquation be not of more then 9 dimensions/. Where {illeg}|{v}| must be negative if y^e number β was greater then S+T+a & subducted from it. so S+T+a {illeg} in {illeg}sed or {illeg} if it was so above.

But if y^e æquation be of more then 9 dimension then Cook in y^e column among the divisors of M for a number w^ch added to or subducted from −S{illeg}\+/T+a gives a number divisible by 24. This number divided by 24 call W & y^e divisor w^ch gave it {γ}. And set down $\begin{array}{l} x^{5} & + & \frac{1}{2} w & x^{4} & + & \frac{1}{2} v & x^{3} & + & T & x x & + & S & x & + & a \\ + & \frac{1}{2} v & - & \frac{1}{2} w & - & \frac{25}{2} v & - & \frac{25}{2} v \\ - & 26 & - & \frac{25}{2} w & + & \frac{25}{2} w \\ + & 25 \end{array}$ to be tried for a divisor if y^e æquation be not{illeg} of more then 11 dimensions or supposed divisible by {illeg}|{−}| Divi{illeg}|so|r of not more {yⁿ} 5 dimensions At ita in infinitum pergitur

Now the trial of these divisors is this suppose y^e divisor be a+bx+cxx+dx³+ex⁴+fx⁵ &c And observe if {illeg} be among y^e divisors of L{,} if {illeg}|th|is Divisor ascend but to {two} dimensions{;} & a+b{illeg}{+c+{illeg}+1}+c+1 among y^e divisors of K if {illeg}ascend but to two or 3 dimensions & a−b+c{−d−1} {illeg} divisors of {illeg} if it ascend but to 2, 3 or 4

<89r> dimensions & a+2b+4c|+d+e+1| among y^e divisors of I if it ascend not to more then 5 dimensions & a−2b+4c{illeg} \−d+e−f+1/ among y^e divisors of N if it ascend to no more then six dimensions & a{illeg}+3b+9c+27d+81e+243f|+|&c among y^e divisor{illeg}s of H if it ascend not {illeg}|to| more then 7 dimensions, & so in infinitum In all w^ch put c=1 & d, e, f = 0 if y^e divisor be but of 2 dimensions or d=1 & e, f, = 0 if but of 3 dimensions, & so on.^[58] And when you have tried all y^e divisors w^ch may be found by this {rale}, {If}|{&}| rejected those w^ch will not hold this trial: if there remain more or if y^e æquation be not reducible divisible by \any of/ those w^ch remaine, you may conclude y^e {illeg}|æ|quation irreducible by {illeg} any rational divisor.

<89v>

Veterum Loca solida componere restituta.

Cartesius de hujus Problematis confectione se jactitat quasi aliquid præstitistet a Veteribus tantopere quæsitum̄, cujus gratia putat Apollonium {tractatum} \libros suos/ de Conicis sectionibus scripsisse. Sed {illeg}|cu|m tanti viri pace r{e}m Veteres: neutiquam latuisse cred{ide}rius{.} Tradit enim Docet enim Pappus modum ducendi Ellipsim per quin data puncta et eadem est ratio in cæteris Con. sect. Et si Veteres norint ducere Conic{illeg}m|{i}s| sectionem per quin data puncta, quis {nori} videt eos cognovisse compositionem loci solidi{.} Imo vero eorum methodus longe elegantior est Cartisiana. Ille {illeg} rem peregit per calculum Algebraicum qui in verba (pro more Veterum scriptorum) resolutus Adeo prolixum et perplexum evaderet ut nauscum crearet ne{illeg} posset intelligi. At illi rem peregerunt per simplices quasdam Analogias, nihil judicantes lectu dignum quod aliter scribereter, & proinde celantes Analysin per quam Constructiones invenerunt. Ut pateat hanc rem eos non latuisse, conabor inventum restituere insistendo vestigijs Problematis Pappiani. In {illeg}|cu|m finem propo{illeg}|no| hæc Problemata.

^[59] 1 Conicam sectionem per quin \tria/ data puncta \ABC/ describere quæ datum centrum \O/ habebit. AO dat A duobus punctis \AB/ ad centrum O age rectas AO BO et {eos} produc \AO/ ad \AO/ Q|O|P ut sit AOP=AO|.| , et OQ {illeg}BO, et puncta P et Q erunt ad curvam. A tertio puncto C age CS parallelam AO et occurrentem OB in S et cape ST ad AO{illeg} $\frac{SB \times SQ}{SC} ∷ {AO}^{q} . {B0}^{q}$ , et erit etiam punctum T ad curvam. Biseca TC in V, et ipsi OV parallelam age CR occurrentem AO in R erit CR ordinatim applicata ad diametrum AP. Et latus rect{um} erit ad AP ut RC^q ad AR×R|A|P{,|.|} Figura existente Ellipsi si punctum R |cadit intra A et P, aliter Hyperbola.|

^[60]

2 Per data quinque puncta \A, B, C, D, E/ Conicam Sectionem describere {illeg}ge Junge quatuor puncta A, B, C D duo puncta AC et alia duo BE sit jungentium intersectio K. Ipsis AC BE age parallelas Dg DG occurrentis AC{,} BE, AC in punctis H et F, et facto $BK \times KE . AK \times KC ∷ \frac{BH \times HE}{DH} . HI ∷ FG . \frac{AF \times FC}{FD}$ , puncta G et I erunt ad curvam. Biseca ergo ID et AC in m et n ut et BE ac GD in p et q et actarum mn, pq intersectio O erit centrum Conicæ sectionis|.| nisi ubi Habito autem centro, describe figuram per puncta A, B, C ut supra, \Quæ quidem ellipsis erit si punctum R cadit inter A et P. Aliter Hyperbola/ Quod si mn et pq parallelæ sint Figura erit Parabola, et rectæ AO Cujus determi{illeg} neutiquam difficilior. Quo casu produc PQ ad V ut sit BP^q.GQ^q ∷PV.QV. et erit PV diameter V vertex et $\frac{{BP}^{q}}{PV}$ latus rectum figuræ {illeg}.

^[61] His præmissis nihil aliud restat agendum in solutio compositio loci solidi quam ut quin puncta quæramus per quæ figur{a} transibit. Id quod in Problemat{illeg}|{e}| Veterum facillimum est{.} Su{illeg} AP, AT, ST, AG, SG quatuor positione datæ rectæ et ad hac ducendæ sint \{illeg} s{um}{illeg}/ in datis angulis a puncto aliquo com{illeg}muni C {a}liæ quat{uor} CD, CF, CB, CH ea lege ut rectangulum duarum primarum CD ×CF dat{a}m habeat rationem ad rectangulum e|r|eliquarum CB×CH {illeg} curva in qua punctum C perpetim reperitur transibit per intersectiones datarum A, G, S, T, nam ubi FC nulla est, rectangul{u}m FC×CD nullum erit adeo et rectangulum CD×CH, {illeg} et {illeg} rectarum CD, CH. Si CD{;} punctum C incidet adT, si CH{;} ad S. At ita ubi CD nulla est punctum C incidet {vel in A vel in G.} Dantur ita quatror puncta A, G, S, T {illeg} et sola restat quinta investiganda. Id /quod\ < insertion from f 90r > facillimum est. Nam per punctum A ag{illeg}|at|ur recta quævis AC {illeg} {illeg} et in ea quaratur punctum C quod Problemati satisfaciet. {Et ergo} Jam adeo ratio DC ad BC, et proinde etiam ratio CH ad FC siquidem ratio DC×CH ad BC×FC detur. Age ergo rectam SC ea lege ut sit CH ad FC in ratione ista data, et hæc secabit rectam AC in puncto quæsito C. {illeg} Eadem lege unum era {puncto} invenire possunt sed uno aliquo invento {habebimus} {illeg} C quin puncta quæ Prob {Conicam Sectionem jaxta} præcedentia determinando Problemati satisfaciunt.

Et hæc videtur methodus naturalissim{e} {illeg} solvend problem{a} {tam quod} non {factum} quod ad{illeg}dum simpl{illeg} {sit} {illeg} pars Problemat{illeg} {illeg} {{illeg}} <90v> ab ipso Cartesio proponitur) est invenire punctum aliquod (data{m} habens conditionem et secunda pars \deinde cum infinita sint ejusmodi puncta/, determinare omnia locum in quo ejusmodi puncta \ista/ omnia reperiuntur. Quid autem magis naturale quam reducere difficultates \hujus/ posteri oris partis ad difficultates prima prioris determinando locum ex paucis punctis inventis. Proinde cum veteribus constiterit rati ducendi conicam sectionem per quin data puncta, nullus dubi{illeg}|ta|verim eos hoc medio loca Solida composuisse.

< text from f 89v resumes >

<90r>

Problema

Data quavis refringente superficie CD quæ radios a{illeg} puncto A divergentes quomodocun refringat: invenire aliam superficiem EF quæ refractos omnes DF convergere faciet ad aliud punctum B.

Junge AB. Eam secent refringentes superficies in C et E. {illeg} Et posito d ad e ut sinus incidentiæ ex aere in medio ACD in medium EDC ad sinum refractionis, cape B produc AB ad G {illeg}|ut| sit BG.CE∷d−e.e, & AD ad H ut sit AH=AG, et DF ad K ut sit DK.DH∷e.d. Junge KB et centro D radio DH describe circulum occurrentem KB productæ in L. Ipsi DL parallelam age FBF et erit F punctum superficiei EF quæ radios omnes ADF convergere faciet ad punctum B. Nam |de| fluxio DF est ad fluxionem AD+BF ut e ad d: adeo CE−DF.AD+BF−AC−BE ∷e.d.

Haud secus Problema resolvitur ubi tres sunt vel plures refringentes superficies.

<91v>

De resolutione Quæstionum circa numeros.

Primo numeri quæsili redigendi sunt ad æquationem secundum conditiones quæstionis, Deinde exponendi sunt per basem et ordinatam curvæ lineæ quam æquatio illa designabit. Sit curva ista DC et numeri AB, BC, curva existente tali ut numerus BC ordinatim applicatus ad basem A numerum AB in dato angulo ABC \semper/ terminetur ad eam. {illeg} Deinde inquirenda erunt puncta curvæ quæ efficient numeros AB, BC rationales. C{illeg}|as|us autem in quibus hoc fieri deprehendo sunt sequentes.

1. Si numeri in æquatione non ultra gradum qquadraticū ascendant ita ut curva sit Conica {illeg}|{s}|actio: et detur aliquod {illeg} punctum F \F/ in \curv{illeg}|{æ}| quo/ efficitur ut numeri \AH. HF./ sint rationales {illeg} ex hoc unico exemplo regula generalis sic elicietur. In AH cape HE cujusvis rationalis longitudinis, age EF, secet hæc curvam in G et demissa GH pararallela {sic} CB numeri AK, KG erunt rationales. Si Si punctum F reperitur in linea AH, FH existente nullâ, tunc cape HN cujusvis rationalis longitudinis, Erige NE parallelam K|B|C, & cujusvis etiam rationalis longitudinis. Age HE occurrentem curvæ in G et erunt AK KG rationales.

Quoniam {illeg}|{ita}| hic unicum s{illeg}|{a}|l|{illeg}|tem exemplum requiritur, primo inquarendum est ejusmodi exemplum, dein regula generalis inde elicietur ut supra. {illeg} Exempl{illeg}|{i}| autem inveniend{i} notand{illeg} est \hec erit methodus{.}/ ni melior occurrat. Sit æquatio axx+bxy+cyy+dx+ey+y=0 ubi x et y designant numeros. Et si terminus f desit, punctt|u|m istud erit A. Si x in æquatione{illeg} axx+b|d|x+f=0 sit rationale punctam erit duo erunt in AB. Si cyy+ey+f sit rationale puncta duo erunt in recta quæ ducitur ab A parallela BC. Si bb−4ac sit quadratus numerus affirmativus tunc curva erit Hyperbola Et recta ducta a puncto A {p} vel B parallela Asymptoto {illeg} secans curvam in G exhibebit numeros AK GK rationales. Sunt alij innumeri casus quibus enumerandis non immoror

2 Si æquatio ascendit ad tres dimensiones, et tria habentur exemplo rationalia non in arithmetica progressione possunt inde innumera alia reperiri {illeg} sint enim P, Q, R puncta Curva ad ista exempla Junge PR, \RQ, PQ/ et punctum S, \T, Y/ ubi PR, \RQ, PQ/ secant curvam dab{illeg}|un|t alios duas \tres/ numeros. Dein junge QS et punctum x dabit alui quo QS secat curvam dabit aluim numerum. et Sic in infinitum.

< insertion from the bottom of the page >

^[62] Punctum B circumferentiam contingit a quo si duæ agantur rectæ una ad datum punctum A altera {illeg}|in| dato angulo c ad rectam positione dat{u}m {illeg}|&| ad dat{u}m punctū terminatam DC, quadratum prioris AB^q æquales sit rectangulo sub posteriore abscissa ac data DC×E

Vel sic. Si AB^q=FC×E punctum B contingit circ.

< text from f 91v resumes >

<92r>

Loca plana.

1. Si datur linea AB, et angulus ACB, punctū est in circulo per A et B transeunte.

2. Si dantur puncta A, B et ratio AC ad BC punctum C est in circulo Et si |C| est in circulo recta BC vergit ad |locum B.|

3 Si a dato puncto A ad rectam positione datam BD ducatur r{{illeg}|ec|}ta quævis AB et in ea {illeg}atur punctum C \{si}/ ea lege ut detur rectangulum BAC punct{u}m C est in circulo, transeunte per A

4 Si detur punctum A et rectangulum BAC vel \BA×DC/ et punctum B \ac D/ est ad circulus, etiam punctum C erit ad circulum.

5 Si detur punctum A et proportio AB ad AC |vel AD ad DC.| et punctum B est ad circulum etiam C erit ad circulum. Sin punctus B est ad rectam erit C a rectam

6 Si dentur puncta duo A, B et differentia quadratorum AC^q−BC^q punctum C est ad rectā

7 Si dentur puncta duo A, B et summa quadratorum AC^q+ BC^q et punct C erit ad Circulum

8 Si dentur puncta plura A, B, D et quadratorum ex lineis AC, DC, BC vel quadratorum quæ ad ips{illeg}|a| sunt in datis, rationibus summa vel quod subducendo aliqua ab alijs restat vel proportio aggregati unius ad agr|g|regatus alterum aliud, punctus C erit in circulo.

10 Si da\n/tur puncta B, D, E et productam BD secat quævis EC in A, et sit rectangulum EAC æquale rectangulo BAD erit punctū C ad circulum.

11. Si dantur puncta E, D et concurrant rectæ CE BD ad rectas positione datam AF ex|t| {detur} rectangul{illeg}|or|us. EAC, DAB differentia vel nulla est vel æqualis rectangulo sub AF et recta data: sit autem punctum B ad circulus erit punctus C ad circulum.

12 Si detur quadrilaterum ADE quod cujus anguli oppo in ci{illeg}|rc|ulo inscribi potest, et a puncto C ad latera ejus ducantur in datis angulis rectæ quatuor CF, CF|G|, CH CI ita ut rectangulus sub duabus CG×CI datam habeat rationem ad rectangulus sub alijs duabus CF×C{illeg}H, et in uno aliquo casu pu{illeg}|nc|tus C est in ci{illeg}|rc|umferentia circuli transeuntis \per/ ABDE, semper erit in circulo illo

A{ff}ini{a} sunt 1, 2, 6, 7, 8. Item 3, 4, 5. Item 10. 11

13 Si dentur puncta A, B. Et centro circulo|u|s centro A radio dato AD descriptus utcun secetu{illeg}r {illeg}|{an}| D et E a ducta AC ac demittatur normalis CF, c|s|it c|a|utem rectangulum DCE=2ABF{illeg} punctum C erit in circulo.

Si trapezij ABDE anguli A ac D recti sint et ad AB et AE{.} demittantur perpendicula CF CI a puncto quovis C secantia BD ac DE in {{illeg}} H et G erit rectan et fuerit rectangulum FCH=GCI erit punctum C in circulo transeunte per puncta ABDE. Et viceversa. {I}dem eveniet si ad latera singula \{illeg} a puncto C/ demittantur perpendicula

<92v>

^[63] Si {illeg}|{i}|n circulo quovis {illeg}|AB|CD inscrib{illeg}|{a}|tur trapezium A B C, D, et a circumferentiæ puncto quovis E ad latera trapezij ducantur lineæ EF, EG, EH, EI constituentes cum lateribus conterminis AB, BC parallelogrammum EFBG et cum alijs duobus lateribus conterminis AD, DC parallelogrammum EHDC. quod sub ductis ad opposita duo latera continetur rectangulum GE×EH æquale est rectangulo EF ×EC sub ductis ad reliqua duo latera contento.

|NB| Idem eveniet si {illeg} puncto E ad latera trapezij demi{illeg}|tt|antur pependicula. Ut et si du{illeg}|ct|{illeg}|{illeg}| ad duo latera \contermina/ ^[64] EH, EC ad AD, CD æquales angulos cum EHD, ECD vel EHA, ECD cum ipsis constituant conficiant et ductæ ad altera duo latera EF EG æquales angul{illeg}|os| cum {illeg}|ipsi|s.

Et hinc si ductæ quosvis angulos con{illeg}|fician|t cum lateribus trap{illeg}|ez|ij, et rectangula GEH, FEC|I| sunt in data ratione facile est cognoscere utrum punctum E {illeg}|si|t in circus ferentia circuli. Nam ad latera duo opposita AB, CD, duc EK, EL in angulis EKB ELD æqualibus illis quos ducta ad altera latera opposita {faciunt cu} in quibus ad altera duo latera duc{illeg}|t{a}| sunt lineæ æqualibus|.| {illeg} \Verbi grati{a}/ EK in angulo AKE=ang CGE et EL in angulo DELE =ang DHE. Et {illeg}|se| ratio rectanguli FEI ad rectangulum GEH componitur ex ratione FE ad KE et EI ad LEL ita ut rectangula KEL GEH æqualia sint, erit punctum E in circulo {illeg}|se|cus non erit in circulo.

^[65] Si ABE sit circulus et detur |A &| rectangulum BAC erit punctum C in recta. |Et si punctus C in recta sit converget recta CA ad datus punctus A. Et in fig 2 si datur rectang BAC erit C in circulo.|

^[66] Si punctarum A linearum AD, BD, CE poli ABC in linea recta sunt, et puncta {illeg}|in|tersectionum duo DE lineaa|s| rectas describunt tertia intersectio F lineam rectam describet. Idem eveve {sic}niet si linea DE parallela est lineæ BC. Ut et si si {sic} AB puncta |A|BC non sint in directum si modo loca punctorum D, E se secant in recta BC.

^[67] Si dati anguli DBC|A|, DCA circa polos B, C volvantur et angulis ABC|D|, ACD æquales capiantur CBF BCF sit punctum D in recta transeunte per punctum F vel etiam in conica sectione transeunte per puncta tria BCF erit punctum A in recta. Et si D in recta sit non transeunte per punctus F aut in con. sectione transeunte per duo E tribus punctis B, C, F, non autem per omnia tria, punctus A erit in conica sectione. Si per unicum tantum E tribus punctu|is| BFC transit conica sectio, {illeg} punctum A erit in curva primi gradus tertij gener{is} Si per nullum erit primi generis quarti gradus.

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^[68] Si circ{illeg} dua \{illeg}E AD/ se secuerint in A et per A agatur recta ACDB et {illeg} CD ad DB {illeg} in circulo transeunte {illeg} intersecti{illeg} {illeg} priorum.

Si recta CD {illeg} {datas} {illeg} {data \{illeg}/} {illeg} et secetur {illeg} cujus {data} est ratio {illeg}

{Si secetur punctum} {illeg} AD, BD, CD {illeg} \vel AB, BC {illeg}/ {illeg} agei B circulum {illeg}

< text from f 92v resumes >

<93r>

^[69] Datis positione lineis AE BE et punctis AB: Si {illeg} recta quævis CD secat alteras in C, D ea lege et rectangulum AC, BD æquætur dato rectangulo AE×BE, compl{e} parall{o}logrammum {illeg}AEBP et locus ad quem recta CP vergit erit punctum P.

^[70] Si AB datur positione et longitudine & AD BC longitudine sint CE DE æquales cape AF.BF∷AD.BC et {illeg} CD verget ad datum {pun} locum puncti F.

^[71] Se|i| a datis punctis A, B ductæ AC.|,| BC datam habeant summam \vel differentiam/ N: d|D|uc CD parallelam AB et in ratione ad AC qual|m| habet N ad AB et punctum D erit in recta quæ perpendicularis est ad AB. Debet vero CD ad plagam versus A duci ubi datur summæ AC+BC, ad plagam versus B ubi datur differentia. AC−BC vel BC−AC.

^[72] Si datur circulus ABD et rectangulum ACA|B| {datam} punctus C erit in circulo idem habens centrum cum circulo ABC|D|

^[73] Si per data puncta AB transit circulus secans \in E/ rectam ipsi AB perpendicularem et arcui BE æqualis sit arcus B{illeg} EF erit F in circulo cujus centrum est A et si F in {illeg} tali circulo sit, {illeg} {illeg} et bisee|c|etur BF in E erit E in recta.

De Loco rectil{í}neo.

Si locus crura duo infinita opposita habet, et non plura, aut rectus est, aut tertij, quinti, septimi vel imparis alicujus generis curva linea.

Si rectæ locum tangentis plaga determinat est rectus est locus.

Si recta nulla ad plagam infiniti infiniti cruris tendens potest locum secare rectus est locus.

Si per datum loci punctum recta transiens non potest locus alibi secare rectus est locus.

Si a loci puncto quovis ad rectas duas \positione datas/ in datis angulis demittantur aliæ duæ rectæ, et progrediendo per additionem subductionem et rationes datas, alterutra demissarum ex altera assumpta vel utra ex assumpta tertia determinari potest rectus est locus.

Si in recta quavis ad datam non infiniti cruris plagam tendente determinabile est loci punctus per simplicem Geometri{illeg} rectus est locus.

Si rectæ per punctum datus extra locum transeuntis et loci intersectio determinabilis est per simplicem Geometriam, rectus est locus.

Si rectæ cujusvis \{illeg} / \cujusvis assignatæ/ et loci intersectio determinabilis est {illeg} per simplicem Geometriam rectus est locus.

<93v>

^[74] 1 A datis punctis A, B \ductæ/ conveniant AC BC \in C/ et si dentur ipsorum A, B, {illeg} summa vel differentia (loc. \C/ solid.) proportio (loc. circ) differentia quadratorum (rect{)} summa quadratorum vel ad|l|iud quodvis compositum ex quadratis (circ) \rectangulum (lineare) Area {illeg}|AB|C (rect)/ angulus ACB (circ) differentia angulorum A|B|{illeg}\−/B|A| ({solid}) vel \summa/ 2A|B|+C (solid \Hyperb &/ /rect\{illeg} diff A|B|−C vel summa {illeg}A|2B|+B|A| \vel 2C+A/ (vel lineare \ellipsis/) \2/A=B \vel {illeg}{et}. 3A+C/ (Hyperb.) 2A=C (Lin {illeg}) 2C=A (Lin {illeg}) A =B (rect) D|B|=C (circ)

^[75] 2 Detur Ang AD positione et ang. DAC et punctus B. Et si dentur etiam ipsorum AC BC summa, differentia (Parab \erit C in/) Proportio (loc. sol.) summa quadrati{illeg}s differentia quadratorum (Parab) summa quadratorum vel aliud quodvis compositum ex quadratis (Loc solid) rectangulum (Lineare)

^[76] $\{\begin{cases} 3 Detur positione DA et punctum P. Et si detur AC (Locus \\ Conchoid) d iff AC ± AP (loc \{\begin{matrix} circ. \\ linear. \end{matrix}) proportio PA ad AC (loc. rect.) rectangulum \\ APC (circ) rectang P AC (lin) rectang PCA (lin.) {AP}^{q} \pm {PC}^{q} . \\ {AP}^{q} \pm {AC}^{q} . {PC}^{q} \pm {AC}^{q} (lin.) \\ 4 Detur circ AD, punctum P et si detur etiam APC \\ vel \frac{AP}{AC} (erit C in loc. Pla n .) \end{cases}$

<94v>

Quæstionum solutio Geometrica.

1 Angulum datum DAB recta datæ longitudinis CB subtendere quæ ad datum punctum P converget Cape PQ=CD et Q erit in circulo cujus centrum P radius PQ. Age QR{∥}∥AD et {Q}PRD∥AD|B| et erit PD.DC=AD−QR∷ ∷PR.QR. Ergo Q{illeg} in conica sectione est. Pone QR infinitū et erit $PR = \frac{PD \times QR}{AD - QR}$ AD−P|Q|R.PQ|QR|∷PD.QPQ|R|. seu PR =−PD. Pone PR infinitus et erit PD+PR.PR∷AD.QR ergo AD=QR. et AB Asymptotos. alteras Cap{er}|{e}| ergo PS=PD et per S parallelam AD age alteram Asymtoton & {his} Asymptotis per punctus P describe Hyperb{o}lam secantem circulum prædictus in Q.

2 Inter circulum PDF et rectam DF ponere rectam datæ longitudinis BC quæ ad punctum P in circumferentia circuli datum converget. Biseca DF in E. Age PD,PE,PF. Cape PQ=BC. Age QR ∥DC & occurrentem PE in R. Et erit PR.{illeg}PQ∷PE.PC. PR.PE∷PQ(BC).|PC∷RQ.EC.| {illeg}Et PQ.|(|BC).FC∷DC.PC Et {PR×PQ} Ergo PQ×PC=FC×DC =EC^q−EF^q PR.PQ Et $BC . \frac{PE \times RQ}{PR} - EF (FC) ∷$ $\frac{PE \times RQ}{PR} + EF (DC) . \frac{PE \times BC}{PR} (PC) .$ Seu BQ|C|,PR.PE,PR|RQ|−PR,EF∷ $\frac{PE \times RQ}{} + PR, EF . PE, BC . {BC}^{q}, PE, PR = {PE}^{q}, {RQ}^{q} - {PR}^{q}, {EF}^{q}$ Si PR infinitum{Q} Ergo Q locatur in Conica sectione cujus diameter PR, ordinata RP|{PR}|. Sit RQ=0, erit PR=0 et $\frac{- {BC}^{q} PE}{{EF}^{q}}$ In EP producta cape ergo $PS . \frac{1}{2} PE ∷ {BC}^{q} . {EF}^{q}$ et erit S centrum et P vertex figuræ. Pone PR infinitus et erit PE^q,RQ^q=PR^q,EF^q, seu PE{:|,|}RQ=±EF,PR. Quare per S ipsis PD,PS age parallelas et hæ erunt Asymptoti figuræ His igitur Asymptotis per punctum P describe Hyperbolam, ut et cent{illeg}|ro| P radio PQ circulum & per eorum intersectionem Q age rectam PC.

Corol. si ang. PEC {illeg}|re|ctus est Problema planus erit. Nam circuli centrum incidit in {illeg}|ax|em figuræ.

3 A dato puncto P rectam PC ducere cujus pars BC inter circulum et recta productam diametrum æquab DF æquabitur \semi/ diametro EF. Age EF ac demitte ⊥ PG,BH. Est EH. HB∷GC(GE+2EH).GP. Ergo punctum B in Hyperbola est. Pone {illeg} EH=0 et erit HB×GE=0 adeo HB=0. Quare Hyperbola transit per punctum E. Pone EH infinitus et erit EH.HB∷2EH.GP. Ergo $\frac{1}{2} GP = HB$ . Pone HB infinitus ergo et erit EH.GE∷HB.\−/GP−2HB∷HB.−2HB Ergo $\frac{1}{2} GE = - EH$ . Quare biseca PE in S et per S age Asymtotos parallelas EH et HB et per punctum E vel P describe Hyperbolam secantem circul{u}m in B|{D}|. Et per B age PC.

Corol. Hinc si ang PEG semirectus erit PE axis Hyper{illeg}|bola| adeo Problema in eo casu planum.

<95v>

Quæstionum solutio Geometrica

1 Datis trianguli cujuvis angulo latere et summa vel differentia {re}liquorum laterum datur triangulum{,} Detur latus AB reliquorum laterum AC+BC summa {illeg}vel differentia AD. Si detur angulus datus dato lateri conterminus est, sit iste A. Et dabitur triangulum DAB. Angulorum vero |C|DB ADBD differentia in priori casu summa in posteriori est ang ABC.

Si{t} angulus datus dato lateri oponitur, sit iste C {illeg} {illeg}datum{illeg} ang DCB dabuntur ang CDB CBD est dabitur tr dabuntur anguli dabitur triangulum CDB specie. In triangulo autem ADB datis lateribus AB AD et ang D datur ang ABD. Unde datur Ang ABC ut ante.

2 Data differentia segmentorum basis uno angulo summa vel differentia laterum et uno angulorum datur triangulum. Nam si datur summa laterum dabitur ratio differentiæ laterum ad basin si differentia debitur ratio summæ laterum ad basin. In utra casu \Ex ratione utravis & uno angulorum {illeg}/ per problema superius datur triangulus specie. Deinde ex data differentia segmentorum basis et ex data ratione differentiæ segmentorum \basis/ ad latera dantur latera.

3 Data summa {illeg}|vel| differentia laterum uno angulorum et ratione basis ad perpendiculum: ex duobus posterioribus dabitur triangulus specie ex priori dabitur etiam magnitudine.

4 Data summa vel differentia laterum uno angulorum et area: \ex area/ rectangulum laterum datum angulum comprehendent{ur}. Si istorum summa \vel diff./ datur ad|a| quadrat{us}|o| summæ aufer duplum rectangulus vel ad quadratum differentiæ \{laterus}/ adde {qu}duplus rectangululum {sic} et habebitur priori casu quadratus {illeg},|differenti{æ}| posteriori quamdratus differentiæ s summæ laterum: Ex dat{illeg}|is| autem summa ac differentia laterum dantur latera. Si angulus datus basi conterminus est problema erit solidum.

5 Datis angulo \{illeg}/ /A\ latera \AC vel BC/ et differentia segmentorum basi|s| AD dabitur triangulum ADC |ut| et angulus B \quod est/ complementus est anguli ADC.

Si detur angulus verticalis C laterum alterutra AC vel BC et segmento basis AC: quiescant BC,AC et punctum D in {circulo} erit radio CB centro C descripto. [Et et AB. $\frac{{AC}^{q} - {BC}^{q}}{AB} = AD$ ergo D determinatur per Geometriam planam simplicicem. Sed et ubi punctum D incidit in B positio rectæ AD|B| determinatur. Ergo locus puncti D conica est sectio] {illeg}|Ut| et in Chonchoide {illeg}|Po|lo {B} {illeg}|asy|mptoto AC intervallo AD descrip{illeg}|ta|.

Vel sic. Dato angulo AC{D|B|} datur summa ang: A+B. seu A+CDB Aufer hoc de duobus rectis ac dabitur differentia ang ADC−A. Unde datur triang. per sequ. Prob.

7 Datis \basi &/ differentia angulorum ad basin una cum basi et \una cum/ latere al{illeg}|tera|tro vel summa differentia ratione \laterum/ aut summa vel differentia {□^lo} laterum aut area, perpendiculo vel segm{en}{illeg}|ta|to basis aut summa vel differentia lateris alterutrius et perpendicula vel segment{i} basis. &c Datur triang. Nam data basi et angulorum ad {illeg}|Basem| differentia, {illeg} C erit ad Hyperbolam; et ex dato tertio, punctum C erit ad recta aut circulus aut conican aliquā sectionem.

8 Ubi datar angulus verticalis et differentia segmentorum basis et tertium aliquod, habebitur aliud triangulum ADC ubi datur {illeg} differentia angulorum ad ba{illeg}|se|m, et tertium aliquod.

9 Dat{illeg} basi ratione laterum et tertio quovis ut ⊥ segment{illeg} basis{,} angulo aliquo{,} ratione ⊥ ad lat{eri} {peh} {illeg} ad segmentum d|D|{tis} {illeg}. Nam {illeg} data {illeg} lat. Dat{ur} circulos {illeg} {illeg}

<96r>

Quæstionum solutio Geometrica.
Prob 1

Circulum \ABE/ per data duo puncta \A, B/ describere quæ rectam FG positione datam continget.

Junge A{,}B. D biseca eam in D. Erige normalem DF occurrentem FG in F. {illeg} Produc AB donec occurrat FG i{n}

|Sit| E punctum contactus. Produc AB donec occurrat FG in G et erit EG medium proprortionale inter datas AG,BG.

Prob. 2

Circulum \ABE/ per datum punctum A describere qui recta duas FE, FH continget. {illeg}

Center F, radi{s} FA describe Recta FD biseca{illeg} angulum HFE. Ad FD demitte normalem AD et produc donec occurrat FE in G. Cape D ad B ut sit DB=AD et per puncta A, D describe circulum ut prius qui contingat rectam FE.

Prob. 3

Circulum \ABE/ per data duo puncta \A, B/ describere qui alium circulum positione datum \EKL/ continget.

\Puta factum/Sit punctum contactus E. Linea contingens EM. et erit AM×BM=EM^q=MK×ML. Divide ergo BK in M ut sit AM.MK∷ML.MB. Cape ME medium proportionale inter AM et BM et centro M radio ME describe circulum. Hic secabit circulum EKL in puncto contactus E. Recta autem BK sic secatur in M. Est AM= AB+MB,|.| MK=BK−BMB.ML=BL−{B}MB. ergo AB+MB.BK−MB∷BL−MB. MB. Et componendo AB+BK(AK).BK−MB∷BL.MB. et inverse AK. BL∷BK−MB.MB. et rersus componendo AK+BL.BL∷BK.MB.

Si AB non secat EKL {illeg} pro MK×ML scribe MG^q+FG^q−KF erit AM×BM{−} AG−MG×BG−{illeg}G(AM×BM)=MG^q+FG^q−KF^q {illeg} est AG×BG−FG^q+KF^q=AG+BG×MG. Seu 2AG.BL∷BK:MB. {illeg} \{Unde}/ cum sit BL.BO∷M{illeg}BN.BK, erit 2AG.BN∷BN.BM. Quæ solutio versalis est.

Prob 4

Circulum \BDE/ per datum punctum \B/ describere qui datum circulus {illeg} & rectam lineam |AD| postione datam continget.

AB est 2CD−AH. NF est \2/CQ−NS posito CQ=CF=CS. Ergo HF est 2CD−{illeg} AB−NF est 2CD−AH−2CQ+NS seu. AB−NF+DQ est Adde 2DQ, erit AB+DQ−HF=NS−AH= $\frac{{AD}^{q}}{AB}$ $= \frac{{NQ}^{q}}{NF} - \frac{{AD}^{q}}{AB}$ . Dividendo|a| est ita \data/ AH in D ita ut $\frac{{DH}^{q}}{NF} - \frac{{AD}^{q}}{AB}$ dato{illeg} æquale sit, nempo dato AB(Hb)+DQ−HF, seu bk. {AD^q}DH^q×AB AD =AB, HK, bL. DH=A{D}{illeg} AH−AD.DH^q=AH^q−2DAH+AD^q.AD×{illeg}K^q{−2A}{illeg}{AH} −2DAH+AD^q−{illeg} $\frac{HK \times {AD}^{q}}{AB} = HK, bk$ . [A{B}{illeg}−HK(bK).AH∷A{illeg}|{B}|.AV.AH^q−2DA{H} $\frac{{AD}^{q}, AH}{AV} = HK \times BL$ ] $AH - 2 DA + \frac{AB - HK}{BAH} {AD}^{q} = \frac{HK, bk}{AH}$ . \Fact/ AB−HK.{illeg}AH∷BA.AV. & AH. $HK ∷ bk . Hp . AH - 2 DA + \frac{{AD}^{q}}{AV} = Hp$ . AD^q−DA{illeg}V+D{illeg} AV^q=AV×PV{illeg}\{P}/AD{−}AV {illeg}PV={DV.}AD {illeg}AV{illeg}PV{illeg}−AD+AV{illeg}=AD{illeg}|{×}|PV =DV. Age ergo BK occurre{nt}em AH in VHK ad {H}A versus A si {HK} {illeg} versus {b} aliter {illeg} A {illeg} ad bk {illeg} {illeg}

<96v>

Nota etiam quod Problematis quatuor sunt sunt {sic} casus quorum duo sunt impossibiles ubi circulus datus et recta data se mutuò secant. Casus impossibiles sunt ubi punctum v cadit inter A et P.

Vel \in {a}ng GED/ agatur GD datam per A transiens posito AE quadrato, quære summam radicum Fd, FD Ad AD erige ⊥ DK age erit CK {illeg} F{illeg} Ergo AK summa illa, et CD^q+CK^q(DK^q)+GD^q=GK^q Aufer BG^q seu CK^q et restabit CD^q+GD^q=BK^q Datur ergo {illeg} summa AK. Quare cum ang ADK rectus; super diametro AK describe circulus secantem FE in D, d

Super datis rectis tribus AB, CD, EF, tria constituere triangula quorum vertices erunt ad \idem/ punctum G et anguli ad vertices AGB, CGD, EGF æquales.Super Junge AD, BC. {illeg}|Bi|seca eas in r, s. Produ AB {illeg} secent AB, CD se mutuo in t. Age tG∥sr. Idem fac in lineis CD FE.

Vel sic. super lineis AB, CD, EF describe similia segmenta quorumvis circulorum satis magnorum \ita ut se mutuo secent compl{illeg} segm. ad circulos {illeg}/. Per intersectionem circulorum AB, CD age rectam, ut et aliam rectam per intersectionem circulorum CD, FE: nam hæ rectæ se secabunt in puncto G

The Problem {illeg} \in Schooten/ de tribus baculis may be solved more easily by supposing y^e Ellipsis to be a circle first & then reducing it to y^e desired circle.

In triangulo DEF \dato ABC/ {illeg} aliud triangulum DEF {illeg} dato def simile inscribere cujus latus EF transibit per datum punctum G. Nemper vi|e|rticis trianguli DEF locus {illeg} est linea recta.

In data conica sectione ABCDE, trapezium ACDB inscribere cujus anguli op duo oppositi CAD CBD dantur et data puncta A et B consistunt. Vizt si locus puncti D est conica sectio locus punctis c erit linea recta.

<97r>

^[77] Ex observationibus proprijs Cometæ anni 168\0/.

A stella major et orientalior duarum in orientali \australi/ pede Persei, B stella minor earundem. AB stellarum distantia 1^gr.46′6″. α, β, γ &c loca Comet{æ}

$\begin{array}{l} Feb 25 hora 8 \frac{1}{2} vesp. Dist Aα 2^{gr} .17′.42″. Ang BAα. \\ F \end{array}$

$\begin{array}{l} Dist. C ometæ stella A. & Angulus \\ Die Feb 25 hora 8 \frac{1}{2} vesp & Aα = 104, \frac{7}{12} part = 2^{gr} . 17'. 42 ″ & BAα 9^{gr} . 17' \\ Feb 25 hora 9 \frac{1}{2} & Aβ = 103 \frac{3}{4} = 2^{} . 16'. 36 ″ & BAβ 9^{} . 27' \\ Feb 27 hora 8 \frac{1}{4} & Aγ = 77 - & BAγ \\ Mart 1 hora 11 & Aδ = 52 = 1^{} . 8'. 30 ″ & BAδ 31^{} . 40' \\ Mart 2 hora 8 & Aε = 43 = 0^{} . 56'. 37 ″ & BAε 43^{} . 8' \\ Mart 5 hora 11 \frac{1}{2} & Aζ = & BAζ \\ Mart 7 hora 9 \frac{1}{2} & Aη = & BAη \\ Mart 9 h o ra 8 \frac{1}{2} & Aθ = 78 \frac{7}{12} = 1^{} . 43'. 30 ″ & BAθ 143^{} . 00' \\ Mart 9 hora 12 & Aι = 80 \frac{1}{3} = 1^{} . 45'. 46 ″ & BAι \end{array}$

<98r>

Observationes Comet{illeg}|æ| habitæ ab Academia Physicomathematica Romana anno 1680 et 1681, a Ponthæo æditæ.

$\begin{matrix} \begin{matrix} Stylo vel. & Stylo novo & Hora. mat & Cometæ Long. & Lat. Aust & L ong. candæ \\ Novemb 17 & Novem. 27 & 6 & 8.30' & 0.40 & 15^{gr} et ultra \\ 28 & 6 \frac{1}{2} & 13 . 0 & 1.20 \\ Decem. 1 & 7 \frac{1}{4} & 27.50 & 1.16 \\ 4 & 7 & 12 . 0 & 0.40 \\ 5 & 7 \frac{1}{4} & 15.50 & 0.40 \\ 6 & 7 \frac{1}{2} & 20 . 0 & 0.35 \\ 7 & 7 \frac{1}{2} & 24 . 0 & 0.30 \end{matrix} & \begin{array}{l} Caput Nov 27 \\ aquabat prim \\ notæ stellas mag \\ nitudine at lu \\ mine ab eis mul \\ tum deficiebat \end{array} \end{matrix}$

Observationes ejusdem cometæ habitæ a R. P Ango in Fleche

Novem 28 \hor. 5 matitin./ in medio erat inter stellas duas exiguas quarum una est minima {illeg} trium quæ sunt in manu australi Virginis altera est in extremitate alce: Adeo longitudo Cometæ jam erat ≏ 13, Latitudo australis 50′.

Decem 1. hora 5 matutina. erat in Libra 27. 45′

Observationes Venetijs habitæ a M. Montenaro.

Novem 30 hora {illeg} post occasum solis duodecima \Cometa/ erat in ≏ 23^gr cum lat Aust. 1^gr 30′

Decem 1, Erat in hor 5 matutina erat in 27. 51 ≏.

Decem 2 erat in {illeg}|♏| 2. 33

Decem 4 erat in ♎ 12. 52

Credidit M. Montenari latitudinem {illeg} ad us finem harum observationem augæri.

Observationes Hevelij destituti instrumentis

Anno 1680 Decem. 2 Cometa erat in ≏ 25 cum lat. Austr. 5^gr

Decem 3 {illeg}|{Arc}|te ortum ☉^is hora sexta erat in ♏ 4 cum lat. austr 4^gr.

Decem 4 mane hor 6 20′ erat in ♏ 10 cum lat austr. 3^gr.

<98v>

Observationes Cometæ Mense Novembri anni 1680

Canterburiæ per Artificem quendam nomine Hill, instrumento cujus radius erat 4 pedum Die Veneris Novemb 11 tempore matutino, |C|ometa inventus est in ♍ 12^gr cum lat. b|B|oreali 2^gr. Locus ☉is ♏ 29^gr. 53′

Romæ per Marcum Antonium Cellium observationes hæ factæ sunt.

$\begin{matrix} Stylo veter. & Stylo novo & Cometæ Longit & Lat austral & Locus ☉^{is} \\ Novemb 17 & Nov 27 mane & ≏ 8^{gr} 30' & 0^{gr} 30' & ♐ {5 .}^{gr} 55' \\ 18 & 28 mane & {13 .}^{} 30 & 1^{} . 00, circiter & {6 .}^{} 56 \\ 21 & Decem. 1 mane & {28 .}^{} 0 & 1^{} + 00, circiter & {9 .}^{} 58 \\ 24 & 4 mane & ♏ {11 .}^{} 40 & 1^{} 00, circiter & {13 .}^{} 1 \\ 25 & 5 mane & {15 .}^{} 47 & 1^{} 00 circiter & {14 .}^{} 2 \\ 26 & 6 mane & {19 .}^{} 45 & 1^{} 00 circiter & {15 .}^{} 3 \\ 27 & 7 mane & {23 .}^{} 35 & 1^{} 00 circiter & {16 .}^{} 4 \end{matrix}$

Romeæ per Galletium hæ

$\begin{matrix} Stylo vet & St. Novo & Comet. Long & Lat. austral. \\ Novem 17 & Novem 27 Hora 18 & 8 . 0 & 0.00 \\ 18 & 28 17 \frac{1}{2} & 13 . 0 & 1.00 \\ 21 & Dec. 1 17 \frac{1}{2} & 0 . 0 & 4.00 \\ 23 & 3 17 & 9 . 0 & 3.00 \\ 25 & 5 18 & 16.15 & 2.00 \\ 26 & 6 18 & 19.30 & 1.00 \end{matrix}$

Cantabrigiæ per juvenem quendam Cometa observatus est Novemb 19 juxta spicam Virginis, quasi duobus gradibus |supra| stellam illam, ad sive ad boream, circa horam quartam vel quintam matutinam. Et cauda extendebatur ad us stellam illam primæ magnitudinis quæ cauda Leonis dicitur.

Observationes Parisijs habitæ Cometæ subsequentis 1680 & 1681.

^[78] $\begin{array}{ccll} Longitudo & Lat. bor & A s cbn. f. & Decl & Long. caud. & Decl. caudæ ab op. O \\ St. vet. Dec. 19 hor 6.30 p.m. & {27 .}^{gr} 4' & 18.53 & 62^{gr} & 2^{gr} . 48'. \\ 24 5.30 & {18 .}^{} 36 \frac{1}{2} & 25.26 & 3^{gr} . 43' \\ 28 6.00 & {8 .}^{} 12 \frac{1}{2} & 28 . 3 & 62^{gr} \\ 29 6 . 6 & {12 .}^{} 55 & 28.16 \\ Jan 4 . 6 . 6 & {5 .}^{} 53 & 26.38 \\ 6 . 6.30 & {11 .}^{} 33 & 25.42 & 5^{gr} . 13' \\ 8 . 6.50 & {16 .}^{} 36 & 24.44 \frac{1}{2} & 4.40 & 29 . 7 \\ Observatio crassa 13 6.20 & {26 .}^{} 7 \frac{1}{2} & 22.20 . \\ 23 6.40 & 28.27 & 31.35 \\ 24 6.20 & 29.30 \frac{1}{2} & 31.36 \\ 23 6.40 & 30.30 & 31.37 \end{array}$

Ejusdem posterioris Cometæ Observationes Grenovici habitæ

$\begin{matrix} \begin{matrix} \begin{array}{l} Hæ æquan \\ tur Perism \\ addendo vel \\ auferado \end{array} \end{matrix} \\ \begin{matrix} Loca^{is} & Tempus verus & Ascentio recta & Declinatio & Longitudo & Latitudo & Longit & in \\ borealis & borealis & caudæ & long. lat \\ 1.53 & St. vet. Decem 12 & 4^{h} 46' p. m. & ———— & ———— & 6^{gr} . 33' & 8^{gr} . 26' & 35 + \\ 11 . 8 & 21 & 6^{} 31 & 302.20 \frac{1}{2} & 2 5 \frac{1}{3} & 5^{} . 08 \frac{1}{5} & 21^{} . 42 \frac{1}{6} & 70 \\ 14.11 \frac{1}{3} & 24 & 6^{} 24 & 313 33 & 9 0 & 58^{} . 52 & 25^{} . 26 & 65 & - 3 . + 2 \\ 16.10 \frac{2}{3} & 26 & 5^{} 16 & 321 15 & 13 19 \frac{2}{3} & 28^{} . 28 \frac{1}{2} & 27^{} . 5 \frac{1}{2} & 60 \\ 19.21 & 29 & 8^{} 0 & 333 27 & 19 21 & 13^{} . 12 & 28^{} . 10 & 50 & + 6 . + 8 \\ 20.22 \frac{1}{2} & 30 & 8^{} 4 & 337 14 & 21 00 & 17^{} . 39 & 28^{} . 12 & 25 \\ 26.23 \frac{1}{3} & Jan 5 & 5^{} 53 & 356 45 \frac{1}{2} & 27 25 \frac{1}{2} & 8^{} . 49 \frac{1}{6} & 26^{} . 15 \frac{1}{2} & 15 & + 1 \frac{1}{2} - 4 \\ 0.30 & 9 & 6^{} 50 & 6 57 & 29 31 & 18^{} . 44 & 24^{} . 12 \frac{1}{10} & + 9 . 0 \\ 1.28 \frac{1}{2} & 10 & 5^{} 56 & 9 2 & 29 51 & 20^{} . 41 \frac{1}{2} & 23^{} . 44 \frac{1}{2} & 10 & + 18 . \\ 4.34 & 13 & 6^{} 55 & 14 52 & 30 37 & 25^{} . 59 \frac{1}{2} & 22^{} . 17 \frac{1}{2} & 5 & + 12 . + 3 \\ 16.45 \frac{2}{3} & 25 & 7^{} 44 & 30 35 & 31 37 & 9^{} . 36 & 17^{} . 57 \\ 21.50 & 30 & 8^{} 7 & 35 2 & 31 41 & 13^{} . 20 & 16^{} . 41 \\ 24.43 \frac{1}{5} & Feb 2 & 6^{} 20 & 37 18 & 31 41 & 15^{} . 14 & 16^{} . 02 \\ 27.50 \frac{1}{2} & 5 & 7^{} 8 & 39 26 & 31 41 & 17^{} . 00 \frac{1}{2} & 15^{} . 27 \end{matrix} \end{matrix}$

<99r>

Observationes de Cauda Cometæ prioris

Novemb 19 Cometa juxta spicam virginis existens caudam projiciebat ad us caudam Leonis, spectante juvenes quodam.

Postea caudam versu{illeg} per meride|i|em versus occidenti|e|m projici {illeg} longam satis & ad horizontem obliquam capite {illeg} vel sub horizonte vel pone ædificia delitescente vidit Humf. Bab. S. T. D.

De cauda Cometæ posterioris

Decemb 8 stylo veteris Hallius noster tempore matutino Parisias versus iter faciens prope Bolonian ante ortum solis Caudam vidit Cometæ quasi perpendiculariter ex horizonte surgentem, ut ipse retulit in epistola quadam cit{illeg}|an|te Flamstedio. Unde Cometa inquit Flamstedius tunc borealem habeb{a}t latitudinem & cum solenondum conjunctus fuerat. |Apparebat autem cauda lat{illeg}|{æ}| divergens et {illeg} ex corpore {illeg} eg{illeg}|re|ssa aer prius {illeg} quam {illeg} {illeg} \{illeg}/ {illeg} {\{illeg}/} {illeg}|

♀ Decemb 10. duabus horis post occasum Solis, {illeg}bat cauda per medium distantiæ inter caudam serpentis Ophi{illeg}{illeg}|cha| et stellam (Bayero δ) in ala austrina Aquilæ. Desinebat vero ad stellas tres exiguas (Bayero Awb) in dorso Aquilæ juxta caudam, eductione caudæ Aquilæ ejusdem, id est in linea jungente stellam|s| lucida{illeg} secundæ magnitudinis in eductione colli Aquilæ, et estellam tertiæ quæ penultima est in cauda ejus, ac stellæ illi penultim{æ} [duplo quidem] propior existebat qu{à}m alteri in eductione colli. {illeg}|F|lamstedius in Epistolis ad nos datis. Desinebat igitur cauda in ♑ $19 \frac{1}{2}$ cum lat. bor. $34 \frac{1}{4}$ circiter

♄ Decemb 11 {Cau} post occasum Solis cauda instar jubaris apparuit ab horizonte erecti et lunâ latioris. Post crepusculi cessationem ex tendebat ad us stellas \duas/ quartæ m{æ}gnitudinis (Bayero α, β,) in capite \seu glyphidæ/ Sagittæ. (Flam{st}. ib.) adeo desinebat in ♑ 26^gr 43′ cum lat bor. 38^gr 34′.

☉ Decemb. 12. Quamprimum non obscura facta est, cauda transibat per medium sagittæ, ne ultra medium longè extendebat. (Flam{st} ib) L{in}quebat igitur stellas 5^tæ et 6^tæ magnitudinis, δ et ζ in tribulo sagittæ, quasi 40′ ad occidentem, et ultra per 3^grad circiter vel fort{æ} 4 extendens desinebat in {♑} ♒ 4 cum lat bor $42 \frac{1}{2}$ circiter vel 42 43 /34\ 43. Desinebat uti e regione superior{is} duarum informium 4^tæ magnitudinis quæ supra sagittam sunt {illeg} non et ultra extendebat{illeg}. Nam cauda ensiformis nobis visa {illeg} sagittam \paul{o}/ longius superare quam Flamstedio, in viam lacteam {illeg} nihil extendens & termino acuto paulatim languescens. Ca|e|ter {illeg} in A{s}trola{b}io Flamstedij, cauda hac nocte {illeg} desinit accurat ad stellas duas exiguas prædictas in tribulo sagittæ.

☿ Decemb 15 hor $5 \frac{3}{4}$ lucida Aquilæ erat in medio caudæ fere {Pergebat vero cauda} Ancon item austrinus Aquilæ erat {illeg} {illeg} \{illeg}/la parte caudæ prope terminum ejus \{medio} caudæ fere prope terminū ejus ad latus australe vergens./ (Ipse \ego & {Bainbro} et ellis/ part{im} ex observa{tione} partim ex circu{s} {stansijs}) Erat autem cauda 50 grad. longa ({illeg} {steed} epist. 1) {nec tutren} \tenuem/ extremitatem ejus propter Lunæ \novel/ splendorem oliquam apparuise probabila est.

< insertion from the left margin of f 99r >

Decemb. 16 hor 5 P.M. Cometa existente in ♑ 17 cum lat. bor. 15^gr circiter(|]| cauda lucidam Aquilæ (quæ nocte superiori erat in medio ejus) latere suo boreali \{illeg}/ tangebat, \aut quasi;/ ut et lucidam in {ancone} austrino cygni tangebat eodem latere |{illeg} aut quasi| Tota Caudæ longitudo erat 60 grad: feré, latitudo 2 gradus. (Observator quidam Scotus.) Unde Cauda terminabatur in long. ♓ 10 vel 12 circiter 9 lat. bor. 53.

Decem. 19 Hor $5 \frac{1}{2}$ P.M. Transibat cauda per Delphini caput dein latere suo boreali stellam penultimam in austrina ala{illeg} Cygni stringebat, tendans inde versus lucidam in Cassiopeiæ cathedra et quasi 60 gradus longa existens ({illeg} observator Scotus) vel potius 63 aut 64 grad ut ex alijs colligo, si non et paullo ultra. Desinebat igitur in ♈ 6 cum lat. bor. 52 |vel {illeg} $81 \frac{1}{2}$ |.

Decem 17 cauda inferiùs duos gradus lata {,} superius non-nihil latior, ad caput Cepher extendebat. Decem 22 cauda $67 \frac{1}{4}$ grad longa ad Cassiopeiam us extendit: ({minor tamem} \& {minor}/ {quam intra} ob D^{illeg} splendorem apparuit. Decem 23 caud{a} tenuis et {illeg} per Cassiosu {illeg} extendit, 72^gr Conica{illeg} existens circiter. Decem 28 {illeg} orta {illeg} fortior et clarior apparuit {illeg} sed 56^gr. {illeg} inter Ala{illeg} {et lum{illeg}} ge{illeg} <99v> in femure Andromedæ ad us Persei caput extendit? (Observator quidam Hamburguesis, qui præ cæteris caudam longam \ad ultimam {illeg}/ descripsisse videtur.

< text from f 99r resumes >

♄ Decem 18 Cauda linquebat stellas Delphini ad dextra{m}. P{enulti}{ma in Ala} austrina Cygni. (qu{illeg}|æ| tertiæ magnitudinis est et in Tabulis Bayeri {illeg} {ζ} diatur) llunat per caudam quarta parte latitudinis cauda{illeg} a latere australi ejus distans. erminus ejus {illeg} habebut multitudinem seu distantiam ab horizonte cum {stellas} quibus {illeg} extrema cauda cygni, Bayero dictis ♊. De{illeg} {igitur} {illeg} ♓ {illeg} lat bor {illeg} 52^gr. 20′.

♂ Dec 21 In cauda stella nulla apparuit sed {cauda} incuvata {illeg} versus {illeg} ad {illeg} onminò {illeg} in loco qui {illeg} <99v> seu pectore Cassiopeiæ et alia tertiæ magnitudinis stella in summa fere cathedra prope brachium dextrum (Bayero β dicta) triangulum æquilaterum constituit, tantum ab utra distans quantum {utra} ab invicem, (Flamsted. epist. {illeg}|{s}|{ubs.}) adeo in long. ♓ 23^gr.54′ lat 47^gr 24′ \desinebat. Cauda jam 70 C||adas \fere/ longa duas Cata tendebat versus intervallum inter {schedir} et lucidam cathedræ({illeg} Epist. 2)/. In Astrolabio vero Flamstedij, Ar{is} caudæ productus secabat ab intervallo inter caput & pectus Cassiopeiæ tertiam ejus partem versus pectus, desinebat autem e regione schedir. Transibat axis ille per caput Delphini, dimidio gradu a stellis duab{us} orientalibus in capite equiculum versus distans. Dein a distantia ultimarum duarum in ala austrina cygni auferebat quin nonas partes distantiæ illius versus stellam ultimam in ala. Postea a spatio inter terminum catenæ Andromadæ & stellam proximam in capite Cephei auferebat tertiam partem distantiæ \illius/ versus terminum caten{æ}

♀ Decem 24 Cauda transibat per medium intervallum stellarum duarum borealium in manu s{illeg}|u|periori Andromedæ et vix ultra Schedir extendebat (Flamst. Epist. post.) Desinebat igitur in long. ♉ 4 vel 5^gr, lat. $43 \frac{2}{3}$ ^gr. In Astrolabio Flamstedij cauda desinebat e regione pectoris Andromadæ. Transibat \autem (sed on rect{æ})/ per medium punctum inter genu dextrum seu australe Pegasi et stellam illam informem ad pedem dextrum quartæ magnitudinis cujus Long ♒ 29 55 lat bor 36. 11. Dein per pr{illeg}|{æ}|dictas st{illeg}|ell|as duas in manu superiori Andromedæ. {Dein per} Juxta Astrolabium erat stellæ duæ {γ}, δ invictu equiculi, et Cometa trian{illeg}|gul|um rectangulum constituebant Angulus rectus erat ad stellam occidentaliorem γ. Cometa boream versus distabat ab hac stella tertia parte distantiæ stellarum.

☉ Dec 26 \{illeg}/ Genu {illeg} sinistrum Pegasi (quæ stella tertiæ magnitudinis est et {sic} Bayero dicitur {η} erat inmedio caudæ \Flamst. Epist. {ult}.)/. Sed hac nocte et præc{æ}denti|{e}|bus caudæ terminus ob Lunæ splendorem haud satis definiri potuit (Flamst. epist. 2.) Unde die 24 gradus unus forte et alter ad caudæ longitudinem addi debet. Cauda vero hactenus semper curva apparuit, sed non valde curva. Convera sui parte austr{illeg}|{um}| respiciebat: qua etiam parte lucidior et {illeg} distinctiùs terminata apparuit quam altera.

♂ Decem 28 Cauda 56^grad longa distantiam inter Alamac et lucidam in femure Cassiopeiæ bisecans ultra pergebat ad us Persei Caput (Observat Hamburgens.)

☿ Decem 29 Cauda tangebat Scheat \sitam/ ad sinistam & intervallum stellar{is} in pede boreali Andromadæ \accur{a}te complebat/ (Flamsteed {illeg} \epist ult/ stella {φ} in femure boreali Andromade erat in medio caudæ, et \Decem 30, hora $8 \frac{1}{2}$ / Situs erat humerus Pegasi seu scheat in latere australi caudæ ita ut per caudam l{a}ceret, a termino caudæ quinta circiter vel sexta \quarta/ \vel quinta sexta/ parte latitudinus caudæ distans. Implebat autem cauda quasi \ $\frac{1}{3}$ vel/ $\frac{2}{5}$ intervalli inter Scheat & genu sinistrum Pegasi (Bayero {illeg}|{η}|) Stella φ in femure boreali Andromedæ erat in medio caudæ. B|b|isecabat axis caudæ intervallum stellarum \{illeg}/ in pede Borel|a|li Andromadæ, {illeg}|&| cauda intervallum illud plusquam implebat. Desinebat vero in medio loco inter stellam τ quintæ magnitudinis in capite Persei & extemam |in| borealis pede Andromadæ, sive inter stellam γ tertiæ magnitudinis in humero {illeg} boreali Persei & punctum \qu{a}/ distantia duarum in pede boreali Andromadæ bisecantur. Un(Ego.) Unde caudæ longitudo tota erat $53 \frac{1}{4}$ ^gr Deflectio caudæ ab oppositione ☉^is, seu angulus quem linea jungens caput et extremitatem caudæ effecit cum {Cabe} linea jungente solem & cometam, 5^gr. Latitudo caudæ juxta duas {illeg}|λ, μ| in {illeg}|pectore| Pegasi (hoc est 5^gr a capite Cometæ) erat {dimidium} distanti{illeg}|{a}| humeri illius duarum illarum stellarum {illeg} una cum triente distantiæ (nempe 1^gr {30′} circiter). Ejusdem juxta humerum Pegasi (seu $10 \frac{1}{2}$ gr a capite) latitudo erat dimidium distantiæ humen illius et \Pegasi/ genu {illeg} sinistri orientalis, Pegasi adeo 2^gr 30′ circitem. Ejusdem inter {illeg}put Andromadæ et annulum qui est in termino catenæ, (hoc est $21 \frac{1}{4}$ ^gr a capite) latitudo caudæ erat quinta par{illeg} distantiæ stellarum illarum \feré/ adeo{illeg} \{illeg}/ {illeg} caudæ latitudo adhuc {illeg} {illeg} <100r> aliquantulum us ad extremitatem fere, ita ut tandem evaderet 5^gr vel paullo major (Ego).

♃ Decem 30 \hor {8} {& hor}/ Scheat sita erat e latere caudæ ad dextram, et australior duarum in boreali pede Andromedæ erat in medio caudæ. {ad dextram} (Flamsteed. ^[79] & Ego \Idem et ego observabam hora 9./) Ultra vero hanc stellam australiorem cauda quasi ad 7^gr vel $7 \frac{1}{2}$ ^gr extendebat circiter.|(|Ego)

☾ Jan 3 Hor $11 \frac{1}{2}$ Cauda transibat per medium intervalli inter Alamac et australiorem in pede boreali Andromadæ & $\frac{4}{9}$ partes distantiæ stellarum (id est $3 \frac{1}{2}$ ^grad) ibi \(hoc est 30^gr a capite)/ lata erat. Tendebat ver{d}|u|s lucidam in latere Persei sed magis accuratè ve{illeg}{d}|rs|us stellam {ι} quartæ magnitudinis in dorso Persei lucidæ proximam \quæ tamen sex vel decem minutis circiter distabat ab axe austrum versus./ Desinebat verò {illeg} quasi \cauda e regione/ {in}|{a}| medij{illeg} luci|e| inter stellam illam quarta magnitudinis, et aliam ejusdem magnitudinis in humero dextro seu clypeo Persei quæ Bayero θ dicitur. Desinebat igitur in ♉ 22^gr. 27′ & lat. bor. 30^gr {illeg} 50′. Si borealiorum duarum μν in angulo Andromadæ distantia dividatur in tres partes \æquales/ & una pars sumatur versus mediam {trium} {illeg} trium in angulo μ, ibi erat medium caudæ (hoc est in ♈ 25^g. 4′ lat 30^gr 52′) et ibi \hoc est 18^gr a capite)/ latitudo ejus æquabat distantiam stellarum illarum, vel paullo superabat adeo erat✝^[80] 2^gr 6′ circiter. Ex his colligitur caudam curvam fuisse & convexo sui latere austrum respexisse concavo boream. Cauda jam haud multò lucidior erat quam partes lucidiores viæ {illeg}|ha|cteæ, si partes capiti proximas excipias, et quidem per ultimos duodecim vel quindecim gradus non erat illis {illeg}|u| lucidior. Caput jam multo magis conspicuum erat quam cauda at Decemb 15 cauda maximè conspicua erat caput vero instar stellæ adeò exiguæ apparuit (a crepusculo; scilicet et luce lunari obscuratum) ut nudis oculis ne quidem videre possem quamvis adstantes {illeg}digitum ad eam intenderent. Longitudo caud{illeg}|æ| 41^gr. Distantia circul termini caudæ a circulo solem et cometas jungente 4^gr 30′ {illeg}′. Delinatio caudæ ab oppositione ☉^is 7^gr.

♂ Jan 4 Hor {illeg}|{9}| Cauda juxta caput Cometæ tendebat versus lucidam in eductione cruris sinistri Persei, sed postea vergebat ad lucid{illeg}|am| in latere Persei et ubi aer admodum {illeg} \defæcatus erat, et meo et \{illeg}{sinetiū} judicio// exte{illeg}|nd|ebat ad \us/ stellam {illeg}|{ι}| in dorso Persei. \Axis caudæ non transibat per stellam {illeg}|{ι}|, sed paucis minutis australior existens, dirigebatur \{illeg} accurate/ versus/. Media {illeg}rium in ci{illeg}|ngulo| /lucidam in eductione cruris Persei Alge{illeg}b. dictam. vel potius versus punctum 5′ aut 6′ australius.\ Andromadæ erat in media|o| caudæ. Latitudo caudæ e regione capitis Andromadæ erat $\frac{4}{5}$ partes distantiæ medij caudæ a capite Andromadæ: Inter Alamac et lucidiorem et|in| altero pede Andromadæ æquabat $\frac{1}{2}$ \{seu}/ vel potius $\frac{4}{9}$ partes {illeg} distantiæ stellarum illarum: Juxta cingulum Andromadæ æquabat distantiam duarum obscuriorem in cingulo. Caudæ limes australis lucidior erat et distinctius terminata, item convexior quam li{mes} borealis. Limes borealis ferè recta erat vel potius nonnihil concava. Caput in centro lucidius, inde ad circumferentiam paulatim languescens, apparebat per tubum duodecim pedum sine stella aliqua vel {ubi} globo lucido in ce{illeg}|ntro|, simill{|i|mum vero stell{illeg}|æ|} alicui vel planetæ per nubem densum lucente ita crassam ut stella distinctè cerni nequeat. Totius lucis in capite diameter erat 12′ \vel 14′/ circiter. Caput nudis oculis instar stell{illeg}|æ|{illeg} quartæ magnitudinis apparebat. (Ego) Hinc Nox hæc superiori clarior erat & Cometa longius distabat ab horizonte. Unde omnia melius definiebam. (Ego) Hinc distantia termini caudæ a {illeg}circulo jungente solem et cometam 4^gr 45′ Angulus quem cauda juxta caput{.} Cometæ efficiebat cum circulo illo $4 \frac{1}{4}$ ^gr, juxta terminum caudæ 10 vel 11^gr, quem chorda caudæ efficiebat cum eodem circulo 8^gr. Longitudo caudæ 42^gr{.} Latitudo e {illeg} capitis Andromadæ (hoc est 4|3|^gr{illeg} a capite cometæ) 1^gr 15′ circiter juxta cingulum Andromedæ (hoc est $16 \frac{1}{2}$ ^gr a capite) 2^gr {illeg} Inter lucid{illeg} stellas in pedibus Androm{e}dæ hoc est (28^gr a capite) $3 \frac{1}{2}$ ^grad.

☿Jan 5. Stella π in pectore Andromadæ {illeg} caudam ad {illeg} \{illeg}/ {illeg} ad dextram Flamsteed epist. ult.

<100v>

♃ Jan 6 hor $8 \frac{3}{4}$ cauda transibat per medium prima et secundæ in cingulo Andromadæ, sed ob aeris crassiliem ultra lineā jungentem Alamach & lucidiorem in altero pede Andromadæ cerni non po{illeg}|ui|t, quamvis aer non ita crassus esset quin stellæ quartæ magnitudinis appare{illeg}|rent|. Caput cometæ cum tota luce sua vix æquabat stellam quartæ magnitudinis.

♄ Jan 8 hor 8 {illeg} Cauda, quæ \ex/ austra{illeg}|li| l{illeg}|at|ere{illeg} lucidior distinctior & nonnihil convexa erat, a capite incipiens primùm tendebat versus Mirach (seu primam in cingulo Andromadæ) quæ sita erat in medio ejus nisi quod sex vel octo minutis circiter distabat ab ipso medio versus austrum: Postea flectebatur versus Alamach qua sita in ipso medio ej{illeg}|us|. Ultra Alamach ad tres vel quatuor gradus luce languescente extendebatur: nec ultra facilè cernebatur quamvis aer adeo clarus esset ut stellæ sextæ magnitudinis apparerent. Aliquando tamen ubi aer solito clarior erat subobscura caudæ vestigia cernebantur us ad lineam jungentem stellas {ι, χ} in tergo et latere dextro Persei & nonnunquam us ad medium locum inter hanc lineam et stellas duas exiguas σ, ψ in cibasi Persei, & semel quidem ul paulo ultra ita ut stellarum illarum exiguarum citeriorem σ videretur attingere. Nam versus stellas illas duas σψ accuratè d|t|endebat. (Ego) Hinc longitudo caudæ minima erat 24^gr circiter, media{illeg} $32 \frac{1}{2}$ , maxima 35 & semel 37|6| vel 37′. Distantia termini caudæ a circulo solem et cometam jungente 5^gr. Inclinatio caudæ ad hunc circulum juxta caput cometæ 7^gr 30′ juxta extremitatem alteram 10^gr{illeg}|40|′ Inclinatio chordæ caudæ ad eundem circulum {8^gr 48′} 9^gr 10′. Caput cum tota sua luce stellis quartæ magnitudinis cassit eas quintæ paullo superavit. Diameter totius lucis circa caput 12′ Lux caudæ semper argentei erat coloris sed jam per totam caudam obscura valde.

☉ Jan 9 hora $9 \frac{1}{2}$ Caudæ longitudo constans erat 15^gr vel 16^gr extendebatur enim paullo ultra pedem sinistrum Andromedæ Alamach seu pedem australem Andromadæ. Aliquando tamen ubi aer erat solito clarior luce tenui {{illeg}} superare visa est dimidiam distantiam inter Alamach & præfatas duas stellas σψ in cibasi Persei ad quaru citeriorem σ nocte superiori semel extendebat, ita ut longitudo ejus tunc esset 24^gr circiter (Ego) Caud{æ} /ad latus boreale tetigit Mirach, desis{illeg} verò ad υτ in femore genu Andromadæ. Flamsteed Epist ult.\

☾ Jan 10 hora \6,/ 8, \10,/ cauda desinebat ad Alamach{,|.|} a|A|liquando tamen ubi aer erat solito clarior, luce suboscura se extendebat ad stellam {illeg}|{χ}| in australi lateri Persei, vel potius ad puctum duodecim vel quindecim minutis borealiorem quàm stella illa. Seribit Flamstedius caudam hac nocte de{illeg}|sy|sse sub Alamech, directam vero fuisse versus stellam illam illam {χ} in latere Persei, id est si recta producas; at ob curvaturas cauda ubi eo us visibilis extitit deflectebat a {χ} ad punctum 12′ vel 15′ borealiorem.

♂ Jan 11 hora 8, 9, 10 cauda satis distincta erat ad us Alamech, et paulo ultra, subobscura ad us stellam præfactam exignam {χ} in latere Persei, per quam axis {illeg}ubi terminabatur axe caudæ per stellam transeunte. Distantia termini caudæ a {illeg} ♃ Jan {illeg} linea sole circulo solem et Cometas jungente {3} erat igitur 3^gr{.} 50′. Inclinatio chodæ caudæ ad circulum illum $8 \frac{2}{5}$ ^gr. At distantia illa et inclinatio pa{illeg}|{u}|llo majores extitissent si modo cauda æqu{e} longe in sig{illeg} Persei visibiliter extendisset ac aute /{illeg} Caput jam\ cum tota sua luce stellas quinta magnitudinis æquabat.

♃ Jan 13 Cauda luce perobscura desinebat e regione stellæ præfatæ {χ} in latere Persei, luce satis sensibili inter Alamach {illeg}t Algol. terminabatur.

{illeg}|{illeg}| Jan 23 & 24{cometam rursus} \{beneficio sensibili impetus}/ vidi {sed} Cauda ejus {ob Linea} splendorem {ne{illeg}tiquam} apparuit caput ejus inter nubecula {illeg}cunda apparuit reliquo ca{illeg}o haud lucidioris ut sentiri ægre {illeg}rit.

{illeg} Jan 25 \{illeg}/ {Luna sub horizonte} cauda cometæ denuò sensibilis {illeg} potuit ad {longitudinem gradus} {illeg} {vel septem.} {illeg} \{illeg} sequente/ ad longitudinem graduū

<101r> 12 aut paullo ultra sed luce obscurissima et {illeg}|{a}|gerrim{à} sensibili: Tali uti luce extendebatur ad lineam jungentem Algol & Pleiadas. Dirigebatur vero axis ejus ad {capillam} \lucidam in humero orientali Auriga/ accuratà. Unda deviatio caudæ ab oppositione solis boream versus 10^gr.|Caput Cometæ cum ommi sua luce stellam septimæ magnitudinis æquare videbatur, aut {quillo} \{non {illeg}}/ superare.|

Symbol (dot surmounted by circle) in text Jan 30 Caudæ non nisi vestigia quædam obscurissima restabant quæ tamen tam is oculis quam armatis sentin potu{illeg}|ete| extrudebantur hæcce caudæ vestigia magis luce \{para}/ magis sensibili \{illeg}/ ad lineam jungentem Algol et stellas informes in nube \{illeg}/ arietis, \luce/ minus sensibili ad us lineam jungentem Algol & Pleiadas {Quinimò} nonnunquam sentire visus sum vestigia quædam lucis rarissim{illeg}|{æ}| ad us lineam jungentem Algol et stellam 3 tertia magnitudinis in australi pede Persei. P|T|endebat vero axis {illeg} reliquiarum caudæ v{illeg}{d|u|}s punctum {inter} lapellam at lucidam in humero orientali Aurigr{a|i|} punctum pa{u}ll \inter genu lucidum Persei &/ lucidam in humero orientali Aurigæ, quam proxim{è} {illeg} \nempe/ versus punctum {illeg} \triente/ gradus australius \quam lucida illa in humero circiter/, adeo ab oppositione solis deflexit 10^gr 4{illeg} circiter. Caput Cometæ cum omni sua luce stellis septima magnitudinis cessit. Ex hoc tempore caudam nudis oculis observare destiti. Telescopio vero septupedali cauda|m| vida us ad Feb 10{illeg} quo tempore duos circiter gradus longa videbatur, & versus punc{illeg} grada uno et altero australius quam lucida in humero orient{a}li Auriga|æ| dirigi, magis et magis ab oppositione solis deflectens. Pos|t|ha|æ| cometam a Feb. 25 ad Mart. 9 demò vidi sed sine cauda. Nam et caput ipsum jam adeo tenue evaserat ut ope Tubi septupedalis cum apertura duarum unciarum cerni vix posset.

Interim ubi me Cometam nudis oculis observasse affirmo nolim credas Myopem vitro concavo c{æ}nisse quo visio redderet distincta. Tali vitro, sed optimo, semper usus sum.

Cæterum cauda quoad directionem, has observabat leges. Ad singulas observationes in globo per caput cometæ et extremitat caudæ \in globo/ due circulos maximos se secantis in A B C D E &c Divide AB{,} segmenta AB, BC, CD &c in duplic{ate} ratione tempo{illeg} inter observationes utrobi factas intercedentium. Per puncta divisio{nus} duc un superficie globi lineam uniformem quæ segmenta illa AB{,} BC, CD &c \in punctis divisionum/ contengat, et in omni casu circulus per caput co{illeg} & extremitatem caudæ ductus tanget linem illam, uniformem {quam} proximè. Unde cauda, dato tempore, quoad positionem duci potest. {illeg} vero quod segmenta AB, BC, CD, DC|E| &c divisa per sum{illeg} temporum duorum observationes utrobi factas intercedentium (AB{illeg} summam temporis pr{e}imi et sedi, BC per summam temporis secun{dam} tertij &c seu AB per tempus inter observationem primam ac tertiam BC per tempus inter secundam at quartam CD per tempus {illeg} tertiam et quintam &c) debent esse in progressione seu geome{tri}ca seu arithmetica alia aut alia {illeg} aliqua quavis regulan. Et {illeg} hinc \collatis inter se observationibus/ cognosci potest an situs caudæ fuerit recta|{è}| observatus.

Si inter capellam et polum eclipticæ sumatur punctum tribus gradibus distans a capella, cauda Cometæ, a Jan Decem 15 ad Jan 8 versus punctum illud satis accuratè, dirigebatur pra{illeg}tim circa J|D|ec 18, 25, Jan 4.

Si in globo ducatur circulus maximus qui sel{illeg} ecliplipticam {illeg} Symbol (x with arrow) in text 10^gr {illeg} 20^gr {illeg} in angulo 54^gr transiens per stellam α {illeg} ala septentrionali sagillæ, dem per stellam θ quarttæ magnitudinis {illeg} orientali brachio cassiopeiæ, deni per stellam {illeg} in tergo Persei aut per punctum $\frac{1}{4}$ gradus circiter australius: Cauda Comed|t|æ {ab imd} ad us Jan {:} 4, imd ad Jan 8 salis accuratè terminabatur. Excipe tantum a {illeg}|De|c 15 ad Jan Dec 26 ubi lux ten{uior} in extremitate caudæ ob Lan{illeg} splendorem videri {suon} potuit. Si tam{illeg} caudæ longitudines {F}lam stedianæ juxta observationes{illeg} Hamburg ens{illeg} no{nn}ibil augeantur. {illeg} upon {circullo} b{illeg} alt{a|e|ri} {illeg} que{illeg} <101v> proxime. A longid|t|tudine caud{illeg}|{a}| aufer dimidiam latitudinem, et habebitur longitudo correcta. circulus \circulus/ termino hujus longitudinis descriptus secat eclipticam in Symbol (x with arrow) in text $20 \frac{1}{3}$ ^gr circite{r} in angulo 54^gr circiter. Via cometæ secat eclipticam in 21 { $\frac{2}{3}$ } in angulo 30^gr circiter. Ubi hæc via præfatum circulus correctum secat, hoc est in 20^gr 4′ {illeg} & latitudine boreali 40′ circiter, ibi erit punctum per quod planum, in quo comet{illeg}|{æ}| movit, transire deb{illeg}|et|{illeg}. Secuit igitur planum illud eclipticam in Symbol (x with arrow) in text 20 vel 20 $14$ circiter.

Cauda Feb Decem 10, 11, 12, &c{illeg} angustior apparuit, Decem 15 paullo latior, Decem 29 & 30 {illeg}ultòlatior, ut et Jan 3 & 4. Uns{/e\} prop{i}or \remoti{illeg}|{o}|r/ a nobi{illeg}|{s}| fuit extremitas caudæ Decem 10, 11, 12 quam Decem 29, 30, & Jan 3, 4, adeo cæteris paribus m{illeg}\in/fore{m} parallaxim habuit imo \imò capila. remotior quam caput,/ a {illeg}|{c}|apite Cometæ \cauda in/ regiones \ulti{illeg} &/ nobis app{illeg} aversas perge{illeg}|n{se}|.

Halleius mihi narravit se iter Parisias instituentem Dec 8 stylo veteri {te{illeg}}p cauda{m} cometæ vidisse perpendiculariter ex horizonte ori surgentem ad instar trabis igne{ce} ad longitudinem decem vel \ma{illeg}iore/ qui{m} decim graduum paulo ante ortum solis. Quod cauda hæc non{ex} corpore solis non prius disperetet quam sol oriens inciperet supra horizontem conspici: ad solis aute{m} fulgorem mo{x} evanesceret. Et quod Cauda e corpore solis exire videretur, ita ut caput cometæ esset soli proximum. Deni quod ipse quid esset ipse quid esset hoc Phænomenon nesciret p{illeg}quam donec Cometa e radijs solis egressus se omnibus conspicuum exhiberet.

Mons^r \Richer sent by y^e French King to make observations/ in y^e Island of Cayenna (north Lat 5^gr) having before he went thither set his clock exactly at Paris, found then ({viz} at Cayenna) that it went too slow {illeg}|so| as every day to loose two minutes {illeg}|&| an half for many days together & after his clock had stood & went again it lost 2 $\frac{1}{2}$ minutes every day as before. Whence M^r Halley concluded that y^e Pendulum was to be shortned in y^e proportion of to to make y^e clock go true at Cayenna. In Goree y^e Observation was less exact. They there found

<102r>

Decem 12 Caput per Telescopium Flamstedio apparuit Iovè minus nec rotundum quidem sed inæquale ad instar quadorati cujus anguli fortuitò & irregulariter diffracti fuissent. Lumen capitis jam fuscum admodum & lumine saturnio multis gradibus deterius.

Decem 21 Caput per Telescopium apparuit ut nubes \locus nubilosus/ in cælo nudis o{f}culis apparere solet: excepto quod per face|i|em ejus puncta quædam lucida sed exigna valdè irregulariter spargebantur. Capitis diameter erat plusquam minuti unius sed non bene terminata nec lucida sed nebulosa

Decem 26 Caput nudis oculis minus apparuit quam Os Pegasi & pallidius, sed per Telescopium ut ante, nisi quod puncta lucida min{illeg} distincta erant. Exinde caput minus & tenuius perpetuò evasit. Hæc Flamstedius epi{illeg}|st|. ult.

<103r>

Ex Hookij Cometa edito ann 1678.

Apr 21 1677 Cometa ab Hookio vesus est \inter basem trianguli et stellas informes in nube Aristis,/ in recta linea jungente{m} Cor c|C|assiopeiæ & Alamak. Distubat ab Alamak { $\frac{4}{5}$ } austr{illeg} versus $\frac{5}{6}$ distantiæ cinguli & pedum Andromadæ. Cauda æquabat $\frac{3}{4}$ distantia ejus ab Alamak, & dirigebatur accurate verus stellam in nasu c|C|assiopei{æ} quartæ magnitudinis. Unde caput dirigebatur non versus solem qui erat in ♉ 11^gr sed versus ♉ 14^gr. Caput æquabat stellam prim{æ} magnitudinis & lumine magis {fi}sco. Stella in medio capitis {æque} (per Telescopium qumidecim pedum conspecta) æque lucida apparebat{illeg} ac ♄ |ubi| prope horizontem versatur. Rotunda erat, sed non distin{illeg} definita. {illeg} Diameter ejus erat 25″. Comæ verò loti{us} caput amp|b|ientis latitudo seu diameter 4′ 10″. id est decuplo major quam diameter capitis. Angustior erat coma et melius terminata solem versus.

Apr 23 Cometa erat in medio puncto inter Algol et l{illeg}idem informium in nube Aristis, nempe in ♉ 14 lat. bor. 17^gr. Unde orientem versus movebat{illeg}|{e}|r sed \in linea/ nonnihil ad austrum deflectente. Caud{illeg} recta erat et versus stellam tertiæ magnitudinis in femore {illeg} Cassiopeiæ dirigebatur quasi 7 vel 8 gr longa existens. Caput it versus ♉ 17 dirigebatur, sole tamen exi{illeg}|st|ente in ♉ 13 .

Capitis lu{illeg}men densum erat et compactum & saturno fer{illeg} æquale, caput tamen limbo æquabili ut saturnus non definitum. Et capitis partes aliquæ lucidiores erant aliæ mius lucidæ. Hæ non pr{o}rsus permanentes sed notabiliter mutabiles sese ostentaban|t|is.

< insertion from the left margin of f 103r >

Imo Hevelius in schemate quidem {v}iam cometæ infra {rostros} corvi {desibit} at in observationibus non{illeg} item. Dicit erum, Decemb $\frac{4}{14}$ , {illeg} 5^h {illeg} mat. cometam properostrum co{illeg}|rv|i a se detectum e{illeg}|ss|e a rostro illo {favonium} versus vix $\frac{1}{2}$ gradu distantur. {Item suo calculo facit} \{Vidit} calculum Hevelius/

Vidit {i}tem Hevelius Cometam Dec $\frac{21}{31}$ {per unam tam} {illeg} <103v> {illeg} {punctorum} Cometa{m}. Et ocul Leporis & se{illeg} Cometam paull{cis} {illeg}|{illeg}| ante posteriorem observationem supra oculus Leporis ad distantiam {illeg} circiter transi{ti}sse, unde {illeg}infra humerum {L}eporis transivit {illeg} & fere tegit. Hevelius præterea cometam pa{{illeg}|rs|}im {omnia} {illeg} quamdiu cometa magnam hab{ui}t latitudinem australē australiorem po{int} quam Auroutius. Fortè quod Auroutius refractiones neglexit vel pro{illeg}oribus habuit.

< text from f 103r resumes >

Cometæ anni 1664 observationes optimæ in lucem edita sunt Hevelij, Ægidij Francisci de Gottignies {illeg} in urb{e} Roma Professoris & M{illeg} Petiti Pariciensis qui Observationes Au{r}antij edidit. Hevelius tamen Gottignies viam Cometæ illius infra{m} stellam in rostro corvi descri{illeg} Petitus autem (quo{cum} consentit Hugenius in observationibus quibus{illeg} ab Hookio visis) viam ejus supra stellam illam seu ad boream status Hevelius præterea ubi fere australiorem facit viam cometæ quam Petitus et Gottignies, & verbi gratia, cùm illi viam sup{illeg}|{r}|am stul{illeg} tertiæ magnitudinis in humero dextro Leporis describunt, {nie} p{o}mit infra. Gottignies in prima sua tabula statuit Cometam in ♊ 4 $12$ lat austr 33 $\frac{2}{3}$ in secunda in ♊ 4 lat austr 34 $\frac{1}{2}$ ut{rum} eodem tempore nempe decem $\frac{21}{31}$ anno 1664.

Coma{m} Cometæ anni 1677 juxta nostrum seu stellam in {illeg}capite, lucid{illeg} {illeg} e latere nuclei quod soli oppo{c}ebitu{r} lucidior er{illeg} {illeg} {illeg}rca reliqos partes nuclei: quæ quidem part{illeg} lucidior {illeg} {illeg}llium {illeg}euda Constituebunt nes capitis {esil}{illeg} <103v> apparuit in c{u}ada ne regio soli {illeg} opposita {obseumor} suit quam regio soli obversa ut opporteret si caput comet{illeg}|æ| corpus opacum esset & lucis expers. Nucleus vero cum coma come{illeg} anni 1664 collatus minorem rationem ad comam obtinebat sub finem ubi Cometa longi{es} a {Sole} recesserat.

Via cometæ anni 1664 & 1665 juxta
delineationem Hookij.

^[81] |Longitudines & latitudines subse
quentium stellarum, ex catalogo
Tychomico, ad annum completum
1664 {suppost} collectæ $\begin{array}{c} Long. & Lat \\ Lucida Arietis & \begin{array}{l} 3 ° & . & 0' & . & {24 ″}^{gr} \end{array} & \begin{array}{l} 9' & . & 57 ″ & B \end{array} \\ Secunda Arietis & \begin{array}{l} 29 & . & 17 & . & 24^{gr} \end{array} & \begin{array}{l} 8 & . & 29 \end{array} \\ Prima Arietis & \begin{array}{l} 28 & . & 31 & . & 24 \end{array} & \begin{array}{l} 7 & . & 8 \frac{1}{2} \end{array} \\ Collum Arietis & \begin{array}{l} 28 & . & 57 & . & 24 \end{array} & \begin{array}{l} 5 & . & 24 \end{array} \end{array}$ |

Distat inquit Auroutius stella A a secundæ ♈^tis dextram versus 45′ vel 46′ a prima vero 1^gr20′ Angulo \recto existente qui/ a lune{illeg}s ad stellam illi|a|m a prima et secunda ♈^tis ductis contint|e|tur. Ait et Hevelius se|t|ellam distare 46′ a secunda ♈^tis & 1^gr 15 vel 20′ a prima. A prima ♈^tis niquit Auroutius Cometa Feb $\frac{7}{17}$ ‡^[82] tanto spatio distitit quanto ab eadem stell{u}la A removetur ho{c} est 1^gr 20′. Unde concludit Auroutius Cometæ Longitudinem {trinc} fuisse 27^gr circ. & Lat{illeg}. boreal. 7^gr 4′ vel 5′.

Feb $\frac{9}{19}$ ait Auroutius Cometa 12 vel 13 movebatur a priori loco et 9 \{pringles}/ propior factus est prima ♈^tis. Feb $\frac{16}{26}$ & $\frac{17}{27}$ aut circiter cometa a prim{illeg}|æ| ♈^tis in minima fuit distantia, qu{illeg}|æ| distantia erat ad summum 50′, {in}quit idem Auroutius, Porro cometa |in| Mart {illeg}7stylo novo, {aid|t|} Auroutius, cometa non ultra 7′ vel 8′ uno die movebatur.

Juxta Observationem R. P Gottignies,& V{illeg}ll{illeg} Aureoutij Cometa Ma{t|r|t{illeg} $\frac{1}{11}$ } jam modo prætergressus fuerat Corn{u} sinistrum ♈^tis quasi spatio qu{i} vel quintæ partis itineris uno die confecti id est 1′ 30″ vel 2{′} circiter: quo{c}um satis consentiunt Hookius et Auroutius. |Ad| Distantia primæ et s{e}dæ ♈^tis, quæ est {illeg} 1^gr 33′, {illeg}Hookius in delineatione qu{i} po{n}it distantiam cometæ a seda ♈^tis e{ab} tem Gottignies in delineation{e} s{ua} {pinil} distantiam cometæ a seda ♈^is esse ^{illeg} ut 4 ad {illeg} 4, Hook ut {illeg}|{4}| ad 45, {illeg} Petitus ut 2 ad 17 sed Petitus in delineationibus suis hand satis asse{c}utus est mente{s} Aurouutij, facil cometam propius {uei}essisse ad stellam A quam ad primam ♈^tis contra qui facit Hooki{{illeg}}os {&} {illeg} Gottignies. Sit ergo distantia illa $\frac{4}{45}$ distanti{æ} primæ et secunda ♈^tis hoc est 8′ 16″ circiter & cometæ longitudos {ea} tempori{bus} {set} Mart {1} Rora 8′ {illeg}esp er{q|i|}t {{illeg}} 1′ {illeg} circiter major quam longitudo prima\secun/da ♈^tis adeo |in| ♈{,} 29^gr 18′ 30″ Latitudo verò {1}8{′} 1{illeg}|{5}|″ m{illeg}quam latitudo ejusdem stellæ adeo{illeg} {illeg} 37′ 15″.

<104r>

Cometæ anni 1661 loca ex Hevelio

$\begin{array}{lrlll} \begin{matrix} Gedani \\ et . novo. \end{matrix} & \begin{array}{r} \begin{matrix} Londini \\ Stylo \end{matrix} & vef \end{array} & \begin{matrix} Tempore \\ matutino \end{matrix} & Long. Cometæ & Lat. Cometæ \\ \begin{matrix} Dec 14 . & 5^{h} . 50' \frac{1}{2} \end{matrix} & \begin{matrix} mediocrit red imò probe Decem & 4 \end{matrix} & 5^{h} . 14' \frac{1}{2} & 7.10 \frac{1}{2} & 21.36' . 14 ″ A \\ \begin{matrix} 15 . & 5.21 \frac{1}{2} \end{matrix} & \begin{matrix} probe & 5 \end{matrix} & 4 45 \frac{1}{2} & 6.24' . 5 ″ & 22.23.26 A \\ \begin{matrix} 18 & 5.58 \frac{1}{2} \end{matrix} & \begin{matrix} probe & 8 \end{matrix} & 5.22 \frac{1}{2} & 3.14.6 & 25.26.0 A \\ \begin{matrix} 21 & 4 . 5 \end{matrix} & \begin{matrix} dub. & 11 \end{matrix} & 3.29 & 27.57.24 & 29.51.49 A \\ \begin{matrix} 28 & 2.43 \frac{1}{2} \end{matrix} & \begin{matrix} dub. & 18 \end{matrix} & 2 . 7 \frac{1}{2} & 3.0 \frac{1}{4} & 49 24 \frac{1}{3} A melius 49^{gr} . 32' A . \\ \begin{matrix} 29 . & 9.33 \end{matrix} & \begin{matrix} dub & 19 \end{matrix} & 8.57' vesp & 28.42 & 45.51 A \\ \begin{matrix} 30 . & 9 46 \end{matrix} & \begin{matrix} mediocr & 20 \end{matrix} & 9.10' vesp & 13.9 . & 39.57 \frac{2}{3} A \\ \begin{matrix} 31 . & 9 49 \end{matrix} & \begin{matrix} optime. probe & 21 \end{matrix} & 9 13 & 1.59 & 33.40 \frac{3}{4} \\ \begin{matrix} Jan 1 & 9 18 \frac{1}{2} \end{matrix} & \begin{matrix} mediocr & 22 \end{matrix} & 8 42 \frac{1}{2} & 24.21 & 27.43 A \\ \begin{matrix} 5 & 7 58 \end{matrix} & \begin{matrix} probe & 26 \end{matrix} & 7.22 & 8.59 & 12.36 A \\ \begin{matrix} 6 & 7.24 \end{matrix} & \begin{matrix} optimè & 27 \end{matrix} & 6.48 & 7.30 & 10.21 A \\ \begin{matrix} 7 & 7.46 \end{matrix} & \begin{matrix} probe & 28 \end{matrix} & 7.10 & 5.24 & 8.23 \frac{1}{2} A \\ \begin{matrix} 9 & 6.11 \end{matrix} & \begin{matrix} dub. & 30 \end{matrix} & 5 35 & 3.3 \frac{1}{4} & 5.33 A \\ \begin{matrix} 10 & 7.57 \end{matrix} & \begin{matrix} probe & 31 \end{matrix} & 7.21 & 2.4 & 4.12 A \\ \begin{matrix} 11 & 6.22 \end{matrix} & \begin{matrix} probe {Jan}_{¯}^{¯} & 1 \end{matrix} & 5.46 & 1.25 . & 3.20 \frac{1}{2} A \\ \begin{matrix} \end{matrix} & \begin{matrix} \end{matrix} \\ \begin{matrix} 17 & 8 14 \end{matrix} & \begin{matrix} opt. prob. & 7 \end{matrix} & 8.38 & 28.24 & 0.54 B or. \\ \begin{matrix} 19 & 6 41 \end{matrix} & \begin{matrix} probe & 9 \end{matrix} & 6 5 & 27.52 \frac{1}{2} & 1.45 \frac{2}{3} bor \\ \begin{matrix} \end{matrix} & \begin{matrix} \end{matrix} \\ \begin{matrix} 20 & 7 59 \end{matrix} & \begin{matrix} prob & 10 \end{matrix} & 7.23 & 27.39 & 2.11 bor \\ \begin{matrix} \begin{matrix} 21 & 9 18 \end{matrix} \end{matrix} & \begin{matrix} prob & 11 \end{matrix} & 8.42 & 27.30 & 2.32 Bor \\ \begin{matrix} 23 & 7 36 \end{matrix} & \begin{matrix} optime & 13 \end{matrix} & 7 . 0 & 27.7 & 3.7 Bor \\ \begin{matrix} 28 & 6.32 \end{matrix} & \begin{matrix} mediocr. & 18 \end{matrix} & 5.56 & 26.40 . & 4.26 Bor \\ \begin{matrix} Feb 2 & 6.36 \end{matrix} & \begin{matrix} opt. & 23 \end{matrix} & 6 . 0 & 26.30 \frac{3}{5} & 5.18 Bor \\ \begin{matrix} 3 & 7 36 \end{matrix} & \begin{matrix} opt. & 24 \end{matrix} & 7 . 0 & 26.29 \frac{2}{3} & 5.24 \frac{1}{3} Bor \\ \begin{matrix} 4 & 7.52 \end{matrix} & \begin{matrix} dub & 25 \end{matrix} & 7.16 & 26.28 & 5.23 \frac{1}{2} Bor \\ \begin{matrix} 12 & 7.15 \end{matrix} & \begin{matrix} dub. {Feb}_{¯}^{¯} & 2 \end{matrix} & 6 39 & \{\begin{matrix} 26.52 \frac{1}{2} \\ item \end{matrix} & \{\begin{matrix} 6.38 \frac{2}{3} mel iu s. \\ pej u s . \end{matrix} \\ \begin{matrix} \end{matrix} & \begin{matrix} \end{matrix} \end{array}$ | $\begin{matrix} Observationes selectæ \\ \begin{aligned} Dec 4 \\ 5 \\ 8 \\ 21 \\ 27 \\ Jan 7 \\ 13 \\ 24 & 26.29 \frac{2}{3} & 5.24 \frac{1}{3} \\ X Feb 7 & 27.12' \\ Mart 1 & 29.18 \frac{2}{3} . & 8.37 \frac{1}{6} \\ Mart 5 \end{aligned} \end{matrix}$ |

Maxima Cometæ latitudo Australis 49^gr 33′ vel 49 $\frac{3}{5}$ ^gr et locus maximæ latitudin{i{illeg}} in ♈|♋| 27 $\frac{3}{4}$ gr. Ut ex {illeg}circulo maximo per loca duo cometæ transeunte {ex} pius {cum} ex pluribus observatiionibus colligitur. Securit autem Cometa eclipticam in {♋}|♈| 28^gr 58′ id Jan 15 hora 9 P. M circiter in angulo. Comet{æ} Decemb 28 pro ratione calurarum observationem latitudinem 2|3|′ vel {illeg}4′ justominorem habere videtur. Cætera quoad latitudinem inter se bene consentiment. Et hinc latitudo maxima cometæ forte 49^gr 35′ vel 36′^gr melius statuitur, quam 49 33′.

Decem $\frac{4}{14}$ Cometa detecta prope rastrus Corvi Favonius versus vix $\frac{1}{2}$ gr ab eo distans.

Decem $\frac{21}{31}$ hor 2 vel 3 mat {illeg} paucis cometa transibat 20′ supra oculum Leporis. $\frac{Decem 22}{Jan 1}$ hor 8 vesp cometa {illeg}infra stellas in eridano (dictam tertiam a primo flera) a semigradum ferè {libe}{ro}tum versus incedebat, sic ut hora 10 com{m}a{in} suam decurta{se}per stellam istam in ea tunc clare emicantem projiceret.

$\frac{Dec 26}{Ja n 5}$ hora 4 $\frac{1}{2}$ vesp. Cometa jam modo mandibulam fere occultarat, non tamen occultabat ommino ub aliqui volunt, nam hora 9 Mandibula in ipsa cauda apparuit. Hora 9 Cometa inter Mandibula et stellam in ore C{illeg}|{a}|ti in linea fere recta apparuit.

$\frac{Dec 27}{Jan 6}$ hor 7 vel 8 \vesp/ circiter Cometa cus Mandibula et illa in {co}re triangulus æquilaterum ferè constituit

$\frac{Dec 28}{Jan 7}$ Et postea Cometam inter mandibulum et Caput Andromedæ existente, Cæpit Hevelius distantiam ab utra stella ut cognosceret an summa distantiarum æquabat distantiam stellarum Inde motum Cometæ quoad progressum in orbita sua exhinc accurat{è} determinavit, præsentim sub initio mensis February.

$\frac{Jan 23}{Feb 2}$ Cometa distabat una, sui diametro ab inferior{illeg}|{e}| duarum stillularum {,} quæ eaudem quo distantiam ferè habebunt ab vivic{è} nempe 2′ $\frac{1}{2}$ circiter vel 3′ {illeg}{d}. $\frac{Jan 24}{Feb 3}$ Bin{ne} illa stellæ limbo orientali comet{illeg} ad hærebant. Inferior et lucidior {binarum} vix {tola}de limbo exi{illeg}{et} superior maxima sui parte suo ten{e}t maxima {illeg} adeita latebat ut ea propt{illeg}{us} etisem mino{rs} et obser{illeg} <104v> videretur{.} Inferior igitur lem {hum} a capite cometico plane {tecta} fuerat superior le{m} hu{m} stri{ns}cerat. $\frac{Jan 25}{Feb 4}$ hor. 7 vesp Cometæ limbus a hinis illis stell{æ} una cometæ diametro distabat Jade H{av}e Hevelius. Unde colligit comet{æ}m vix ultra 6 jam per diem movisse. Movebatur autem, inquit, sursum ita ut longitudo ejus vix quicquam mutaretur.

Di{x}i latitudinem Cometæ Decemb 18 insto minorem esse per 3{′} vel 7′, imò non consentit cum cæteris observationibus nisi 8{′} {minus} augeatur. Pro 49^gr24′ $\frac{1}{3}$ lat. Aust. {scribe} igitur 49^gr32′ lat austr. et maximam Cometa latitudinem 49^gr 4{illeg}|{0}|′ in ♋ 27^gr45′. <105r> A, B, C, D, E, F, G are y^e stars in y^e greater {ev}ain. L, m, n, s, t stars in y^e Bears bright hinder leg. H, J, K stars in his head & {illeg} neck. Anno 1682 Sept

On Satturday at {illeg}|{1}|^h 20′ after midnight I saw y^e comet in V in a right line w^th y^e stars F & s, distant from y^e star s twice as far as that star was from y^e star t. The tayle pointed directly towards the star K in y^e eye or cheek, & was about six degrees long reaching $\frac{1}{3}$ of y^e way to that star.

Sunday at 9^h 20′ before midnight the {illeg}|co|met was in X. Xs & sn were equal & a little greater then Xn. ms, {illeg} mX & 1 $\frac{1}{2}$ Xs were equal. Xs was equal to 3 $\frac{1}{2}$ st The tai{l}e ended over against the middle of st & produced cut of $\frac{1}{4}$ or $\frac{1}{5}$ of qr towards q.

Munday {illeg} at 8^h 40′ at {n}ight vYo were in a right line Yo=1 $\frac{1}{2}$ st. The taile ended over against m{illeg}n or alittle beyond those stars suppose about a degree beyond. & pointed towards a little star p not noted (I think) in y^e globe.

Tuesday {illeg} at 9^h. 0′ {illeg} The comet was in Z. oZ was a little greater then DE almost as great as CD. The comet passed about 8′ or 10′ above y^e star o w^ch is a little scarce {illeg} noted in y^e globes. The tayle was crooked, the convex side southward was sensibly brighter{,} then y^e concave side. The head in {illeg} this & y^e former observations scarce so luminous as a star of y^e first magnitude but more luminous then one of y^e 2^d. The taile went exactly in y^e middle between the stars m & L or a very little nearer to m & pointed almost at y^e Pole star, vizt as much below it as y^e middle star in y^e little b|B|eares taile was above it & reached up within a degree or two to over against it or very nearely. The tayle produced would have wiped the star A w^th its north concave side.

<108r>

Problemamm solutiones juxta sequentes
Regul{a}s

Reg. 1. Circumspicera quid ex datis consequatur ut ex pluribus datis facilius assequamur quo|d| propositum est. Item crc|ir|cumspicere quomodo schemata constmantur ut en datis aliquid colligamus. In hune cognoscendæ sunt proportionalium legès et transmutationes, eo quod Geometria{,} proportionales ob simplicitatem magis quam per æquationes amat progredi. Cognoscendæ s{u}nt etiam Figurarum proprietates quæ in elementis sunt & determinationes simpliciores: Et quando {illeg} triangula vel quadrangula dantur specie, quando specie et magnitudine Determinatæ item sectiones vete{r}e{m} quæ sunt æquationes recentio{r}us in promptu ess{a|e|} debent. Ut {illeg}|et| Locorum determinationes. Nam Geometria tota nihil {a}loud est quam inventio punctorum per intersectiones Locorum.

^[83] Sectio{illeg} determinatæ dici potest simplex duplex triplex &c {proud} in uno, duobus, tribus punctis &c fit, vel ut recentis loq{uca}ntur prout æquatio unius duarum {illeg}|tri|us dimensionum est.

Si secanda sit recta data AB in x ita \sit/ ut \Ae.Ax \&c/ ∷Bx.De. vel/ rectangulum AxB sequetur dato rectangulo AeD: sit angulus D|B|AD {illeg}|re|ctus. Biseca BD in C {illeg} Radio C. Centro C radio Ce {illeg} describe circulo secantem AB in x. At hoc modo construi potest omnis æquatio quadratica. Sed rem longius prosecutus est Apollonius.

Igitur is in recta aliqua A{illeg} dantur tri{o} puncta A, B, F et secanda sit recta in x ita ut sit Ae.Ax∷Bx. Fx componendo vel dividendo erit Ae.F|e|x∷Bx.BF unde solvetur Problema uti prius.

Si d|i|n recta dentur quat{r}or puncta A, B, F, G et seconda sit recta in x ita ut rectangulū AxB, \sit ad rectangulus/ FxG in data ratione AH ad HG, erige perpendiculum {illeg}HQ quod sit medium proportionale inter AH et HG. I{u}nge AQ, GQ. Super BF constitue trian In angulo A{e}g AFP æquali AGQ et GBR æquali GAQ age rectas FP, BR occurrentes AQ et GQ in P et R super BF constitue triangulum BSF simile triangulo{,} AQG et ad easdem partes {illeg} rectæ AG si punctum \x/ quæritur \vel/ inter A et B vel inter F et {illeg}QG, {illeg}|{A}|liter ad partes contrarias. Super diametro PR describe circulum secantem rectam AG in x

Addo si recta {QG} secanda sit in x ita ut rectangulus AxB sit ad {illeg}{cessum} quo \diferentiam inter/ rectangulam FxG {illeg}{separat} \&/ rectangulum datum mXn: seca{illeg} rectam illum in T et V ita ut {illeg}rectangula{illeg} {\{illeg}/}{illeg} {illeg}T{G} {illeg} F{QG} {{illeg}} \{secunda}/ {æquater} dato {illeg}rectangulo M{illeg} <108v> Dein {r}ursus seca in x ita ut rectangulum AxB sit ad rct|ec|tangulum TxV in ratio ratione illa data.

Si recta secanda est in X ita ut rectangulum AxB sit ad summam rectanguli FxG et rectanguli dati mXn. Erit dividendo AxB÷mXn ad FxG in ratione data et inverse FxG ad AxB÷mX{illeg}n in rad|t|ione data. erg Qui casus est superioris propositionis.

Ad ho{illeg}|{c}| casus facie|l|{e} est cæteros reducere.

<127v>

"Signo verò ± additionem et subductionem abigua|è| denoto.

<128r>

Geometria.
Lib. 1.

^[84] Problemata pro numero solutionum quas admittunt distingu{n}ntur in gradus. Qu{æ} unit|{c}| tantum admittunt solutionem sunt primi gradus, quæ secundi, quæ tres tertij, {illeg}|&| sic {deincepe} {illeg} {deincepo} in reliquis. Ut si {recta} data recta AB producenda est ad D ita ut punctum D dao{illeg} {habent} \intervallo/ ^[85] distantiam \distet/ a puncto \aliquæ/ C quod in sublimi datur: Solvetur problema si centro C et intervallo isto dato describatur circulus datam \illam/ rectam secans. Et ista solutione prodibit ostendit Problema secundi gradus esse. \Et duplici intersectione \in D et d/prodibit/ duplex respon \ejus/ solutio, |{Producend}|{blematis}. Ad utrum vis enim punctum D vel d produci potest recta AB. Quod ostendit Problema secundi gradus esse.

^[86] Quantitates autem quibus quæstioni respondetur aliquando directæ & positivæ sunt aliquando retrorsæ vel subductitiæ quas et negativas vocant. Ut si \datum illud/ intervallum {illeg}|{B}C| detur majus \sit/ quam distantia BC ita ut circulus rectam \illam/ sec|t|et in Δ et δ, respondebitur quæstioni vel directe producendo AB ad δ vel \et/ contrario modo ducendo BΔ retrorsum. Directas quantitates {signum{illeg}} \notamus/ præfigendo {notam} \signum/ + retrorsas præfigendo signum −: e|u|t {illeg} in his, + ABBδ & BΔ. Et ut|b|i neutrum signum proæfigitur quantitas directa est \est/. Quantitates per quas {æquar} |His signis etiam ad additionem et subductionem significamus. Un|t| in AD|B| + Bδ & AB − BA ubi Bδ addi, BΔ subduci intelligitur. ^IISigno {illeg}|

^[87] Hœ quantitates \quibus {sestijoni } respondemus/ aliquando etiam impossibile evadunt; Ut in hoc casu quantitates BD et Bd ubi intervallum CD minus assignatur quam ut circulus rectam AB secare possit. Impossbilium ver{è} numer{illeg} semper est par Et quando duæ vel \forte/ quatuor vel {illeg} \aut plure{illeg} {etiam servant plures}/ impossibiles sunt (nam numerus impossibilium semper est par) {illeg} gradus Problematis non æstimabitur ex numero \solarum realium/ prossibil{iu}m sed ex omnium numero omnium, id est omnium qui in quocun casu Problematis generaliter propositi possibiles \r{e}abes/ evade{illeg}|re|{illeg}t \possunt/. Problema verò generaliter proponi dico in quo quantitates nullæ ita. {limitantur datum possun} limitantur quin possint additis vel{illeg} subductis datis \in datis {retienibus} dat{i}s different{ij}s/ \additis vel subductis datis/ majores vel minores sumi \in uno termino quam in alia/. Ut si inter A et B inveniendæ sint duæ mediæ proportionales x et y, unica{illeg} tantum est \Problematis hujus/ solutio realis, nec tamē ideo primi erit gradus Problema. Nam si omnes ejus termini{.} exprimantur & datis quibusvis \{deminantur} singulæ in datis rationibus, $\frac{}{e}, \frac{q}{e}, \frac{h}{e}$ datis differentijs datis quibusvis CD/ |quotquot aliquo modo limitantur|, ut C, D, E{,} F{,} G, {H} \indefinite/ /&c\ augeantur vel diminuantur, \Problem{a}/ generaliter enuni{illeg}|ci|abitur hoc modo Invenire {q}uas quantitates x et y ita ut sit|n| AC + A ad B + x ut A − \+/ {C} ad x − \+/ {C} & {+} x − \+/ C ad {illeg} {illeg} et {illeg}y ± D ad B {illeg} in eadem ratione. Hujus generalis \s{er}pe/ problematis casus est ubi C, D, E, F, G et H nulla sunt id est ubi sæp{illeg}ro tres sunt \possunt esse/ solutiones {illeg} reales, adeo casus ejus ubi C, D, E, F, G et {H} nulla sunt id est ubi x et {illeg} med{o}sunt proportionales inter A et B, problema {illeg} grad{es} quamvis duæ ex solutionibus hic evaserint impossi{illeg} {illeg} in {omtres} ejusdem sunt gradus cum genera{illeg} {illeg} {illeg}forte per conditiones {quesdein de quibus} {illeg} {illeg} ad gradum \aliqu{aor}/ inferiorem a

<129r>

Ubi Si linea A ducitur in lineam B rectangulum genitum signamus scribendo A×B vel AB et si id rursus ducatur in lineā C parallel{i}pipedum genitum signat{rus} scribendo A×B×C vel ABC Latus vero quod orit oritur applicando rectangulum vel parallelipipedum{,} illud \illud/ ad lineam quamvis D sic notamus $\frac{A B}{D}$ . Et sic in reliquis. Sed et exposita linea aliqua ad quam tanquā mensuram universalem referantur aliæ omnes lineæ referant{ur} scribimus A×B vel AB \at{it} AB/ designandum quartam proportionales ab hac linea ubi \duæ/ mediæ \{propor}/ sunt A et B et A× \'/B× \'/C ad designandam etiam quartam ab eadem linea ubi A× \'/ et C sunt duæ meda|i|æ et sic in infinitum. Et si {ab eod} linea illa sit prima {Et et} continuè proportionalium et {illeg} alia quævis A secunda tertiam {illeg} desi designamus AA vel A9|1|vel A² quartam \quintam {&} sextam {et} sequentis sic A3, A4, A5, {&}c./ sic A^{c} vel A³ quintam sic A99 vel A⁴ {illeg} {illeg} per ha{illeg} notas A9, A^c, A99, A9^c, A^cc, &c \ubi A^{c}{,} A³{,} A^{4} {{illeg} \A²/} diei possunt A duarum, tri{sun}, quatuor \{q}uim/ {f}rationum et sic A $\frac{1}{2}$ est/ |[| intelligendo A \A dimidiæ rationis seu media proportion inter mensura{m} universalem et A/ quadratum, cubum, quadrato-quadratus, quadrato-cubum, cubocubum de A et sic in superioribus infinitum. Nam et quadratum et cubum super latere A constitutum designamus ijsdem notis {.} Unde et ac tertium quartum{,} proportionalem,. Unde et reliquis proportionalibus per analogiam nomina {derivantur} dantur qu{æ} |.| Quæ et {illeg} \analogicè/ etiam {dicun} dicuntur dimensiones ac potestates lineæ A. Sit|c| A⁴ dicitur A \quo/ quatuor dimentionus vel A potestatis quadrato-quadraticæ quamvis revera nihil utra trinam dimensionem et potestatem cubicam in g|G|eometria reperiatur. e|E|t simili analogia dicimus proportionales A9, A^c, A99 generari ducendo A in se et A9 in A et A^c in A. Et {B}quartam proportionalem AB generari ducendo A in B. Et vicissi A99 applicatum ad A producere A^c et AB applicatum ad B producere A, applicatum vero ad C producere $\frac{A' B}{C}$ .|]| Deni ad designandum \tum/ latus quadrati æqualis {retrigo} areæ A'B tum medium proportionale inter mensuram universalem et A'B scribo $\sqrt{} A' B$ vel $\bar{A' B} | \frac{1}{2}$ et ad designandus tum latus cubicum solidi A,|'|B,|'|C−B,|'|C9 {c}|t| perimum e duo bus medijs proportionalibus inter mensuram universalem et A'B'C−B'C9 scri{icas}|o| $\sqrt[3]{A' B' C - B' C 9}$ vel $\bar{A' B' C - B' C 9} | \frac{1}{3}$ & et|ad| hujus quadratum designandum scribo{illeg} $\bar{A' B' C - B' C 9} | \frac{2}{3}$ . Eadem notarum ratio in magis compositis tenenda est.

Terminis Arithmeticis multiplicandi dividendi et extrahendi radices non utor quod {{illeg}|ho|s} non opus est \impropri{é}|æ| sint et minimè necessari{é}|æ|/ et scientias diversi generis confundere {no}lui. Quantitas per aliam quantitatem multiplicare absurdè dicitur. Solus numerus est \per/ qui|e|m potest multip possumus multiplicare. Tr{à}s homines {p} multiplicari possunt per 4 non at|u|tem per {illeg}|ua|tuor homines et linea tri{illeg}|{u}|m pedum per quatuor non autem per lineam quatuor pedum. Si linea exponantur et Multiplicatio non fit per lineas nisi equatimus hæ per numeros exponi ha{c} deciman {illeg}antur il est pede propria natura in quantitates {illeg}meticas convertantur. Est autem Arithmetici exponere et speculare quantates omnis generis per numero Geo{illeg} {illeg} lineas superficies et solida {illeg} <129v> Quæ de causæ \Certe/ Veteras ut Geometriam servarent incontaminatas a terminis illis exoticis maxima abstinuerunt{{illeg}|{Trite}|} Inter has scientias {tritam} \maximam/ esse affinitatem animadverterunt.|,| ita ut ex analogia termino{;} geometricos quadrati cubi et similium in Arithmeticam introducerent et Euclides scripta Geometrica libros Arithmeticos miscerea{illeg} Geometricis; sed tamen Geometriam \tamen/ quæ scientiarum Mathematica{rū} regina est terminis exoc|t|icis contaminare noluerunt. Inventa est uti Geometria ut \ejus succinctis operationibus/ in terris metiendis effugeremus tæ{dium} computi Arithmetici Proinde ut {est \sed/} computis quantum fieri potest vacare debet, sic etiam a computi nominibus ne horum usu {et|s|} \ad rem significatam plus nomio invitenur/ ad {illeg}{caleputa} et sic sei entiam nobilissimam \contra institu{t|c|id} ejus/ cum Arithmetica \tandem/ confund{em}|amus|. Hae igitur in re {si veteres sequar} reprehendi non debiam si v|V|eteres sequar.

<130r>

^[88] Geometria per intersectiones linearum solvit omnia Problemata, singulas \ejusdem problematis/ solutiones per totidem intersectiones una vice exhibens Nam solutionum omnium eadem est lex et natura ita ut una exhi Geometricè exhiberi non possit qu{a|i|n} reliquæ eadem constructione simul {exhibi} prodeant. Unde fit ut ad constructionem cujus Problematis lineæ \duæ/ adhiberi debent quæ se mutu{ò} in tot punctis secare possunt quot Problema admittit solutiones. Ad constructiones \omnium/ problematum primi gradus sufficiunt lineæ rectæ ad eas secundi requiruntur recta et circulus vel duo circuli ad eas tertij requiritur linea magis complexa quæ rectam aut circulum in tribus punctis ad minimum secare possit et sic in infinitum.

^[89] Et hinc pro numero punctorum in quibus lina|e|a quævis secari potest a linea recta \secari potest,/ oritur distinctio linearum in gradus. Primi gradus vel generis \ordinis/ est linea quam recta in unico tantum puncto secare potest vel cujus intersectionē cum imperata \quavis/ recta determinare Problema est primi gradus. \et hujusmodi {illeg} s{u{illeg}t|n|t} solæ rectæ le|i|neæ./ Secundi gr gradus linea est cujus intersa|e|ctionem cum {data} \recta/ quavis recta determinare Problema est secundi gradus et sic in Ter hujusmodi sunt circulus et reliquæ lineæ illæ \omnes/ quas Comicas Sectiones appellant. Tertij verò generis \gradus/ linea est cujus intersectionem cum recta determinare problema \est/ tertij gradus \est {lissoida} Veterum/ est et sic in infinitum. {Sic et} Quas quidem lineas omnes sic licebit exprimere. |Line{illeg}|æ| vero quas recta in punctis infinitis secare potest (qualls {illeg} sunt Spiralis Quadratrix Trochoides & similes) meritò dicētur{illeg} \ordinis/ ultimi {gradus} ordinis|

Datis positione quotcun rectis AB, \mF,{,}/ HP IQ, KR concipe rectam BC super dat{o} recta AB in dato angulo AC incedere et interea |productam secare \reliquas/ positione dat{a}s |in| FP, Q, R, {&} {&}c.| termino suo c|C| lineam cCc describere ea lege ut {sit s} \semper/ sint AB ad BC, PC ad BD, QD ad BE, RE ad BF in eadem ratione.

^[90] Concipe rectam BC parallelo motu ad latus ferri et interea secare {datos} \rectas/ p|q|uotcun positione dat{a}s AB in B, \mF in F/ HP in P, IQ in Q, KR in R net|c| non curvas {nec} in C, dd in D ee in E et punctis \in ea mobilibus/ {suis} C, D, E, F {illeg} \alias/ lineas cc, dd, ee, |ff| describere. Sit Determinari autem concipiantur longitudines BC, BD, BE, BF hac semper lege ut sint AB ad BC et PC ad BD et QD ad BE et RE ad BF in eadem ratione. Et |si| linea d{illeg}D ad quam ratio secunda desinit recta est, tunc linea cC ad quam prima desinit erit secundi generis {illeg} {illeg} et aliquando primi \nes ulla est linea secundi generis quæ non potest hoc modo exhiberi/. Sin linea eE ad quam ratio tertia desinit recta assumitur{;} linea vel ad quam prima desinit erit tertij generis et aliquando secundi vel primi ne ulla est linea tertij generis quæ ne{u} potest hoc designari \{sic}/. Quod si linea |f|F ad quam quarta ratio desinit recta {illeg}\statuatur/ \tunc/ linea cC ad quam prima de{sinit e}rit {illeg} \aut qua{rt}i {aut} inferioris alicujus/ generis. Et sic \{illeg}/ |novas| in infinitum {illeg} lin{eas designa}{illeg}\re licet/, et numerus rationum gradum \altissimum/ lineæ cC semp{er æqua}bit. Tot enim punctis &

<131r>

non pluribus possibile est Curvam illam cC a recta BC secari quot sunt rationes. Nam si verbi gratia tres sint rationes et linea BC detur positione dabuntur puncta AB BE, BP, BQ et BC invenienda erit ea lege ut sit AB.BC ∷BC+BP.BD∷BD+BQ.BE quod Problema triplicem admittere solutionem ex superioribus constat adeo linea BC triplex est. {Secu} Tria sunt igitur igitur et non plura possunt esse puncta C in quibus recta BC occurr{i}tt Curvæ cC, proinde Curva illa tertij est generis.

Facilius autem imaginamur has curvas ubi \per/ motus locales linearum inter se cohærentium tanquam per organa quædam {delineari} describi concipimus. Ut si regulæ PC PD {illeg} datum angulum |C|PD continentes volvantur circa datu{r} datum punctum P quod in anguli illius vertice est et similitur regulæ QC QD \circa/ datum angulum Q \continentes circa punctū Q/ ea semper lege ut regulæ PQ, QD se mutuo semper secent ad rectam aliquam lineam positione datam AD et interea reliquarum regularum PC, QC intersectio C motu suo lineam cC describat \designet/: Erit hæc linea cC linea secundi gradus et aliquando primi. Et hac ratione possunt omnes lineæ secundi generi|gradus| describi \designari/. Deinde si loco rectæ AD substituatur curva \linea/ aliqua secundi gr|e|neris \per neutrum punctorum PQ transiens/ et inter regularum intersectio D in hac movere cocipiatur, altera intersectio C, describet lineam \designabit lineam quarsi gradus aut etiam/ tertij. gradus. Qua ratione et omnes hujus \tertij/ gradus lineæs quarum commoda aliqua descriptio organica hactemus reperta fuit describere \designare/ liceat. At ita ad lineas superiorum generum pergitur, licet omnes non possunt hoc modo describio Quod idem fiet si regularum {.}duæPC, PD non volvantur circa polum P sed parallelo motu ferantur ita ut concursus earum P movent \pergat/ in recta aliqua data positione data. |Sed| Et ad majorem describer|ignan|di copiamvice \rectarum/ regularum {illeg} adhiberi possunt curvæ.

^[91] Lineam vero ut cC in qua punctum aliquod indeterminatū ut C perpetuò reperitur veteres dixerunt puncti illius locus et quoniam problematum solutiones \constructiones pendebant a descriptione/ ab {inventiam} duorum locorum puncti quæsiti pendebant {quasi s} in quorum intersection{es} situm esset \inveniretur/, ideo. Veteres ad \{illeg}/ \hujusmodi/ locorum inventiorum ac determinnationem quam compositiones eo \compositionem ut loquebantur id est ad eorum inventiorum ac determinationem/ summis viribus intebuntur. Duo autem hic requiruntur. Primum ut sciamus {illeg} sciamus sa{illeg} datis loci conditionibus, {illeg} \sciamus/ {æqualis} sit et quomodo describendus deinde ut in {illeg} quocun d{illeg} in quolibet problemata loca{illeg} simplicissima inveniamus quæ {simplicissima sunt} et facillis{illeg}à determinari ac describi {possint} sed {illeg} quam {hec de} his agamus proprietates {curvarum cognoscendæ sunt.} Insigniores \{autem}/ sunt hæ.

^[92] Si para{illeg} {illeg} \quotcun quavis A{G}, DF, G{S} {illeg}/ agantur {illeg} curvam \{illeg}/ in {tot} punctis A, B, C ac D, E, F, \G, H,/ quot curva {illeg} {illeg} a recti secari potest Dein tertia agatur recta {illeg} {illeg} {secantis} \prioribus/ ita secans \in K et {of}/ <132r> utrius parts vel summa partium \ad curvam extensarum/ ex uno latere æqualis sit parti vel summæ partium ad curvam extensarum ex altero latera, viz \vizt/ KA+KB=KC et LD=LE+LF: tunc recta partes \etiam/ reliquarum parallelarum hinc inde æquales erunt MG=MH +MI. Linea{illeg} vero quæ parallelas ita secat \in partes/ Diametrum Curvæ appellamus et partes parallelarum {quæ} \ipba/s ordinatim applicatas ad Diametrum, {ut} puncta item |ut R, S, {T}| ubi Diameter secat curvam \ut R, S, T/ Vertices ejus, et conjugatam Diametrum quæ ordenat{u}m applicatis parallel{a} est siqua ta{illeg}|tis| si{tu} conjugabam Diametrum quæ ordinatim applicatas parallela est siqua {illeg}|{tat}|is si{t}. Nempe \Porro/ ubi Diameter curvam \verticum duarum quarumvis intervallum latus rectum transversus et partem Diametri inter vert verticem quamvis et ordinatis applicatam tum Diamtri. Quibus nominibus analogi{a}m int{er} Conicas Sectiones et superiores/ |insinuare {illeg} solui. Porrò ubi diameter curvam {illeg}| in tot punctis R S T secat quot ipsam recta secare potest et ab aliqua ordinatim applicatarum ita quæ \ita/ divib|d|itur {in} {illeg} ut pars vel \summa/ part{illeg}|iu|m ex una {latera} ad {ve} \segmentum vel summa segmenr|t|orum/ ex uno latere ad \puncta illa seu/ vertices exte{n}s{illeg}|{a}|{m} conjunctim æqualis pat \sit {illeg}/ \segmento vel/ vel {sic} summæ {latium partium} \segmentorum/ ex altero latere, \{illeg} illa/ ordinatim applicata dicetur Diameter conjugata, Diametrorum \verò/ intersectio centrum \vel unum ex centris/ & summa partium \primæ diametri/ ex utra latere secundæ \jacentium summa/ Latus transverum Figuræ et linea ill{illæ} \quævis/ lat{illeg} rectam quæ ita sunt ad latera transversa ut content{us} sub oni|m|nibus ordinatim applicatis AK, {illeg} BK, {illeg} CK, ad contentus sub omnibus segmentis diametri RK, {illeg} SK, {illeg} TK verbi gratia ut contentum sub M{illeg}g /G,\ MH M, mg M \MG, MH, Mg/ ad contentus sub MK|R|, K MS, {illeg} MT. {K}. Nam |in omnibus figuris| contentum sub \ordinatim/ applicatis est ad contentum sub segmentis \diametri/ in data \ratione/ si modò {tot sunt} segmenta tot \applicata {illeg}&/ segmenti|o|rū quot ejus generis {illeg} numerus pro genere figuræ ple{illeg} est.

Quinetiam si datis \positione/ rectis vx, xy utrun ducantur parallela \du{æ}/ IG, RT \utrun ducantur/ secantes se mutuo in M curvam vero in tot punctis quot rectæ curvam ejus generis secare potest {puncta} in G, H, I et R, S, T {illeg} Rectangul{illeg} contentum sub \omnibus/ partibus unius rectæ inter curvam et alteram rectam sitis MG, MH, MI erit ad contentum sub omnibus ejusmodi partibus alterius rectæ {in} MR, MS, MT in data ratione. Et hinc recta duci potest ^[93] quæ curvam quamvis de{illeg}scriptam in puncto imperato tanget{.} \{secetve} in dat angulo/ Sit illud punctum P|{R}|. Per illud \quod/ age duas quasvis rectas {P}|R|P, {P}|R|T, |se secantes in| secantes curvam in {illeg} \pleno numero/ puncti|o|rū qu{æ}, I, M et uni earum \{P}|R|I/ parallela {G}|{I}|G \secantem altera RT in M/ quæ omnes /etiam\ secent Curvam in pleno numero punctorum {I}|R|{illeg}, G{;} M, {E}, T{;} {illeg} P, R, Q; R, S, T; I|G|, H, I. Sit In IG cape MN ita ut sit rectangulum s{illeg} contentum sub PR, QR, MN ad contentum sub RS, RT, RM ut {illeg} contentum Sub GM, HM, IM ad rectangulus sub ut contentum sub RS, RT ad contentum sub MS, MT et acta RN tanget curvam in R, si modo RN capiatur in in eo angulo PRM qui|{e}|m curva secat. Nam concipe {illeg}g\m/{,}ξ parallelam {ess}{illeg} PQ et ad eam accedere {illeg} \interea d{illeg}/ secat{illeg} curvam in g, h, {illeg}i et evanescentium, RM, Mh|N| {illeg} erit ea quæ est {illeg} RM ad M{illeg}N ubi RN {illeg} curvam. {Est autem } {rectangulum , {illeg}} contentum sub gM{illeg}hM /,\ {illeg}{iN} ad contentum sub GM, HM, IM ut contentum sub {illeg} MR, MS {illeg} MT ad contentum sub MR, MS, MT. Hic pro ratione RM ad MR substitu{atur} {illeg} {inter} rationem NM ad MR et coalescentibus {lineas} PQ {et} ig sub scrib{o} R pro M, Q pro g et P pro i et {illeg} {hat} \incides/ {illeg} \in/ proportione qu{ar} tangentem RN determinavimus.

<133r>

^[94] Linea{e} {illeg} Cæterum hæ lineæ utplurimum crura habent in infinitus serpentia quæ aut Hyperbolici sunt generis aut Parabolici. Concipe punctum B secundum lineæm \curv{æ}m crus/ AB delatum abire in infinitum et interea curvam a linea mobili BC rectas GD secante in semper tangi. Sicet autem tangens illa BC rectam positione datam GD Incidat autem \semper/ a puncto atiquo G in tangentem illam perpendiculum GC, et ubi punctum B in infinitus abit si G{C}C fit infinite longum \tangente BC prorsus evanescente/ crus illud AB {Hyper} Parabolicum est, sed si GC non fit infinitè longum, crus est Hyperbolicūm est et tangens in ultima positione seu recta illa {illeg}uti DE qu{illeg}(uti DE) quacum tangens ultimò convenit, cruris illius Asymptotos A|a|ppelatur. Crus vero infinitum semper habet socium suum \qui/ nunc ad eandem nunc ad opposit{u}m plagam tendit. Et paris Hyperbolici semper eadem est Asymptotos. Est hæc Assymptotorum insignis proprietas, quod si curva cujusvis generis p{illeg}|len|è Hyperbolica secetur a recta in pleno numero punctorū et recta G, H, I et recta illa secet etiam omnes Asymptotos \puta AB, AC, BC in D, E, et F/ segmentum \pars/ vel summa segmentorū \partium/ rectæ \Asymptoto vel/ ab Asymptotis ad curvam \crura {illeg}|tot|idem/ versus unam plagam tendentes extens{a}rū æqualis est parti vel summæ partium similium a reliquis Asymptotis ad \versus/ alteram plagam \ad reliqua crura/ tendentium DG=EH+FG, vel EG+FH=DI. Figuram \Curvā/ verò {illeg} plenè Hyperbolicam voco qua alia ejus generis cruva non potest habere plura paria crurum Hyperbolicorum. Habet autem tria paria si sit tertij generis quatuor si quarti et sic deinceps. Et par crurum Parabolicoram æquipollet duabus paribus crurum Hyperbolicorum.

^[95]Ex crurum infinitorum n{illeg}|um|ero et et diversitate {}endet distinctio curvarum in gradus {illeg} Qu species principales. Sunt autem alia crura conspirantia seu ad eandem plagam tendenti{æ}, alia opposita \divergentia/ seu p|v|ergentia in oppositas plagas et utra rursus {vel} ad easdem partes convexa vel ad contrarias. Par crurum \Parabolicorū {illeg}/ Hyperbolicorum æquipolle conspirantium æquipollet duobus paribus \{illeg}/ Hyperbolicorum divergentium, et Par {illeg} Parabolicorum divergentium {illeg} contrari{e}s partes convex{illeg} æquipollet tribus: saltem in curvis tertij generis. Unde sci{{illeg}ri} potest quot crura cujusvis generis curva quævis habere potest Ut si curva terti generis habebt par Parabolicum conspirans sive Parabolicum sive Hyperbolicum conspirans: non habebit misi aliud par Hyperbolic{u}m divergens. Sed et Ellipses conjugatæ considerandæ sunt quæ et aliquando in puncta \conjugata/ contra{punctur} quæ puncta conjuga cujusmodi et polus Conchordis, aliquando prorsus evanescente: {illeg} Asymptotorum item situs an parallelæ sunt vel {inclinatus et aliæ} {illeg} quædam differentia {illeg} notæ {illeg} {illeg} est. De De curvis sup e|l|ineis autem enim superiorum generum fase disserere non est instituti. {illeg} \nec/ in demonstrandis quæ dicta \sunt/ tempus non {illeg} tantum quæ de lineis secundi {illeg} ab Apollonio {illeg} demonstram habentur

<134r> ita commemorare ut {line} eadem lineis etiam superiorum generū competere insinuarem. C

^[96] Cæterum quoniam cognitio determinatio {illeg}|ac| linearum \curvarum/ maximè pendet a crubribus infinitis, hæc autem cum eorum Asymptotis noscuntur ex tangentibus; tum etiam qui{a} tangentium inventio post{hec} alijs inservient usibus: sub methodum jam subjungam ducendi rectas quæ curvas \quasvis/ nondum descriptas postquam describuntur tangent. Sed notæ operationum quibus in hujusmodi operationibus tam Analyticis quam syntheticis {illeg} quam \Geometricis/ utimu{s|r|} sunt prius expicandæ.

^[97] Ubi linea aliqua AB ducitur in aliam lineam BCD rectangulum genitum significamus scribendo AB'B|C|D {illeg} {AB} et si id rursus ducatur, in {quamvis} \tertiam/ lineam C|E|F \ad experimendum/ parallelipipedum genitum {illeg} \scribimus/mus {sic} AB'C vel ABC AB'CD'EF. Latus verò quod oritur applicando quadratur \rectangulum/ illud ad lineam quamvis DGH sic notamus $\frac{AB' CD}{GH}$ ,|.| a|A|t ita in alijs. Sed et exposita linea {illeg} linea aliqua ad quam tanquam mensuram universalem aliæ omnes |(|ut fit in decimo Elementorum|)| referantur scribimus AB'CD ad designandam quartam proportionalem ab hac linea ubi \ubi per/ media|æ|s s|d|u{a|æ|}s sunt \sunt/ AB et CD et AB'CD'EF ad designandam etiam quartam \ab eadem linea per ubi/ ubi medi{illeg}|æ| du{illeg}|æ| sunt AB'CD et EF et sic in infinitum. Et si linea \linea/ \{illeg} mensura/ \linea/ illa sit prima continuè proportionalium et alia quævis AB secunda, tertiam sic designamus AB², quartam sic AB³, quintam sic AB⁴ at ita deinceps. Et {illeg} inter lineam illam et aliam quamvis AB notamus mediam proportionalem sic AB^{$\frac{1}{2}$}, primam e duabus medijs \proportionalibus/ sic AB^{$\frac{1}{3}$} secundam sic AB^{$\frac{2}{3}$}. {illeg} {Inter} lineam illam {et} AB'CD similiter ${\bar{AB' CD}}^{\frac{1}{2}}$ est media propor denotat tum latus quadrati æqualis rectangulo AB'CD tum mediam proportionale inter mensuram illam universalem et AB'CD vel quod perinde est {inter} {illeg} medio proportionalem inter AB et CD. Sed AB'CD^{$\frac{1}{2}$} est quarta proportionalis {inter} {illeg} a mensura illa ubi \{pem}//ubi\ medi{illeg}|æ| sunt /sunt\ AB et CD^{$\frac{1}{2}$}. Et has quantitates nominibus usitatis significamus præterquam quod a vocabi|u|lis Arithmeticis certas ob rationes cum veteribus abstinendum esse duximus. Porro quantitates compositæ eodem lege \modo/ signantur. Si {illeg}|{c}| A+B'C+D {illeg} A+B² denotat quadra A+B'C{illeg}−D denotat rectangulum sub A+B et C−D et A|R|±B|S|² quadratum ipsius A±B. Quæ quantitates juxta p{illeg}am propositionem \etiam partibus juxta/ secund{á}|ā| Elementorum in se ductis possunt et sic scrib|intur| A'C+B'C−A'D−B'D et A|R|²±2A|R|'B|S|+S². Ubi notes quod pars positiva ducta in subductitiam vel subductitia in positivam producit subductitiam du{illeg}{illeg}|æ| vero subductitiæ in se ductæ producunt poss{illeg}|it|ivam. Sit AB {cujumvis} seu |line{o} seu| men{illeg}ara l{i}nearum omnium ad quam {ubiræ} omnes referuntur secentur AB AD parallelis BD, EG ita ut sit AB ad AD ut AE ad AG et poss{illeg}|it|a AB mensu{illeg}ra illa ad quam lineæ omnes referuntur AG erit AD'AE. Diminuatur AE donec evanes{cet} et postea evadat retrors{a} Ae et A G simul ca|di|minuetur evanescet & convertitur in retror{s}am A{G}. Diminuatur

<135r> etiam AD donec evenescat et postea retrorsa evadat Ad et retrorsa Ag simul diminuetur evanescet et evadet directa Aα et convertetur in directam Aα. Duæ igitur retrorsæ Ae, Ad \by retrorsam id est/ directam Aα efficiunt.

^[98] {F{illeg}} {illeg} Præter F{illeg} Notis intellectis \proscognitis/ præmittenda est etiam methodus determinandi fluxiones linearum {f} et fluxionum plagas. Per Fluxionem intelligo celeritatem incrementi vel decrementio quan lineæ cujusvis indeterminatæ ubi lineæ aliquæ super alias in descriptione curvarum {illeg} moveri concipiantur aut quomodocun \et inter movendum/ augeri vel diminui aut motu punctorum describi. Unde et quantitates illas fluente \indeterminati|a|s illas quantitates/ fluentes nominare licebit. Proponantur dat{i|æ|} aliquot quantitates a, b, c, d, et fluentes x, y, z {et} sit ea fluentiū relatio inter se ut datur semper \quæ in {illeg} hac æquatione/ hoc aggregatum \complexio exprimitur/ $a x - \frac{b}{a} x x$ {+yx} \=/ +zz|−yz.| Fluat|n|t x y et z donec et |in| ea fluxione fiat sit sit {illeg}\r/o augmentum ipsius x, {illeg} \so/ ipsius y & $\frac{to}{}$ ipsius |z| ita ut x y et z jam fiant x+|{r}|o, $y + \frac{so}{}$ , $z + \frac{to}{}$ et perinde \has/ pro x y et z scrib \respectivè/ scribendo {illeg} æquatio superior fiet $a x + a r o - \frac{b}{a} x x - \frac{2 b}{a} x r o - \frac{b}{a} r r o o = z z + 2 z t o + t t o o$ −yz−yto−zso−stoo. Quantitates se per æqual vi primæ æquationes æquales dele et restab restabunt $a r o - \frac{2 b}{a} x r o$ $- \frac{b}{a} r r o o = 2 z t o + t t o o - y t o - z s o - s t o o$ . Applica omnia ad o \{illeg}/ et {illeg} fiet $a r - \frac{2 b}{a} x r - \frac{b}{a} r r o = 2 z t + t t o - y t - z s$ −sto.Proponantur dat{illeg}|æ| aliquot quantitates A, B, C, D, et fluentes |V| X, Y, Z quarum fluxiones respectivè designent{ur} {illeg} minusculæ |v,| x, y, z. et requiratur fluxiones linearum \alicujus/ quæ ex his genera{illeg}tur \fit/ ut linea X'Y. < insertion from lower down f 134v > Maneat primum X et Y fluendo fluat \Y/ donec ipsa fiat Z et XY fiat XZ et quia Y et XY fluendo non mutant rationem fluxiones earum erunt ut ipsæ hoc est ut 1 ad X. Unde cum fluxio Y sit y fluxio XY erit Xy. fluat jam Maneat jam Z et fluat X donec ipsa fiat V et XZ fiat VZ et fluxio XZ eri{illeg}|{t}| Zx ut in casu superiorem priore. Fluant jam X et X|Y| simul {donec} \motu/ celeritate priore donec XY fiat {illeg} ut XV una vice fiat VZ et quia fluxio Xy sufficit ad mutandum XY in XZ et fluxio Zx ad mutandum XZ in VZ Fluxio tota qua XY mutatur in VZ erit Xy+Zx. Pone V æqualem X et Z æqualem Y ut fit ipso fluendi initio et fluxio \initialis/ ipsius XY erit Xy+Yx.

Proponatur jam \factum/ XYZ et ponendo {V}{illeg} {=} XY=V erit XYZ=VZ. Ergo Cujus fluxio {illeg} jut|x|ta {illeg} casū priore{s} est Vz+Zv.|Sed|{illeg} et ob XY=V est Xy+Y{X}=v. Pro V et v scribe {sublimus} æquipollentia et Vz+Zv hoc \est/ fluxio ipsius XYZ fiet XYz+XZy+YZx. Et progressionis modum observando colligitur universalitur quod facti cujuscun {, ut}VXYZ{,} fluxio {illeg} semper invenietur substituendo sigillatim in facto illo pro unoquo{} factore {fluxionem} ejus et {sumendo resublantium} terminorum aggregatum. Qu{ot} regula < text from f 135r resumes > x'y'z', x², y³ et simili{us} ^[99] Fluant X et Y donec evadent x+ro et in ea fluxione {illeg} sit \ro/ augmentum {x+} ipsius X et so augmentum ipsius Y ita X et Y jam evaserint X+ro & Y+so et X'Y fiet X'Y+Xso+Yro+s{illeg}rosroo adeó augmentum ipsius x|X{'}Y| erit Xso+Yro+sroo. Jam {cum} fluxiones Defluant jam quantitates X+ro et Y+so donec {evase} iterum ad X et Y redierint et {erit} ultimæ rationis partius|m| evanescentium ro, so, Xso+ Yro+sroo quæ est {erit} {illeg} \ quæ est |eæ {e}runt quæ| sunt/ fluxiorum x, y et xy \{x, y, xy}/ quantitatum fluentium X, Y et XY Sunt igitur fluxiones illæ ut r, s, Xs+Yr+sro Sunt igitur x, y, xy ut {id} ut {illeg} Rationes ill{e} sunt ipsorum r, s, Xs+Yr+sro Applica partes illas ad communem factorem o, et {illeg} ia|od| pars ro jam nulla est \evanuit/ dele sro et fluxiones erunt |ut| r, s, Xs+Yr, {illeg} hoc est si r denotet fluxionem ipsas|is|s X et s eam ipsius Y, Xs+Yr denotabit eam ipsius XY. Pro r et s substitue æquipollentes x et y et Xy+Yx denotabit fluxionem ipsius XY.

Simili argumentatione fluxio ipsius XYZ {illeg} invenietur XYz+XyZy+{illeg}YZx et fluxio ipsius VXYZ invenietur VXYz +VXZy+VYZx{illeg}+{Y}|X|{V}|Y|Zv et sic in infinitum flu{illeg}|{xio}| fasti semper invenietur substituendo \sigillatim/ pro unoquo factore fluxionem ejus et sumendo resultantiun terminorum aggregatam. Quæ regula

<136r> etiam obtinc|e|t ubi aliqui factores æquales sunt. Ut si X et y æquales $\frac{sint}{ita}$ i|u|t XY valeat X²{illeg} erit ejus fluxio XY{+} Xy+ Yx id est \fiet/ 2Xx. Et similiter fluxio \|3|X²{x erit}/ ipsius X³ erit |3|X²x et fluxio eri|st|t 3X²x et ipsius X²{x}X²Z fluxio X²z+|2|Xz|Z|x. At ita in compositis{,} fluxio ipsius AX−3X² eri|s|t Ax−|6|Xx Nam fluxiones partium simul sumpt{illeg}|æ| sunt fluxio{illeg} totius.

In lateribus applicatorum ad fluentia methodus hæc est Proponatur \latus/ $\frac{X^{2}}{Y}$ . Pone ipsum æquale V et erit XZ|X|=YV adeo |2|Xx=Yv+Vy nam æqualium fluentium semper æqualium fluxiones æquales sunt. Aufer utrobi Vy et reliquum divisum per Y nemper $\frac{2 X x - V y}{Y}$ erit v. hac est \Est autem v ipsius V id est/ ipsius $\frac{X^{2}}{Y}$ fluxio quam invenire oportuit.

Similis est methodus in {illeg}|me|dijs proportionali{illeg}|bus|. Proponatur X^{illeg}Y^{$\frac{1}{2}$}. Pone ipsum æquale |V|Z|Y| et erit XY=X²V². Adeo Xy+Yx=|2|Vv et $\frac{X y + Y x}{2 V} = v$ .

Similis est methodus in med proportionalibus \{illeg}/{.} Proponatur /med{illeg} proportionalibus lateribus æquilaterorum\ \late{illeg}|ribu|s {illeg}|{quadraticis}|, {illeg}|{cubicin}| alijs./ {medium P} medium AX-X²|{illeg}lateribus æquilaterorum. Proponatur \quadrati abicujus/ latus AX-X²|^{$\frac{1}{2}$}. Pone ipsum æquale V, et erit AX-X²=V² adeo Ax−|2|Xx=|2|Vv, et $\frac{A x - 2 X x}{2 V} = v$ .

< insertion from lower down f 135v >

Ad AC demitte normalem BG et quia BG²+GA²=ABA² et BG datur erit |2|GA'ga=|2|ABA'ba seu GA'ga=BA'ba. Sed fluxiones ga et ca æquat{illeg} æ ædem sunt ergo |2|GA'ca =|2|ABA'ba et $\frac{GA' ca}{AB} = ba$ .

Fluat jam CA donec evas{erit} Ca. Dein \maneat{illeg} Ca/ fluat etiam CB donec evaserit Cb et demisso ad AH normali CB linea AB fluxione sua $\frac{GA' ca}{AB}$ {conver}{fit} linea convertitur in lineam {illeg}aB et similiter linea aB deinceps fluxione sua $\frac{HB' cb}{aB}$ convertitur in ab Ergo utra fluxione conjunctim convertitur AB in ab. Id est si CA et CB simul fluant erit $\frac{GA' ca}{AB} + \frac{HB' cb}{aB}$ fluxio ipsius AB, {illeg} \it/ quia Ab et aB initio æquales sunt, erit ead|{em}| $\frac{GA' ca + HB' cb}{AB}$ .

< text from f 136r resumes >

In figuris hæc est methodus. In triangulo ABC dentur angulus C et latus CB, fluant cætera. Et ubi CA fluendo evasit Ca simul BA evadat Ba. In Ba cape BD æqualem BA, et age A partes genitæ erunt Aa et Da. Defluant jam lineæ Ca Ba donec{illeg} ad priorem magnitudinem et positionem CA et BA redierint et partium {illeg} si ad CA demittatur perpendiculum BG, ultima ratio partium eva{n}escentium Aa, B|D|a erit ea quæ est linearum AB AG eo quod in eo casu AD perpendiculare fit ad Ba adeo triangul{illeg}|ū| aAD simile sit triangulo aBG id est triangulo ABG ea|v|adit simile. Quare cum fluxiones sint in ultima ratione partium per evanescentium erit AB BG ut AB ad B|A|G ita {illeg}fluxio ca ad fluxionem ba seu $\frac{AG' ca}{AB} = ba$ .

Fluant jam ruirsus CA et BA donec evaserint Ca et Ba et dein fluat etam CB donec ipsa evaserit CB et Ba evaserit ba et hac recta secund demisso ad CB normali AH fluxio ipsius aB hac secunda vice ex jam inventis erit $\frac{BH' cb}{aB}$ . AB fluxio{ne} \sua/ priori convertitur in aB{a} et \illud/ aB{a} fluxione posteriori convertitur in ab{a} Ergo utrà \fluxio{s}l{e}/ conjuxtim B{illeg}A{illeg} convertetur \AB/ in ab.|:| id est

<137r> Si CA et CB simul fluant ut|ita| ut AB {fiat it} evadall ab fluxio ipsius AB erit $\frac{BG' ca}{AB}$ AH $\frac{AG' ca}{AB} + \frac{BH' cb}{aB}$ seu quia initio fluxionis AB et aB æquales sunt, erit ea $\frac{AG' ca + BH' cb}{AB}$ .

Ubi CA et CB fluendo evaserint {illeg}Ca et Cb fluat insuper angulus ACB et fluendo fiat ACB. In aB cape aF=aβ et junge bF, bβ. Et ipso fluxionis initio bF perpendiculare erit ad ab vel AB adeo perallelum KI et bβ perpendiculare erit ad Cb adeo perpendiculare parallelus AK. Unde triangula βFb & AIK æquales habentia angulos ad b et K et rectos ad F et I similia erunt et bβ erit ad Fβ id est augmentum lineæ ab ipso fluxionis initio ad augmenttum arcus subtendentis angulum ACB radio Cb descripti vel quod perinde est fluxio lineæ ab ad fluxionem arcus illius ut AI ad AK, hoc est ut CI ad CB.

Cæterum ad fluxionem arcus angulum ad as \circuli dati/ significandam pono angulum literis minisculis nominando \illud ejus/ latus {illeg} \ej{us} ultimo/ loco quod radius est. Aut si neutrum anguli \ejus/ latus radius est ubi angulum literis minusculis posui subjungo radium literis majusculis cum litera {r} ijsdem prefixa \suffixa/. Est it fluxio lineæ ab \fluxio/ ad \arcus præfati fluxionem/ a{illeg}ca|b| ut CI ad BC. {illeg}|Adeo| {sunt} $\frac{CI' acb}{BC}$ vel quod perinde est $\frac{CI' acbABr}{AB}$ fluxio est ip{illeg}m \lineæ/ ab. Hanc fluxionem quia ab convertitur in aβ adde fluxione qua AB conversa fuit in ab et habebis totam fluxionem nempe AG'{c} $\frac{AG' ca + BH' cb + CI' acbABr}{AB}$ qua AB per fluxionē laterū CA, CB et anguli C convertitur in aβ. Est ita AG'ca +BH'cb+CI'c{illeg}ABr=AB'ab. {illeg}relatio{n} inter fluxio relatio{n} inter fluxiones {illeg}|tr|ium laterum et anguli trianguli cujuscun, cujus beneficio si tres ex his fluxionibus cognos cantur possumus quartam invenire.

In figuris pro significanda fluxione lineæ alicujus pono lineam ileam literis minusculis.|:| ut ab p bc pro{illeg} significanda fluxione lineæ BC. Angulorum verò fluxiones expono per fluxiones arcuum quibus subtenduntur ad datam distantiam. A Et distantiam illam quæcun tandem assumatur designe per literam R; fluxionem arcus per angulum literis minusculis serptum, lineam {tam} quæ {illeg} |opposit{a}m ordinatat{i}m applicatam in in hoc circuloid est {s}| sinum cujus arcus per literam s angulo præfixam \distantiam ordinatæ {illeg} {centro}seu id est/ sinum complemen{illeg} \{rationis}/ per literam s angulo præfixam & horum senuum fluxio s per literas easdem s it {'s}angulo min{illeg}|us|culis litera|i|s not{illeg}o {præteras}. Sit ABC{illeg} angulus quilibet \{flucus} {illeg}/ DC arcus \{s}uo/ ad {datam} {distantis} Bc subtend{illeg}|it|{{illeg}c{illeg}} & DE [ordenatim applicata est sinus {illeg} ad latus angulis BC sinus ejus et {vel} significabit {datum illam} sustantiam BC illam \datai|m|/ BC vel BD, {I} vel ab{illeg}c fluxio{n}e arcus CD, sB lineam DE, s'B lineam BE{&}sb s'b fluxi{orum} earum fluxiones

<138r>

Illud etiam præmittenduni est, fluxionem arcus esse ad fluxionem sinus ejus ut Radius ad sinum complementi et ad defluxionem com sinus complementi ut radius ad sinum. {illeg} ad sb ut R ad s'B et b ad−s'b ut R ad sB. Fluant enim omnes \aliquantulum/ donec HI fiat HM, IK fiat MN et CD fiat CF, DE fiat FG et B|C|E fiat B|C|G et ipso fluendi initio fluxiones erunt ut augment{e}|a| incipientia IM, ML, FD, FH, HD id est ut BD, BE, DE.

His præmissis proponatur triangulum aliquod ABC et demissis ad latera singula perpendiculis AE, BF, CG erit ex erit (ut e demonstratis trig{illeg}|on|ometricis notum est) sA.sB {illeg} \ut/ sA ad sB ut ut sA ad sB ita BF, ad AE & ita BC ad AC adeo sA'AC=sB'BC. Ergo sA'ac+AC'sa =sB'bc+BC'sb. Sed sa.a{illeg}|∷|s'A.R∷AG.Ac Ergo AC'sa=AG'a. Et eodem modo est BC'sb= BG'b. Quare sA'ac+AG'a=sB'bc+BG'b. Quo theoremate conferre possumus fluxiones angulorum duorum et laterum oppositorum trianguli cujuscun et ex tribus cognitis invenire quartam.

* < insertion from f 137v > * Et cum summa trium angulorum detur adeo aggregatum fluxiorum omnium nullum sit, vel quod perinde est duorum fluxio æqualis sit defluxioni tertij si pro +BG'b scribas + \−/BG'a+c, & \idem utrobi/ cons {illeg} AG'{illeg} \{addæ} BG'{illeg}/ BG' \auferas/ fiet sB{illeg}sA'ac+AB'a+BG'c=sB'bc, Theorema ad comparandas fluxiones duorum angulorum totidem laterum ubi quorum unum angulis illis in terijcitur. < text from f 138r resumes >

Rursus \*/ est R.s'A∷AC.AG ergo seu s'A'AC=R'AG. Ergo s'A'ac+AC's'a=R'ag. Est et (per Præmissa) s'A{illeg}'a ac∷sA R a.−s'a vel −a.s'a∷R.sA∷AC.CG. Ergo pro AC's'a scribendo CG'{illeg}−CG'a, fit s'A'ac−{illeg}CG'a=R'ag. Eodem modo est s'B'bc−CG'b=R'bg. {illeg}Et æqualibus æqualia addendo fit s'A'ac+s'B'bc−CG'a+b=R'ab. Ob datam summa trium angulorum pro a+b scribe −c et fit s'A'ac+s'B'bc+CG'c=R'ab. Theorema ad comparandas fluxiones trium laterum et anguli cujusvis.

< insertion from f 137v >

Simili argumentatione possunt alia Theoremata colligi ^[100] ubi perpendicula triangulorum et segmenta basium \aliæve lineæ/ considerantur. Sic AC'ac−BC'bc+BG'ab=AB'ag Theorema est ubi latera \tria/ et segmentum basis considerantur, et, {illeg}posito X commmunni trium perpendiculorum intersectione, est BX'ac +AX'bc=GX'ab+AB'gc Theorema ubi agitur de lateribus ut perpendiculo. Sed hæc et similia {illeg} necessaria sunt et a Geometris ubi usus eorum inciderit, ex inventis haud d{illeg}|if|ficulter colligentur. Sed hæc non prosequor. Satis est investigandi methodum aperuisse.

^[101] Horum verò Theorematum beneficio possumus in propositus {illeg} \quibus{illeg}/ figuris fluxiones flinearum et angulorum haud secus ac in computo trigonometric{os} lias ab alijs colligere donec ad quæsitam pervenimus. Ut si dentur positione lineæ AB, AD, DE et BC data longitudinis \moveatur perpetuo/ subtendæt{illeg}s angulum /A\ {illeg} <138v> et producta secans rectam ED in E, |et| ex cognita vel desideretur fluxio lineæ EC: primum in trianguli ABC per secundum \i{illeg}/ Theorematum invenietur sA'ac+BG'c=0 evanescunt enim fluxio termi{ni}

< text from f 138r resumes >

Simili argumentatione possunt varia Theremata colligi ad {datur} ubi seg{illeg}a perendicula triangulorum et segmenta bai|s|ium condiderantur. Ut si perpendiculorum communis intersectio sit x, erit Sic AC'ac−BC'bc+|−|BG'ab=AB'ag Theorema est ubi latera et segmentum basis consideratur et si trium perpendiculorum communis intersectio sit x, erit BX'ac+AX'bc {illeg}=GX'ab+AB'gc Theorema ubi latera et perpendiculum in quæstione sunt.

Porro in triangulis ACG, ACE ubi fluxio{illeg} anguo|i|rum recto|i|rum {illeg} E null{illeg}|æ| est {illeg} per Theorema in tribus novissimis primum {illeg} sA'ac+AG{'a}=R'gc {illeg} \mutatis {mutandir sit}/ sC'ac+CE'c= R'ae. Est et in triangulo ABC, per Theorematim secundum, sA'ac+AB=sB'bc−AB'{c}−BG'c. Quod h{a}c epsi sA'ac æquale est substitue in prioribus et orie{illeg}tur sB'bc+CE−BG'c−AB'a =R'ae, Theorema ubi Perpendiculum, angulus verttical{illeg}|e|s, basis et angulus \alter{uter}/ ad b{illeg}m in quæstione sunt Ubi si pro −a sub

<139r> stituas æquipollentem b+c, fit sB'bc {illeg}{AGc} \+AG{+}BG'c+CE+AG'c/ +B|A|B'b=R'ae. Quæ dei|u|o Theoremata casus omnes determinant ubi perpendiculum basis et anguli duo \quilibet/ in quæstione sunt. Sic et ubis Perpendiculum basis et latera duo in quæstione{illeg} sunt, {illeg} est \pos{illeg}|it|o/{.} X communi{illeg} intersectio|ne| trium perpendiculorum, et invenietur hoc theorema BX'ac+AX'bc=GX'ab+AB'GC. Alia {e}jusmodi Theoremata ubi bas perpendicula et segmenta basium {illeg} \aliæve lineæ/ considerantur Geometra, p{illeg} {re recta}{illeg} {illeg} \quoties usus eorum inciderit,/ insistendo vestigijs methodi hic patefactæ, facilè inveniet; \adeò/ ut rem plenius presequi super{illeg}acaneum ducam.

Cæterum Theorematum quæ frequentius usui fuerint et ex quibus cætera, siquando opus erit, \licebit/ derivare licebit, seriem sub convenit subjungere ut promptius citori possint.

\De Proportionalibus/ Th. 1. Si quotcun \continue/ proportionalium unum S, T, A, V, X, Y, unum A datur, fluxiones eorum erunt ut ipsa multiplicata per numerum locorum quibus distant a dato termino s.t.v.x.y∷−|2|S.−T.v|V|.2X.|3|Y.

Th. 2. Positis tribus continuè proportionalibus V.X.Y, si summa extremorum datur, erit ut excessus unius extremi supra alterum ad duplum medij ita fluxio medij ad fluxionem extremi illius alterius vel ad defluxionem prioris. V−Y. |2|X∷x.y∷x.−v. Et si differentia extremorum datur, erit ut summa extremorū ad duplum medij ita fluxio medij ad fluxionem alterutrius extremi. V+Y. {|2|}X∷x.y∷x.v.

Th. 3. Iijsdem positis si summa medij et extremi datur, a duplo medio aufer alterum extremum & residuum erit ad extremum prius ut fluxio alterius extremi ad fluxionem medij vel ad defluxionem extremi prioris Detur V+X, erit |2|X−Y.V∷y.x∷y.−v. Et si differentia medij et extremi datur, ad duplum medium adde alterum extremum, et summa erit ad extremum prious ut fluxio alterius extremi ad fluxionem tam medij quam extremi prioris. Detur V−X vel X−V, erit |2|X+Y.V∷y.x∷y.v.

Th. 4|2|. Positis quatuor proportionalibus XV.X∷Y.Z fluxiones extreorum mutuò ductæ in extremas æquantur flu{illeg}|xi|onibus mediorum mutuò ductis in medias. Vz+Zv=Xy+Yx.

De P{illeg}|Trian|gulis Contentis

Th. 5|3|. Contenti fluxio {ut quæ fit} \et quæ fit/ ducendo sigillat{um|i|} lateri{s} cujus fluxionem in contentum illud applicatum ad latus illud et summam productorum capiendo. Et fluxio aggregatio ex content{is} componitur ex fluxionibus parti{um} Sic fluxio XY est Xy+Yx et fluxio AY est Ay. Unde conjunctim fuxio AY+XY est Ay+Xy+Yx.

<140r>

De Triangulis

Th. 6. In triangulo quovis acutangulo, perpendiculo ad basem demisso, differentia fluxionum laterum ductarū in sinus conterminorum angulorum ad basem, æqualis est differentiæ fluxionum arcuum subtendentium angulos ad basem ductarum in contermina segmenta

De Triangulis.

^[102] Th. 6|4| In triangulo quovis \ad basem/ acutangulo, si perpendiculis ad basem demittatur, et fluxio lateris utrius ducta in sinum anguli \sibi/ contermini ad basem, seorsim addatur fluxioni arcus subtendentis angulum illum ductæ in conterminum segmentum basis; æquales erunt summæ. In triangulo ABC, demisso ad basem AB perpendiculo CG, est sB'bc+BG'b=sA'ac+AG'a. |Ad quod Theorema re{illeg}|{cu}|rrendum est quoties anguli duo et latera duo opposita in quæstione sunt.|

Th. 75. Fluxio In triangulo quovis \ad basem/ acutangulo, |si| perpendiculum demittatur ad basem, fla|u|xio lateris alterutrius ducta in sinum contermini anguli ad basem æqualis est summæ fluxionis lateris alterius ductæ in sinum contermini anguli ad basem & fluxionis arcus subtendentis angulum illum ductæ in basem et fluxionis arcus subtendentis angulum ad verticem ductæ in segmentum basis lateri primo conterminum. In triangulo ABC demisso perpendiculo CG est sB'bc =sA'AGac+AB'a+BG'c. |Ad quod Theorema recurrendum est ubi anguli duo et latera totidem interjectum et oppositum in quætione sunt.|

Th. {8}|6| In triangulo quovis \ad basem/ acutangulo, fluxio basis ducta in radium æqualis est summæ fluxionum laterum seorsim ductarum in {illeg}sinus complementorum conterminorum angulorum ad basem & fluxionis arcus angulum vertical{illeg}|{e}|m subtendentis ductæ in perpendiculum ab angulo isto ad basem demissum. In triangulo ABC demisso perpendiculo CG est R'ab=s'A'ac +s'B'bc+CG'c. {Adeo ad} Theorema {illeg} et ubi angulus et latera tria in queatione {sunt}.

Scholium {illeg} casus \{illeg}/ triangulorum \{illeg} habent/ a{illeg}|{illeg}|angulo{r}|s|{us} ad basem, {illeg} Theoremata extendu{i}r ad omnes casus. Nempe si angulus alterate ad basem obtusus sit, {illeg}|{s}|inus {illeg} complementi ejus et conterminam segmentum basis pro retrorsis haberi

<141r> debent et perinde signa eor{illeg}|um| de + in − mutari, terminis \positivis/ Theorematum \terminis/ in quibus {positive ponuntur} \{repericintur}/ transformatis in subductitios. Et quamvis hæc Theoremata de quatuor fluentibus proponuntur, continent tamen omnes casus triangulorū ubi tria vel duo tantum fluentia sunt. Numerus enim qua ternarius ex datis terminis trianguli implen semper implendus et in Theoremata delendæ fluxiones datorum. Ut si in triangulo ABC duratur angulus A et lat{illeg} {BC} oppositus et ex fluxiones alterius anguli B invenire vellem fluxionem lateris huic oppositi AC: quatuor termini quæestionem ingredient{e}s erunt tam {illeg} anguli duo datus A et fluens B et latera \duo/ opposita datum BC et fluens AC. {Vide} Consulo q{illeg} \igitur/ Theorema sextum \quartum/ quod hunc casum includit, et ibi deleta fluxione Anguli A dato datorum terminorum A et BC {prod} invenio BG'b=sA'ac; id est fluxionem lateris AC esse ad fluxionem arcus subtendentis angulum B ut BG ad sinum anguli A. Et eodem modo ubi in quovis trianguli{illeg} {illeg} \alicujus/. dantur duo termini quilibet, Theoremata se \semper/ resol{ē}|u||nt| in proportiones. Unde et proportiones illas nomine Theorematum tunc licebit citare perinde ac si \eædem/ in Theorematis expressæ fuissent. In triangulis autem rectangulis angulus rectus pro dato termino semper habendus est, {illeg}|et| numerum quatuor terminorum implet. Hoc modo semper incides in quatuor terminos;|.| q|Q|uibus cognitis in promptu Theoremata consulere |{estest}|per annotata in calce {illeg}cujus Theorematum legitimum Theorema consulere. Nec minori promptitudine recurretur ad Triangulorum easus sequentes.

De motuum plaga

{illeg}

^[103] Cas. {illeg}. De {illeg}erpendiculo bas{illeg} et lateribus.

De motuum plagis et fluxione
Curvarum.

Th. 7 Si recta mobilis BD rectæ positione datæ {BD} et ad datum punctum A terminatæ AB tanquam basi insistat et habeantur fluxiones basis illius AB rectæ insistentis BD et arcus subtendentis angulum quem h{illeg}|æ| rectæ comprehenurit exponantur fluxiones illæ \tr{e}s/ per \totidem/ rectus {illeg}S, T, V respectivè.

<142r>

De mottum plaga et celeritate

Th. 7. Ubi puncti alicujus motus pendet a diversis linearum et angulorum fluxionibus colligenda sunt sigillatim loca in quæ fluxiones singulæ \seorsim vel etiam plures earum conjunctim/ temporibus æqualibus punctum illud transferrtet si modo plagas et celeritates servarent inmutatas servarent inmutatas quas habent ipso fkuendi initio. Et lo{cos} ultimus locorum ultimus vel ultimorum intersectio ipse erit locus ad quem omnes fluxiones conjunctim eodem temporis spatio punctum illud rectà transfer{illeg}ent: adeo si a loco primo ad hunc locum recta linear ducatur, hæc et Curvam motu puncti descriptam tanget et puncti describentis celeritatem exponet longitudine suaexhibebit \seu fluxionem curvæ/ exponet.

Exempli gratia si linea mobilis {illeg}|in|mobili AB feratur et cognoscantur fluxiones lineæ illius mobilis AB, arcus angulum ABC ad datam distantiam subtendentis et lineæ |mobilis| BC: exponantur fluxiones illæ tres per totidem lineas S, T, V: In BC producta cape CD æqualem S, erige normalem DE quæ sit ad T ut BC ad r|R|adium, et {ipsi A{illeg}} age CF parallelam AB et æqualem V. Et eodem tempore quo fluxio lineæ BC si sola esset faceret punctum C transferri ad D, fluxio anguli B si sola esset ipsum transferri faceret ad E, et fluxio lineæ AB si sola esset, ad F. Duco igitur DG parallelam et æqualem CE et GH parallelam et æqualem CF, et concipiendo quod æqualibus temporibus punctum C sola fluxione prima transferretur de C in D \dein/ sola secunda de D in G, \postea/ sola tertia de G in H concludo quod omnibus conjunctis rectà perget{illeg} eodem tempore|i|s spatio de C ad H. Comple parallelogramma ECDG & CGFH si {illeg} motus transferre valet punctum C de linea CE ad lineam DG et alius \eodem tempore/ de linea CD ad lineam EG uter conjunctim transferent in diagonali CG {illeg}|ab| utrius \linea/ concursuus G. Et ex motu de C ad G et de C ad F rursus componetur motus in diagonali CD. Quare linea CD curvam puncto C descriptam tanget & puncti illius celeritatem seu fluxionem curvæ exponet. Duo Lotus CD, CE componunt motum CG in diagonali parallelogrami ECDG, et motus ille cum tertio motu CF componit motum CH in diagonali parallelogrammi FCGH. Erit igitur hæc linea CH tam tangens curvæ motu punctis C quàm expone{illeg} fluxionis ejus.

Rursus si a {illeg} rectis d{e}cabus positione datis AB, AI ad idem curv{illeg}|æ| alicujus punctum C conveniant line{illeg}|æ| duæ mobiles BC, IC, et ex cognitis fluxionibus anguli B et linearum <143r> BC, AI, IC determinandus esset motus puncti C motum quem punctum C haberet si linea BC solum modo flueret exporo quoad plagam et celeritatem per lineam CD;|.| d|D|ein motum quem idem punctum haberet si linea {illeg} \angulus B/ solummodo flueret expono similiter per lineam {illeg} etiam quoad plagam et celeritatem per lineam DG: Demi {illeg} quia fluxio lineæ AB ignoratur, motum puncti C ab ea oriundum expono quoad plagam tantum per lineam indefinitam GH ductam parallelam AB. Et considero lineam GH ut locum ultimum indefinitum puncti C. Tum perge|ns| ad alteram lineam mobilem IC {illeg}|a| motum puncti C a fluxione lineæ IC expo oriundum expono per lineam CK et motum ejus a fluxione lineæ AI oriundum per lineam KL, motum et ultimò motum ejus ab ignota fluxione anguli I ex pono quoad plagam per indefinitam lineam LM. Cum igitur {illeg} hi otus faciant ut punctum C ultimo locetur alicubi in linea LM, et priores motus ut ultimo locetur alicubi in linea GH, necesse est ut ultimo locetur in harum linearum communi intersectione H, |adeo| recta CH motum ejus exponet et Curvam ab eo descriptum tanget.

Ne res difficilior est si detur fluxionum duarum summa fifferentia vel proportio. Detur fluxio anguli B et summa fluxionum linearum AB, BC. Motum puncti C a fluxione anguli illius oriundum expone per lineam CG, dein positionem rectæ OQ ea lege quære ut si summa fluxionum linearum AB et BC dividatur utcun in duas partes et motus duo puncti C \qui/ ab his fluxioni partibus {illeg} seorsim orirentur exponantur per quod plagam et quartitatem per per GP et PQ, punctum Q semper incidat in hanc rectam. Et erit hæc recta locus ille ultimus indefinitus puncti C cujus intersectione cum alio ejusdem ultimo loco ex alijs quibusvis datis inveniendo determinabitur tangens. Determinabitur autem positio rectæ OQ p{illeg}roducendo GP ad C ut sit PO æqualis PQ et jungendo OQ.

Qum etiam si plurium linearum fluxiones sigillatim ducerentur in cognitas quantitates et productorum aliquorum summa poneretur æqualis sum {\{illeg}/} reliquorun, problema nihil minùs solvi posset. A mobili puncto C ducantur tres lineæ CA ad datum punctum A, CB et CD in datis angulis ad rectas positione datas BI ac DK, et fluxionibus earum respectivè

<144r> ductis in tres datas lineas L, M, N æquentur producta duo priora producto tertio, nempe L'ac+M'bc=N'dc.. |Et| Curvæ a puncto C descriptæ tangens ita ducetur. Ad AC erigo normalem CE quæ plagam motus puncti C circa |A| gyrantis exhibeat, et rectis BI DK parallelas CF duco CF CG quæ plagas motuum puncti C secundum rectas BI DK exhibeat. Dein quæro punctum aliquod H ita est si ad CE ducatur HE parallela AC, et ad CF {aut} autem HF parallela BC et ad CG HG paralella CD, fiat L'HE+M'HCF=N'GH et fluxiones linearum AC, BC, DC motus \relativi/ puncti C inde oriundi exponentur per lineas HE, HF, HG et motus absolutus puncti C per actam lineam CH. Hæc igitur erit tangens C|q|urv|æ|a|s|ita. Quomodo vero {illeg} \inveniri potest/ punctū aliquod H {ita} a quo \si/ ad rectas \quotcum/ positione datas CE, CF, CG totidem aliæ in datis angulis ducgantur quarum \agantur earum/ aliquæ ductæ in datas lineas \ductæ/ æquentur alijs in datas etiam ductis{.} patebit e sequentibus.

Sunt et alij casus difficiliores, ad quæ pergere liceret sed ex his credo sensus et vis \et vis/ Theorematis satis constabit Quapropter pergo jam exemplis aliquot methodum hic propositam illustrare

Proponatur Hyperbola Ellipsis vel Hyperbola ADB cujus centrum sit G, vertices A et B, \diameter AB/ ordinatim applicata ad diametrum CD, et latus rectum N et ex natura figuræ erit {illeg}ut ACB ad CD{illeg} ut ut AB ad N ita ACB ad CD² et (ab datam hanc rationem) ita \prioris/ fluxio {illeg} \prioris/ AC'cb+CB'ac, {illeg} id est {illeg} AC'cb−CB'cb seu 2GC'cb{illeg}, ad \posterioris/ fluxionem \{illeg}/ 2CD'cd. Exponantur (juxta Theor 7) fluxiones {illeg}cb et cd per FC et CD ita ut FD tangens fiat Ellipseos ad Desconver{i}|e|tur 2GC'cb{illeg}GC'FC et 2CD'cd in in {sic} GC^'|2|FCG' et 2CD²: quorum ita dimidia sunt ut ACB ad CD², adeo GC'CG et ACB æqualia sunt. Cape igitur FC ad AC ut CB ad CG {u}t acta {illeg}FD figuram tangent in D.

Propo

<145r>

In figuris hæc est methodus. Puncti mobilis considero semper motus diversos juxta diversas plagas quarum principalis sit via puncti. Et hos motus expono vel saltem exponi imaginor describendo per punctum illud circulum quemvis cujus centrum sit in via illa et in singulis plagis ducendo rectas us ad hunc circulum. Ut si punctum A moveaur in linea BA, per \illud/ A decribo circulum quemvis cujus centrum sit in BA et cui illa BA aliæq lineæ CA quævis CA, DA, EA occurrant in {illeg}F, G, H, I, et linearum partes intra circulus AF, AG, AH, AI erunt inter se ut motus puncti A in illarum plagis. Adeo ut si motus puncti A a B exponitur per AF, motus ejus a C exponatur per AG et sic in reliquis: Aut quod perinde est si fluxio lineæ A|B|A ex parte termini A exponitur per AF, aliarum linearum CA, DA EA ad idem mobile punctum A semper desinentium fluxiones ex parte termini illius A exponantur per AG AH, et lineæ EA defluxio per AI. Unde ex cogniis motibus duorum punctorum ad quæ linea quævis utrin terminatur, cognoscetur et exponi potest ejus fluxio absoluta: quippe quæ summa est fluxiorum ejus ad utrum terminum, vel excessus fluxionis ad unum terminum supra defluxionem ad alterum. Porro

Porro motus punctorum circa polos quosvis ijdem sunt |et| easdem habent exponentes cum ac motus in plagis perpendicularibus ad radios. Sic motus puncti A cir{illeg}|um| quemvis in linea CA{illeg} situm exponens est normalis AK circulo occurrens in K. Expositis vereo duorum punctorum rectæ cujusvis motibus circumpolaribus, recta alia per terminos exponentium acta secabit rectam illam in Polo suo. Et per harum exponentium rationem ad radios id est ad distantias suas a Polo, exponere licebit motum angularem lineæ \hujus rectæ/ seu fluxionem angulorum quos hæc linea \ea/ cum rectis positione datis continet.

Et ut ex motibus punctorum invenire et exponi possunt fluxiones linearum et angulorum sec vice versa ex horum fluxionibus colligere licet motus punctorum. Nimirum considerando lineam AF in qua punctum quodvis A movetur et ut exponentem motus ejus, et exponentis illus terminum ulteriorem F ut metam ad quam punctum illud A tendit{,} et lineas omnes FA, FG, FH, FI per metam transient{i}s ut {loca} rectæ ex inventione duorum ejusmodi locorum, meta quas {illeg} utras inter intersectione est determinabitur. Loca verò si invenientur. < insertion from lower down f 144v > Quando mobile punctum ex assumptione duarum quarum vis vel plurium {illeg} determinatum et stabile redditur, invenien{dus} est motus ill{illeg} quem punctum illud haberet si una quantitatum assumeretur et alterius tantum vel reliquarum fluxio maneret et motûs illius quoad plagam et quantitatem tam {designas} \exponens/ ducenda est. {Detero} Cognoscenda est etiam plaga motus <145v> quem punctum idem haberet si \vice versa/ illa una quantitas flueret et altera vel reliquæ assumerentur. Et in plaga illa per terminum exponentis acta recta loc{illeg} erit metæ unus {illeg} locis metæ. < text from f 145r resumes > Quando motus puncti ex diversas \{darbus} vel pluribus/ quantitatum fluxionibus certas ac determinatus redditur, quarum una ignota est vel ut ignota spectatur, ex \altera vel/ reliquis invenendus est motus ejus qualis foret si fluxio illa

<146r> ignota esset nulla. Et si motus inventi exponens ducatur & per terminum ejus in plaga qua punctum vi solius fluxionis ignot{illeg}|æ| pergeret, recta agatur, erit hæc unus e locis metæ. Metâ vero ex duobus locis inventa, simul habetur {illeg} exponens motus quæsit{illeg}. {illeg}

Hoc modo a motibus punctorum ad fluxiones quantitatus et vicissim ab fluxi harum fluxionibus ad illorum motus pergelicebit donec quoadus libuerit perventum sit. Et ubi exponens motus puncti curvam propositam describentis inventa est, hæc et cuevam in puncto illo tanget et exponens erit fluxionis ejus. Sed res exemplis clarior fiet.

^[104] A mobili puncto A qua linea curva describit ad rectam KL positione datam in dato angulo ALK.

Super recta KL positione data et ad datum punctum A terminata incedit recta AL in dato angulo ALK et termino suo A curvam lineam {illeg}KA describit. Datur relatio linearus KL et AL ad invicem et recta ducenda est quæ cuevam hanc tangat in A.

Quoniam {illeg} punctum A ex assumptione fluentium KL et LA determinatur, postquam earum fluxiones exposin per LM et AN motum quem punctum A haberet si ALA assumeretur et fluxio solius KL maneret expono quoad plagam et {illeg} quantitatem ducendo AD parallelam et æqualem LM, eo quod punctum illud A moveret per hanc AO eodem tempore quo KL evaderet KM. Deinde quoniam plaga motus puncti A foret AN si vice versa flueret ALA et altera KL assumeretur in hac plaga per exponentis terminum O duco indefinitam lineam OF et concludo hanc OF esse unum locum metæ. Et simili argumento quoniam AN exponens est tam motus puncti A ubi KL assumitur quam fluxionis lineæ K{illeg}LA per terminum ejus N in plaga motus quem punctum A ex sola fluxione ipsitus KL haberet duco lineam indefinitam NF pro altero loco metæ. Et ad {illeg} locorum concursus F meta erit ad quam tangens quæsita AF ducenda est. Quam conclusionem ut concinniorem reddas produc AF donec occurrat LK \etiam/ productæ in Q, et ob similitudinem triangulorum QLA, AOF erit QL ad LA ut AO ad OF vel \seu LM ad/ AN adeo vice exponentium AG et AN ad LM et AN adhibe {illeg} \{si} possunt/ QL et LA, qu{illeg} ratione \eo ut/ determina{illeg} \invenietur/ punctum Q. Comple parallelogrammum KLAH cujus latus HK tangentem secet in I & ob proportionales AH, HI et LM AN {,} vice exponentium LM et AN adhibere licet AH, HI eo ut invenietur punctum I. Utrumvis punctum Q vel I proet commodum videbitur quære.

<147r>

A mobili puncto A qua curva quævis EA describitur ad rectas duas positione datas DB, DC in datis angulis ducuntur rectæ duæ AB AC et ductarum relatio ad invicem habetur. Ducenda est recta quæ curvam hanc tangat in A.

Ut hot fiat exponentur ductarum fluxiones per AG et AH. Jam quia punctum A, assumptione fluentium DB, BA determinatur, et ubi earum una DB assumitur et altera BA solummodo fluit, linea AG exponens est tam motus puncti quam fluxionis lineæ BA, ubi vero vice versa altera BA assumitur el prior DB fluit punctum A movetur in plaga lineæ DB, recta GF quæ per exponentis terminum G in plaga lineæ DB parallela ducitur erit unus locus Metæ. Et simili argumento recta HF quæ per exponentis AH terminum H in plaga lineæ DC ducitur erit alias locus Metæ. Et locorum intersectio F metam dabit ad quam tangens quæsita AF ducenda est. Quam conclusionem sic concinnare licebit. Lecet tangens rectam DB in M et ipsi DC parallela agatur MN occurrens AC in N et AB, AM, AN erunt inter se ut AG, AF, AH, adeo vice exponentium AG, AH adhiberi possunt AB, AN: qua ratione longitudo AN at adeo punctum M ad quod tangens duci debet invenietur.

Ut is relatio inter AB et AC sit quod rectangulum sub AC et data quavis recta R æqual{illeg}|es| \sit/ quadrato AB², æquales erunt etiam horum fluxiones R' ac & 2AB'ab{.} |Hic| Pro fluxionibus ac et ab substitue earum exponentes AG AH, AH|vel| potius harum vice lineas AN, AB, et fiet R'AN=2AB². Unde R'AN et 2R'AC æquales sunt utpote eidem 2AB² æquales; adeo AN=2AC. Cape ergo CN={illeg}AC|&|Per N age ipsi CD parallelam age NM occurrentem DB in M et ra|e|cta AM curvam propositam tanget in A.

Hand secus si ad definendam relationem inter AB et AC ponatur R'AC−AC² esse ad AB² in data ratione, colligentur horum fluxiones R'ac−2AC'ac & 2AC'ab & 2AB'ab, et inde R'AN−2AC'AN in eadem ratione. Unde R'AC−AC² & $\frac{1}{2} R' AN - AC' AN$ æqualia erunt, utpote eandem rationem ad AB² habentia. Cae|i||atur| ergo AN ad AC ut R-AC ad $\frac{1}{2} R - AC$ et, actâ MN parallelâ CD, habebitur tangens AM. {Quod} \Porrò/ Curvæ EA h{e}c sit propietas ut si inter datos duos circulos EL \{illeg} a dato circulo/ FK per data puncta P, Q ducantur rectæ duæ LI, LK, concurrentes ad idem curva {illeg} ad \dat{illeg}/ circulum EL, ponatur AB æqualis LG et AC æqualis LK: ducantur circulorum tangentes IM KN, LR et fluxio arcus ER exponatur per LR cujusvis{.} longitudinis. Super diametro LR describatur circulus secans PL productam in S et QL in T{,} et erit LS exponens fluxionis {illeg} rectæ PL et LT exponens fluxionis retrogradæ rectæ QL. Erigantur normales LV ad LP et LX ad LQ

<148r> occurrentes circulo LTR in v|V| et X et erunt hæ exponentes motuum puncti L circa polos P et Q. Erigantur etiam normales YI ad PI et ZK ad QK ita ut sit YI ad IP ut VL ad LP et ZK ad KQ ut XL ad LQ et erunt hæ exponentes motuum punctorum I et K circa polos eosdem P et Q. R{illeg} punct{illeg} AI IY et KZ erige normales YM, & ZN occurrentes IM et KN in M et N et erit YM exponens fluxionis ipsius PI eo quod æqualis sit segm{en}to {quoad} parti lineæ PI productæ quæ \{illeg}/ intra circulum per puncta I et Y descript{am} et cujus centrum sit in IM circa centrum constitutum in IM descriptum caderet. Et simili argumento NZ exponens est fluxionis retrogradæ ipsius QK Conciper per puncta I et Y circulum describi cujus centrum sit in tangente IM et pariter per puncta K et Z alium circulus cujus centrum sit in tangente KN, et horum circulorum diametri IM KN exponentes erunt motuum punctorum I et K in circumferentia circuli JK: item YM exponens \æqualis/ erit exponenti fluxionis lunæ PI et N{illeg} ZN æqualis exponenti fluxionis totius IL et TL+ZN exponens fluxionis retrogradæ totius KL. Cape ergo AG=SL+YM et AH=LF+ZN, sed ob|f| fluxionem retrogradam ipsius LK vel AC cape AH ad partes ipsius A versus C, et HF acta parallela DC secabit GF actam parallelam DB in Meta F ad quam tangens quæsita AF duci debet.

Quod si vice rectarum LI, LK adhibeantur circulorum arcus EL, FK ponendo AB æqualem arcui EL et AC æqualm|e|m arcui FK, tunc AG sumenda erit æqualis arcui LR et AH {illeg}|æ|qualis KN, eo quod LR et exponens sit fluxionis arcus EL et KN exponens |de|fluxionis arcus FK, et act{illeg}|æ| GF, HF ut prius tangentem determinabunt. Ne problema difficilius esset \erit/ si vice circulorum EL FK adhibeantur aliæ quævis curvæ lineæ L quarum tangentes LR, KN ductæ habentur. Sed et alijs modis innumer{illeg}|ris| relatio inter AB et AC exprimi potest, \imò/ et vice rectarum DB, DC curvæ {illeg}|qu|vis adhiberi ad quas quam tangentes{sent} DB et DC \AB, AC ducantur in datis plagis et quarum tangentes DB et {illeg}/ ad puncta B et C sint DB et DC. et quad quas AB, AC ducantur i datis plagis.

Ducatur verò jam linea DB, DC non in datis plagis sed ad data puncta B et C, et earum fluxiones exponantur per AG et AH. Et quoniam assumptione anguli ABC et longitudinis BA determinatur punctum A, et ubi angulus ille \solummodo/ assumitur exponens motus puncti A est linea AG, ubi vero e{illeg} tra angulus ille fluit et longitudo BA ass{mi}tur plaga motus puncti A perpendicularis est ad BA, recta GF in plaga illa per exponentis terminum G ducta erit unus locus {metæ}. Et simili argumento recta HF per {t}erminum exponentis AH in plaga perpendiculari ad CA ducta erit alius locus

<149r> Metæ. B Et meta erit in locorum intersectione F Metæ. Et meta in utro loco consistens erit in eorum inter sectione F, adeo AF ad intersectionem illam ducta curvam motu puncti A descriptam tanget in A.

Ut si ea sit natura curva hujus ut summa vel differentia linea fluentis AB et datæ cujusvis R sit ad fluentem AC in data ratione (qui casus est quatuor Ovalium Cartesij) fluxiones illarum AB et AC erunt in eadem data ratione, adeo si \in plagis fluxionum illarum/ capiātur AG {ad} AH vel quod perinde est AN ad AC in illa \si/ in plagis contrarijs \capiantur/ An et AC in illa ratione et ad terminos captarum erigantur perpendicula concurrentia in F vel M acta AF vel AM curvam propositam tanget in A. Unde si ratio illa {illeg}est æqualitatis (qui casus est Hyperbolæ et Ellipsis) tangens bisecabit angulum CAN.

Ponamus jam super plano immobili in quo puncta{illeg} P et K et recta {illeg}infinita KD positione data habentur, planum {illeg} mobile B{illeg}CA motu parallela ita ferri ut puncta du{o} B, C in s{illeg} BCA curva \aliqua/ CA terminatum, ita ferri{,} ut puncta duo {illeg} recta BC in eo data semper coincidat cum linea in recta KD, et interea secum trahere regulam PB per punctum suum B perpetuo transeuntem et circa polum P rotantem {illeg}cujus {illeg} |,| & ejus intersectione cum termino suo curvilineo CA describere curvam lineam PAL in plano immobili{,} et requiratur hujus curvæ tangens ad punctum quodvis A. Quoniam assumptione recta {illeg} KD et DA rectæ KC et curvæ CA det{u}rminatur punctum A assumatur solummodo curva AC et sit CQ exponens fluxionis punct{illeg} lineæ KC et huic parællela et æqualis AG exponens erit motus puncti A, et GF duct{a|o|} parallela tangenti curvæ AC ad linea in plaga motus quem punctum A haberet si vice versa {illeg} KC assumeretur et curva CA solummodo flueret id est ducta parallela rectæ AD qu{æ} curvam AC tangit in A, erit unus locus metæ. Rursus quoniam punctum A assumptione longitudinis {illeg}|KB| et proportionis PA ad PB determ\in/atur, assumamus solummodo proportionem illam et punctum movebit in linea AG erit motus ejus ad motum puncti B ut PA ad PB. Exponatur ergo motus ejus per AH quæ sit ad alterius exponentem id est ad CQ vel AG ut PA ad PB, et per punctum H in plaga motus

<150r> quem haberet punctum A si vice versa KB assumeretur et ratio PA ad PB flueret, id est parallela PB acta recta HF erit alter locus metæ. Et Habetis autem duobus metæ locis habetur Meta in eorum intersectione F una cum tangente AF quæ ad metam duci debet. Quæm conclusionem si concinare animus est, produc tangentem donec secet BK in N, et ob similes figuras AFGH, NADB erit BN ad BD ut AH ad HG hoc est ut AP ad AB

Ut si Curva CA Parabola sit qu cujus vertex C diameter CK ordinatim applicata AI, |(|quo casu AL Parabola erit Cartesij|)| imprimis ducenda erit AD quæ Parabolam CA tangat in A quod fiet si capiatur CD æqualis CI, dein capiend{à} est BN ad BD ut AP ad AB et acta AN tanget curvam AL in A.

Quod si AC circulus sit centro A|B| descriptus, quo casu AL Conchoides erit Veterum, erigenda est ad AP normalis AD occurrens BN in D, hàc enim circulum illum tanget. Dein capienda est BN ad BD ut AP ad AB. Vel brevius capienda{illeg} \est/ BM=AP et erigenda normalis MN occurrens BD in N et acta AN figuram AL tanget in A.

<153r>

^[105] pag 130' post verba [ - Meritò dicentur ordinis ultimi] adde ^{[Editorial Note 3]}

|Genera Line{a}rum ejusdem {O}rdinis| Si linea aliqua \oculo extra planum ejus sito/ spectetur per planum translucidum, et in plano illo locus ejus apparens vel (ut voce mathematica utamur) projectio notetur, erit linea projecta ejusdm ordinis cum projiciente. Si projiciens est recta projectio erit recta, si curva est quæ rectam secare potest in duobus vel pluribus punctis, projectios ejus projectionem rectæ in totidem punctis secare potest. Et hinc habita linea aliqua cujusvis ordinis possunt aliæ plures ejusdem ordinis inde derivari. Sic Veteres ex circulo derivarunt omnes secundi ordinis figuras et inde Conicas sectiones nominarunt, considerantes spatium illud \solidium/ quod radijs per circuli \spretati/ perimetrum transeuntibus terminatur ut conum quem planum figuræ projectæ secat. S{illeg}t|ic| et figuræ superiorum ordinum possunt omnes a simplicioribus quibusdam ejusdem ordinis figuris \per successivas projectiones/ derivari{,} et inde distingui in genera coordinata pro numero figurarum quæ ad omnium projectionem {requiri{illeg}tur} \sufficiunt/ positis illis ejusdem esse generis quæ ab eadem figura derivantur. Nam hæ {illeg} omnes & solæ in se mutuò per projectiones transeunt et ea ratione cognotæ sunt, a cæteris verò in quas non transeunt alienæ. Hac lege unicum tantum est genus {f} linearum secundi ordinis, eo quod omnes derivantur a circulo: at ordinis tertij genera sunt quin.

^[106] In recta infinita EAB dentur puncta duo A, E et ad tertium quodvis ejus p punctum B in datos ang \in dato angulo/ erigatur {illeg} \ordinata./ BC ejus longitudinis cujus quadratum, si præterea dentur rectæ duæ M et N, {illeg}|æ|quale fuerit rectangulo sub N et AB una cum cubo ex AB applicato ad M $\frac{{AB}^{cub}}{M} + N \times EB$ . Et curva lin{i}|e|a ad quam hujusmodi recta omnis BC terminatur erit Parabola a{illeg}ta Parabolæ casus sunt quin principales; primus \et simplicissimus/ ubi linea N nulla est: Secundus ac tertius ubi N negativè ponitur et præterea AE est $\frac{2}{3} \sqrt{\frac{MN}{2}}$ , et secundus quidem ubi AE capitur ab A versus D seu versus alas figuræ, tertius verò ubi AE capitur ad contrarias partes ipsius A: Quartus et quintus sunt ubi AB est atrius{,} cujusvis longitudinis {illeg} ubi Parabola illa secas lineam utrin {illeg} puncto quintus verò {illeg} tribus{.} Primo casu habetur Parabola Neiliana cujus uti longitudinem \ubi {illeg}/ {Neilius noster primus} invenit: secundo haetur, Parabola {illeg} \{illeg} longitudinis/ parabola {campani} formis \ellipsin conjugatam {ab} {illeg} punctu{m} {illeg} quam punctum conjugatum/ {illeg} /{um}\ habens \habens/ conjuga{illeg} /{um:}\ {puncto parabola cum jam formis} solitaria quinto Parabola {illeg} {illeg} \{sum}/ Ellipsi{illeg} \habens/ conjugata{illeg} \quæ si in punctum {contragitur} cons{illeg} {illeg} illud conjugatum in casu tertio./ Et hæ qum figuræ cum profi{illeg} {illeg} cons{illeg} qum {fen}era curvarum tertij ordinis quinta nulla {ipsius} generis pro{si}cit aliquam alterius omnes verò quæ {illeg}generis per successivas projecti{illeg} in se mutuò {illeg} {illeg} curvæ omnes tertij ordinis comprehenduntur et eadem ratione curvæ superiorum ordinum subdistinguuntur in genera.

<154r>

^[107] Quinetiam per casus Projectionum distinguuntur genera {curva} linearum in species. Nomine{nuis} planum illud Horizontem quod per oculum transit et plano figuræ \lineæ/ projectæ parallelum est, et lineam illam Horizontalem in qua Horizon secat planum figuræ \lineæ/ projicientis. Et linea omnis \projiciens/ dubit tot projectionum species quot sunt \casus/ positionis lineæ Horizontalis casus respectu lineæ projicientis {illeg} \{species}/ positionum lineæ Horizontalis|.| ad respecta projicientis Si linea Horizontalis alicubi secat projicientem intersectio illa generabit in projectione cruræ duo Hyperbolica|i| generis cirva eandem Asymptoton ad oppositas plagas in infinitum tendentia, et linea quæ projicientem in puncto intersectionis tangit projiciet Asymptoton id ex eodem Asymptoti latere si intersectio sit in puncto flexus contrarij, aliter ex latere diverso, et Asymptotos erit projectio rectæ quæ curvam projicientem tangit in puncto intersectionis, tot ejusmodi crurum paria quot sunt in projectione quot sunt intersectiones lineæ Horizontalis cum projiciente. Unde curva \linea/ secundi ordinis non nisi duo paria crurum Hyperbolicorum habere potest, curva ter l|L|inea tertij ordinis non trisi tria paria {L}inea quarti quatuor &c; et ei|a|rum Asymptoti \tris vel plures/ se secabunt in uno punct{o} si tangentes se secant in uno, {illeg} \{illeg}bus vel/ pluribus; & si Projicit{u}r {illeg} \{senel} vel {illeg}/ {illeg} secat in puncto dec{crissatioris,} {illeg} duæ \{illeg}/ parallelæ {erunt}.

Si Linea Horizontalis {illeg} ge{n}erunt crura duo Parabol{illeg} generis ad eadem pla{illeg} in infi{tum} {illeg} et concavis partibus {illeg} {illeg}picientia, {insi} ubi cont{illeg} {illeg} est {illeg} puncto {illeg}casa crura Parab{illeg} ad mod{illeg} celerum {illeg} oppo{illeg} et ex eodem latere {concave} erunt {illeg} vertice {cento} curva alicujus {illeg} quum obliquissimè tangit Projicie{n}tem seu ut {projiriè} {illeg} {illeg} \angulo/ contactus crura Parabolica {illeg} ad pla{gas} oppositas {ut si} di{illeg} latere concavæ erunt, at si tangit ipsam {illeg} angulo qui rectilineo æqualis <155r> sit contactas ille generabit crura duo Hyperbolica ex eodem latere ejusdem Asumptoti ad eandem plagam in infinitum tendentia.

{D} Si linea Horizontalis et Asymptotos Projicientis crura Hyperbolica quæ circa Asymptoton illam sunt, convertentur in Parabolica:{illeg} Et vice versa si linea Horizontalis tendit ad plagam crurum Parabolicorum crura illa convertentur in Hyperbolica. Omnia vero crura infinita quæ non tendunt ad plagam lineæ horizontalis in omni casu evanescunt.

Si \deni Linea/ Horizontalis linea transit per punctum conjugatum, generabitur \curva/ linea cujus punctum conjugatum in infinitum abijt. Et ne punctum conjugatum infinite distans absurdum videatur scias projectiones hujus curvæ {projectione} curvæ haberæ puncta conjugata finitè distantia quæ sunt projectiones puncti illius infinitè distantis projectiones.

At hæ sunt mutationes linearum \li/ curvarum \linearum/ quæ projectione fiunt: quarum casus omnes \et eorum complexiones/ siquis ad curvam aliquam projicientem enumeraverit, is simul enumerabit figurarum \linearum/ species omnes quæ sunt ejusdem generis cum projiciente: salti|e|m si in lineis altiorum {illeg} ordinum Projicius \satis/ latè {illeg} sumitur.

^[108] Sic ubi Projiciens est circulus, Linea Horizontalis hunc circulum aut secabit in duobus punctis aut tanget in uno aut tota cadet extra circulum, et perinde Projectio aut quatuor habebit crura infinita crura Hyperbolica aut duo Parabolica aut nullum. Unde hujus ordinis tres erunt species Hypebo|rb|la Parabola et Ellipsis præter Circulum. At in generibus linearum tertij ordinis casus sunt plures

In primo Genere

^[109] 1. Si oculus infinitè distat, vel |si| planum project{illeg}|ion|is plano projicientis parallelum {illeg} est, projectio erit Parabola ejusdem speciei cum projiciente id est Parabola \cuspidata quam/ Neilianam quam et \nominavimus/. cuspidatam nominave licet.

2. Si Linea Horizontalis Projicientem in vertice cuspidata secat \transit per verticem cuspidatum Projicientis id/ in angulo contactus, Projectio erit Parabola corcumflexa \Wallisiana/, habens crura duo Parabolica ad oppositas plagas in infinitum tendentia et ex diverso latere concava. |et centrum in puncto flexus contrarij.|

3. Si linea illa Horizontalis transit per verticem cuspidatum et tendit ad plagam infinitorum crurum Projicientis Projectio erit Crux Hyperbolica {cuspida} \par/ \b{illeg}{ijnga} ad Diametrum librata/, habens duas Hyperbolas ex eodem habere unius Asymptoti ex diverso alterius. B{illeg} vero \{Parem}/ v{illeg}co {illeg} \lineam/ cujus area \comprehensa{illeg}/ {ita} {illeg} {æquales} partes a recta aliqua {illeg} ut ordinata ad {illeg} ex utro distantis {illeg}: {illeg} \{illeg}/ {non potest ita} secari{Brijngam} \Libratam/ vero {vaco} {illeg} curvam quæ diametrum \{illeg}/ rectie|i|neā habet ordinat{a}s h{illeg} ad{illeg} ordinatas {illeg}inde æquales terminantem: non {illeg} \{illeg}/ quæ {illeg}

4. Si linea illa transi{bis} {illeg} cuspidatum {illeg} tendit ad aliam quamvis plagam: Projectio {illeg} hyperbolica non {brijnga} \lineata/ habens Hyperbolas duas {duarum} duo crura ex diverso latere unius {illeg} ad {eandem plagam} altera duo ex diverso latere {deterius} {illeg}proti ad plagas oppositas tendunt.

5. Si tendit ad plagam crurum infinitorum et Projicientem nec secat nec tangit Projectio erit Cisso{i}s librata, et uno casu Cissois

<156r> {Veterum}

6. Si tendit ad plagam crurum infinitorum et secat Projicientem in duobus punctis Projectio erit Hyperbola triplex librata cuspidad|t|æ. Hyperbolarum una qu{illeg}|æ| cuspidata erit jacebit extra angulum Asymptotorum alteræ duæ jaceb non cuspidatæ jacebunt intra.

7 Si secat Projicientem in unico tantum puncto et non transit per cuspidem ejus Projectio erit Cissois circa Asympotom torta.

8 Si tangit Projicientem extra cuspidem, at adeo in alio etiam puncto secat Projectio erit Crux Parabolica cuspidata. Ejus crura duo Parabolica tendunt ad eandem plagam et concavitate se mutuò respiciunt, in vertice verò non junguntur sed postquam convergendo unum eorum processit in cuspidem, divergunt {denuò} et ad plagas oppositas \cruribus Hyperbolicis/ ex diverso latere ejusdem Asymptoti cruribus Hyperbolicis in infinitum tendunt.

9 Si secat Projicientem in tribus punctis Projectio erit Hyperbola triplex cuspidata non librata. Hyperbolarum illa quæ cuspidata est jacebit extra angulum Asymptotorum suarum, altera jacebit intra, tertia uno crure jacebit {i}ntra altero extra.

In secundo Genere.

1. Si oculus infinite distat vel si plana Projectionis et Projicientis parallela sint, Projectio erit ejusdem species cum Projiciente id est Parabola nodosa.

2. Si linea Horizontalis tendit ad plagam crurum infinitorum Projicientis et præterea secat projicientem \nullibi secat nec tangit/ Projicientem nec secat nec tangit Projectio erit Cissois nodosa {illeg}if librata.

3. Sin Projicientem tangit in vertice Projectio erit Crux Parabolica nodosa librata

4. Si secat eam inter vertiem et nodum projicietur Hyperbola triplex librata cum nodo in pari Hyperbolarum.

5. Si secat {a}lta nodum versus in ipso nodo, Projectio erit Hyperbola triplex librata duas ex tribus asymptotis parallelas habens.

6 Si secat ultra nodum versus crura infinita Projectio erit Hyperbola tri{illeg}|plex| librata cum nodo in impari Hyperbola.

7. Quod si linea Horizontalis non tendit ad plagam crurum infinitorum et se{cet} occurrit Projicienti in unico tantum puncto, Projectio erit Cissois nodosa circa Asymptoton torta

8. Si præterea tangit Projicientem inter verticem et nodum projicientur Crux Parabolica nodosa {,} non {librata,} clausa in vertice.

9 |11|. Si secat {eam bis} ad partes {nodi} versus vertic{e}m {et} semel alicubi projicietur Hyperbolæ triplex {non} librati cum nodo in pari Hyperbolarum

10|2|. Si secat {illeg} ad {illeg}versus verticem et bis in nodo projicietur Hyperbola triplex non {illeg} Asymp{totos} parallelas habens, \{illeg}Hyperbolam concavo {illeg} habens {illeg}/ et præterea {illeg} in {illeg} contrario {illeg}modo \si{illeg} modo linea/ Horizon \{illeg}/ secat Projicientem in ipso vertice.

11 |13|. Si secat \eam/ bis in nodo et semel cum sum versus crura infinita Projectio erit Hyperbola triplex non librata duas ex Asumptotis parallelas habens et inter eas Hyperbolam \ad easdem partes/ ommino concavam.

<157r>

Si secat unam Projicientis lineam in nodo et {ibidem} tangit alteram, Projectis erit Tridens Parabolica sive Parabola illa cujus proprietates {Cartesis}in Geometria, expli{mit}.

Si Secas {illeg} nodum \Projicientem/ in tribus punctis extra nodū versus crura infinita, Projectio erit Hyperbola triplex non bifida nodum habens in{illeg} {illeg}Hyperbola{illeg}

{Si} {illeg} secat {Parabolas}

{Si} tangit {illeg} eodem in crure infinito

9 Si tangit {projicientem} \eam {bis}/ {illeg} nodo Projectio erit Parabola Carte{illeg}.{illeg}

10. Si {tangit} eam {illeg}|ul|tra {illeg} projectio erit Crux Parabolica nodosa{,} non librata sep{illeg} {illeg}vertice

<158r>

In tertio genere.

1. Si oculus infinite distat vel si plana projectionis et projicientis parallela sint, projectio est, ejusdem speciei cum projiciente id est Parabola campaniformis cum puncto conjugato.

2. Si Linea Horizontalis vel tendit ad plagam crurum infinitorum vel transit per flexum contrarium Projicientis et præterea transit ultra punctum conjugatum Projectio erit Concha librata punctum habens conjugatum ad convexitatum verticis.

3 Sin transit per punctum conjugatum, orietur Concha librata cum puncto conjugato ad infinitam distantiam.

4 Si transit inter punctum conjugatum et Projicientem Projectio est Concha librata punctum conjugatum habens ad concavitatem verticis

5 Si tangit Parabol{am} Projicientem fit Crux Parabolica {punctotis} librata cum vertice aperto et puncto conjugato ultra verticem.

6 Si secat Parabolam \Projicientem/ inter verticem et puncta flexus contrarij fit Hyperbola triplex librata cum flexibus contrarijs in pari Hyperbolerum et flexu con puncto conjugato inter tres Asymptotos.

7 Si secat Projicientem in pun utro flexu contrario fit Hyperbola triplex, trifariam librata, sine flexu contrario, cum puncto conjugato in centro trianguli Asymptotis inclusi, quod centrum est figuræ projectionis.

8 Si secat Parabolam \Projicientem/ ad alteras partes \alterutrius vel utrius/ flexus{illeg} contrarij{rer} fit Hyperbola triplex librata cum flexibus contrarijs in impari Hyperbola et puncto conjugato int{illeg}|er| \tres/ Asymptotos.

9 Quod si linea Horizontalis nec tendit ad plagam crurum infinitorum nec transit per {punc} flexum contr{illeg}|ar|iū{;} {illeg}transit verò ultra \per/ punctum cong|j|ugatum, fit Concha \flexu contrario/ circa Asymptoton torta cujus punctum conjugatem in infinitum abit quæ insuper centrum habebit in flexu contrario ubi \si modò/ linea Horizontalis secat Projicientem in Vertice transit per verticem Projicientis.

10 Sin transit ultra vel citra punctum conjugatum et Projicientem secat in unico tantum puncto \extra flexus contrarios/ fle \Projectio erit/ Concha flexu contrario circa Asymptoton torta cum puncto {illeg} conjugato ad finitam distantiam.

11 Quod si transiens ultra vel citra punctum conjugatum tangit Projicientem projicientem \{Et} habebitur/ {autem} verticem ex fluxi{illeg} ultrariorū projicitur Crux Parabolica \non librata/ aperta in ver{sus} Asymp{toto} conjugado

12 Si deni secat Projicientem in tribus punctis projicitur Hyperbola triplex non {librata} {cum puncto} conjugato {inter} tres Asympto{tos.} Et una Hyperbolarum {illeg} ultra Asymptotos \sua/ altera {secet} uno {illeg} {ultra} {illeg}

In quart{illeg} {genere}

Species 1. 2. 3. 4. 5. 6. 7. 8. 9 {illeg}dem \{serit}/ {illeg} {describuntur} ac in Genere tertio species 1. 2. 5. 6. 7. 8. 10. 11{illeg}. 12 respectivé, nisi quod projectio\nes hic/ non |(|habent punctum conjugatum. Et {illeg} 4. 5. 6. 9 casus sunt {illeg} implicissimi ubi tres Asymptoti in unico puncto concurr{u}nt.

<159r>

In quinto genere.

Species 1. 2. 6. 7. 8. 9. 10. 11. 14. 15 ijsdem verbis describuntur ac in Genere tertio species 1. 2. 4. 5. 6. 7. 8. 10. 11. 12 nisi quod loco puncti conjugati Ellipsis conjugata ponenda est.

3. Si linea Horizontalis \vel/ ad plagam infinitorum crurum tendeus vel per punctum flexus contrarij transiens tangit Ellipsin ad partes|m| {illeg} exteriorem Projectio erit Parabola librata cum Concha quæ convexitate sua Parabolam respicit.

4 Sin secat Ellipsin Projectio erit Hyperbola {illeg} \duplex/ duplex cum triplex librata, quarum Hyperbolarum una est Conchordatis inter alias duas sita \cum concha interjecta: cujus casus est simplicissimus ubi tres/ se {illeg} concurrunt in eodem puncto.

5 Quod si tangit Ellipsin ad partes interiores seu versus Parabolam campaniformem Projectio erit Parabola librata cum concha quæ Parabolam concavitate sua Parabolam respicit.

12. Si tangit Ellipsin et non tendit ad plagam infinitorum crurū nec transit per flexum contrarium Projectio erit Parabola non librata cum concha flexu contrario circa Asymptoton torta.

13. Si secat Ellipsin in duop|b|us punctis et alibi {extra{illeg}} in tertio extra flexum contrarium; Projectio erit Hyperbola duplex non librata cum c|C|oncha flexu contrario circa Asymptoton torta: et præterea centrum habebit \in flexu illa contrario/ si linea horizontalis per tres figuræ \Projicientis/ vertices transit; quo casu tres \etiam/ asymptoti per centrum illud transibunt.

At hæ sunt species linearum tertij ordinis quarum formas et particulares proprietates conditiones fusius describere non operæ pretium duxi quoniam has ubi opus est Geometræ speculando formam situm et conditiones Lineæ Projicientis haud difficulter colligent. Mal{u}i propositionibus quibusdam porist{eis} \paucis/ inventionem

{Cae}terum qua ratione generaliores proprietates linearum inventis particularibus {eruantur} non pigebit paucis insinuaresg{illeg}|ene|raliorum proprietatum linearum aperire. U

Considero igitur quod quæ conveniunt duabus linearum speciebus convenire solent generi et quæ conveniunt duobus generibus convenire solent ordini et quæ conveniunt duobus ordinibus observato progressionis tenore convenire solent ordinibus universis: demide quod combinatio duarum simpliciorum linearum quarū ordines conjuncti efficiunt {illeg} ascendunt ad ordinem teneæ minus simplicis, \superiorem vicem obire potest lineæ illius ordinis superioris/ {superior} {vicem obira} er{illeg}die vicem {obira} {illeg} obit generis generis linearum illius ordinis superioris. \Ut Combinatio/ Duarū rectarū \combinat{illeg}/ vicem obit{illeg} bis generis linearum secundi ordinis \U combinatio duarum linearum primi ordinis vicem lineæ secundi/ {,} {illeg} combinatio trium, quatuor vel pluris vicem generis {linearum} Vertij quarti {{illeg}ut} superioris ordinis Et recta & linea \combinatio {unius linea primi superiores }/ secundi ordinis vicem generis {illeg} tertijs ordinis et sic in {illeg}lineas. Nam linea superioris ordinis {sæpe} transit {illeg} combinationem {illeg} {illeg}|sim|pliciorum \et combinatio cujusvis ordinis {illeg} \{illeg}/ recta {illeg} linea {tenævis} ejusdem ordinis/ {illeg} igitur proprietates combinatio {illeg} {incipienatus} combinationibus rectarum simplicioribus, et in serie rectarum \incipiendo {a simplicioribus}/ {illeg} considero proprietates {rectarum} combinatorium {in infinitorum deinde} in {illeg} in proprietates circuli vel alterus cujusvis non curvæ {illeg}{: cum} rectis in infinitum. {Nam quæ} {illeg} duabus generibus combination combinationum {con}{illeg} {solent} curvis {illeg} \{illeg}/ {per ordinis universos} \inveni{illeg}/, {fieri} vix potest {illeg} conveniant lineis et linearum combinationibus universis.

<183r>

Porismata

^[110] 15 Si punctis a duobus datis punctis A, B, C ad rectam Dz positione datam inflectantur duæ rectæ Bz, Cz secantes rectam A{illeg}y ^[111] ipsi positione datam \sit Ay/ {illeg}|et| parallelam Dz, habebunt Ax, Ay, xy datas rationes ad invicem.

Est enim Ax.Dz∷AB.DB∷dat.dat et Dz.Ay∷DC.AC∷ dat.dat. Ergo ^a^[112] Ax.Ay∷dat.dat et ^b^[113] Ay.xy∷dat.dat. Q.E.D.

26 Si a duobus datis punctis A, B, C ad rectam Dz positione datam in {illeg} \punctum tertium z concurrant/ duæ recta Bz, Cz d{a}tus secantes rectam Ay positione datam \in x et y/ et habeat Ax ad Ay datam rationem parallel{a} erunt Ay Dz tanget punctum z rectam positione datam.

Nam quia Ax est ad

Agatur enim per punctum Dz ipsi Ay parallela. Et quia Ax est Dz∷AB.DB et Dz.Ay∷DC.AC et conjunctis rationibus Ax.Ay∷ AB×DC DB×ACAB.DB{illeg}|+|DC.AC datur ratio {illeg} AB×DC.DB×AC. datur ratio AB×DC ad DB×AC sed datur etiam ratio AB ad AC ergo datur ratio DC ad DB \et divisim ratio DC ad {illeg} datam BC/ at{illeg} adeo datur punctum D. Datur autem puncto illa et angulo {illeg} etiam angulus D {illeg} et pr\o/inde rect{as} Dz quam punctum z tangit datur positione. Q. E. D.

37 Si a duobus datis punctis A B,C ad rectam positione datam Dz inflectantur rectæ duæ {illeg} Bz, Cz secantes rectam Ay ipsi Dz parallelam in punctis x et y et detur ratio Ax ad Ay datur Ay positione.

Nam ob parallelas Ay, Dz sit Ax.Ay∷AB×DC.DB×AC ut supra. Sed Ergo datur ratio AB×DC∷DB×C|A|C sed datur etiam ratio DC ad DB et|r|go et ratio AB ad AC, |ut| et dat{illeg} AB ad BC . Et inde of datam BC datur AB. dat{illeg} Dato autem tum puncto A tum {illeg}|an|gulo BAy datur positione Ay. Q. E. D.

47|8| Si a dato puncto B agatur recta Bzz secare parallela duas positione datas in x et z capiatur autem Ay et Ax in data ratione et jungatur zy converget zy ad datum punctum C.

Es Est enim Ax ad Dz ut AB ad DB hoc est in dat{a} ratione et Ax ad Ay in data ratione adeo^a^[114] Dz ad Ay in dat{a} ratione ergo \{illeg}/ /sed est\ DC ad AC in dat{a} ratione {est D{illeg} ad Ay} {in eadem ratione ergo} dat{illeg} /divisim\ ratio D{C} ad AD dat{illeg} {illeg} datur {illeg}et punctum {ergo} \et inde/ {illeg} {punctum} {illeg}

{5|8|} 9. {illeg}sdem {positios dantur} {illeg} zy {illeg} yC.

Nam{illeg} zy.yC∷DA.AC. {illeg}

610. {Easdem positios dantur ratione {illeg}} \{illeg}/ yxz, AxB, DzB, DAxz,\DAyz,/ BzC, ByC, ACy seu Dz {in datum.}

<184r>

^[115] D|P|

D|P|orism. 1 Si a datis duobus punctis B, C ad rectam D|A|z positione datam ^[116] concurrentes rectæ secent \in punctis x, y/ rectam Ay a dato puncto A ipsi BC parallelam ductam, erit Ax ad Ay in data ratione

N Nam si Az producta occurrat BC in D erit Ax.xy∷DB.BC ∷dat.dat Q{illeg}. E. D.

Porism 2 Et si a datis duobus punctis B, C ductæ rectæ Bz, Cz secent Ay in data ratione, erit punctum z tanget rectam positione datam.

Nam pro Age rectam zAD etoccurrrentem BC in D et erit Ax.xy∷DB.BC ergo {illeg} ergo datur ratio DB ad BC. Ergo datur punctus D.

Porism 3 Et si a dato puncto B agatur Bxz occurrens rectis positione datis Ax, Az in x et z detur autem ratio Ax ad xy. inclinabit zy ad datum punctum C. Est {illeg}

|Per B| Ipsi Az|y| parallela, agatur DBC occurrens rectis zA, zy in D et C. Ergo ^a^[117] datur punctum \linea/ DB. Est BC.DB{illeg}.BC∷Ax.xy .{illeg}dat{illeg} Ergo datur BC. Ergo datur punctum C. Q. E. D.

^[118] Porism 4 Si a datis punctis B, C concurrentes rect{illeg}|æ| Bz Cz secent{illeg} \in v et y rectas a datis punctis In/ rectas positioine datas et ipsi BC parallelas \ductas/ Iv, ny Sit Iv ad ny in data ratione tanget punctum z rectam positione datam.

Age BI occurrentem {illeg} ny in L et erit Iv ad Lx ut IB ad LB hoc est in data ratione Ergo Lx est ad ny in data ratione. In eadem ratione capiatur KL ad Kn et erit erit {sic} Kx ad Ky in eadem data ratione. Ergo (per Porism 2) punctum z tangit rectam zK positione datam. Q. E. D.

^[119] Porism. 11.{illeg} Si a datis punctis {A,} B, C conveni{u}nt \concurrant/ rectæ duæ Bz Cz secantes rectas positione datas Ax Ay in dat{a} ratione jaceant autem puncta ABC in directum, punctum z tangit rectam positione datam.

|Cu |Cas. 1.|| Junge xy et triangulum Axy dabitur {illeg}. Jam si xy parallela {illeg} {Se} sit ipsi BC, junge xy {secundum} Az in E, et produc{illeg} xy ad E {ut} sit E{x} ad xy {et} AB ad BC {illeg} et concurrent{es} {illeg} {rectæ} A {illeg} {illeg} puncto {illeg}z Atqui {illeg} {datam} {illeg} Ax ad xy {illeg} {Ex} ad {illeg} datur ratio A Ex {Ax}{illeg}{. Ergo} {illeg} \{illeg}/ {illeg} Ax {illeg} {et} recta AE positione. {Ergo} {illeg} {illeg}positione datas. Q. E. D.

^[120] Cas. 2. Sin {illeg} {perfectio} {illeg}. {illeg} due vy ipsi BC par{illeg} {et ob } {illeg} {rationem {illeg}x ad xy {illeg}} vx ad xy d{illeg}t{illeg} ratios A {speciefiguram Avx} {illeg} ratio Az|v| ad Av|x| In {ista} rationes {cape} DC ad EB ad A{B} et debitur punctum ad {D}C ad AE fac{illeg} {ut} sit AC ad EC nec non in ratione{m} A{illeg} \EC/ ad {illeg} \Av/ {ut} sit DC ad DB et dabitur punctum D. Ipsis Ax Ay {age} parallelas DC Dn occurrent{es} Bz Cz in {illeg} et n. {illeg} Converte{m} rationem {nomissimam} et fiet EC DC∷AB.DB∷Ergo EC ad Ax.DO Ergo {illeg} sit Av.Ax∷AC.EC et Ax.DO.∷{illeg}.DC erit \{denægno}/ Av.DO∷AC.DC. Sed {in} eadem ratione est Ay.Dn ergo {illeg} Avn {illeg} DOn . Ergo On parallela est BC. Ergo et

<185r> ^[121] ratio DO ad Dn datur. Ergo (per cas 1) punctum z tangit rectam positione datam. Q. E. D.

Cas 3. Si datur A ratio Ax ad {illeg}T \ay/ age Ay|T| ipsi {illeg}T \ay/ parallelam et dabitur ratio Ay|T| ad {illeg}T /ay\: quippe quæ eadem sit rationi AC ad aC. Ergo datur ratio Ax ad AT. Ergo \per cas 2/ punctum z tangit rectam positione data. Q. E. D.

Porisma 12

Definitio

Q Magnitudo P magnitudine major est sui parte {illeg} quam in ratione quando ablata \sui/ parte, reliqua ad eandem habet rationem datam

Porisma

I{y}sdem positis est si Ax est ad A \a/y ut datum data parte ipsius Ax auctum vel diminutum A|a|d datum, tangit punctum z rectam positione datam.

<186r>

The pricked circle is the Moon according to the parallex of M.C. 46′.20″. And so the digits by the type are 11.43′. which were observed 11.22. The luminous part alwayes seemeth broader than it is.

{In this Type the Sun standeth as in the former, for the time is the same. Now because then was the greatest Observation, it is manifest that the ☾ was then at s, k not at q where the Tables place it. the Tables gô give the ☾ 9'{illeg} too must in Longitude, as you may measure with your compasses in this Type.}

^{[Editorial Note 4]}

$\begin{array}{l} - 3 a x x & + 3 a x x & - a^{3} & = \frac{q q}{p^{3}} . a = \frac{2}{3} . \frac{8}{27} + \frac{}{27} - \frac{18}{27} \\ + 2 x x & - 4 a x & + 2 a a \\ + x & - a \end{array}$ $x - \frac{2}{3} = y y . x^{3} * - \frac{4}{3} x - \frac{2}{27} = \frac{q q}{p^{3}}$ $\begin{array}{l} x + a = y y . & x^{3} & + 3 a x x & + 3 a x x & + a^{3} & = \frac{q q}{p^{3}} . a = \frac{2}{3} \\ - 2 x x & - 4 a x & - 2 a a \\ + x & + a \end{array}$ $x^{3} * - \frac{4}{3} x + \frac{2}{27} = \frac{q q}{p^{3}}$

^{[Editorial Note 5]}

Eclipse of the Sun observed at Ecton A.D. 1652. marrs 29.h.10.32′. mins tempore apparente; sed tempore æqualis 10.26′. Digits eclipsed 11.22′.

This type agreeth with the Tables of M. lunitia & the Rudolphim. the other type repesenteth the observation

$\begin{matrix} By the tables & . & 0^{s} & . & 10 & . & 47 & ′ & . & 10 & ″ \end{matrix}$ $\begin{matrix} distance from — & 8 & . & 30 & . & 34 & . & 8 & . & 41 & ′ & . & 06 & ″ & . \end{matrix}$ $\begin{array}{l} right Asc. & 17 & . & 48 & . \\ Horarius – & 2 & . & 27 & . & 36 & . & 56 & . \\ domid: – & 15 & . & 12 & . & 16 & . & 10 & . \\ parall. hor. & 1 & . & 00 & . & 62 & . & 53 & . \end{array}$ $\begin{array}{l} \begin{matrix} Oriturim \{\begin{matrix} Æquatore \\ Eclipticâ \end{matrix} \\ Angulus Orions — \\ Dist. a Descend. – \end{matrix} & \begin{matrix} 85 & . & 48 & . & 15 & . \\ 114 & . & 12 & . & 26 & . \\ 42 & . & 01 & . & 30 & . \\ 85 & . & 05 & . & 18 & . \end{matrix} \end{array}$

For the Altitude of the Sun

Ut rad. ab sim. anguli orientis: ita sinus distantiæ ☉ ab ang a Decendente ad sim alt. ☉ 41.50′.

In the same maner {sic} I find the altitude of the next superiour degree in the Ecliptic to be 41.54′. & the altitude of the next inferiour degree to be 41.45′. The one being 4′ more than the Suns altitude, & the other 5′ lesse; I take the meane &c. $4' \frac{1}{2}$ for the distance of the Almicenters, sc. of the Almic. of the Sun, & the Almic. that cutts the Ecliptic either one degree before the Sun, or one degree behind him. & this number $4 \frac{1}{2}$ I keep

Now \I/ trace a line (AB) for the moones Orbit. & because the Eclipse hapneth in the 9^th degree from ♌ I prick that degree behoren 8 & 9 from a scale of one degree, or from my Sector set for the purpose, by where I can measure with my compass\es/ to the 6 part of a minute. The Lat. of 8°. is 43′.58″. I take it into my compasses from my Scale or Sector & setting one foot in 8 of the Orbit with the other I draw the arch about h. & the Lat. of 9° being 49′.20″ I take likewise & setting one foot in 9 of the Orbit I draw a second arch below the Orbit. about & by the outsides of these arches I draw the Ecliptic in his true situation.

Then from 8 in the Orbit I let fall a perpendicular (8h) upon the Ecliptic. which perpendicular falleth short of 8 in the Ecliptic by the quantity of the Reduction, which here is h 8 being 2′.03″. set 8 therefore in the Ecliptic {illeg} so much to the left hand (s.s.s.) from the perpendic. & 60 minutes furtherest 9.

Then I prick the center of the Sun upon the Ecliptic 30′.34″. from 8 toward 9. I & I prick the ☾ in her Orbit 41′.06″ from 8 toward 9.

From the Sun measure one degree in the Ecliptic bd. Take in your compasses the $4' \frac{1}{2}$ . before {illeg}ned for this purpose, & setting one foot in d, with the other draw the arch at e. & laying a ruler to the Sun & to the outside of this arch draw a strait line which shall be the Almicenter of the Sun. Then from the Sun {illeg} raise a perpendicular at right angles with the said Almicenter, & it shall be the Azimuth \of the Sun/. & draw a parallell thereto through the Moones {illeg}in her Orbit, & that shall be the Azimuth of the Moones.

I measure {illeg} from the Ecliptic downward in the Suns Azimuth so much as his parallax of {altit.} comes to (which here is 45″) & there set the apparent center of the sun (as at c) & there upon with his semidiameter 15′.12″ I draw his circle. Also from the Moones place in her Orbit {her} Azimuth I measure her parallax of alt. and {illeg} (m n {illeg} being 46′.20″. according to my Tables & {illeg} {illeg} where the parallax ends prick the moones apparent center (at n. {illeg}) & there{illeg} with her semidiameter (16′.10″.) describe her circle.

{illeg} you a ꝑfect type in which you may measure with your compasses what you will and if you would know the posture of the Luminaryes an hours or half or quarter {illeg} or before prick the points of 8 & 9 into another paper. & by these points draw {illeg} {illeg} then rectifie the places of the Sun & moone by adding or subtracting the {illeg} {illeg}here you must a\{illeg}/ {illeg} orient angle, \&/ altitude of the Sun {illeg} {illeg} the distance of the {illeg} {illeg} labour of {illeg} {illeg}

<1191r>

ed=x=distantiæ solis a planeta Rad=gb=a. abg=medio motai. b=ghb sini vel cosinui medii motus ab aplelio. af=q= diametro màximo ellipseos=be+ed. bc bd=c=distantiæ focorum. eb=q−x. qq−2qx+xx−zz{−}=xx−zz+\2/cz−cc. $\frac{q q + c c - 2 q x}{2 c} = z = b c$ . $b : a ∷ \frac{q q + c c - 2 q x}{2 c} : q - x$ . & therfor{{illeg}} 2bcq−2bcx=aqq+acc−2aqx. $x = \frac{a q q + a c c - 2 b c q}{2 a q - 2 b c} = e d$ . $x = q \frac{+ a c c - a q q}{2 a q - 2 b c} = \begin{matrix} distantiæ inter \\ solem et plan \end{matrix}$ $c d = b d - b c = \frac{c c - q q + 2 q x}{2 c}$ . $c d = \frac{c}{2} + \frac{q q}{2 c} \frac{+ a q c c - a q^{3}}{2 c a q - 2 b c c}$ . $c d = \frac{2 a q c c + b c^{3} - b c q q}{2 c a q - 2 b c c}$ $c d = \frac{2 a c q - b c c - b q q}{2 a q - 2 b c}$ . That is $c d = c + \frac{b c c - b q q}{2 a q - 2 b c}$ . $b e = q - x = \frac{a q q - a c c}{2 a q - 2 b c}$ . Therefore $2 a q c - b c c - b q q : a q q + a c c - 2 b c q ∷ a : \frac{a a q q + a a c c - 2 a b c q}{2 a c q - b c c - b q q}$ soe is cd to de. & $\frac{a a q q + a a c c - 2 a b c q}{2 a c q - b c c - b q q}$ is y^e secant of the angle edc.

In y^e Ellipsis of y^e Earths motion, ad:df∷ diameter of ☉ at f: Diamet of him at a. &|O|r {illeg} ad:de∷Diam{illeg} of ☉ at e:Diam of ☉ at a &c. & by this meanes y^e foci of y^e Ellipsis may be found.

Haveing y^e Aphelion viz ∠akl, y^e middle motion of y^e ☉ viz ∠eba+∠akl & y^e ☉s apparent place viz: ∠edp, taking any quantity for af to find y^e distances of y^e foci bd {illeg} Na{illeg} y^e given quantitys bg={r}d=Rad=a. gh=b. rs=c. bd=x. eb=y. af=q. bh=e. d{s}=d. Then, $a : b ∷ y : \frac{b y}{a} = e c$ $e d = q - y$ . $a : c ∷ q - y : \frac{c q - c y}{a} = e c = \frac{b y}{a}$ . $\frac{c q}{b + c} = e b = y$ . $\frac{b q}{b + c} = e d$ . $a : e ∷ \frac{c q}{b + c} : \frac{e c q}{a b + a c} = b c$ $a : d ∷ \frac{b q}{b + c} : \frac{d b q}{a b + a c} = c d$ . $\frac{d b q + e c q}{a b + a c} = x = b d$ .

Or if y^e angle (edb) bee right, & af=q. bg=a. bd=x. eb=y. gh=c. yⁿ $a : c ∷ y$ . $\frac{c y}{a} = e d$ . $a y + c y = a q$ . $\frac{a q}{a + c} = e b = y$ . $\frac{c q}{a + c} = c d$ . $\frac{\sqrt{a a q q - c c q q}}{a + c} = \frac{q}{a + c} \sqrt{a a - c c} = b d = \frac{a f \times b h}{g b + g h}$ . As for example if y^e greatest difference twixt y^e \{illeg}/ middle & apparent place of ☉ y^t is y^e \when he is at e y^t is y^e/ ∠bhg bee 2^degr..2′.54″.{illeg}. The signe of it 3574{illeg}|25|, y^e cosine 9993609{illeg}|=|gh, & y^e rad=gb=af=100000000. Then is /& ∠edb=90 degrees.\ $b d = \frac{a f \times b h}{g b + g h} = \frac{3574500000000}{19993609} = 178770$ . y^t is, af:bd∷10000000:1787{illeg}70. And this is y^e exactest way to find y^e Ellipsis of ☉. For in March {illeg}|&| September when ☉ is about 90^d {illeg} \2′ 54″/ of his meane motion from his Apogæ heeplace may perhaps be observed to bee {illeg} 90^d from his Apogæ of his apparent motion. That is y^e ∠bgh=2^d.2′.54″. when ∠edb=90^d.

Having y^e middle motion of a planet in its orbe {illeg} viz {illeg}, {to find} abe+fka. ba+ad=af=q=Radio. bh=b. {illeg} {illeg}. bg=a. be=x Then $b : a ∷ x : \frac{a x}{b} = b e$ . $\frac{q b - a x}{b} =$ $a : b ∷ x : \frac{b x}{a} = b c$ . $c d = \frac{c a - b x}{a}$ . $e d = q - x$ . $x x - \frac{b b x x}{a a} = q q - 2 q x + x x - \frac{c c a a + 2 c b a x - b b x x}{a a}$ . Or $\frac{a q q - a c c}{2 a q - 2 c b} = x = b e$ . &, $e d = q \frac{+ a c c - a q q}{2 a q - 2 c b} = \frac{a q q + a c c - 2 b c q}{2 a q - 2 c b} =$ to y^e distance of a planet from ☉. also {illeg} $b c = \frac{b x}{a} = \frac{b q q - b c c}{2 a q - 2 c b}$ , & $c d = c \frac{+ b c c - b q q}{2 a q - 2 b c} = \frac{2 a c q - b c c - b q q}{2 a q - 2 b c}$ . Making af=bg={illeg}q=ra{illeg}|di|o for brevitys sake, yⁿ $c d : e d ∷ 2 a c q - b c c - b q q : a q q + a c c - 2 b c q ∷ Rad = q : \frac{q^{4} + q q c c - 2 q q b c}{2 q q c - b c c - b q q} = r d =$ to y^e secant of y^e angle eda. Or{illeg} thus, $e d : c d ∷ q^{3} + q c c - 2 q c b : 2 c q q - b c c - b q q ∷ q : \frac{2 c q q - b c c - b q q}{q q + c c - 2 c b} = s d =$ to y^e cosine of y^e angle eda. Note y^t after y^e first operacon y^e calculacōn will bee very short. for haveing \once/ found 2cqq {illeg} & cc+qq I call, cqq=m. & $\frac{c c + q q}{2}$ . Soe y^t in all other opera{illeg}|co|ns {illeg} y^e wherein m & n vary not \as in y^e same planet/ y^e equation is $\frac{m - n b}{n - c b} = s d$ , soe y^t y^e middle motion {illeg}& consequent{e} (b) being given {illeg}sd{illeg} the cosine of eda is readily found. By this meanes y^e ☉s place in y^e Ecliptick may always bee found | $e d = q$ This equation may be ordered so y^t n, or {e} be a decimall {illeg}|

^{[Editorial Note 1]} Newton numbered the first two paragraphs to indicate their order within the text. These paragraphs have been moved in our transcription according to Newton's numbering.

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^[2] November 8^th 1665.

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^[21] May. 1665.

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^[24] {No}vemb^r y^e 13^th

^[25] {To} find y^e velocitys of bodys by the lines they de{scr}ibe.

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^[28] Of tangents to Geometricall lines.

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^[31] Of tangents to Mechanicall lines.

^[32] Of y^e crookednesse of Geometricall lines.

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^[34] May 30^th 1665

^[35] The resolution of plaine problems by y^e Circle.

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^[38] The construction of sollid — & Linear Problems

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^[40] A Generall rule wherby any Probleme may bee resolved.

^[41] By what lines a Problem{illeg} may bee resolved.

^[42] In how many points two lines may intersect.

^[43] How by the Parabola plaine problems are resolved.

^[44] How sollid ones.

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^[47] Constructions performed by a Parabola of y^e 2^d kind. x=y³

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^{[Editorial Note 2]} Written upside down at the bottom of the page.

^[51] † juxta proxima præce{illeg}us, erit VF−2gT {ad}{illeg} VF+2TV {ut} sinus refrationis ad sinum inciden{tiæ}.

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^[58] {illeg} equall, otherwise {illeg} they differ by {illeg} 5 let the differen{ce} of y^e numbers be {illeg} R & their sum {illeg} tenth ꝑ^t of their Summ S.

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^[78] {illeg} Jan: 6. 5^hor. 34′

^[79] {Decem} 1^st Epist. 2)

^[80] ✝ 9?

^[81] Nota. distantia Cometæ {illeg} stella B Feb 7 juxta Hookium fuit $\frac{1}{15}$ pars distantiæ primæ et {Sedæ} Arietis id, est {5}′ 12″ seu $\frac{93}{15}$ . Situs autem erat cometa in medio inter stellas B et C.

^[82] ‡ Hookius facit distntiam duabus minutis

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^[84] I Gradus Problematum:

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^[86] II Quantitates positivæ et subductitiæ cum earū notis

^[87] III Quantitates im{p}ossibiles.

^[88] IV Quibus line{illeg}|ijs| problemata solvuntur{.}

^[89] V Genera|Ordines| Linearum.

^[90] Modus exprimendi lineas

^[91] Locus \linearis/ puncti vagi.

^[92] Curvarum proprietates generales.

^[93] Tangentes ad Curvas descriptas duca|e|nture.

^[94] De cruribus infinitis et Asymptotis curvarum.

^[95] Quomodo curare in species distinguendæ

^[96] De curvarū tangentibus

^[97] Notarum quarundam explicatio.

^[98] Fluxiones quantitatum.

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^{[Editorial Note 3]} This edit was later made to the text on f. 130r by Newton.

^[106] Exemplum in {lineis} terij|t|ij ordi{s}.

^[107] Species Linearum ejusdem Generis.

^[108] Exemplum in lineis secundi ordinis ac tertij ordinis.

^[109] Exemplum

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^{[Editorial Note 5]} The following paragraph is written upsidedown at the bottom of the page

Newton's Waste Book (Part 3)

How to Draw Tangents to Mechanicall Lines

Lemma.

To resolve these and such like Problems these following propositions may bee very usefull.

To resolve Problems by motion ye 6 following prop: are necessary & suffcient.

To draw a tangent to ye Ellipsis

To draw a Tangent to ye Concha.

To find ye point c wch distinguisheth twixt ye concave & convex portion of ye Concha.

Concerning Equations when their rootes relation twixt \ratio of/ their rootes is considered.

Reductions of Equations may bee very often & readily \perhaps/ performed by this method

The same done otherwise.

Resolution.

Demonstration.

By helpe of ye preceding probleme divers others may bee readily resolved.

[28] 1. To draw tangents to crooked lines (however they bee related to sreight {sic} ones).

Resolution

[31] 2. Hitherto may bee reduced ye manner of drawing tangents in mechanicall lines. see Fol 50.

[32] 3. To find ye quantity of crookednes in Geometricall lines.

{illeg} Resolution

Of the construction of Problems.

Theoremata Optica.

Hujus autem Theorematis inventio totis est.

Probl.

Probl.

Telescopij novi delineatio

Some Problems of Gravity & levity &c

Descriptio cujusdam {illeg}|ge|neris curvarum {terty} \secundi/ ordinis.

A Method Whereby to Square lines Mechanichally.

Octob. 1676.

Prob.

Cas 1

Cas. 1.

Cas 2.

Prob Curvam Geometricam describere quæ per data quotcun puncta transibit.

Cas 1

Cas. 2.

Of the nature of Equations

Prob{l}

Of Equations,

To reduce sev{{illeg}}erall equations of four \by divisors of {three}/ dimensions.

How numeral æquations are to be reduced by divisors of 3 or 4 \or more/ dimensions

Veterum Loca solida componere restituta.

Problema

De resolutione Quæstionum circa numeros.

Loca plana.

De Loco rectil{í}neo.

Quæstionum solutio Geometrica.

Quæstionum solutio Geometrica

Quæstionum solutio Geometrica. Prob 1

Prob. 2

Prob. 3

Prob 4

[77] Ex observationibus proprijs Cometæ anni 168\0/.

Observationes Comet{illeg}|æ| habitæ ab Academia Physicomathematica Romana anno 1680 et 1681, a Ponthæo æditæ.

Observationes ejusdem cometæ habitæ a R. P Ango in Fleche

Observationes Venetijs habitæ a M. Montenaro.

Observationes Hevelij destituti instrumentis

Observationes Cometæ Mense Novembri anni 1680

Observationes Parisijs habitæ Cometæ subsequentis 1680 & 1681.

Ejusdem posterioris Cometæ Observationes Grenovici habitæ

Observationes de Cauda Cometæ prioris

De cauda Cometæ posterioris

Ex Hookij Cometa edito ann 1678.

Via cometæ anni 1664 & 1665 juxta delineationem Hookij.

Cometæ anni 1661 loca ex Hevelio

Problemamm solutiones juxta sequentes Regul{a}s

Geometria. Lib. 1.

De P{illeg}|Trian|gulis Contentis

De Triangulis

De Triangulis.

De motuum plaga

De motuum plagis et fluxione Curvarum.

De mottum plaga et celeritate

In primo Genere

In secundo Genere.

In tertio genere.

In quart{illeg} {genere}

In quinto genere.

Porismata

Porisma 12

To resolve Problems by motion y^e 6 following prop: are necessary & suffcient.

To draw a tangent to y^e Ellipsis

To draw a Tangent to y^e Concha.

To find y^e point c w^ch distinguisheth twixt y^e concave & convex portion of y^e Concha.

By helpe of y^e preceding probleme divers others may bee readily resolved.

^[28] 1. To draw tangents to crooked lines (however they bee related to sreight {sic} ones).

^[31] 2. Hitherto may bee reduced y^e manner of drawing tangents in mechanicall lines. see Fol 50.

^[32] 3. To find y^e quantity of crookednes in Geometricall lines.

Descriptio cujusdam {illeg}|ge|neris curvarum
{terty} \secundi/ ordinis.

Prob
Curvam Geometricam describere quæ per
data quotcun puncta transibit.

How numeral æquations are to be
reduced by divisors of 3 or 4 \or more/ dimensions

Quæstionum solutio Geometrica.
Prob 1

^[77] Ex observationibus proprijs Cometæ anni 168\0/.

Via cometæ anni 1664 & 1665 juxta
delineationem Hookij.

Problemamm solutiones juxta sequentes
Regul{a}s

Geometria.
Lib. 1.

De motuum plagis et fluxione
Curvarum.