Mathematical Notebook

Author: Isaac Newton

Source: MS Add. 4000, Cambridge University Library, Cambridge, UK

Published online: October 2011

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  - 14 February 2011
    - Catalogue information compiled from CUL Janus Catalogue by Michael Hawkins
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<2r>

Of y^e extraction of \Pure/ Square Cubick. Square-square & square-cubick rootes &c.

Let y^e number whose roote is to bee extracted bee pointed bec makeing y^e first point under y^e {ut|n|ite} & comprizeing soe many numbers under each point as y^e number hath dimensions as if y^e number be square-cube tis thus pointed $570 \underset{.}{8} 6352 \underset{.}{4} 1080 \underset{.}{2}$

Then then then t out of y^e figures of y^e first point next y^e left hand extract y^e \greatest/ roote proper to y^e power of y^e number & set y^t downe in y^e m{illeg}|Quo|tient w^ch is y^e firt{illeg} {sic} side & is called A. (as y^e roote quintuplicate of 5708 is $(5)$ , & $(5)$ quintuplicate is 3125 ) yⁿ takeing y^t roote duely multiplied out of y^e number (as 3125 out of 5708 ) w^th y^e rest of y^e numbers to y^e next point. |seeke| {illeg}|y|^e seacond side w^ch is found by divideing y^t number by another number made out of y^e first side (w^ch is called y^e Divisor) & this second side I name E. (thus by divideing 258363524 by $5 A$ /qq\\ $+ 10 A c + 10 A q + 5 A$ / after such a mane {sic} y^t $5 A qq E + 10 A c E q + 10 A q E c + 5 A E qq + E qc$ may be conteined in y^e number y^e product of y^t division shall be E =

<2v>

The extraction of y^e sqare {sic} roote

$\begin{matrix} The square to be resolved 2 \underset{.}{9} & 1 \underset{.}{6} (54 The Product \\ The square of y^{e} first side 25 & be taken away. \\ The rest of y^{e} sqare to be 4 & 16 resolved. \\ The divisor for finding y^{e} seacond 1 & 0 sidie. which is y^{e} first side doubled \\ \begin{matrix} The rectangle by 2A & E 4 \\ The square of E \end{matrix} & \begin{matrix} 0 \\ 16 \end{matrix}} to be substracted \\ The su \bar{m} e of y^{e} rectangles 0 & 16 to be subducted \\ 0 & 00 The remainder \end{matrix}$

The extraction of y^e cube roote

$\begin{matrix} The cube to be resolved 15 \underset{.}{7} & 46 \underset{.}{4} (54 \\ The cube to be subducted 125 & whose roote is A = 5 \\ The remainder for y^{e} finding 32 & 464 of E \\ \begin{matrix} \begin{matrix} The divisors for y^{e} finding \\ of (E) y^{e} seacond side. \end{matrix} & {\begin{matrix} 7 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 5 \\ 15 \end{matrix} & \begin{matrix} 3 A q \\ 3 A \end{matrix} \end{matrix} \\ The sume of y^{e} divisors 7 & 65 \\ \begin{matrix} \begin{matrix} Sollids to be substracted \end{matrix} & {\begin{matrix} 30 \\ 2 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 40 \\ 64 \end{matrix} & \begin{matrix} 3 A q E \\ 3 A E q \\ E c \end{matrix} \end{matrix} \\ The sume of those 32 & 464 sollids \\ The remainder 00 & 000 \end{matrix}$

The extraction of y^e square square roote

$\begin{matrix} The square-square 3 \underset{.}{3} & 177 \underset{.}{6} (24 \\ The square-squ: to be subduc: 16 & = A q q \\ Remainder. 17 & 1776 \\ \begin{matrix} \begin{matrix} Divisors for finding y^{e} \\ seacond side E. \end{matrix} & {\begin{matrix} 3 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 2 \\ 24 \\ 8 \end{matrix} & \begin{matrix} 4 A c \\ 6 A q \\ 4 A \end{matrix} \end{matrix} \\ Theire su \bar{m} e 3 & 448 \\ \begin{matrix} \begin{matrix} Squ-Squares to be sub= \\ =ducted \end{matrix} & {\begin{matrix} 12 \\ 3 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 8 \\ 84 \\ 512 \\ 256 \end{matrix} & \begin{matrix} 4 A c E \\ 6 A q E q \\ 4 A E c \\ E qq \end{matrix} \end{matrix} \\ Theire Su \bar{m} e 17 & 1776 \end{matrix}$

<3r>

The extraction of y^e Square-Cube roote

$\begin{matrix} The squ: cube to be resolved 7 \underset{.}{9} & 6262 \underset{.}{4} (24 \\ Substract 32 & A qc \\ Remain d^{e} 47 & 62624 \\ \begin{matrix} \begin{matrix} Divisors \end{matrix} & {\begin{matrix} 8 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 80 \\ 40 \\ 10 \end{matrix} & \begin{matrix} 5 A qq \\ 10 A c \\ 10 A q \\ 5 A \end{matrix} \end{matrix} \\ The Sume of y^{e} divisors 8 & 8410 \\ \begin{matrix} \begin{matrix} Plano-Sollids to be \\ substracted \end{matrix} & {\begin{matrix} 32 \\ 12 \\ 2 \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 0 \\ 80 \\ 560 \\ 2560 \\ 1024 \end{matrix} & \begin{matrix} 5 A qq E \\ 10 A c E q \\ 10 A q E c \\ 5 A E qq \\ E qc . \end{matrix} \end{matrix} \\ Theire Su \bar{m} e 47 & 62624 \\ Remainder 00 & 00000 \end{matrix}$

Note y^t y^e 3^d 4^th 5^th & other figures are found by y^e same mann{illeg}|e|r y^t y^e seacond figure is found onely makeing all y^e figures found to stand for A y^e first side & {illeg} y^e figure sought for e or y^e 2^d side

And if y^e number {propounded} be {t} roote is found inexpressible in whole numbers y^t|n| adding ciphers & pointing them f{illeg}rom y^e unite towards y^e right \{kind}/ as was before exp{illeg}|l|ained & soe hold on y^e worke in decimalls.

As for y^e Divisors they are easily found by y^e 2^d Table of Powers from a Binomial roote.

If y^e {illeg} Number bee of 6.7.8.9.10 &c dimensions The roote may be extracted after y^e same manner

<4r>

Of y^e Exta{illeg}|c|tion {sic} of R{illeg}|o|otes in Affected powers.

The manner of y^e extraction of rootes in pure & affected powers is verry {sic} much alike, \especially/ when y^e affected powers are decently prepared, y^t is, when theire affections are not over large & those altogether either affirmative or negative, & y^e power affirmative, affirmations & negations so mixt y^t there be noe ambiguity & all fractions & Asymmetry taken away

All y^e figures in y^e coeffents {sic} & affected power are to be pointed (af{illeg}|t|er y^e manner before eplained {sic} in y^e Analisis of pure powers) according to y^e degree of theire dimensions & the worke onely differs from y^t in pure powers y^t in y^t {illeg}|y^e| coefficients enter into y^e divisors

Let y^e first side be called A. y^e 2^d be called E. y^e Roote of y^e equation B{L} y^e coefficients $B . C q . D c . F qq . G qc .$ $H cc$ &c y^e Power $P . P q . P c . P qq$ &c & y^e Operation follows

The analysis of Cubick Equations.

The equation supposed $L c^{*} + 30 L = 14356197$ . $L c + C q L = P c$
$\begin{matrix} \begin{matrix} \begin{matrix} \underset{}{} & The square coëfficient \\ \underset{}{} & The cube affected to be \end{matrix} & \begin{matrix} \underset{}{} \\ 1 & \underset{.}{4} \end{matrix} \end{matrix} & \begin{matrix} \underset{}{} & 3 \\ 3 & 5 & \underset{.}{6} \end{matrix} & \begin{matrix} \begin{matrix} \underset{.}{0} & \underset{.}{} & \underset{.}{} \\ 1 & 9 & \underset{.}{7} \end{matrix} & \begin{matrix} (243 \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollids to bee substracted \end{matrix} & {\begin{matrix} 8 \end{matrix} \end{matrix} & \begin{matrix} 6 \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} & \begin{matrix} = A c \\ = A C q \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Theire su \bar{m} e \end{matrix} & \begin{matrix} 8 \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & 6 \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Rests \end{matrix} & \begin{matrix} 6 \end{matrix} \end{matrix} & \begin{matrix} 3 & 5 & 0 \end{matrix} & \begin{matrix} \begin{matrix} 1 & 9 & 7 \end{matrix} & \begin{matrix} for finding y^{e} 2^{d} side \end{matrix} \end{matrix} \\ The extraction of & y^{e} seac & ond side \\ \begin{matrix} \begin{matrix} \underset{}{} & Coëfficient \\ \underset{}{} & The rest of y^{e} cube to be \end{matrix} & \begin{matrix} \underset{}{} \\ \underset{}{} & 6 \end{matrix} \end{matrix} & \begin{matrix} \underset{}{} \\ 3 & 5 & \underset{.}{0} \end{matrix} & \begin{matrix} \begin{matrix} 3 & \underset{.}{0} & \underset{.}{} \\ 1 & 9 & \underset{.}{7} \end{matrix} & \begin{matrix} or superior divisor & \underset{}{} \\ resolved & \underset{}{} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} The inferior divisors \end{matrix} & {\begin{matrix} 1 \end{matrix} \end{matrix} & \begin{matrix} 2 \\ 6 \end{matrix} & \begin{matrix} \begin{matrix} 0 \end{matrix} & \begin{matrix} 3 A q \\ 3 A \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Theire su \bar{m} e \end{matrix} & \begin{matrix} 1 \end{matrix} \end{matrix} & \begin{matrix} 2 & 6 & 0 \end{matrix} & \begin{matrix} \begin{matrix} 3 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollids to be sub= \\ stracted \end{matrix} & {\begin{matrix} 4 \end{matrix} \end{matrix} & \begin{matrix} 8 \\ 9 & 6 \\ 6 & 4 \\ 1 \end{matrix} & \begin{matrix} \begin{matrix} 2 & 0 \end{matrix} & \begin{matrix} = 3 A q E \\ = 3 A E q \\ = E c \\ E C q \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Theire su \bar{m} e \end{matrix} & \begin{matrix} 5 \end{matrix} \end{matrix} & \begin{matrix} 8 & 2 & 5 \end{matrix} & \begin{matrix} \begin{matrix} 2 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \end{matrix}$

<4v>

$\begin{matrix} \begin{matrix} \begin{matrix} The superior part of y^{e} divisor \\ The remainder for finding \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 5 & 2 & 4 \end{matrix} & \begin{matrix} \begin{matrix} 3 & \underset{.}{0} \\ 9 & 9 & \underset{.}{7} \end{matrix} & \begin{array}{l} or y^{e} square coefficient \\ y^{e} third side \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} The inferior part of y^{e} \\ divisor \end{matrix} & {\begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 1 & 7 & 2 \end{matrix} & \begin{matrix} \begin{matrix} 8 \\ 7 & 2 \end{matrix} & \begin{matrix} 3 A q that is 3 \times 24 \times 24 \\ 3 A or 3 \times 24 . \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} The su \bar{m} e of y^{e} divisor \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 1 & 7 & 3 \end{matrix} & \begin{matrix} \begin{matrix} 5 & 5 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollids to be taken \\ away \end{matrix} & {\begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 5 & 1 & 8 \\ 6 \end{matrix} & \begin{matrix} \begin{matrix} 4 \\ 4 & 8 \\ 2 & 7 \\ 9 & 0 \end{matrix} & \begin{matrix} 3 A q E \\ 3 A E q \\ E c \\ E C q \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Theire Su \bar{m} e \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 5 & 2 & 4 \end{matrix} & \begin{matrix} \begin{matrix} 9 & 9 & 7 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Remaines \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 & 0 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \end{matrix}$

But y^e Coëfficient maybe greater yⁿ y^e Power soe y^t it cannot be substracted from it w^ch argues y^t y^e Cube {illeg} more propperly affects yⁿ is affected. In this case y^e coëfficient must descend towards y^e unite soe many points untill it may be substracted, & soe many points as y^e coëfficient is devolved soe many pricks {illeg}|mu|st be blotted out towards y^e left hand in y^e power affected. As y^e Example shews
$L c + 95400 L$ = {illeg} / $= 1819459$ . \

$\begin{array}{l} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} \underset{}{9} \\ \underset{.}{1} \end{matrix} \end{matrix} & \begin{matrix} \underset{}{5} & 4 & 0 \\ 8 & 1 & \underset{.}{9} \end{matrix} & \begin{matrix} \begin{matrix} \underset{.}{0} & \underset{.}{} & \underset{.}{} \\ 4 & 5 & \underset{.}{9} \end{matrix} & \begin{array}{l} Coefficiens & \underset{}{} \\ The Power & \underset{}{} \end{array} \end{matrix} \end{matrix} \\ Since 9 is greater y^{n} 1 make a devolution thus. \\ \begin{matrix} \begin{matrix} \begin{matrix} \underset{}{} \\ \underset{}{} & The Quote (19 \end{matrix} & \begin{matrix} \underset{}{} \\ \underset{}{} & \underset{}{1} \end{matrix} \end{matrix} & \begin{matrix} \underset{}{9} & \underset{}{5} & \underset{}{4} \\ 8 & 1 & \underset{.}{9} \end{matrix} & \begin{matrix} \begin{matrix} 0 & \underset{.}{0} & \underset{.}{} \\ 4 & 5 & \underset{.}{9} \end{matrix} & \begin{array}{l} The Coefficient & \underset{}{} \\ The affected power & \underset{}{} \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollids to be substracted \end{matrix} & {\begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 9 & 5 & 4 \\ 1 \end{matrix} & \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & \begin{array}{l} A C q \\ A c \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Su \bar{m} a \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 9 & 5 & 5 \end{matrix} & \begin{matrix} \begin{matrix} 0 & 0 \end{matrix} & \begin{matrix} substrahenda \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Divisor \bar{u} superior pars \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 9 & 5 \end{matrix} & \begin{matrix} \begin{matrix} 4 & 0 & \underset{.}{0} \end{matrix} & \begin{matrix} Coefficiens Planum \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 8 & 6 & 4 \end{matrix} & \begin{matrix} \begin{matrix} 4 & 5 & 9 \end{matrix} & \begin{matrix} Potestas reliqua \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Divisor \bar{u} pars inferior \end{matrix} & {\begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 3 \\ 3 \end{matrix} & \begin{array}{l} 3 A q \\ 3 A \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Divisor \bar{u} su \bar{m} a \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 9 & 5 \end{matrix} & \begin{matrix} \begin{matrix} 7 & 3 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollida ablativa \end{matrix} & {\begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 8 & 5 & 8 \\ 2 \\ 2 \end{matrix} & \begin{matrix} \begin{matrix} 6 & 0 & 0 \\ 7 \\ 4 & 3 \\ 7 & 2 & 9 \end{matrix} & \begin{array}{l} E C q \\ 3 A q E \\ 3 A E q \\ E c \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Eor \bar{u} Summa \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 8 & 6 & 4 \end{matrix} & \begin{matrix} \begin{matrix} 4 & 5 & 9 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} Restat \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 0 & 0 & 0 \end{matrix} & \begin{matrix} \begin{matrix} 0 & 0 & 0 \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{array}$

<5r>

To place y^e unite of y^e coefficient in its right place in respect of y^e power make so many pricks above as there are under y^e power begining at y^e unit, & if y^e coefficient be one dimension lesse yⁿ y^e power make a prick on every figure if 2 dimensions les yⁿ every other figure of 3 dimensions lesse make it one each third figure &c

If there be many coefficients in y^e equation each must be placed according to this rule.

Sometimes y^e coefficient is under a negative sine as $L c - 10 L = 13584$ & y^e Analysis is as follows

$\begin{matrix} Coëfficiens planum - & 1 \underset{.}{0} sublaterale \\ Cubus resolvendus + 1 \underset{.}{3} & 58 \underset{.}{4} (24 \\ \begin{matrix} \begin{matrix} Solida ablativa \end{matrix} & {\begin{matrix} + 8 \\ - \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 20 \end{matrix} & \begin{matrix} A c \\ A C q \end{matrix} \end{matrix} \\ Su \bar{m} a + 7 & 80 \\ Restat + 5 & 78 \underset{.}{4} resolvendum \\ Divisor \bar{u} {\underline{p}}^{s} superior & - 10 coëfficiens planum \\ \begin{matrix} \begin{matrix} Divisor \bar{u} {\underline{p}}^{s} inferior \end{matrix} & {\begin{matrix} + 1 \\ + \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 2 \\ 6 \end{matrix} & \begin{matrix} + 3 A A \\ + 3 A \end{matrix} \end{matrix} \\ Suma divisor \bar{u} + 1 & 25 \\ \begin{matrix} \begin{matrix} Solida ablativa \end{matrix} & {\begin{matrix} + 4 \\ + \\ + \\ - \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 8 \\ 96 \\ 64 \\ 40 \end{matrix} & \begin{matrix} 3 A A E \\ 3 A E E \\ E E E \\ E C C \end{matrix} \end{matrix} \\ Eor \bar{u} su \bar{m} a + 5 & 784 \end{matrix}$

But sometimes y^e square coëfficient hath more paires of figures yⁿ y^e cube to be analysed, \hath/ & yⁿ \there is/ præfixing so many ciphers to y^e cube as figures are wanting, y^e first side will not much differ from y^e square roote of y^e coefficient. as $L c - 116620 L = 352947$

$\begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} - & 1 & \underset{}{1} \end{matrix} \end{matrix} & \begin{matrix} 6 & 6 & \underset{}{2} \end{matrix} & \begin{matrix} \begin{matrix} \underset{.}{0} & \underset{.}{} & \underset{.}{} \end{matrix} & \begin{array}{l} Coefficiens plan \bar{u} \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Cubus resolvendus \end{matrix} & \begin{matrix} 0 & \underset{}{0} \end{matrix} \end{matrix} & \begin{matrix} 3 & 5 & \underset{}{2} \end{matrix} & \begin{matrix} \begin{matrix} 9 & 4 & \underset{.}{7} \end{matrix} & \begin{array}{l} (343 \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollida Ablativa \end{matrix} & {\begin{matrix} + & 2 & 7 \\ - & 3 & 4 \end{matrix} \end{matrix} & \begin{matrix} 9 & 8 & 6 \end{matrix} & \begin{matrix} \begin{matrix} \end{matrix} & \begin{array}{l} A c \\ A C q \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Restat auferend \bar{u} \end{matrix} & \begin{matrix} - & 7 \end{matrix} \end{matrix} & \begin{matrix} 9 & 8 & 6 \end{matrix} & \begin{matrix} \begin{matrix} \end{matrix} & \begin{array}{l} \end{array} \end{matrix} \\ \begin{array}{r} \begin{matrix} Reliquum resolvendi \end{matrix} & \begin{matrix} + & 8 \end{matrix} \end{array} & \begin{matrix} 3 & 3 & 8 \end{matrix} & \begin{matrix} \begin{matrix} 9 & 4 & 7 \end{matrix} & \begin{array}{l} Cubi \end{array} \end{matrix} \end{matrix}$

<5v>

$\begin{matrix} \begin{matrix} \begin{matrix} Divisor \bar{u} {\underline{p}}^{s} superior Coeff: \end{matrix} & \begin{matrix} - & \underset{}{1} \end{matrix} \end{matrix} & \begin{matrix} 1 & 6 & \underset{}{6} \end{matrix} & \begin{matrix} \begin{matrix} 2 & \underset{.}{0} & \underset{.}{} \end{matrix} & \begin{array}{l} planum. \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Reliqu \bar{u} resolvendi cubi \end{matrix} & \begin{matrix} + & \underset{}{8} \end{matrix} \end{matrix} & \begin{matrix} 3 & 3 & \underset{.}{8} \end{matrix} & \begin{matrix} \begin{matrix} 9 & 4 & \underset{.}{7} \end{matrix} & \begin{array}{l} negative affecti \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Divisor \bar{u} {\underline{p}}^{s} inferior \end{matrix} & {\begin{matrix} + & 2 \\ + \end{matrix} \end{matrix} & \begin{matrix} 7 \\ 9 \end{matrix} & \begin{matrix} \begin{matrix} \end{matrix} & \begin{array}{l} 3 A A . \\ 3 A . \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Su \bar{m} a Divisorum \end{matrix} & \begin{matrix} * & * \\ + & \underset{}{1} \end{matrix} \end{matrix} & \begin{matrix} * & * & * \\ 6 & 2 & \underset{}{3} \end{matrix} & \begin{matrix} \begin{matrix} * & * & * \\ 8 & \underset{.}{} \end{matrix} & \begin{array}{l} ************ \\ 3 A A + 3 A + C q \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollida ablativa \end{matrix} & {\begin{matrix} + & 1 & 0 \\ + & 1 \\ + \\ - & 4 \end{matrix} \end{matrix} & \begin{matrix} 8 \\ 4 & 4 \\ 6 & 4 \\ 6 & 6 & 4 \end{matrix} & \begin{matrix} \begin{matrix} 8 \end{matrix} & \begin{array}{l} 3 A A E \\ 3 A E E \\ E E E \\ C C E \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Eorum summa \end{matrix} & \begin{matrix} + & 7 \end{matrix} \end{matrix} & \begin{matrix} 6 & 3 & 9 \end{matrix} & \begin{matrix} \begin{matrix} 2 \end{matrix} & \begin{array}{l} \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Restat Resolven \bar{d} \end{matrix} & \begin{matrix} + \end{matrix} \end{matrix} & \begin{matrix} 6 & 9 & \underset{}{9} \end{matrix} & \begin{matrix} \begin{matrix} 7 & 4 & \underset{.}{7} \end{matrix} & \begin{array}{l} pro 3^{o} latere \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Divisoru \bar{m} {\underline{p}}^{s} superior ~ \end{matrix} & \begin{matrix} - \end{matrix} \end{matrix} & \begin{matrix} 1 & 1 & \underset{}{6} \end{matrix} & \begin{matrix} \begin{matrix} 6 & 2 & \underset{.}{0} \end{matrix} & \begin{array}{l} C C \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Diviso \bar{ru} {\underline{p}}^{s} inferior \end{matrix} & {\begin{matrix} + \\ + \end{matrix} \end{matrix} & \begin{matrix} 3 & 4 & 6 \\ 1 \end{matrix} & \begin{matrix} \begin{matrix} 8 \\ 0 & 2 \end{matrix} & \begin{array}{l} 3 A A \\ 3 A \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} *********** \end{matrix} & \begin{matrix} * & * & * \end{matrix} \end{matrix} & \begin{matrix} * & * & * \end{matrix} & \begin{matrix} \begin{matrix} * & * & * \end{matrix} & \begin{array}{l} * \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Eorum Summa \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} 2 & 3 & 1 \end{matrix} & \begin{matrix} \begin{matrix} 2 & 0 & 0 \end{matrix} & \begin{array}{l} = 3 A A + 3 A + C C \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Sollida ablativa \end{matrix} & {\begin{matrix} + & 1 \\ + \\ + \\ - \end{matrix} \end{matrix} & \begin{matrix} 0 & 4 & 0 \\ 9 \\ 3 & 4 & 9 \end{matrix} & \begin{matrix} \begin{matrix} 4 \\ 1 & 8 \\ 2 & 7 \\ 8 & 6 & 0 \end{matrix} & \begin{array}{l} 3 A A E \\ 3 A E E \\ E E E \\ E C C \end{array} \end{matrix} \\ \begin{matrix} \begin{matrix} Eorum Summa \end{matrix} & \begin{matrix} + \end{matrix} \end{matrix} & \begin{matrix} 6 & 9 & 9 \end{matrix} & \begin{matrix} \begin{matrix} 7 & 4 & 7 \end{matrix} & \begin{array}{l} \end{array} \end{matrix} \end{matrix}$

Sometimes though there be as many 2 figures in the coefficient as 3 figures \in/ y^e cube affected yet y^e coëfficient may be so greate as to deceive an uwary {sic} Analist As in this $L c - 6400 L = 153000$ . where y^e roote of 64 is 8 w^ch cubed is 512 w^ch added to 153 makes 665 thē whose \roote/ y^e number immediately greater is 9 w^ch make is y^e first sid {sic} $= A$ .

But if y^e coefficient had beene affirmative, yⁿ not y^e aggregate of y^e facts but y^e difference must be taken as in this. $L c + 64 L = 1024$ .

Since y^e roote of 64 is 8 . w^ch cubed is 512 . & $1024 - 512 =$ $512$ . y^e roote of w^ch is $8 = A$ . The like is observable in equations of higher powers

If y^e Cube be affected w^th a negative sine as $13,104 L - L c = 155,520$ . Then y^e Equation is expressible of 2 rootes: whereof is less y^e other greater yⁿ {i} y^e square of one is <6r> lesse & the square of y^e other is greater ther|n|e $\frac{13104}{3}$ . & therefore one roote is lesse y^e other greater then $\frac{155520}{13104}$ . & in this equation $27755 L - L qq = 217944$ are two rootes whereof one is greater y^e other lesse then $\frac{217944}{27755}$ .

☞

Suppose in y^e former \cubick/ equation y^e lesse roote be 12. yⁿ $\frac{155520}{12} = 12960$ . or else $13104 - 12 \times 12 = 12960$ . & $L q + 12 L = 12960$ . where $L = 108$ is y^e greater roote.

And in y^e latter equation if y^e lesse \greater/ roote be 27. & $\frac{217944}{27} = 8072$ , c. or $- 27 \times 27 \times 27$ $=$ | $+$ | $27755 = 8072$ . {illeg} $27 \times 27 = 729$ . If there be 4 cubes continully {sic} proportionall whose greate extreame is $27 c = 19683$ . & y^e aggregate of y^e 3 rest is 8072 & $L c$ y^e lesse extreame, therefore $L c + 27 L q + 729$ {illeg} | $L$ | $= 8072$ . y^e roote of w^ch ./is\ 8 y^e other roote of y^e equation

☞ Or haveing one roo{illeg}|t|e of an equation y^e e|E|quation may be lessoned {sic} by division thus $13104 l - l c$ {illeg}| $=$ | $155520$ or $l^{3}$ {illeg} $- 13$ {illeg} | $1$ | $04 l + 155520 = 0$ . & one roote is 12 . therefore divide this equation by $l - 12$ & the Quote is an equation conteing {sic} y^e ot{illeg}|h|er roote viz: $l q + 12 l = 12960$ .

<7v>

To find two meane proportionalls twixt BC & IK

<8r>

Propositiones Geometricæ. Franc: Vietæ.

prop 1

$ab ∶ ac ∷ ce ∶ bd$

pro{illeg}|p| 2

& if $ab ∶ ac ∷ ac ∶ bd$ : then $ac ∶ ab ∷ ab ∶ ce$

prop 3. If $ab \times ac = bd \times ce$ . then $bd ∶ ac ∷ ac ∶ ab ∷ ab ∶ ce$ ∺

prop 3. To find two mene {sic} proportionalls {twixt} Bc & IK . On y^e center a w^th the Rad ai describe the circle ibck . inscribe b {illeg} | c | $= cd$ . draw da through y^e center & bg parallel to it. draw hk through A soe y^t $gh = ab = (= ai)$ . & $ik ∶ hb ∷ hb ∶ hi ∷ hi ∶ bc$ . ∺

Prop: 4

I{illeg}f $ad = db = cb$ . yⁿ y^e Angle {illeg} / c \ be is tripple to y^e Angle abd .

Prop 5

{illeg}|I|f $ab = bd = Rad$ . $3 Ang: bad = cde$

Prop 6

If $3 rpq = spq :recto$ . that is If $2 qr = pr$ . then $3 or \times or = sp \times sp + op \times op + px \times px$

Prop 7

Iaf If $ad = dc = ce = ef$ . then $ecf = efc = 3 dac = 3 dca$ & ${AC}^{3} = 3 AC \times {ad}^{2} + Cf \times {ad}^{2}$ . $Z^{3} = a a z + b^{3} . ~ ~ ~$

prop 8. If $op = pr = qr = qs$ . yⁿ, $prt = 3 qsr$ &c and ${sr}^{3} = 3 sr \times {qr}^{2} - or \times {qr}^{2}$ . $Z^{3} = a a z - b^{3}$

prop 9 If{illeg} $op = pr =$ $ah = hb = bf = fd$ . & $ch = 2 eh$ or $ceh = 3 hce$ . then $\begin{matrix} {bg}^{3} \end{matrix}$ $\begin{matrix} {ac}^{3} & = & 3 ac & \times & {ah}^{2} & - & db \times {ah}^{2} \\ & & {cb}^{3} & = & 3 cb & \times & {ah}^{2} & - & db \times {ah}^{2} \end{matrix}} Z^{3} = a a z - b^{3} .$

<8v>

Prop 10

If $de = ea$ & $db ∶ da ∷ ab \times ab ∶ dc \times dc$ . yⁿ be is a side of a 7 equall sided & angled figure. or $7 eab = 4 right angles$ .

prop 10

If $ac = ef$ . & aef a right angle & ab pas {sic} through y^e center then $cd ∶ de ∷ de ∶ df ∷ df ∶ db$ . And if $cd ∶ de ∷ de ∶ df ∷ df ∶ db$ then ae is perpendicular to ef . $2 hd$ is y^e difference of y^e extreames & $2 do$ is y^e difference of y^e meanes. w^ch given y^e proportionall lines may be found &c.

prop 11

\Pse{illeg}|u|domesolabium wherby/ To find 2 meane proportionalls. {illeg} If, $ae ∶ ec ∷ ec ∶ ed ∷ ed ∶ eb$ . they be inscribed in the {illeg}|c|ircle acbd the diam : being $ae + eb$ . If twixt g {illeg} | i | & {illeg} | i | h two meane proportionalls are sought on y^e same center f w^th y^e Rad : $\frac{gi + ih}{2}$ describe gkhl & inscribe a line kl parallel to cd cutting ab in y^e point e | i | & $gi ∶ ki ∷ ki ∶ il ∷ il ∶ ih$ . Examine it.

^[1]

prop 12

If $do = dh$ & ac bisected in b & bd bee drawne rd is i|y^e| side of a pentagon w^ch may be inscribed in defcro

prop 13.

If rd be the side of a ${\begin{matrix} octogon \\ decagon \end{matrix}}$ & pd y^e side of an ${\begin{matrix} hexagon \\ octogon \end{matrix}}$ y^e arch rp divided equally in o , od will be {illeg}|th|e side of an ${\begin{matrix} heptagon \\ enneagon \end{matrix}}$ to be inscribed in y^e circle ord & y^e arch RP is bisected \rightly divided/ by Bisecting y^e Line ac . /Examine it\

<9r>

Of Angular sections.

prop 14

If $ead = cab$ . Then $ab ∶ {ab}^{2} ∷ cb ∶ eb \times ad - ae \times db ∷ ac ∶ ae \times ad + eb \times db$ or, $ab ∶ aq \times ab ∷ op ∶ eb \times an - ae \times nq ∷ ao ∶ ae \times an + eb \times nq$ . But y^e angles anq , aop are right ones and {illeg} | e | $an = oap$ {illeg} | = | $eab - dab$

prop 15

If \y^e angle/ $cab + dab = eab$ . or $naq + oaq = eab$ . & anq , aop are right ang\les/ then $Ab ∶ {ab}^{2} ∷ Eb ∶ ad \times bc + ac \times db ∷ ea ∶ ad \times ac - db \times cb$ . or the triang: unequall. $ab ∶ ap \times aq ∷ eb ∶ an \times op + ao \times nq ∷ ea ∶ an \times ao - nq \times op$

prop 16.

In 2 rectang: triang: acb & aed , if y^e first have an acute angle cab submultiple to the acute angle eab of y^e 2^d triang aeb y^e si{illeg}|d|es of y^e seacond have this proportion. Suppose y^e Hypoten of y^e first tri: be z . y^e base b . y^e Cathetus c .

$\begin{matrix} \begin{matrix} If y^{e} acute angle of \\ y^{e} seacond triangle be \\ to y^{e} acute angle of \\ y^{e} first triangle in \\ a proportion \end{matrix} & {\begin{matrix} Hypoten: & Base & Perpendicular \\ \begin{array}{r} Duple, Z^{2} . \end{array} & \begin{matrix} B^{2} . \\ {-C}^{2} . \end{matrix} & \begin{matrix} 2 B C . \end{matrix} \\ \begin{array}{r} Triple, Z^{3} . \end{array} & \begin{matrix} B^{3} . \\ -3 D D C . \end{matrix} & \begin{matrix} 3 B B C . \\ {-C}^{3} . \end{matrix} \\ \begin{array}{r} Quadruple, Z^{4} . \end{array} & \begin{matrix} B^{4} . \\ -6 B^{2} C^{2} . \\ {+C}^{4} . \end{matrix} & \begin{matrix} 4 B^{3} C . \\ -4 B C^{3} . \end{matrix} \\ \begin{array}{r} Quintuple, Z^{5} . \end{array} & \begin{matrix} B^{5} . \\ -10 B^{3} C^{2} . \\ +5 B C^{4} . \end{matrix} & \begin{matrix} 5 B^{4} C . \\ -10 B^{2} C^{3} . \\ {+C}^{5} . \end{matrix} \end{matrix} \end{matrix}$

Prop 17. I{illeg}|f| {\∠/} $ab = bc = ce = eg$ &c: & $ac = cd$ . & $ae = ef$ &c then $ab ∶ ac ∷ ac ∶ ad ∷ ae ∶ af$ &c /& $ed = ab$ & $ac = gf$ &c. {nam} △ cde & cba , efg & eac &c: = & sim. \

Prop.18.

If $bd = dg = gh = hk = pq = pw$ &c Then $al ∶ ak ∷ pe ∶ pc ∷ ed ∶ do ∷ pd ∶ pc + do ∷$ $rg ∶ gs ∷ rq ∶ qo ∷ qg ∶ qo + gs ∷$ &c & if $(\frac{lf}{2} = lʒ)$ from ʒ to y^e center be drawne cʒ then $al ∶ ak ∷ di ∶ dv ∷ iq ∶ qx ∷ dq ∶ dv + qx ∷ gz ∶ gx ∷ wz ∶ wx$ &c & Ergo $ac ∶ ak ∷ ab ∷ aδ + ad ∶ ad ∶ ab + ag ∶ ag ∶ ad + ah ∶ ah ∶ ag + ak$ &c.

<9v>

Prop 19

If $fa = ab = be = eh$ &c. &. $af + ab + be + eh$ are greater y^t
|n| y^e semiperiphery: then, $Rad ∶ ge (= ec) (= ad) ∷ bd (= bc)$ $∶ cd (= cd - de) ∷ de ∶$ {illeg} $dh - bd ∶$ & dh is y^e greatest, db y^e least line drawn from d to these points $a$ , $b$ , $e$ , $h$ . yⁿ $rad ∶$ {illeg} | $dh$ | $∷ db ∶ da - de$ .

Prop 20

Out of y^e 18^th & 19^th {illeg} Prop: To divide An angle into any number of pts in y^e figure of y^e 18^th prop: $al = diam =$ \ $2$ / z . ah is y^e greatest of y^e inscribed lines $= B$ : now z /{illeg}\ ∷ | ∶ | $B ∷ ah +$ {illeg} | $2 z$ . | therfore $b b = ah in z + 2 z^{2}$ . & $\frac{b b - 2 z z}{z} = ah$ . And $z ∶ B ∷ \frac{b^{2} - 2 z^{2}}{z} ∶ b + ag$ . therfore $\frac{b^{3} - 2 z z b}{z z} - b = ag$ Likewise $\frac{b^{4} - 4 z z b b^{2} + 2 z^{4}}{z^{3}} = ad$ . & $\frac{B^{5} - 5 z z B^{3} + 5 z^{4} B}{z^{4}} = ab$ $\frac{B^{6} - 6 z z B^{4} + 9 z^{4} B B - 2 z^{6}}{z z z z z} = aδ$
$\frac{B^{7} - 7 z z B^{5} + 14 z^{4} B^{3} - 7 z^{6} B}{z^{6}} = to a seaventh line$
$\frac{B^{8} - 8 z z b^{6} + 20 z^{4} b^{4} - 16 z^{6} b b + 2 z^{8}}{z^{7}} = to an eight$ {illeg} $line \binom{}{}$ $\frac{B^{9} - 9 z z b^{7} + 27 z^{4} b^{5} - 30 z^{6} b^{6} + 9 z^{8} b}{z^{8}} = a$ n $nineth line \binom{}{}$ $\frac{B^{10} - 10 z z b^{8} + 35 z^{4} b^{6} - 50 z^{6} b^{4} + 25 z^{8} b b - 2 z^{10}}{z^{9}} =$ {illeg}/tenth &\

Prop 21

out of y^e 17^th Theor.: in y^e figure whereof if ab {\:/} y^e least inscribed line $= z$ . & ac y^e next line bee B . then $z ∶ B ∷ B ∶ z + ae$ . & $\frac{b b - z z}{z} = ae$ & $\frac{B^{3} - 2 z^{2} B}{z z} = ag$ & $\frac{B^{4} - 3 z^{2} b b + z^{4}}{z^{3}} = to a fift line.$ $\frac{B^{5} - 4 z z b^{3} + 3 z^{4} b z^{4}}{z^{4}} = a sixt.$ & $b^{6} - 5 z z b^{4} + 6 z^{4} b b - z^{6} = seaventh$ {sic} $\frac{b^{7} - 6 z z b^{5} + 10 z^{4} b^{3} - 4 z^{6} b}{z^{6}} = to an eight line$
$\frac{B^{8} - 7 z z b^{6} + 15 z^{4} b^{4} - 10 z^{6} b b + z^{8}}{z^{7}} = to a nineth line$ $\frac{B^{9} + 8 z z b^{7} + 21 z^{4} b^{5} - 20 z^{6} b^{3} + 5 z^{8} b}{z^{8}} = to a tenth line$ $\frac{B^{10} - 9 z z b^{8} + 28 z^{4} b^{6} - 35 z^{6} b^{4} + 15 z^{8} b b - z^{10}}{z^{9}} = eleventh$ .

<10r>

Prop 22.

If $aq = ab = bd = dc = ch = hk = kl = lf$ . Then $GK Rad ∶ kl ∷ kl ∶ el (= al - ah) ∷$ And $hl ∶ hm (= qh - qd =$ $ak - ac) ∷ dl ∶ do (= qd - qa = ac - ab) ∷ lc ∶ cn (= qc - qb =$ $ah - ad$ &c ) Soe that Rad{illeg} y^e Periph: divided into any number of pts. $gl ∶ lk ∷ lk ∶ al - ak ∷ lh ∶ ak - ac ∷ lc ∶ ah - ad ∷ dl ∶ ac - ab$ &c. & $gl ∶ lk ∷ ah ∶ lc - lk ∷ ac ∶ ld - lh ∷ ad ∶ lb - lc ∷ ab ∶ al - ah$ &c.

hence Prop 23.

In y^e former scheame If al {illeg} $= 2 x = hypotenusa$ . $kl = b$ . $x ∶ b ∷ b ∶ 2 x - ah$ & $\frac{- b b + 2 x x}{x} = ah$ / $x ∶ b ∷ \frac{- b b + 2 x x}{x} ∶ \frac{2 b x x - b^{3}}{x x} (= lc -$ {illeg} | b | ) lc{−}kl{=}dl{−}\h/ therefore $\frac{3 b x x - b^{3}}{x x} = lc$ . & $\frac{2 x^{4} - 4 b b x x + b^{4}}{x^{3}} = ad$ y^e base of y^e 4^th triang: &{c} $\frac{5 b x^{4} - 5 b^{3} x x + b^{5}}{x^{4}} =$ y^e perpendicular ( {illeg} | bl | ) of y^e 5^t tri: &{c} $\frac{2 x^{6} - 9 x^{4} b b + 6 x x b^{4} - b^{6}}{x^{5}} =$ base of y^e 6^t triang. $\frac{7 x^{6} b - 14 x^{4} b^{3} + 7 x^{2} b^{5} - b^{7}}{x^{6}} =$ perpendic of the 7^th tri $\frac{2 x^{8} - 16 x^{6} b b + 20 x^{4} b^{4} - 8 x x b^{6} + b^{8}}{x^{7}} =$ base of y^e 8^th tri. $\frac{9 x^{8} b - 30 x^{6} b^{3} + 27 x^{4} b^{5} - 9 x x b^{7} + b^{9}}{x^{8}} =$ perp: of y^e 9^th tri:

<10v>

Prop 24:

If {illeg}| bd | $= dh = hk = kl =$ b {illeg} g &c: {illeg}|t|hen $bh = gd$ & $bk = gh$ & $bl = hm$ &c: & then
$xt ∶ bx ∷ ac ∶ ag ∷ ce ∶ eh ∷ ei ∶ em ∷ io ∶ ok ∷$ $op ∶ on ∷ pq ∶ ql ∷ qr ∶ qy ∷ rs ∶ sx ∷ st ∶ sf ∷ ba ∶ ad$ therefore $xt ∶ bx ∷ bt ∶ dg + hm + kn + ly + xf$ . againe $xt ∶ bt ∷ ab ∶ bd ∷ ac ∶ cg ∷ ce ∶ ch ∷ ei ∶ im$ &c Therefore $xt ∶ bt ∷ bt ∶ bd + gh + mk + ml + yx + ft$ . & since, as $xt ∶ bt ∷ bx ∶ dg + hm + kn + ly + xf$ Therefore $xt ∶ bt ∷ bx + bt ∶ bd + dg + gh + hm + mk + kn +$ $+ nl + ly + yx + xf + ft$ . And $xt ∶ bt ∷ bx + bt + xt ∶$ to all y^e perpen dicular & transverse line $+ bt$ . that is
| ( {illeg} \5/ {illeg} ) | $xt ∶ bt ∷ xt + bt + bx ∶ 2 bd + 2 bh + 2 bk + 2 bl + 2 bx + 2 bt$ .

Prop 24

If in y^e circle cfgh be inscribed y^e helix bedc a|&| ac touch it in y^e point c then $ab =$ to y^e circumference.

Prop 25

If apcr be les yⁿ halfe y^e circle. & $vt = tp$ . & $vo =$ to y^e vrap : then $\frac{rq \times po}{2} =$ 4 times y^e section /rapc\

Prop 26

If $ab = bd = ad$ & bh perpendicular to ad fro {sic} y^e angle {illeg} | b |. $ce = ed$ . yⁿ $aed = adi = 3 dae$ . & ed is y^e side of a heptagon

Prop 27.

If a line be cut by extreame & meane proportion y^e lesse {illeg}|s|egment almos{illeg}|t| is to y^e whole line as y^e diameter is to $\frac{5}{6}$ of 5 times the periphery divided by 6 .

Prop 28

Si secetur linea per extremam & mediam proportionem er{illeg}|i|t proximè, ut tota line{illeg}a plus minori segmento ad {tot} bis totam lineam, ita quæ potest quadrato sesquialterum semidiametri, ad latus quadrati circulo equalis.
linea secta sit 100,000 . minus segmentū 38,197 . Semidiametrū 100,000 , quæ potest quadrato sesquialterū semidiametri paulo maior est quam 122,474 . Radix Peripheriæ, 177,245 .

<11r>

Prop 28.

If $er = rh = or$ . & $ao = fc =$ to y^e side of a decagon; & fn parallell to cd yⁿ en shall be almost equall to y^e fourth ꝑ^te of a circle for af {sic} is divided in extreame & meane propor in y^e point c . & $ec ∶ ef ∷$ hf $ef ∶ \frac{5}{12}$ Perimeter|hbk|fa $∷ hr ∶ \frac{5}{24} Perim ∷ \frac{6}{10} ef (= de) ∷ \frac{1}{4} Perim$ ; by y^e 27^th prop: & $ec ∶ ef ∷ ed ∶ en (= \frac{1}{4} Perim)$ .

Prop 29.

If $os = 2 cp$ . & co is divided by extreame & meane proportion in r . & od parallell to rp then db is y^e side of a square = to o|t|he area of y^e circle. for be|y| y^e 28^th prop: As $br (= to line + less s \bar{egm}) ∶ bo (= twice y^{e} line) ∷ bp$ ({illeg} $(= \sqrt{\frac{3}{2} of the square of y^{e} semidiameter}) ∶ bd (= to y^{e} roote$ $of a square equall to y^{e} area of a circle$ .

Prop 30

If y^e line dc touch y^e helix in y^e line ag . & y^e line hf toucheth y^e beginning of it in y^e center a & $4 ac = af$ then $2 ad$ shall bee equall to perim: asr . & ac being y^e Diam: y^e tri: acd area of y^e triang acd = to y^e area of y^e circle asr

Prop 31

If bed be a square of one revolution of an helix & y^e angle $dbe = dba$ & through y^e points a , d , in y^e helix be drawne y^e line adk & through y^e points ed {sic} in y^e Helix be drawne edg . & y^e angle kdg bisected by dh ; then dh shall almost touch y^e helix in d . {illeg}|&| it shall be soe much y^e nigher a touch line by how much y^e angles
ebd dba are lesser.

<11v>

Prop 32

If many Polygons be inscribed in a circle y^e number of theire sides increaseing in a double proportion. & theire apotomies, or y^e t|b|ase \of/ a tri: whose cathetus is a leg of y^e Polygon & hypotenusa is y^e {illeg}|D|iam (as y^e apotome of y^e Polygō cgp is ce . of pacegi is ae &c) if y^e Apotome of y^e sides of y^e first Polygō be called b . |of| y^e 2^d $= c$ . |of| y^e 3^d $= d$ . of y^e 4^th $= f$ . of y^e 5^t $= g$ of y^e Sixt $= h$ . & y^e diameter be z then And y^e first Pol{illeg}ygon be $= p$ . y^e 2^d $= q$ . y^e 3^d $= r = abcdefghiopq$ . y^e fourth $= s$ . y^e 5^t $= t$ y^e sixt $= v$ . y^e {illeg} 7^th $= w$ &c then
$p ∶ q ∷ b ∶ z$ . & p{illeg} $p ∶ r ∷ b c ∶ z z$ . & $p ∶ s ∷ b c d ∶ z z z$ . & $p ∶ t ∷ b c d f ∶ z^{4}$ . & $p ∶ v ∷ b c d f g ∶ z^{5}$ . & $p ∶ w ∷ b c d f g h ∶ z^{6}$ &c

To know how many elections may bee made of \divers ways/ things, whereof some of y^m are equall, \may bee ordered./ . as of . $a . b . b . c . c . c . d . d .$ doe thus { $\begin{matrix} a & b b & c c c & d d \\ \frac{1}{1} & \times & \frac{2 \times 3}{1 \times 2} & \times & \frac{4 \times 5 \times 6}{1 \times 2 \times 3} & \times & \frac{7 \times 8}{1 \times 2} & = 112 \end{matrix}$ } { $\frac{\overset{a}{1}}{1} \times \frac{\overset{b}{2} \times \overset{b}{3}}{1 \times 2} \times \frac{\overset{c}{4} \times \overset{c}{5} \times \overset{c}{6}}{1 \times 2 \times 3} \times \frac{\overset{d}{7} \times \overset{d}{8}}{1 \times 2} = 112$ } the number of changes, in order.

To know how many elections may bee made doe thus $\overset{a}{2} \times \overset{bb}{3} \times \overset{ccc}{4} \times \overset{dd}{3} \times \overset{eeee}{5} = 360 = \overset{a}{2} \times \frac{\overset{b}{2} \times \overset{b}{3}}{2} \times \frac{\overset{c}{2} \times \overset{c}{3} \times \overset{c}{4}}{2 \times 3} \times \frac{\overset{d}{2} \times \overset{d}{3}}{2} \times \frac{\overset{e}{2} \times \overset{e}{3} \times \overset{e}{4} \times \overset{e}{5}}{2 \times 3 \times 4}$ therefore there are $359 = 360 - 1$ elections in $a b b c^{3} d d e^{4}$ .

<12r>

Propositiones Geometricae Ex Schootenij
Sectionibus miscellaneis.

Sectio 1^ma

To know how many changes 6 Bells, $a b c d e f$ or how divers conjuctions y^e 7 planets can make ♄ ♃ ♂ ☉ ♀ ☿ ☾. or how many divisors $a b c d e$ hath, or how man{e {sic}} divers compositions y^e 24 le{illeg}|t|ers {sic} can make &c the examples following show.
$1 .$ $a 1$
$2 .$ $b . a b 3$
$4 .$ $c . c b . c a b . a c . 7$
$8 .$ $d . d a . d b . d a b . d c . d a c . d c b . d c a b . 15$
$16 .$ $e . e a . e b . e a b . e c . e a c . e c b . e c a b . e d . e d a . e d b . e d a b . e d c . e d a c .$ \ $e d c b . e d c a b .$ \ 31 / /
$32 .$ $f . f a . f b . f a b . f c . f c b . f a c . f c a b . f d . f d a . f d b . f d a b &c 63 .$
$64 .$ $g . g a . g b . g a b . g c . g c b . g a c . g c a b . g d . g d a . g d b &c 127$
w^ch shows y^t in 7 letters 127 el{illeg}e|ec|tions may be made. y^t 7 Planets may be conjoyned 120 divers ways. y^t $a b c d e f g$ . hath 128 divisors for an unite is one of {y}& $1 \times 2 \times 3 \times 4 \times 5 \times 6 . = 720$ ; are y^e number of changss {sic} in six bells.

Sec 2

To know how many things & of w^t sort they are w^ch may be chosen 15 ways. $15 + 1 = 16 . \frac{16}{2} = 8 . \frac{8}{2} = 4 . \frac{4}{2} = 2 . \frac{2}{2} = 1$ . & {illeg}4{illeg} \ $2 - 1 . 2 - 1 . 2 - 1 . 2 - 1 . = 1.1.1.1 . = 4$ . that 4 things all unequall/ m{illeg}|a|y be varyed 15 ways. also. $\frac{16}{4} = 4 . \frac{4}{2} = 2 . \frac{2}{2} = 1$ $4 - 1 . 2 - 1 . 2 - 1 = 5$ {illeg}|&|. 5 things whereof 3 are equall viz: $a . a . a . b . c .$ {illeg}|&| $\frac{16}{4} = 4 . \frac{4}{4} = 1$ . $4 - 1 . + 4 - 1 = 3 + 3 = 6$ . & 6 thins {sic} whereof 3 & 3 are eaquall {sic} as $a a a b b b$ . may be varied 15 ways. & $\frac{16}{8} = 2$ $\frac{2}{2} = 1$ . $8 - 1 . 2 - 1 = 8 = 7 + 1$ . & 8 things whereof 7 are = may be varyed 15 ways. as $a a a a a a a b$ . $\frac{16}{16} = 1$ . $16 - 1 = 15$ . 2 wherefore 15 alike things &c |as a \15/|. 2 w^t things vary 23 ways. $23 + 1 = 24$ & 24 admitts a {illeg}|7| fold divisor therefore y^e multitude of things sought may be 7 fold but since 43 is a primary number (viz w^ch cannot bee divided) $42 + 1 = 43$ . $\frac{43}{43} = 1$ $43 - 1 = 42$ . therefore onely 42 like things can be varyed 42 ways as $a^{42}$ .

<12v>

Sec 3

Every quantity hath one divisor more y^t it hath aliquote pts (y^t is pts greater yⁿ an unite \of whole numbers./). How to find a quantity haveing a given multitude of divisors or ali{illeg}|q|uote{illeg} pts: suppose its aliq: pts must be 15. $15 + 1 = 16$ & \soe/ by y^e former section $a b c d . a^{3} b c . a^{3} b^{3} . a^{7} b . a^{15} .$ may be varyed 15 ways. therefore they shall have 15 aliquote pts & 16 divisors. but since onely 42 like things (as $a^{42}$ ) ca{illeg}|n| be varyed 42 ways therefore oenely {sic} $a^{42}$ hath 42 aliquote pts & 43 divisors. &c

Sec 4

To find y^e least numbers haveing a given multitude of divisors & aliquote pts instead of soe many letters in the former sec: put soe many p{illeg} least primary numbers & take y^e least result from y^m. as from y^e former example: $a b c d . a^{3} b c . a^{3} b^{3} . a^{7} b . a^{15} .$ that is $2.3.5.7 .$ or | $2 .$ | $2.2.3.5 .$ &c. {illeg} now. $2 \times 3 \times 5 \times 7 . = 210$ . & $2 \times 2 \times 2 \times 3 \times 5 = 120$ . &c therefore $2 \times 3 \times 5 \times 7 .$ \ $= 210$ / are \is/ y^e least number{s} haveing 16 divisors.

Sec: 5 conteines a table of Primary numbers.

Sec 6

To find progressions constituteing rectangular triangles w^th sides rationall y^e examples \following/ shew. take two numbers as $1.2$ . yⁿ $1 \times 2 = 2$ since y^e product is eaved|n| double it viz: $2 \times 2 = 4$ . & 4 is y^e numerato\r/ yⁿ $1 + 2 = 3$ & since 3 is od multiply it by the difference of y^e termes: $1 \times 3 = 3$ & 3 is y^e denominator. & y^e first terme $\frac{4}{3}$ . yⁿ since (1) y^e difference of y^e termes is od multiply it by 4 . $4 \times 1 = 4$ & $4 \times$ per 2 majore terminum. $4 \times 2 = 8$ $8 + 4$ (the former numerato\{r}/) $= 12 =$ numerator 2^d. yⁿ 3 (y^e former denom) added to. {illeg}| 2 | (y^e double square of y^e diff: of y^e termes because y^e square (1) is o{illeg}|d|d) $= 5$ y^e 2^d denominator. I ad anoth^e example take $1.3 .$ yⁿ $1 \times 3 = 3 =$ 1^st numerator. yⁿ $1 + 3 = 4$ & since 4 is eaven halfe 4 {illeg} 1^st denom. & y^e 1^st {illeg} \halfe/ is $\frac{4 \times 2}{2}$ (diff: of y^e termes) $= 4$ & y^e first denom is 4. y^e first terme{illeg} $\frac{3}{4}$ . yⁿ becaus y^e diff of y^e termes is eaven 2 $2 \times 2 = 4$ & $4 \times 3 = 12$ & $12 + 3 = 15$ . yⁿ $2 \times 2 = 4 . 4 + 4 = 8$ . & $\frac{15}{8}$ y^e 2^d terme & now {illeg}|t|er{illeg}|m|es may be had by Arithmeticall proportion. thus. $\frac{4}{3} . \frac{12}{5}$ or $1 \frac{1}{3} . 2 \frac{2}{5} . 3 \frac{3}{7} . 4 \frac{4}{9} . 5 \frac{5}{11} . 6 \frac{6}{13} 7 \frac{7}{15} 8 \frac{8}{17} 9 \frac{9}{19} 10 \frac{10}{21}$ &c |&| $\frac{3}{4} . \frac{15}{8}$ or $\frac{3}{4} 1 \frac{7}{8} . 2 \frac{11}{12} . 3 \frac{15}{16} . 4 \frac{19}{20} . 5 \frac{23}{24} . 6 \frac{27}{28} . 7 \frac{31}{32} . 8 \frac{35}{36}$ &c thus may other progressions be obteined. For y^e use take y^e numerator for one leg & y^e denom for another & y^e Hypoten: will be rationall as in $2 \frac{2}{5}$ $5 \times 2 = 10 . 10 + 2 = 12$ or $\frac{12}{5}$ $\sqrt{144 + 25} = \sqrt{169} = 13$ . & in this $1 \frac{7}{8}$ or $\frac{15}{8}$ $\sqrt{225 + 64} = 17$ .

<13r>

If y^e {illeg}|s|uposed number{illeg}|s| be $2.5$ . yⁿ $2 \times 5 = 10$ . $10 + 10 = 20$ . & $2 + 5 = 7$ . $3 \times 7 = 21$ . so y^t $\frac{20}{21}$ . yⁿ {illeg}be $4 \times 3 = 12$ . $12 \times 5 = 60$ . {illeg} $60 + 20 = 80$ . |&| $3 \times 3 = 9$ . $9 doubled = 18$ . $18 + 21 = 39$ . & y^e 2 first termes $\frac{20}{21} . \frac{80}{39}$ or $2 \frac{2}{39}$ . Agai{illeg}|n|e, if y^e numbers be $3.4$ $3 \times 4 = 12$ . $12 \times 2 = 24$ . & $3 + 4 = 7$ . $1 \times 7 = 7$ . therefore $\frac{24}{7}$ . yⁿ $4 \times 1 = 4$ . $4 \times 4 = 16$ $16 + 24 = 40$ . & $1 \times 1 = 1$ . $2 \times 1 = 2$ . $7 + 2 = 9$ therfore $\frac{40}{9}$ is y^e 2^d & y^e progres may be continued, as $\frac{20}{21} . 2 \frac{2}{39} . 3 \frac{5}{57} . 4 \frac{8}{75} . 5 \frac{11}{93}$ . & $3 \frac{3}{7} . 4 \frac{4}{9} . 5 \frac{5}{11} . 6 \frac{6}{13}$ &c.

Sec 7

To find a {number} w^ch divided by 7 leaves 2 . by 11 leaves 1 . by 13 leaves 9 . the least common divisor of $7.11.13 .$ is $7 \times 11 \times 13 \times =$ {illeg} \1001 /. divide {illeg} \1001 / twice by each & consider y^e remainder of y^e {illeg}|s|eacond division thus.
$\begin{matrix} \frac{210}{2} & \frac{105^{(1}}{2} & 52 \end{matrix}$ . because 1 is left 105 is y^e multiplier.
$\begin{matrix} \frac{210}{3} & \frac{70^{(1}}{3} & 23 \end{matrix}$ . since 1 is left 70 is y^e multiplier.
$\begin{matrix} \frac{210}{5} & \frac{42^{(2}}{5} & 8 \end{matrix}$ since more yⁿ 1 is left (viz: 2 ) multiply 2 till it divided by 5 leaves ( 1 ) $\frac{2 \times 3}{5} = 1 \frac{1}{5}$ . therefore $42 \times 3 = 126$ is y^e multiplyere. $\begin{matrix} \frac{210}{7} & \frac{30}{7} & (4 \frac{2}{7} \end{matrix}$ since more yⁿ 1 is left

1 Since more yⁿ 1 is left (viz 3) multiply 3 till it divided by 7 leavs 1 . $\frac{5 \times 3}{7} = 2 \frac{1}{7}$ therfore $5 \times 143 = 715$ y^e multiplier $| \frac{1001}{7} (\frac{143}{7} (20 \frac{3}{7}$ .

2 Since more yⁿ 1 is left (viz: 3 ) $\frac{3 \times 4}{11} = 1 \frac{1}{11}$ therfore $4 \times 91 =$ $= 364$ y^e multipl: $| \frac{1001}{11} (\frac{91}{11} (8 \frac{3}{11}$ .

3 If but 1 had ben|e|ne left 77 had beene divisor but now $\frac{12 \times 12}{11} = 13 \frac{1}{11}$ . therfore $12 \times 77 = 924$ is multiplyer. $\frac{1001}{13} (\frac{77}{13} (5 \frac{12}{13}$ . now the number sough {sic} is thus found.

< insertion from the center right of f 13r >

$\begin{matrix} \begin{matrix} Divisor. & Reliq: & Multip. \\ 7 . & 2 & \times & 715 & = & 1430 . \\ 11 . & 1 & \times & 364 & = & 364 . \\ 13 . & 9 & \times & 924 & = & 8316 . \end{matrix} \\ The Su \bar{m} e 10110 . \end{matrix}$

< text from f 13r resumes >

Lastly did|v|{illeg}|i|de
by y^e least com. divis: $\frac{10110}{1001} (10 \frac{100}{1001}$ whe{illeg}|r|efore 100 y^e number left is y^e number sought.

Sec 8.

Touching y^e Method of weights suppose a man have weigs {sic} of $1.2.3.4.5$ $1.2.4.8.16.32$ pounds &c by y^m all intermediate pounds may be thus weighed $\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}$ $\begin{matrix} 1 . & 2 & 3 . & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ 1 . & 2 & 1 + 2 & 4 & 1 + 4 . & 2 + 4 . & 1 + 2 + 4 . & 8 & 8 + 1 & 8 + 2 . & 8 + 1 + 2 . & 8 + 4 . & 8 + 4 + 1 . & 8 + 4 + 2 \end{matrix}$ &c or is|f| his wights {sic} be $1.3.9.27.81 .$ all weights may be supplyed thus. $\begin{matrix} 1 . & 2 . & 3 . & 4 . & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ 1 & 3 - 1 . & 3 . & 3 + 1 . & 9 - 1 - 3 . & 9 - 3 . & 9 + 1 - 3 . & 9 - 1 . & 9 . & 9 + 1 . & 9 + 3 - 1 . & 9 + 3 . & 9 + 3 + 1 . & 27 - 9 - 3 - 1 . \end{matrix}$ &c Note y^t weight marked w^th − signifie y^e wigh {sic} to be put in y^e opposite scale ballance.

<13v>

{illeg}|S|ec. 9.

To find numeri amicabiles that is 2 numbers whose aliquote pts are mutually equall to theire wholes. take this Des-Cartes his rule

If $(2)$ , or any other number producc|e|d out of 2 {illeg} \a{illeg}|s|/ $2 \times 2 . 2 \times 2 \times 2$ &c (viz $2.4.8.16.32$ &c .|)| bee such a number y^t 1 taked|n| out of it triple there rests a primary number{,} & y^t if {illeg} 1 taken f{illeg}|r|om it sextuple there rests a primary number, & if 1 taken from its square octodecuple a primary number rests: yⁿ multiply this last prime number by y^e assumed number doubled & y^e product is one amicable number & y^e aliquote pts of {illeg} it make y^e other Example. if 2 be taken. $2 \times 3 - 1 = 5$ numero primario primo. $2 \times 6 - 1 = 11$ numero primario scdo. $2 \times 2 \times 18 - 1 = 71$ numero primario tertio. $4 \times 71 = 284$ , one amicable number, & y^e 2 former prime numbers \ × one another & y^e product/ $\times 4$ y^e double of y^e assumed number viz $5 \times 11 = 55 55 \times 4 = 220$ . Thus from 8 . & 64 &c. may be deduced amicable numbers.

Sec 1{illeg}|0|

To find triangles whose sides, segments of theire bases, & Perpendiculars are expressible by rationall numbers
1^st if y^e perpendic: is without y^e tri: let $ac = z$ . $bd = x$ $cd = y$ . $ad = z + y$ . {illeg} $ad = y + b$ . $x x + y y = y y + 2 b y + b b$ . $y = \frac{x x - b b}{2 b}$ . & $cd = z + y + a$ . $x x + z z + 2 y z + y y = z z +$ $y y + a a + 2 z y + 2 z a + 2 a y$ . $2 a y = x x - a a - 2 z a = \frac{a x x - a b b}{b}$ . $b x x - b a a - 2 z a b = a x x - a b b$ . $\frac{b x x - b a a - a x x + a b b}{2 a b} = z$ . puting any numbers for a , b , & x ; y & z may be found. then $ad = z + y =$ $= \frac{x x + b b}{2 b}$ . $cd = z + y + a = \frac{x x + a a}{2 a}$ . w^ch reduced to y^e common denominator $2 a b$ ; & y^t cast away. $cd = b x x + b a a$ . {illeg} $ad = a x x + a b b$ . $de = 2 a b x$ . $ae = a x x - a b b$ . $ce = b x x - b a a$ . $ac = b x x - a x x + a b b$ $- a a b$ .

In like manner if y^e perpendicular fall w^thin side. $ab = b x x + b a a$ . $bd = 2 a b x$ . $ad = b x x - b a a$ . $dc = a x x - a b b$ . $bc = a x x + a b b$ . $ac = b x x + a x x - a b b - b a a$ .

Also by y^e conjunction & disjunction of 2 triangles it may be found y^t $ab = b b x + a a x$ $ad = b b x - a a x$ . {illeg} $ac = b b x - a a x - a a x$ $+ a b b$ . {illeg} $bc = a x x + a b b$ . {illeg} $db = 2 a b x$ . $dc = a x x - a b b$ . For if $bd = x$ $dc = \frac{x x - b b}{2 b}$ . $bc = \frac{x x + b b}{2 b}$ . that is $bd = 2 b x$ . $dc = x x - b b$ . $bc = x x + b b$ . Likewise $bd = 2 a b$ . $ad = b b - a a$ . $ab = b b + a a$ . $2 a b x$ y^e least quanti{illeg}|t|y divisible by $2 b x$ & $2 a b$ , being divided by y^m, leaves a & x w^ch must {illeg} multiply y^e bases & hy{illeg}|p|otenusas. If y^e perpendic: fall w^thout y^e legs may be thus exprest $cd = a c c + a y y$ . $da = y y c + a a c$ <14r> $ca = a c c - a y y + c y y - a a c$ . $ae = y y c - a a c$ . $ce = a c c - a y y$ . $ad = 2 a c y$ .

Sec 11

To ma{illeg}|k|e y^t two such tri: be of y^e same base & altitude. Suppose an equation twixt y^e bases & pependiculars {sic} of y^e 2 last tri: as $2 a b x = 2 a c y$ . $x = \frac{c y}{b}$ . $x x = \frac{c c y y}{b b}$ . $b b x - a a x - a x x + a b b = a c c - a y y + y y c - a a c$ or $\frac{b b c y - a a c y}{b} - \frac{a c c y y}{b b} + a b b = a c c - a y y + y y c - a a c$ & $y y = \frac{+ b^{3} c y + a a b b c - a a b c + a b^{4} - a b b c c}{b b c + a c c - a b b}$ . Suppose $a a b b c + a b^{4} = a b b c c$ . {illeg}|o|r $a = c - \frac{b b}{c}$ . let $c = 3$ greater yⁿ $b = 2$ . $a = \frac{5}{3}$ . $y = \frac{22}{61}$ . $x = \frac{33}{61}$ & consequently

Sec 1{illeg}|4| differs not from Cap 19: prob 18 Ou{illeg}|g|htred.

Sec: 15 Of P{illeg}|o|lygons or multangular num{illeg}|b|ers

\The sum̄e of all y^e tearms|e|s in/ an arithmet: progres: increasing from {illeg} \an/ unite by 1 compseth {sic} tra|i|{illeg} triangles. by 2 , composes □s. by 3 , composes pentangles. by 4 , hexang: &c {illeg}ke as $1.2.3.4.5.6 .$ compos {sic} y^e triangles $\begin{matrix} 1 \begin{matrix} ● \end{matrix} & 3 \begin{matrix} ● \\ ● ● \end{matrix} & 6 \begin{matrix} ● \\ ● ● \\ ● ● ● \end{matrix} & 10 \begin{matrix} ● \\ ● ● \\ ● ● ● \\ ● ● ● ● \end{matrix} & 15 \begin{matrix} ● \\ ● ● \\ ● ● ● \\ ● ● ● ● \\ ● ● ● ● ● \end{matrix} \end{matrix}$ &c likewise $1.3.5.7.9 .$ compose $\begin{matrix} 1 \begin{matrix} ● \end{matrix} & 4 \begin{matrix} ● ● \\ ● ● \end{matrix} & 9 \begin{matrix} ● ● ● \\ ● ● ● \\ ● ● ● \end{matrix} & 16 \begin{matrix} ● ● ● ● \\ ● ● ● ● \\ ● ● ● ● \\ ● ● ● ● \end{matrix} & 25 \begin{matrix} ● ● ● ● ● \\ ● ● ● ● ● \\ ● ● ● ● ● \\ ● ● ● ● ● \\ ● ● ● ● ● \end{matrix} \end{matrix}$ &c So $1.4.7.10.13 .$ compose y^e quintangles $1.5.12.22.35.51.70 .$ &c. If $a = 1 =$ y^e first term{e} y^e excess {illeg}|o|f the progression $= x$ . The sume of y^e termes $= z =$ \to y^e polygon/ y^e multitude of y^e termes $= t =$ to y^e side of y^e Polygon. {illeg} given to find {illeg} Suppos {sic} t given to find {illeg}z. $z = \frac{1 t t + 1 t}{2}$ or $z = \frac{t t + t}{2}$ in trigons. $z = t t$ in 4^gons. $z = \frac{3 t t - t}{2}$ in 5^gons. $z = 2 t t - t$ in 6^go $z = \frac{5 t t - 3 t}{2}$ in {illeg}|7|^{{illeg}|g|ons}. $z = 3 t t - 2 t$ in 8^gons. $z = \frac{7 t t - 5 t}{2}$ in 9^gons. &{c} {illeg}|&| z given t is found thus $t = \frac{- 1 + \sqrt{1 + 8 z}}{2}$ in tri. $t = \frac{\sqrt{0 + 16 z}}{4}$ in 4^go $t = \frac{+ 1 + \sqrt{1 + 24 z}}{6}$ , in 5^gons. { $t = + 2 + \sqrt{4 + 32 z}$ } { $t = \frac{+ 2 + \sqrt{4 + 32 z}}{8}$ } in 6^gons &c. As {illeg} y^e side 12 of a tri given. y^e $tri = z = \frac{12 \times 12 + 12}{2} = 78$ &c & if $z = 21$ be octangled. $t = \frac{+ 4 + \sqrt{16 + 48 z}}{12} = \frac{4 + \sqrt{16 + 48 \times 21}}{12}$ $t = \frac{4 + \sqrt{1024}}{12} = 3$ .

<14v>

July 4^th 1699. By consulting an accompt of my expenses at Cambridge in th{illeg}|e| {illeg}|y|ears 1663 & 1664 I find that in y^e year 1664 a little before Christmas I being then Senior Sophister, I borrowed bought Schooten's Miscellanies & Carte's's Geometry (having read this Geometry & Oughtred's \Clavis/ above half a year before) & borrowed Wallis's works & by consequence made these Annotations out of Schooten & Wallis in winter between the years 1664 & 1665. At w^ch time I found the method of Infinite Is series. And in summer 169|6|5 being d|f|orced from Cambrid{illeg}|g|e by the Plague I computed y^e area of y^e Hyperbola a{illeg}|t| Boothby in Lincolnshire to I{illeg} two & fifty figures by the same method. Is. Newton

<15r>

Annotations out of D^r Wallis his Arithmetica infinitorum.

|1| A primanary series of {illeg}|q|uantys {sic} is arithmetically proportionall, as $0, 1, 2, 3, 4$ . & its index is 1

|A| Secundanary series i|a|re those whose rootes are arithmetically proportionall; as $0, 1, 4, 9, 16$ . & its index is 2

A Tertianary, quartanary, quintanary series of quantitys are those whose cube, square square, square cube rootes are Arithmetically Proportionall as $0, 1, 8, 27, 64$ . / $0, 1, 16, 81, 156$ . / $0, 1, 32, 243, 624$ . &c Their indices being $3, 4, 5$ &c.

3 Subsecunda\na/ry, sub{illeg}|t|ertianary, series &c: are those whose squares, cubes, &c are arithmetically proportionall, as $\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}$ . $. \sqrt{c : 0}, \sqrt{c : 1}, \sqrt{c : 2}, \sqrt{c : 3}$ &c. Theire indices being {illeg} $\frac{1}{2}, \frac{1}{3},$ &c.

2 Primary Secundanary, tertianare|y| series &c are said to bee reciprocally proportionall ({illeg} y^t is to y^e sam {sic} se{illeg} increasing) which continually decrease as. $\frac{1}{0}, \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},$ . $\frac{1}{0}, \frac{1}{1}, \frac{1}{4}, \frac{1}{9}, \frac{1}{16},$ . $\frac{1}{0}, \frac{1}{1}, \frac{1}{8}, \frac{1}{27}, \frac{1}{64},$ . Their indices being negative as $- 1, - 2, - 3$ .

4 The indices of compound \or mixt of rationall & irrati{illeg}{onall} \series// series, by multiplying or dividing y^e indices of y^e simple series may bee found as in a subsecundanary progression cube{illeg}|d| $\sqrt{0}, \sqrt{1}, \sqrt{8}, \sqrt{27}, \sqrt{64}$ y^e index is $\frac{1}{2} \times 3 = \frac{3}{2}$ . So in y^e cube rootes of a secundanary progression, $\sqrt{c : 0}, \sqrt{c : 1}, \sqrt{c : 4}, \sqrt{c : 9}$ &c. y^e index is $\frac{1}{3} \times 2 = \frac{2}{3}$ . so in irrationall reciprocal progressions
$\sqrt{qq : \frac{1}{0}}, \sqrt{qq : \frac{1}{1}}, \sqrt{qq : \frac{1}{2}}, \sqrt{qq : \frac{1}{3}}$ , y^e index is $- 1 \times \frac{1}{4} = - \frac{1}{4}$ .

<15v>

Now su{illeg}|p|pose y^e line ac be divided into an infinite number of equall pts $ad, de, ef, fg$ &c, from each of w^ch are drawne perpendiculars \parallels/ $nd pe$ $qf$ &c. w^ch w^ch ino|c|rease continually in so{illeg}|m|e of y^e foregoeing prg|o|gressions or in some progresion {sic} compounded of y^m, all those lines may be taken for y^e surface bqnac , & to know w^t proportion \all/ those lines have that superficies hath to y^e superficies ambc y^t is w^t proportion all those lines have to soe may equal to y^e greatest of y^m, I say as y^e index of y^e progression increased by an unite is to an unite soe is y^e square abcm to y^e area of y^e crooked line. As if abc is a parabola y^e lines $ad, pe, qf,$ &c ao|r|e a subsecundanary series (for $y = \sqrt{r x}$ ) whos {sic} index is $\frac{1}{2}$ w^ch added to an unite is $1 + \frac{1}{2} = \frac{3}{2}$ Therefore $\frac{3}{2} ∶ 1 ∷ 3 ∶ 2$ so is y^e square ambc to y^e area of y^e Parab. ({illeg}|y^e| names of y^e lines are $(ad),$ {illeg} $ae, af$ {illeg}|&|c $= x$ . $dn, pe, qf$ &c $= y$ . $ac = p$ . $bc = q$ .) T{illeg}|h|e case is y^e same if abc bee supposed a sollid, as suppose it{illeg}|s| {nature} t a Parabolicall conoides. yⁿ since y^e nature of it is $r x = y y$ . $y y$ designes y^e squares $nd, pe, qf$ &c: all w^ch taken together are equivalent to y^e Sollid. & those □s increase in y^e same proportion w^ch $r x$ . or x doth. y^t is they are a primanary series whose index i{illeg}|s| 1 to w^ch (according to y^e rule I ad an unite & tis 2. Therefor 1 {illeg} $∶ 2 ∷$ soe are all y^e □s of y^e Primary series to soe many □s equall to y^e greatest of y^t series. & soe is y^e conoides to a cilinder of y^e same altid|t|ude.

<16r>

Also if a superficies be compounded of 2 or more of these series, Their area is as easily found as if y^e nature of y^e line bee {illeg}, {illeg} y {illeg} $= a a - x x$ , or {illeg} $y = a^{4} - 2 a a x x + x^{4}$ or {illeg} $y = a^{6} - 3 a^{4} x x + 3 a a x^{4} - x^{6}$ . &c. Their areas will bee to y^e parallelograms {ar|b|o{illeg}|u|t} them as {illeg}|2 | to {illeg}|3 |, as 8 to $\frac{1}{5}$ 15 , as 48 to 105 &c. but if I put in y^e intermediate termes in these last named lines th{illeg}|e|ir order will bee $y = \sqrt{a a - x x}$ , $y = a a - x x$ , $y = \bar{a a - x x \sqrt{a a - x x}}$ , $y = a^{4} - 2 a a x x + x^{4}$ . $y = \bar{a^{4} - 2 a a x x + x^{4} \sqrt{a a - x x}}$ . $y = a^{6} - 3 a^{4} x x + 3 a a x^{4} - x^{6}$ ; &c: & since these lines observe a geometricall progression their areas must observe some kind of progression. of w^ch every other terme is given viz $1 . □ . \frac{2}{3} . ∗ . \frac{8}{15} . ∗ . \frac{48}{105} . ∗ . \frac{384}{945} . ∗ . \frac{3840}{10395}$ . T{b}{l}wixt w^ch {illeg}|t|ermes if y^e intermediate termes $□ . ∗$ can bee found y^e 2^nd □ will give y^e area of y^e line $y = \sqrt{a a - x x}$ , y^e circle. Soe likwise {sic} in this progression of lines {illeg}| y | $= 1$ . {illeg}| y | $= \sqrt{a x - x x}$ . $y = a a - x x$ $y = \bar{a x - x x \sqrt{a x - x x}}$ . $y = a a x x - 2 a x^{3} + x^{4}$ &c: y^e progression of their areas {illeg} is $1 : □ : \frac{1}{6} : ∗ :$ $\frac{1}{30} : ∗ : \frac{1}{140} : ∗ \frac{1}{630} :$ &c. y^e 2^nd if it can bee found givs {sic} y^e area of y^e ○ for as its denominator to its numerator so is y^e □ of y^e diameter to y^e area of a semicircle. If th{illeg}|i|s last progresion {sic} bee multiplyed by y^e respective termes in y^e progress $1.2.3.4$ & it may bee diminished y^e reslut {sic} being $1 . 2 □ . \frac{1}{2} . 4 ∗ . \frac{1}{6} . 6 ∗ . \frac{1}{20} . 8 ∗ . \frac{1}{70}$ soe y^t in this progression {illeg} $1, b, \frac{1}{2}, c, \frac{1}{6}, d, \frac{1}{20}, e, \frac{1}{70}, f,$ &c: if <16v> b can be found yⁿ, {illeg} y^e □ of y^e diameter to y^e area of y^e circle is as y^e denominator of b to its numerator. Likewise y^e 1^st series of areas may be diminished by multiplying each terme by its correspondent terme in this progression $1, 2, 3, 4, 5, 6,$ &c: & it will become, $1, a, 2, b, \frac{8}{3}, c, \frac{48}{15},$ $d, \frac{384}{105}, e, \frac{3840}{945}$ . &c. In w^ch if a can bee found yⁿ as y^e denom of a to its num: so y^e □ of y^e Radius is a semicircle, y^t making y^e radius $= q$ . $2 a q = ○$ . The same \kinds of/ changes may bee performed |by| other p any other progressions, as \by division/ by y^e geometricall progression $1, 2, 4, 8, 16,$ & y^e {illeg}|fi|rst series of areas becomes $1, g, \frac{1}{6}, h, \frac{1}{30}, k, \frac{1}{140}, l, \frac{1}{630},$ &c viz y^e same w^th y^e 2^d series. Also these changes may be done by addition or substraction of mutuall termes in 2 proportions. Soe y^t y^e most convenient way \may be/ be chosen, {illeg}|w|he|r|by to reduce any series of proportions to y^e most convent {sic} forme.

Now if it be propounded to find thes{illeg}e mi{illeg}|d|dle termes, f|i| |I|t w{illeg}|i|ll {illeg}|b|ee convenient to d|f|ind how the given proportion may bee deduced from an Arithmeticall, Geometricall, or some other familiar proportion, viz whose meane termes may be found, as this progression $1 . \frac{2}{3} . \frac{8}{15} . \frac{48}{105}$ deduceth its originall from this {illeg}| A | $\times \frac{0 \times 2 \times 4 \times 6 \times 8}{1 \times 3 \times 5 \times 7 \times 9}$ & in w^ch A is an infinine {sic} number $= \frac{1}{0}$ .

It will also be convenient to find what relatiō all y^e other meanes have to y^e first soe y^t if y^e first bee had all y^e other may bee deduced thence. As in this case suppose y^e 1^st meane to bee a . The progression will bee <17r> $\frac{1}{2} a : 1 : a : \frac{3}{2} : \frac{4 a}{3} : \frac{15}{8} : \frac{8 a}{5} : \frac{105}{48} : \frac{64 a}{35} : \frac{945}{384} : \frac{640 a}{315} :$ deducing its originall from $A \times \frac{0 \times 2 \times 4 \times 6 \times 8 \times 10}{1 \times 3 \times 5 \times 7 \times 9 \times 11}$ {illeg}|&| from this $A (= \frac{1}{2}) \times \frac{2 a \times 4 a \times 6 a \times 8 a \times 10 a}{1 \times 3 \times 5 \times 7 \times 9}$ . &c {+} (note y^t y^e proportions of th{illeg}|e|se te|m|eane termes to oneano {sic}ther, or to $(a)$ , are found f by finding y^e proportion of y^e circle $y = \sqrt{a a - x x}$ to y^e line $y = \bar{a a - x x \sqrt{a a - x x}}$ &c).

In this case to find y^e quantity a : it may be considered y^t $\frac{a}{\frac{1}{2} a} = 2$ . {illeg} $\frac{3}{1 \times 2} =$ {illeg} $\frac{3}{2}$ . $\frac{4 a}{3 \times a} = \frac{4}{3}$ . Naming y^e termes in y^e progress: $\begin{matrix} \frac{1}{2} a & : & 1 & : & a & : & \frac{3}{2} & : & \frac{4 a}{3} & : & \frac{15}{8} & : & \frac{8 a}{5} & : & \frac{35}{16} & : \\ b & c & d & e & f & g & h & k & . \end{matrix}$ 1^st observe y^t $\frac{d}{b} = 2 . \frac{e}{c} = \frac{3}{2} . \frac{f}{d} = \frac{4}{3} . \frac{g}{e} = \frac{5}{4} . \frac{h}{f} = \frac{6}{5}$ &c y^e proportions still {illeg}|de|creasing & therere|f|ore {sic} y^t in $\frac{c}{b} . \frac{d}{c} . \frac{e}{d} . \frac{f}{e} . \frac{g}{f} . \frac{h}{g} . \frac{k}{h}$ &c: y^e latter terme is lesse yⁿ y^e former; & therefore $\frac{d d}{c c} {is \begin{matrix} less y^{n} \frac{d}{c} \times \frac{c}{b} = \frac{d}{b} = 2 . \\ greater y^{n} \frac{d}{c} \times \frac{e}{d} = \frac{e}{c} = \frac{3}{2} . \end{matrix}$ or / $a =$ \ $d is {\begin{matrix} less y^{n} 1 \times \sqrt{2} = \sqrt{1 + \frac{1}{1}} \\ greater y^{n} 1 \times \sqrt{\frac{3}{2}} = \sqrt{1 + \frac{1}{2}} \end{matrix}$ . Also
$\frac{f f}{e e} is {\begin{matrix} lesse y^{n} \frac{f}{e} \times \frac{e}{d} = \frac{f}{d} = \frac{4}{3} \\ greater & \bar{y} & \frac{f}{e} \times \frac{g}{f} = \frac{g}{e} = \frac{5}{4} \end{matrix}} = a : \frac{8 a}{9} = \frac{2 \times 4 a}{3 \times 3}$ . Therefore $3 \times 3 \sqrt{\frac{4}{}}$
Therefore $a is {\begin{matrix} lesse y^{n} & \frac{3 \times 3}{2 \times 4} \sqrt{\frac{4}{3}} \\ greater y^{n} & \frac{3 \times 3}{2 \times 4} \sqrt{\frac{5}{4}} \end{matrix}$ . And So by y^e /same reasoning.\ $a is {\begin{matrix} less y^{n} & \frac{9 \times 25 \times 49 \times 81 \times 121 \times 169}{2 \times 16 \times 36 \times 64 \times 100 \times 144 \times 14} \sqrt{\frac{14}{13}} & . \\ greater y^{n} & \frac{3 \times 3 \times 5 \times 5 \times 7 \times 7 \times 9 \times 9 \times 11 \times 11 \times 13 \times 13}{2 \times 4 \times 4 \times 6 \times 6 \times 8 \times 8 \times 10 \times 10 \times 12 \times 12 \times 14} \sqrt{\frac{15}{14}} & . \end{matrix}$ &c. Thus {illeg}|W|allis doth it. but it ma{illeg}|y| bee done thus. $a is {\begin{matrix} greater & y^{n} & 1 & . \\ less then & \frac{3}{2} & . \end{matrix} |$ $\frac{4 a}{3} is {\begin{matrix} greater then & \frac{3}{2} \\ less then & \frac{15}{8} \end{matrix}$ Therefore <17v> $a is {\begin{matrix} greater y^{n} \frac{3 \times 3}{2 \times 4} \\ lesse then \frac{3 \times 3 \times 5}{2 \times 4 \times 4} \end{matrix}$ . $\frac{8 a}{5} is {\begin{matrix} greater y^{n} \frac{15}{8} \\ lesse y^{n} \frac{35}{16} \end{matrix}$ y^t is $a is {\begin{matrix} greater then & \frac{3 \times 3 \times 5 \times 5}{2 \times 4 \times 4 \times 6} = \frac{3 \times 5 \times 5}{2 \times 4 \times 4 \times 2} \\ lesse then & \frac{3 \times 3 \times 5 \times 5 \times 7}{2 \times 4 \times 4 \times 6 \times 6} = \frac{5 \times 5 \times 7}{2 \times 4 \times 4 \times 4} \end{matrix}$ &c. By y^e same reasoning
$a is {\begin{matrix} greater y^{n} \frac{3}{2} \times \frac{3}{4} \times \frac{5}{4} \times \frac{5}{6} \times \frac{7}{6} \times \frac{7}{8} \times \frac{9}{8} \times \frac{9}{10} = \frac{5 \times 49 \times 81}{2 \times 16 \times 4 \times 64 \times 2} . \\ lesse y^{n} \frac{3}{2} \times \frac{3}{4} \times \frac{5}{4} \times \frac{5}{6} \times \frac{7}{6} \times \frac{7}{8} \times \frac{9}{8} \times \frac{9}{10} \times \frac{11}{10} = \frac{49 \times 81 \times 11}{2 \times 16 \times 4 \times 64 \times 4} . \end{matrix}$ Or
$a is {\begin{matrix} greater \bar{y} \frac{11 \times 169 \times 225 \times 289 \times 369 \times 441}{2 \times 16 \times 4 \times 64 \times 4 \times 144 \times 4 \times 256 \times 4 \times 400 \times 2} \\ lesse y^{n} \frac{169 \times 225 \times 289 \times 369 \times 441 \times 23}{2 \times 16 \times 4 \times 64 \times 4 \times 144 \times 4 \times 256 \times 4 \times 400 \times 4} \end{matrix}$ . Note y^t a is greater yⁿ $\frac{1}{2}$ these two summes.

<18r>

Having y^e signe of any angle to find y^e angle or to find y^e content of any segm^nt of a circle

Suppose y^e circle to be aec its \se{illeg}|m|i/diameter $ap = pc = 1$ . y^e given sine $pq = x$ , viz: y^e signe of y^e angle epa . y^e segment sought eapq . abcp the □ of its Radius. & y^t, $qi : qk : qg : qf : qe : qd : qh : ql : qr :$ &c are continally {sic} proportionall. Then is $eq = \sqrt{1 - x x}$ . $fq = 1 - x x$ . $gq =$ {illeg} $\bar{1 - x x in \sqrt{1 - x x}}$ . $hq = 1 - 2 x x + x^{4}$ . $iq = 1 - 2 x^{2} + x^{4} \sqrt{1 - x x}$ $dq = 1$ . $kq = \frac{1}{\sqrt{1 - x x}}$ . $lq = \frac{1}{1 - x x}$ . $rq = \frac{1}{1 - x x in \sqrt{1 - x x}}$ . &c & since all the{illeg} ordinately applyed lines in these figures $abcq, aecq, afcq, agcq,$ &c are geometr{illeg}|i|cally proportionall their areas $adqp, aeqp, afqp, agqp, ahpq$ &c will observe some proportion amongst one another. To find w^ch proportion, 1^st $adqp = 1 \times x x = x$ . 2^dly afc is a parab: therefore $afqp = x - \frac{x^{3}}{4}$ . also since tis $qh = 1 - 2 x x + x^{4}$ , therefore $ahqp = x - \frac{2}{3} x^{3} + \frac{1}{5} x^{5}$ . |Also| $vq = 1 - 3 x x + 3 x^{4} - x^{6}$ , therefore $avqp =$ {illeg}/ $x - x^{3}$ \ $+ \frac{3}{5} x^{5} - \frac{1}{7} x^{7}$ . & by y^e same proceeding y^e proportion may bee still continued after this manner <18v> $x : x - \frac{1}{3} x^{3} : x - \frac{2}{3} x^{3} + \frac{1}{5} x^{5} : x - \frac{3}{3} x^{3} + \frac{3}{5} x^{5} - \frac{1}{7} x^{7} :$ $x - \frac{4}{3} x^{3} + \frac{6}{5} x^{5} - \frac{4}{7} x^{7} + \frac{1}{9} x^{9} : x - \frac{5}{3} x^{3} + \frac{10}{5} x^{5} - \frac{10}{7} x^{7} + \frac{5}{9} x^{9} - \frac{1}{11} x^{11} :$ $x - \frac{6}{3} x^{3} + \frac{15}{5} x^{5} - \frac{20}{7} x^{7} + \frac{15}{9} x^{9} - \frac{6}{11} x^{11} + \frac{1}{13} x^{3} : x - \frac{7}{3} x^{3} + \frac{21}{5} x^{5} - \frac{35}{7} x^{7} + \frac{35}{9} x^{9} - \frac{21}{11} x^{11}$ $+ \frac{7}{13} x^{13} - \frac{1}{15} x^{15}$ . &c.
And if y^e meane termes be inserted it will bee
$x : x - : x - \frac{1}{3} x^{3} : x - \frac{3}{6} x^{3} + : x - \frac{2}{3} x^{3} + \frac{1}{5} x^{5} : x - \frac{5}{6} x^{3} + \frac{2}{5} x^{5} -$ or $x : x - \frac{1}{6} x^{3} : x - \frac{1}{3} x^{3} : x$ The first letters \ x / run in this {illeg} progression $1.1.1.1.1 .$ &c. y^e 2^d \ $x^{3}$ / in this $\frac{- 1}{3} . \frac{0}{3} . \frac{1}{3} . \frac{2}{3} . \frac{3}{3} . \frac{4}{3} . \frac{5}{3}$ &c y^e 3^d $x^{5}$ in this $0 + 1 . 1 + 2 . 3 + 3 . 3 .$ \ $6.3.1.0 .$ / $0 + 1 = 1 . 1 + 2 = 3$ $3 + 3 = 6 . 6 + 4 = 10 . 10 + 5 = 15$ . y^e 4^th $x^{7}$ th{illeg}|i|s $\begin{matrix} \begin{matrix} 1^{st} & . & + x \times 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . \\ 2^{d} & . & - \frac{x^{3}}{3} \times 0 & . & 0 + 1 = 1 & . & 1 + 1 = 2 & . & 2 + 1 = 3 & . & 3 + 1 = 4 & . & 4 + 1 = 5 & . & 6 & . & 7 & . & 8 & . & 9 & . & 10 & . \\ 3^{d} & . & + \frac{x^{5}}{5} \times 0 & . & 0 + 0 = 0 & . & 0 + 1 = 1 & . & 1 + 2 = 3 & . & 3 + 3 = 6 & . & 6 + 4 = 10 & . & 15 & . & 21 & . & 28 & . & 36 & . & 45 & . \\ 4^{th} & . & - \frac{x^{7}}{7} \times 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 1 = 1 & . & 1 + 3 = 4 & . & 4 + 6 = 10 & . & 20 & . & 35 & . & 56 & . & 84 & . & 120 & . \\ 5^{} & . & + \frac{x^{9}}{9} \times 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 1 = 1 & . & 1 + 4 = 5 & . & 15 & . & 35 & . & 70 & . & 126 & . & 210 & . \\ 6^{} & . & - \frac{x^{11}}{11} \times 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 0 = 0 & . & 0 + 1 = 1 & . & 6 & . & 21 & . & 56 & . & 126 & . & 252 & . \\ 1^{st} . & ^{∗} 2^{d} . & ^{∗} 3^{d} & ^{∗} 4^{th} & ^{∗} 5^{th} & ^{∗} 6^{th} & 1 & . & 7 & . & 28 & . & 84 & . & 210 & . \\ ^{∗} & 7^{th} & . & 1 & . & 8 & . & 36 & . & 120 & . \\ ^{∗} & 8^{th} & 1 & . & 9 & . & 45 & . \\ ^{∗} & 9^{th} & 1 & . & 10 & . \\ ^{∗} & 10^{th} & 1 & . \\ ^{∗} & 11^{th} & . \end{matrix} \end{matrix}$ Now if the meane ter{illeg}|m|es in these progressions can bee callculated y^e first of y^m gives y^e area aeqp. Which is thus done $\begin{matrix} \begin{array}{c} 1^{st} & + x \times 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . \\ 2^{d} & . & - \frac{x^{3}}{3} \times 0 & . & \frac{1}{2} & . & 1 & . & \frac{3}{2} & . & 2 & . & \frac{5}{2} & . & 3 & . & \frac{7}{2} & . & 4 & . & \frac{9}{2} & . & 5 & . & \frac{11}{2} & . & 6 & . \\ 3^{d} & . & + \frac{1}{5} x^{5} \times 0 & . & - \frac{1}{8} & . & 0 & . & \frac{3}{8} & . & 1 & . & \frac{15}{8} & . & 3 & . & \frac{35}{8} & . & 6 & . & \frac{63}{8} & . & 10 & . & \frac{99}{8} & . & 15 & . \\ 4 & . & - \frac{1}{7} x^{7} \times 0 & . & + \frac{1}{16} & . & 0 & . & - \frac{1}{16} & . & 0 & . & \frac{5}{16} & . & 1 & . & \frac{35}{16} & . & 4 & . & \frac{105}{16} & . & 10 & . & \frac{231}{16} & . & 20 & . \\ 5 & . & + \frac{1}{9} x^{9} \times 0 & . & - \frac{5}{128} & . & 0 & . & \frac{3}{128} & . & 0 & . & - \frac{5}{128} & . & 0 & . & \frac{35}{128} & . & 1 & . & \frac{315}{128} & . & 5 & . & \frac{1155}{128} & . & 15 & . \\ 6 & . & - \frac{1}{11} x^{11} \times 0 & . & \frac{7}{256} & . & 0 & . & - \frac{3}{256} & . & 0 & . & \frac{3}{256} & . & 0 & . & - \frac{7}{256} & . & 0 & . & \frac{63}{256} & . & 1 & . & \frac{693}{256} & . & 6 & . \\ 7 & . & \frac{1}{13} x^{13} \times 0 & . & - \frac{21}{1024} & . & 0 & . & \frac{7}{1024} & . & 0 & . & - \frac{5}{1024} & . & 0 & . & \frac{7}{1024} & . & 0 & . & - \frac{21}{1024} & . & 0 & . & \frac{231}{1024} & . & 1 & . & \frac{3003}{1024} & . \end{array} \end{matrix}$ Soe y^t $1 \times x - \frac{1}{2} \times \frac{1}{3} \times x^{3} - \frac{1}{8} \times \frac{1}{5} \times x^{5} - \frac{1}{16} \times \frac{1}{7} \times x^{7} - \frac{5}{128} \times \frac{1}{9} \times x^{9}$ &c. is y^e area, apqe y^t is $\frac{0}{0} \times x - \frac{0}{0} \times \frac{1}{2} \times \frac{1}{3} x^{3} - \frac{0}{0} \times \frac{1}{2} \times \frac{1}{4} \times \frac{1}{5} x^{5} - \frac{0}{0} \times \frac{1}{2} \times \frac{1}{4} \times \frac{3}{6} \times \frac{1}{7} x^{7} - \frac{1 \times 3 \times 5 x^{9}}{2 \times 4 \times 6 \times 8 \times 9}$ &c: The progression may be deduced from hence $\frac{0 \times 1 \times - 1 \times 3 \times - 5 \times 7 \times - 9 \times 11}{0 \times 2 \times 4 \times 6 \times 8 \times 10 \times 12 \times 14}$ . &c <19r> Soe y^t if {illeg}|y^e| given sine bee $pq = te = x$ . & if y^e Radius $pc = 1$ . Then is y^e superficies $ape = x - x \sqrt{1 - x x} - \frac{1}{6} x^{3} - \frac{1}{40} x^{5} - \frac{x^{7}}{112} - \frac{5 x^{9}}{1152} - \frac{7 x^{11}}{2816}$ &c: And y^e area $ade = \frac{x^{3}}{6} + \frac{x^{5}}{40} + \frac{x^{7}}{112} + \frac{5 x^{9}}{1152} + \frac{7 x^{11}}{2816} + \frac{21 x^{13}}{13312}$ $+ \frac{11 x^{15}}{10240} + \frac{429 x^{17}}{557056} + \frac{715 x^{19}}{1245184} + \frac{2431 x^{21}}{5505024}$ &c. By w^ch meanes y^e angle ape is easily found for $aecpa ∶ ∠ apc = 90 ∷ ape ∶ ∠ ape$ .

The same may bee thus done.

$adp = \frac{x}{2}$ . Or $2 adp = x$ . $2 afp = x + \frac{x^{3}}{3}$ . $2 ahp = x + \frac{2 x^{3}}{3} - \frac{3 x^{5}}{5}$ . And $2 avp = x + x^{3} - \frac{9 x^{5}}{5} + \frac{5}{7} x^{7}$ . &c. as in this order $x . x + \frac{1}{3} x^{3} . x + \frac{2}{3} x^{3} - \frac{3}{5} x^{5} . x + x^{3} - \frac{9}{5} x^{5} + \frac{5 x^{7}}{7} . x + \frac{4 x^{3}}{3} - \frac{18 x^{5}}{5} + \frac{20 x^{7}}{7} -$ $- \frac{7}{9} x^{9} . x + \frac{5}{3} x^{3} - \frac{30}{5} x^{5} + \frac{50 x^{7}}{7} - \frac{35 x^{9}}{9} + \frac{9 x^{11}}{11} . x + \frac{6 x^{3}}{3} - \frac{45 x^{5}}{5} + \frac{100}{7} x^{7} - \frac{105 x^{9}}{9} +$ $+ \frac{54 x^{11}}{11} - \frac{11 x^{13}}{13} . x + \frac{7 x^{3}}{3} - \frac{63 x^{5}}{5} + \frac{175 x^{7}}{7} - \frac{245 x^{9}}{9} + \frac{189}{11} x^{11} - \frac{77 x^{13}}{13} + \frac{13 x^{15}}{15}$ . &c Which progression w^th their intermediate termes {illeg}|may| bee thus exhibited. By{illeg} w^ch it may appeare y^t if $pe = 1$ . $pq = x$ . yⁿ $aep = \frac{1}{2} x + \frac{x^{3}}{12} + \frac{3 x^{5}}{80} + \frac{5 x^{7}}{224} + \frac{35 x^{9}}{2304}$ &c. And y^e area aep given gives y^e angle ape for $apce ∶ ∠ apc = 90^{d} ∷ ape ∶ ∠ ape$ Likewise y^e angle ape given its sign ma{illeg}|y| bee found hereby |&c| $\begin{matrix} \begin{matrix} 2 \times adpa = & 2 \times aep = & 2 afp = & 2 agp = & 2 ahp = & 2 aip = & 2 avp = \end{matrix} \\ \begin{array}{c} + x in 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . \\ \frac{x^{3}}{3} in 0 & . & \frac{1}{2} & . & 1 & . & \frac{3}{2} & . & 2 & . & \frac{5}{2} & . & 3 & . \\ - \frac{3}{5} x^{5} in 0 & . & - \frac{1}{8} & . & 0 & . & \frac{3}{8} & . & 1 & . & \frac{15}{8} & . & 3 & . \\ + \frac{5}{7} x^{7} in 0 & . & \frac{1}{16} & . & 0 & . & - \frac{1}{16} & . & 0 & . & \frac{5}{16} & . & 1 & . \\ - \frac{7}{9} x^{9} in 0 & . & - \frac{5}{128} & . & 0 & . & \frac{3}{128} & . & 0 & . & - \frac{5}{128} & . & 0 & . \\ + \frac{9}{11} x^{11} in 0 & . & \frac{7}{256} & . & 0 & . & - \frac{3}{256} & . & 0 & . & \frac{3}{256} & . & 0 & . \end{array} \end{matrix}$ Note y^t $\sqrt{1 - x x} = \frac{x}{2} - \frac{2 x^{3}}{6} - \frac{5 x^{5}}{80} - \frac{7 x^{7}}{224} - \frac{45 x^{9}}{2304} - \frac{77 x^{11}}{5632}$ &c that is $\sqrt{1 - x x} = \frac{x}{2} - \frac{x^{3}}{3} - \frac{x^{5}}{16} - \frac{x^{7}}{32} - \frac{5 x^{9}}{256} - \frac{7 x^{11}}{512} - \frac{21 x^{13}}{2048}$ {illeg}|&|c. According to this progression $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{4} \times \frac{3 \times 5 \times 7 \times 9 \times 11 \times 13 \times 15 \times 17}{6 \times 8 \times 10 \times 12 \times 14 \times 16 \times 18 \times 20}$ &c. Note also y^t y^e segment $ae = \frac{x^{3}}{12} + \frac{3 x^{5}}{80} + \frac{5 x^{7}}{224} + \frac{35 x^{9}}{2304}$ . &c. $aep = \frac{x}{2} + \frac{x^{3}}{12} + \frac{3 x^{5}}{80} + \frac{5 x^{7}}{224} + \frac{35 x^{9}}{3304} + \frac{63 x^{11}}{5632} + \frac{231 x^{13}}{26624} + \frac{143 x^{15}}{20480} + \frac{6435 x^{17}}{1114112} + \frac{12155 x^{19}}{2490368} + \frac{46189 x^{21}}{11010048}$ .

<19v>

If $pq = a$ . $qd = x$ . $pc = 1 = pb$ . $db = \sqrt{1 - a a - 2 a x - x x}$ yⁿ y^e areas of y^e lines in this progression. $1 - \sqrt{1 - a a - 2 a x - x x} . 1 - a a - 2 a x - x x . \bar{1 - a a - 2 a x - x x}} \frac{3}{2} .$ $1 - 2 a a - 4 a x - 2 x x + a^{4} + 4 a^{3} x + 6 a a x x + 4 a x^{3} + x^{4} . * .$ $1 - 3 a a + 3 a^{4} + 8 a^{3} x + 8 a a x x$ & (supposeing also $\overline{b - 2 a x - x x}} \frac{3}{2} . \begin{matrix} b b - 4 a b x - 2 b x x + 4 a x^{3} + x^{4} \\ + 4 a a \end{matrix} . *$ $\begin{matrix} b^{3} - 6 a b b x - 3 b b x x + 8 a b x^{3} + 3 b x^{4} - 6 a x^{5} - x^{6} \\ + 12 a a b x^{2} - 8 a^{3} x^{3} - 12 a a x^{4} \\ + 4 a b x^{3} \end{matrix}$ . &c

<20r>

To squar{illeg}|e| y^e Hyperbola.

Epitome Geometriæ

So if nadm is an Hyperbola. & $cp = 1 = pa$ . $pq = x$ . $qd, qe, qf, qg$ &c $= y$ . & $\frac{1}{1 + x} = y = dq$ . $1 = y = eq$ . $1 + x = y = qf$ . $1 + 2 x - x x = y = qg$ . $1 + 3 x + 3 x x + x^{3}$ . $1 + 4 x + 6 x^{2} + 4 x^{3} + x^{4}$ . &c. There {sic} squares are. $* . x . x + \frac{x x}{2} . x + \frac{2 x x}{2} + \frac{x^{3}}{3} . x + \frac{3 x x}{2} + \frac{3 x^{3}}{3} + \frac{x^{4}}{4} .$ $x + \frac{4 x x}{2} + \frac{6 x^{3}}{3} + \frac{4 x^{4}}{4} + \frac{x^{5}}{5} . x + \frac{5 x x}{2} + \frac{10 x^{3}}{3} + \frac{10 x^{4}}{4} + \frac{5 x^{5}}{5} + \frac{x^{6}}{6}$ . &c As in y^e following table. By whose first terme is represented y^e square of y^e Hyperbola, viz y^t it is $\begin{array}{l} \begin{matrix} \begin{matrix} \begin{array}{l} \begin{array}{l} a & a & a & a \\ p & p & p & p \\ q & q & q & q \\ d & e & f & g \\ = & = & = & = \end{array} \\ \begin{array}{l} x \times 1 . & 1 . & 1 . & 1 . & 1 . & 1 . & 1 . & 1 . & 1 . \\ \frac{x^{2}}{2} \times - 1 . & 0 . & 1 . & 2 . & 3 . & 4 . & 5 . & 6 . & 7 . \\ \frac{x^{3}}{3} \times 1 . & 0 . & 0 . & 1 . & 3 . & 6 . & 10 . & 15 . & 21 . \\ \frac{x^{4}}{4} \times - 1 . & 0 . & 0 . & 0 . & 1 . & 4 . & 10 . & 20 . & 35 . \\ \frac{x^{5}}{5} \times 1 . & 0 . & 0 . & 0 . & 0 . & 1 . & 5 . & 15 . & 35 . \\ \frac{x^{6}}{6} \times - 1 . & 0 . & 0 . & 0 . & 0 . & 0 . & 1 . & 6 . & 21 . \\ \frac{x^{7}}{7} \times 1 . & 0 . & 0 . & 0 . & 0 . & 0 . & 0 . & 1 . & 7 . \end{array} \\ \begin{array}{l} 0000.00000.00000 . \\ 3333.33333.33333.33333.33333.3 \\ 0000.00000.00000.00000.00000.0 \\ 8571.42857.14285.71428.57142.8 \\ 1111.11111.11111.11111.11111.1 \\ 9090.90909.09090.90909.09090.9 \end{array} \end{array} & \begin{matrix} \begin{matrix} x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} - \frac{x^{6}}{6} + \\ + \frac{x^{7}}{7} - \frac{x^{8}}{8} + \frac{x^{9}}{9} - \frac{x^{10}}{10} . &c. \end{matrix} \\ \begin{matrix} \begin{array}{r} \begin{array}{r} \begin{matrix} As if x = \frac{1}{10} .
or cq = \frac{11}{10} = 1,1 . \\ y^{n} is dapq = \frac{1}{10} - \frac{1}{200} + \frac{1}{3000} - \\ - \frac{1}{40000} + \frac{1}{500000} - \frac{1}{6000000} . &c \end{matrix} \end{array} \end{array} \\ \begin{array}{r} y^{t} is apqd = 0,10000.00000.000000 \\ \frac{1}{3} x^{3} = + 0,00033.33333.333333 \\ \frac{1}{5} x^{5} = + 0,00000.20000.000000 \\ \frac{1}{7} x^{7} = + 0,00000.00142.857142 \\ \frac{1}{9} x^{9} = + 0,00000.00001.111111 \\ \frac{1}{11} x^{11} = + 0,00000.00000.009090 \\ \frac{1}{13} x^{13} = + 0,00000.00000.000076 \\ \underline{\frac{1}{15} x^{15} = + 0,00000.00000.000000} \\ 0,10033.53477.310755 . \\ Summa \end{array} \end{matrix} \end{matrix} \end{matrix} \\ \begin{array}{l} \begin{array}{l} 9230.76923.07692.30769.23076.9230769230 \\ 6666.66666.66666.66666.66666.6666666666 . \\ 0058.82352.94117.64705.88235.2941176470 . \\ 0000.52631.57947.36842.10526.3157947368 \\ 0000.00476.19047.61904.76190.4761904761 . \\ 0000.00004.34782.60869.56521.7391304347 . \end{array} \end{array} \end{matrix} \\ \begin{array}{l} \begin{array}{l} 0000.00000.04000.00000.00000.0000000000 \\ 0000.00000.000 == ===== ===== . \\ 37.03703.70370.3703703703 \\ 0000.00000.00000.34482.75862.0689655172.4137931034.48275862 \\ 0000.00000.00000.00 === ===== = \\ 322.58064.5161290322.5806451612.90322580 \\ 0000.00000.00000.00003.03030.3030303030.3030303030.30303030 \\ 0 02857.1428571428.5714285714.28571428 \\ 0000.00000.00000.00000 00027.0270270270.2702702702.70270270 \\ 0000.00000.00000.00000 00000.2564010256.4010256401.02564010 \\ 0000.00000.00000.00000 00000.0024390243.9024390243.90243902 \end{array} \\ \begin{matrix} = 8063.57265.52007.40736.63159.415063 &c = summæ \end{matrix} \\ \begin{array}{r} - 0.00500.00000.00000.00000.00000.00000.00000.00000.00000.00000.0 \\ 2.50000.00000.00000.00000.00000.00000.00000.00000.00000.0 \\ 1666.66666.66666.66666.66666.66666.66666.66666.66666.6 \\ 12.50000.00000.00000.00000.00000.00000.00000.00000.0 \\ 10000.00000.00000.00000.00000.00000.00000.00000.0 \\ 83.33333.33333.33333.33333.33333.33333.33333.3 \\ 71428.57142.85714.28571.42857.14285.71428.5 \\ 630.55555.55555.55555.55555.55555.55555.5 \\ 5045.45454.54545.45454.54545.45454.5 . \\ 41666.66666.66666.66666.66666.6 \\ 384.61538.46153.84615.38461.5 \\ 3.57142.85714.28571.42857.1 \\ 3364.58333.33333.33333.3 \\ 29411.76470.58823.5 \end{array} \\ - 0.00502.51679.26750.72059.17744.28779.27385.30147.14044.12586 . \end{array} \end{array}$

<20v>

cui addendum $\begin{matrix} \begin{array}{r} - 277.77777.77777.7 \\ - 2.63157.89421.0 \\ - 02523.80952.4 \\ - 22727.3 \\ - 217.4 & that is \\ 2.1 \end{array} \\ \begin{matrix} - 280.43459.71098.0 \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix}$ And so y^e summe will bee $\begin{matrix} \begin{array}{r} + 0.10033.53477.31075.58063.57265.52007.40736.63159.41506.3 \\ - 0.00502.51679.26750.72059.17144.28779.27385.30427.57503.8 \end{array} \\ \begin{matrix} 0.09530.01798.04324.86004.40121.23228.13351.32731.84002.5 \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix}$ w^ch is y^e quantity of y^e area adpq . If $cpab = 1$ . & $cp =$ $ab = 10 pq$ & $qde || ap || bc = ap$ . In like manner if I make $x = \frac{1}{100} = pq$ . The opperacon followeth. $\begin{matrix} \begin{array}{r} 0, 01000.03333.33333.33333.33333.33333.33333.33333.33333.3 \\ + \frac{1}{5} x^{5} + \frac{1}{7} x^{7} = 33333.20001.42857.14285.71428.57142.85714.28571.40 \\ == ===== ===== ===== ===== = \\ \frac{1}{9} x^{9} + \frac{1}{11} x^{11} = 11.11202.02020.20202.02020.20202.0 \\ \frac{1}{13} x^{13} + \frac{1}{15} x^{15} = 769.23076.92307.69230.7 \\ \frac{1}{17} x^{17} = 6666.66666.66666.6 \\ \frac{1}{19} x^{19} = 58823.52941.1 \\ \frac{1}{21} x^{21} = 5.26315.7 \\ 47.6 \end{array} \\ \begin{array}{r} + 0, 01000.03333.53334.76201.58821.07551.40422.38870.97309 . \end{array} \\ \begin{array}{r} - 0, 00005.00025.00166.66666.66666.66666.66666.66666.66666.6 . \\ - \frac{1}{8} x^{8} - \frac{1}{10} x^{10} - \frac{1}{12} x^{12} = - 1250.10000.83333.33333.33333.33333.3 \\ - \frac{1}{14} x^{14} = - 7.14285.71428.57142.8 . \\ - \frac{1}{16} x^{16} - \frac{1}{18} x^{18} = - 62.50555.55555.5 \\ ===== ===== = \\ - \frac{1}{20} x^{20} = - 5000.4 . \end{array} \\ \begin{array}{r} - 0, 00005.00025.00166.67916.76667.50007.14348.21984.17699 . \end{array} \\ \begin{array}{r} + 0, 00995.03308.53168.08284.82153.57544.26074.16886.79610 \end{array} \\ \begin{matrix} \end{matrix} \end{matrix}$
w^ch is y^e quantity of y^e area apqd if $100 p = cp$ . and $abcp = 1$

<21r>

$y = db$ . $x = ba$ $a a y = x^{3}$ . a |b | + y | z| $= y$ {illeg} $z = bc$ $a a b + a a$ y |z| $= x^{3}$ . a |b| − y |z| $= y =$ {illeg} $z = bf$ . $a a b - a a$ y |z| $= x^{3}$ . y |z|− a |b| $= y =$ {illeg} $z = bg$ $a a$ y |z|− | $a$ | $a b = x^{3}$ . {illeg} $x = d + ξ$ {illeg} $ξ = ah$ . {illeg} $a a$ {illeg}|y| $- d^{3} - 3 d d ξ - 3 d ξ ξ - ξ^{3} = 0$ ξ $\begin{matrix} a a b + a a z \\ a a b - a a z \\ a a z - a a b \end{matrix}} = \begin{matrix} d^{3} + 3 d d ξ + 3 d ξ ξ + ξ^{3} \end{matrix}$ . $x = b - ξ$ . $aq = ξ$ $\begin{matrix} a a y \\ a a b \pm a a z \\ a a z - a a b \end{matrix}} = \begin{matrix} d^{3} - 3 d d ξ + 3 d ξ ξ - ξ^{3} \end{matrix}$ . $x = ξ - b$ . $ak = x$ $\begin{matrix} a a y \\ a a b \pm a a z \\ a a z - a a b \end{matrix}} = \begin{matrix} ξ^{3} - 3 d ξ ξ - 3 d d ξ - d^{3} \end{matrix}$ .

$na = ξ$ . $nd = z$ . |Or.| $ξ^{3} - a^{2} z + b b ξ$ $ab ∶ an ∷ a ∶ b$ . &. $ab ∶ nb ∷ a ∶ c$ then $ξ^{3} = \frac{b^{3}}{a} z - \frac{b b c}{a} ξ$ . or if abn is \a/ right angle. & | $ξ = d + ζ$ {illeg} $ζ = mn$ | $d^{3} + 3 d^{2} ζ$ / $+ b b c ζ$ \ $+ 3 d ζ ζ + ζ^{3} - \frac{b^{3}}{a} z + \frac{b b c d}{a}$

<21v> <22r>

$db = x$ . $ba = y$ . $a a x = y^{3}$ {illeg}. $x = b + z$ . $z = bc$ . $y = c + ξ$ . $ξ = ah$ $\begin{matrix} a a b + a a z = \\ (x = b - z . z = bf .) \\ a a b - a a z = \\ (x = z - b . z = bg) \\ a a z - a a b = \end{matrix}} \begin{array}{l} ξ^{3} + 3 c ξ^{2} + 3 c c ξ + ξ^{3} . y = c - ξ . ξ = aq \\ c^{3} - 3 c c ξ + 3 c ξ^{2} - ξ^{2} . y = ξ - c . ξ = ak . \\ ξ^{3} - 3 c ξ^{2} + 3 ξ c^{2} - c^{3} . \end{array}$ $na = y$ . d {illeg}| n | $= x$ . $ab ∶ an ∷ a ∶ b$ . $ab ∶ nb ∷ a ∶ c$ . & $y^{3} = \frac{b^{3}}{a} x - \frac{b b c}{a} y$ . Or $d^{2} x = ε ε y + y^{3}$ . $\begin{matrix} (x = z + o . z = nc) \\ d^{2} z + d^{2} o = \\ (x = z - o . z = gn) \\ d d z - d d o = \\ (x = o - z . z = nf) \\ d d o - d d z = \\ d d z \end{matrix}} \begin{array}{l} ε ε y + y^{3} . (y = n + ξ . ξ = na) \\ ξ^{2} n + ε ε ξ + n^{3} + 3 n n ξ + 3 n ξ^{2} + ξ^{3} . \\ (y = n - ξ . ξ = as) \\ ε ε n - ε ε ξ + n^{3} - 3 n n ξ + 3 n ξ^{2} - ξ^{3} . \\ (y = ξ - n . ξ = av .) \\ ε ε ξ - ε ε n + ξ^{3} - 3 n ξ^{2} + 3 ξ n^{2} - n^{3} . \end{array}$ $al = y$ . $dl = x$ . $ab ∶$ {illeg} \ al / $∷ a ∶ b$ &. {illeg} $al ∶ bl ∷ b ∶ c$ . whence $y^{3} = \frac{x b^{3} + y b^{2} c}{a}$ or $y^{3} = \frac{y b^{2} c}{a} =$ {illeg} / $y^{3} - ε ε y = d d x$ \ &c: as before onely varying y^e signes at εεn & εεξ . $ao = y$ . $do = x$ . $a ∶ b ∷ bd ∶ do$ . $b ∶ c ∷ do ∶ ob$ . $\frac{a^{3}}{b} x = y^{3} - \frac{3 c x y y}{b} + \frac{3 c c x x y}{b b} - \frac{c^{3} x^{3}}{b b}$ .

<22v>

D^r Wi|a|llis in a letter to {illeg}|S|^r Kenelme Digby promiseth y^e squareing of y^e Hyperbola by finding a meane propor{illeg}|t|ion twixt 1 , & $\frac{5}{6}$ in the progression $1, \frac{5}{6}, \frac{31}{30}, \frac{209}{140}, \frac{1471}{630}, \frac{10625}{2772}$ &c.

<23r>

The resolution of cubick equations out of D^r Wallis in his dedication before Meibomius confuted

suppose $x = ♉ a ♉ e$ . yⁿ $x^{3} = ♉ a^{3} ♉ 3 a a e ♉ 3 a e e ♉ e^{3}$ . or $x^{3} =$ \+ /{illeg} $3 a e x ♉ a^{3} ♉ e^{3}$ . that is ma{illeg}|k|ing $a^{3} + e^{3} = q$ {illeg}. yⁿ $x^{3} = 3 a e x ♉ c^{3}$ . & $3 a e = p$ . yⁿ $x^{3}$ × |=| $+ p x ♉ q$ . Againe suppose {illeg}| y | $= ♉$ b | a | $\underset{\cap}{O}$ {illeg}| e | Then {illeg}| $y^{3}$ | Againe suppose $x = a - e$ . yⁿ $x^{3} = a^{3} - 3 a a e + 3 a e e - e^{3}$ . y^t is making $a^{3} - e^{3} = ♉ q$ , & $3 a e = p$ , yⁿ $x^{3} = - p x ♉ q$ .

Then in the first o{illeg}|f| th|e|se $p = 3 a e$ . or $\frac{p}{3 e} = a$ . or $\frac{p^{3}}{27 e^{3}} = a^{3} = q - e^{3}$ . Therefore $e^{6} = q e^{3} - \frac{p^{3}}{27}$ . {illeg}|&| $e^{3} = \frac{1}{2} q ♉ \sqrt{\frac{1}{4} q q - \frac{p^{3}}{27}}$ . & by y^{illeg}|e| \same/ reason $a^{3} = \frac{1}{2} q \underset{\cap}{O} \sqrt{\frac{1}{4} q q - \frac{p^{3}}{27}}$ where y^e irrationall quantitys have. divers signes other{illeg}|w|ise $a^{3} + e^{3} = q$ would bee false. Soe that
$x = ♉ a ♉ e = ♉ \sqrt{c : \frac{1}{2} q \underset{\cap}{O} \sqrt{\frac{1}{4} q q - \frac{1}{27} p^{3}}} ♉ \sqrt{c : \frac{1}{2} q ♉ \sqrt{\frac{1}{4} q q - \frac{1}{27} p^{3}}}$ . is a rule for resolving y^e equation $x^{3} * - p \underset{\cap}{O} q = 0$ , when it hath but one roote y^t is when it may be generated according to the supposition $x = ♉ a ♉ e$ . &c. By y^e same reason $x^{3} * + p x \underset{\cap}{O} q$ . may be resolved by this rule $x = a - e = \sqrt{c : \frac{1}{2} q ♉ \sqrt{\frac{1}{4} q q + \frac{1}{27} p^{3}}} - \sqrt{c : \frac{1}{2} q \underset{\cap}{O} \sqrt{\frac{1}{4} q q + \frac{1}{27} p^{3}}}$ .

But here of|b|serve y^t D^r Wallis would h|A|rgue y^t since in the first of these two cases fsometimetimes (viz when y^e equation hath 3 \reall/ rootes) y {sic} first rule faile|th| as it were imposs|i|ble {sic} for y^e equation to have rootes when yet it hath, therefore y^e fault is in Algebra. & therefore when Anal{illeg}|i|ses Analysis leads us to an imposs|i|bility {sic} wee ought not to conclude y^e thing \absolutely/ imposible, untill wee have tryed all y^e ways y^t may bee.

But let me answer y^t y^e fault is {illeg}i{illeg}|not| in y^e Analysis in this example, but \in/ his opperation. for {illeg} when y^e equation $x^{3} * + p x ♉ q = 0$ , hath 3 root{illeg}|s| hee supposeth it to have but one roote viz $x = ♉ a ♉ e$ . but sin{illeg}|c|e y^e Equation cannot be then generated according to y^t supposition it is impose|i|ble {sic} it should be relolved {sic} by it.

<23v>

In like manner hee s{illeg}|a|yeth y^t Algebra representehteth a thing possible when tis not so as in this examl|p|le, in y^e △ abc, make $ab = 1$ . $bc = 2$ $ac = 4$ . Then to find $dc = x$ , worke |thus,| {illeg} $ad = 4 - x$ . $bd \times bd = 1 - 16 + 8 x - x^{2} = 4 - x^{2}$ {illeg}|t|herefore $8 x = 19$ . or $x = \frac{19}{8}$ . In w^ch oppera{illeg}|c|on all things proceede as possible though they are not soe for ac is greater yⁿ $ab + bc$ .

yet I answer y^t if y^e opperation & conclusion be compared together y^e absurdity will appeare. for in y^e equation $bd \times bd = 4 - x x = 4 - \frac{361}{64} = \frac{256 - 361}{64}$ or $bd \times bd = \frac{- 105}{8}$ . but it is impossible y^t a □ number should be negative.

Thus $x = \sqrt{- b}$ is impossible. square it & tis $x x = - b$ . Againe, & tis $x^{4} = b b$ . Extract y^e roote & tis $x x = b$ or $x = \sqrt{b}$ . w^ch is possible. The reason of this proceeding Event is y^t $x^{4} - b b = 0$ hath two possible rootes viz $x = \sqrt{b}$ . $x = - \sqrt{b}$ . & two impossible viz: $x = \sqrt{- b}$ . $x = - \sqrt{- b}$ .

Thus y^e valors of $x^{8} - a^{8} = 0$ are $x = a$ , $- a$ , $\sqrt{- a a}$ , $- \sqrt{- a a}$ , $\sqrt{4 : - a^{4}}$ , $- \sqrt{4 : - a^{4}}$ , $\sqrt{- \sqrt{- a^{4}}}$ , $- \sqrt{- \sqrt{- a^{4}}}$ .

<24r>

D^r Wallis in a letter to S^r Kenelme Digby teacheth how to find y^e center of gra{illeg}|v|ity in divers lines first when their position is as in this figure.

Suppose ad y^e Axis, a their vertex & as 1 to y^e series of y^e progresion cons. Then saying, as 1 to y^e index of y^e line increased by an unite (vide pag {illeg}|2|^da) so cd to ca Then c is their center of gravity.

The Demonstracon.

Let p bee y^e index of y^e series according to w^ch y^e odinately aplyed lines (parallell to db) increase, yⁿ {illeg} {illeg} $1 ∶ p + 1 ∷$ area of y^e line ∶ to nmbq . y^e distances of those ordinate lines from y^e vertex a are equall to y^e intercepted diameters & therefore a primanary series \(whos index is 1. / & since supposing a y^e center of y^e ballance y^e whole weight of y^e surface or figure is composed of its magnitude & distance from y^e center and therefore y^e index of all its moments or whole weight is $p + 1$ , viz: y^e aggregate of y^e other two. Therefore as all its mom moments (or y^e \weight of the/ figure in its site in respect of y^e center a are to soe many of y^e greatest (or to y^e weight of y^e ▭ nmbq hung on y^e point d ) soe is 1, to $p + 2$ . and if $ap ∶ ad ∷ 1 ∶ p + 2$ , then nmbq hung on y^e point q shall counterballance y^e figure in its site &c therefore if {illeg} $ac ∶ cd ∷ p + 1 ∶ 1$ , {illeg}|c | shall be y^e center of gravity of those figures.

Also as the figure is now put extending infinitely towards δ if $- 2 p + 1 ∶ -$ {p } $+ 1 ∷ am ∶ ac$ . {illeg}|m | being y^e center of qnbd yⁿ c shall bee y^e center of gravity of y^e whole figure qndbδ .

Demonstration

<24v>

sincce {sic} y^e lines parallell to aδ increase in series reciprocl|a|lly proportionall their index is $- p$ & since y^e halfe{illeg}|s| of those lines increas {sic} in y^e same proportion their index is $- p$ . whose extremitys or middle points of y^e whole lines (suposing a y^e center of y^e ballance) are theire centers of gravity, their distances from a being proportionall to y^e lines whose centers they are & consequently their index is $- p$ & since all y^e moments (or whole weight of y^e figure) increase in a proportion compounded of y^e proportion of y^e magnitudes & distances of y^e lines from y^e center a, they will be in a du{illeg}|p|licate proportion of y^e lines magnitudes that is a series reciprocall series whose index is $- 2 p$ . Therefore y^e figure is t{illeg}|o| y^e inscribed parallelogram as 1 to $1 - p$ . & all its moments or w{illeg}|h|ole weight of the Para in this its site to {illeg}|t|he weight of y^e pgr as 1 to $1 - 2 p$ . Therefore if, $am ∶ ap ∷ 1 - 2 p ∶ 1$ , the paralelogra {sic} hanging on y^e point p s{illeg}|h|all counterballanc{e} y^e whole figure in its site &c: whence y^e point c may be found easily, {illeg} viz $am ∶ ac ∷ 1 - 2 p ∶ 1 - p$ .

<26r>

Of Refractions.

1 If y^e {illeg} ray ac bee refracted at the center of y^e circle acdg towards d & $ab ⟂ be ⟂ gc ∥ ed$ . Then suppose $ab ∶ ed ∷ d ∶ e$ . {illeg}|S|ee Cartes Dioptricks

2 If there be an hyperbola whose \the/ distance of whose foci bd are to its transverse axis hf as d to e . Then y^e ray $ac ∥ bd$ {illeg}|i|s refracted to y^e exterior focus $(d)$ . See C: Dioptr

3 Having y^e proportion of d to e, or. $bd ∶ hf$ . The Hyperbola{illeg} may bee thus described.

1 Upon y^e centers a, b let y^e instrument adbtec bee moved in w^ch instrum^nt observe y^t ad {illeg} $⟂ de ⟂$ c {illeg} et & y^t the beame cet is not in y^e same plane w^th adbe but intersects it at y^e angle tev soe y^t if $tv ⟂ ev$ , then $d ∶ e ∷ et ∶ tv$ . Or $d ∶ e ∷ Rad ∶ sine of ∠ tev$ . Also make $de = \frac{q}{2}$ , i.e half y^e t{illeg}|r|ansverse diamet{er.} Then place \the fiduciall side of/ plate chm in the same plaine w^th ab . & moving y^e instrument adbect to & fro its edge cet shall cut or weare its {illeg} into y^e shape of y^e desired Parabola. Or the plate chm may bee filed away untill y^e edg {sic} cet exactly touch it everywhere.

2 By the same proceeding Des=Cartes concave Hyperbolicall wheele may bee described by beeing turned w^th a chissell {illeg}| d |tec whose edge is a streight line inclined to the edge \axis/ of the mandrill by y^e ∠ tev w^ch angle is found by making $d ∶ e ∷ et ∶ tv ∷ Rad ∶ sine of etv$ .

3 By the same reason a wheele ma{illeg}|y| be turned Hyperbolically concave y^e Hyperbola being convex. Or a Plate may bee turned Hyperbolically concave

<26v>

Also Des=Cartes his Convex wheele \ B / may be turned {illeg}|o|r {ground} trew a concave wheele \ A / being made use of instead of a patterne

5 In turning the concave wheele A it will \perhaps/ bee best to weare it w^th a stone p & let {illeg}|t|he streight edged chissell d serve for a patterne. And it may bee convenient to grind y^e stone (or iron &c {illeg}) p into y^e fas{illeg}|h|ion of a cone S That it may fit y^e hollow of the wheele A. The angle of w^ch conce {sic} being a right one or something greater it will almost grind the wheele to a {illeg}tre{illeg} figur Same done by helpe of a Cone.

6 Draw $ab = q$ ; ac & cb of any length or intersecting one another at any angles. to make up the triangl{illeg}{e} abc . [Suppose y^t ac bee called b, & that $e = q =$ {illeg} then is d the distance of y^e foci]. produc{e} ac to d soe that $ad = \frac{d d - e e}{b}$ . Then draw bk through y^e point d, & draw eh parallell to bc , lastly w^th the sides he , ek & angle hek describe the cone heklm . Then produce ba to g &c \indefinitely/ & ag being y^e axis of a section mal shall be y^e sought Par Hyperbola

7 Since the proportion of cb to ab & ∠ cba is not deter{illeg}|m|ined it will be most convenient to make <27r> $cb = ab = e = q$ . & $cb ⟂ ab ⟂ ah$ . And then there will bee little danger of error at y^e vertex of the {illeg}|Hy|perbola. And y^e calculation is readier for drawing $bp ⟂ ca$ , Then is $cp = bp = ap = \frac{e}{\sqrt{2}} = \sqrt{\frac{e e}{2}}$ & $pd = \frac{d d}{e \sqrt{2}}$ . Soe that $e e ∶ d d ∷ cp = bp ∶ pd ∷$ $Rad ∶ tangen$ $∷ Radius ∶ Tangent of pbd$ soe y^t y^e $∠ cbk = hek$ is {illeg} easily found.

9 Halving such a cone smo|o|thly pollished w^thin & w^thout, by the helpe of a square set y^e plate perpendicular {illeg}|t|o one side hae the fiduciall edge being distant from y^e vertex the length of $ae = \frac{e d d - e^{3}}{d d + e e}$ & if y^e edge of y^e plaine every where tou{illeg}ch the cone, tis trew.

10 The exact distance \ $(ae)$ / of y^e plate from the vertex of y^e cone neede not bee much regarded for that changeth onely the shape \bigness/ not y^e shape of y^e figure.

[By y^e broken lookinglasse I find in glasse refraction, y^t $d ∶ e ∷ 43 ∶ 28 ∷ 1000 ∶ 651 + ∷ 1536$ {illeg} $∶ 1000$ . These are insensib almost insensibly different from truth $d ∶ e ∷ 20 ∶ 13 ∷ 100 ∶ 65 ∷ 153 - ∶ 1000$ . Or $d ∶ e ∷$ $23 ∶$ {illeg} / $23 ∶ 15$ \ $∷ 100 ∶ 652 +$ $d ∶ e ∷ 66 ∶ 43 ∷ 100 ∶ 651,5151 +$ . Or $d ∶ e ∷ 100 ∶ 651$

For y^e Ellipsid|s| $\frac{d d - e e}{d} x + \frac{e e - d d}{d d} x x = y y$

<27v>

The former ${\begin{matrix} descriptions \\ propositions \end{matrix}}$ demonstrated.

Lemma. If in y^e Opposite Hyperbolas one of abc edf (one of w^ch are to bee described) supposing $bd = d$ . $hf = e$ . $gh = x$ $gc = y$ . $gc ⟂ ghd$ & {illeg} point & gc terminated by y^e hyperbola Then is $\frac{d d - e e}{e e} x x + \frac{d d - e e}{e} x = y y$ . b d | h | $= \frac{d - e}{2}$ . $dh = \frac{d + e}{2}$ . d |b| $g = \frac{2 x - d + e}{2}$ . $gd = \frac{2 x + d + e}{2}$ . ${dc}^{2} = \frac{4 x x + 4 d x + 4 e x + d d + 2 e d + e e + 4 y y}{4} = gd \times gd + gc \times gc$ . ${bc}^{2} = \frac{4 x x - 4 d x + 4 e x + d d - 2 e d + e e + 4 y y}{4} = {gb}^{2} + {gc}^{2}$ . And since af $dc = bc + hf$ . /Or ${dc}^{2} = {bc}^{2} + 2 bc \times hf + {hf}^{2}$ \ Therefore $2 d x + e d - e e = e \sqrt{4 x x - 4 d x + 4 e x + d d - 2 e d + e e + 4 y y}$ . Both ꝑts of w^ch □ed & ordered y^e result is $4 d d x x - 4 e e x x + 4 e d d x - 4 e^{3} x - 4 e e y y = 0$ . That is $\frac{d d - e e}{e} x + \frac{d d - e e}{e e} x x = y y$ .

Desciption {sic} y^e 1^st demonstrated \Synthetically/. See y^t Scheame

Naming y^e quantitys $ed = \frac{e}{2} = dh$ {illeg} $d ∶ e ∷$ {illeg} ∶ e{illeg} tv Nameing y^e quantitys \{illeg}/ $ed = dh = \frac{e}{2}$ . $gh = x$ . $gc = be = y$ . $dg = x + \frac{e}{2} = nc$ . $cg ⟂ dhg$ . ${cd}^{2} = x^{2} + e x + \frac{e e}{4} + y$ ${ce}^{2} = x x + e x + y y$ . ${eg}^{2} = x x +$ {illeg} $e x$ . $d ∶ e ∶$ Also $d ∶ e ∷ et ∶ tv ∷ ce ∶ eg$ , therefore $d d x x + d d e x = e e x^{2} + e^{3} x + e^{2} y$ That is $\frac{d d - e e}{e} x + \frac{d d - e e}{e e} x x = y y$ . As in y^e lem̄a

The Same demonstrated synthe \Analy/tically.

Nameing y^e quantitys, $de = dh =$ {illeg}|a |. $gh = x$ . $gc = y$ . $dg = a + x$ ${dc}^{2} = a a + 2 a x + x x + y y$ . ${ce}^{2} = 2 a x + x^{2} + y y$ . ${eg}^{2} = x x + e x$ . Supose y^t {illeg} $b ∶ c ∷ et ∶ tv ∷ ce ∶ eg$ .

<28r>

Then is $b b x x + b b e x = 2 c c a x + c c x x + c c y y$ . That is $\frac{b b - c c}{c c} x x + \frac{b b e - 2 c c a}{c c} x = y y$ . Therefore y^e line chm is a Conick Section & since $(b b)$ is greater yⁿ $(c c)$ tis an Hyperbola, w^ch y^t it may bee {illeg} y^e same w^th y^t in y^e lemma, Their correspond^nt termes are to bee compared together & soe I find y^t $\frac{b b - c c}{c c} x x = \frac{d d - e e}{e e} x x$ . & $\frac{b b e - 2 c c a}{c c} x = \frac{d d - e e}{e} x$ by y^e 1^st =^tion $b b = \frac{c c d d}{e e}$ . Or $b = \frac{c d}{e}$ . y^t is $b ∶ c ∷ d ∶ e$ . by y^e 2^nd $c c e e - c c d d + b b e e = 2 c c e a$ . And by substituting $\frac{c c d d}{e e}$ into the place of $b b$ {illeg}|A|nd ordering it tis {illeg}2{illeg} $c c e e = 2 c c e a$ . Or $\frac{e}{2} = a$ . Therefore if I take $\frac{e}{2} = a = de$ . & $d ∶ e ∷ b ∶ c ∷ ct ∶ tv$ . then shall chm bee y^e Para|Hyp^er|bola desired Q:E:D.

The 2^d 3^d 4^th & 5^th Propositions are manifest from this

The 6^th Description demonstrated Syntheticaly The quantity named are $ab = e$ .

☞ Instead of y^e 6^th & 7^th Descriptions w^ch are false use these

6 Draw 2 concentrick circles (na & cd) w^th y^e Radij e & d. Then from y^e comon center b draw 2 lines bc & ${\begin{matrix} ba \\ bd \end{matrix}}$ at the given angle ${\begin{matrix} bae = abc \\ aed = cbd \end{matrix}}$ of ${\begin{matrix} section \\ y^{e} Cone \end{matrix}}$ then draw a line cad thr from c throug by y^e end of y^e Rad ${\begin{matrix} ba \\ bd \end{matrix}}$ & to y^e intersection of y^t line w^th y^e circle ${\begin{matrix} cd \\ na \end{matrix}}$ draw ${\begin{matrix} bd \\ ba \end{matrix}}$ & so the ∠ of ${\begin{matrix} y^{e} Cone, hek = cbd \\ section, eab = abc \end{matrix}}$ is found.

<28v>

Or w^ch is the same {illeg} make $ab = e$ . $bd = d$ & then if y^t cone is sought make $cd = 2 r$ the ∠ cba being given, make $ac = a$ . Then is $cd = \frac{d d - e e + a a}{a}$ . & soe y^e $∠ cbc = aed$ is knowne & also $ae = ed = \frac{d^{3} - d e e}{d d - e e + a a}$ , & $ad = \frac{d d - e e}{a}$ . Buf But if y^e $∠ bae = abc$ of y^e section is sought y^{illeg}|e| cone b|e|ing given yⁿ make $cd = 2 b$ . And it will bee $ac = b + \sqrt{e e - d d + b b}$ . & soe $∠ abc = bae$ is given also $ad = b - \sqrt{e e - d d + b b}$ . & $ae = \frac{db - d \sqrt{e e - d d + b b}}{2 b}$

{illeg}|I|n generall observe y^t in any cone cut any ways $bd = be + ea = d$ . & {illeg}|b| $a = e$ .

7. DesCartes his wheele \thus described/ cut by any plaine produceth one of y^e Conick=Sections.

Description y^e 6^th Demonstrated. Synthetically.

Call, {illeg} $bd = d$ . $ba = e$ . {illeg} $cp = pd = a$ . $bp = \sqrt{d d - a a}$ . $ag = x$ $ap = \sqrt{e e - d d + a a}$ . $ac = a + \sqrt{e e - d d + a a}$ . $ad = a - \sqrt{e e - d d + a a}$ . $ba ∶ ac ∷ ag ∶ gh = \frac{a x + x \sqrt{e e - d d + a a}}{e}$ . $ba ∶ ad ∷ bg ∶ gk = \frac{e a + a x}{e} \frac{- e - x}{e} \sqrt{e e - d d + a a}$ . \ $gk \times gh = {gm}^{2} = y^{2}$ ./ Therefore $\frac{d d - e e}{e e} x x + \frac{d d - e e}{e} x = y y$ . by ordering y^e result of / $gk \times gh$ . \ w^ch is like y^t in the lemma.

The 7^th Proposition may be easyly demonstrated {illeg} after the same manner.

If the two \equall/ cones bad bcd intersect the one the other soe y^t $ab = bc$ their intersection $(bf)$ shall bee one of y^e Conick sections as they had each beene intersected by the plane bf .

<29r>

To describe y^e Parabola (& other figures after y^e same manner) pretty exactly.

Thake a squire cbe , soe y^t $cb = \frac{r}{2}$ (for then the{illeg} circle described by $(bc)$ will bee as crooked as y^e Parabola at the vertex d ). Divide y^e other leg $(be)$ of y^e Squire into any number of pts, Then get a plate of Br{illeg}|a|sse &c: lkfd streight & eaven. And taking one point d for y^e vertex of it & another point c for y^e Squire to move{illeg}n soe y^t $cd = cb = \frac{r}{2}$ , & wearein{illeg}|g| away y^e edge of the plate untill (y^e {illeg}|S|quire being erected) $ab = qd$ . the squire touching y^e plate at a . thus shall y^e edge adf become Parabolicall. w^th y^e Rad: ab describe a circle adg & by that {meanes} it may bee knowne when $ab = a$ . \Instead of y^e leg be a/ Demonstraco\circle may be used/ Supose $aq = y$ . $cd = cb = \frac{r}{2}$ then {illeg}. Then is {illeg} $\sqrt{\frac{r r}{4}} +$ {illeg} $z z$ {illeg} cq {illeg} $ad = \sqrt{x x + y y} = ab$ . & $ac = \sqrt{\frac{r r}{4} + x x + y y}$ . {illeg}|A|nd $cq = \sqrt{\frac{r r}{4} + x x}$ . & $dq = x$ . Demonstracon. $qd = x$ . $cd = \frac{r}{2} = cb$ . $cq = \frac{r}{2} - x$ . $aq = \sqrt{r x} = y$ . ${ac}^{2} = \frac{r r}{4} + x x$ ${ab}^{2} = \frac{r r}{4} + x x - \frac{r r}{4}$ & $ab = x$ . Q.E.D.

Another description of y^e Parabola y|w|^e {sic} y^e compasses. Make $ab = bc = \frac{r}{4}$ . Make $ce = cd$ & $ce ⟂ bd$ . Make $af = ae$ , & $bf = bd$ then shall f be a point in y^e Parabola.

Another. {illeg}|M|ake $ab = \frac{r + x}{2} = ac$ . | $eb = x ⟂ ce$ | & y^e point c shall bee in y^e parabola. This like y^e first by calculation may bee made use of in other lines.

<29v>

The manner whereby any kind of \little/ lines may {illeg}be described very accurately. And that the same Instrument serve for all lines (though never so small) differing in quantity but not{illeg} in qual{illeg}ity.

Make y^e plate d of y^e figure requ{illeg}|ir|ed (by some of y^e former meanes) the larger the better. Then hold the streight \steele/ staffe b again{illeg}|st| the center a & {roule}{route} i{illeg}|t| to {illeg}|&| fro it shall grind c into y^e same figure but soe much lesse as ac is lesse yⁿ ad.

Soe if y^e glass c bee fastened upon y^e mandrill f, it may be ground acording to y^e sollid f{illeg} figure d by y^e helpe of a stick of steele (as a cilind cone) {illeg} whose cuspis is in y^e hole a upon w^ch it is moved as on a center. when y^e stick cone b leanes uppon y^e vertices of d & c it must be perpendicular to the mandrill f. Perhaps it may be convenient to cause y^e cone b to turne about its axis. Or it may bee {illeg} better {illeg} instead of y^e nutt at a w^th a hole in it to make a sharpe pointed nutt, & instead of y^e cilinder /cone\ b to make use of a broad plate to cover a, c & d & move every way upon them

<30r>

Another way to describe lines on plates

Suppose y^e plate bee abc, whose edg {sic} boc is to be made into y^e fashion of a given crooked line suppoes {sic} $(o)$ is its vertex & y^t a circle described w^th y^e {illeg}|R|adius eo would bee as crooked as y^e given line at its vertex. A{illeg}gaine suppose two streight rulers mn & pq to bee very trew & steddyly fastened together w^ch must a \very/ little incline y^e one to y^e other, soe as that being produced they would meete at ar. Then are y^e lines $pn = a$ {illeg}, & $pr = b$ given.

Suppose yⁿ y^e point d in y^e crooked line is to bee found yⁿ is dc given by supposition, & {illeg} consequently (supposing dk to bee a tangent) $dg = y$ . $gc = x$ . $fg = v$ . $fd = s$ $ec = c$ . $fk = v + \frac{y y}{v}$ {illeg}. {illeg} $ef = v + x - c$ . $ek = c - x + \frac{y y}{v}$ . & (if $eh ⟂ dk ⟂ df$ ) then is $eh = \frac{c v - x v + y y}{\sqrt{v v + y y}} = d$ . $(eh)$ being thus foun{illeg}|d|, supposing y^t $pn = a = ec$ , then I take $re = \frac{b d}{a}$ . that is $pe = b - \frac{b d}{a}$ . {illeg} haveing thus found y^e point e lay y^e plate twixt two rule the two rulers so y^t y^e point of it, fall upon y^e point e yⁿ should y^e line mn touch y^e plate in d. But note y^t $pn ⟂ mn$ .

In both telescopes & microscopes tis most convenient to have a convex glasse next y^e eye for by that meanes y^e angle of vision will bee much greater yⁿ it will bee w^th a concave one (though both doe magnifie alike). If y^e convex glasse be Hyperbolicall (&c) make it soe bigg y^t y^e penecilli may crosse in y^e pupill; y^t is, y^e exterior focus will be as far distant from y^e vertex as y^e eye is. let y^e glass bee as thinn y^t y^e as may bee y^t y^e eye bee not to{illeg}|o| farr {sic} from y^e vertex y^et {sic} it should bee about as thick as y^e distance of y^e interior focus from y^e vertex.

And by this meanes also, (y^e focus of y^e objectglasse being within y^e telescope twixt y^e glasses) there may bee placed at {illeg}|t|hat {illeg}|f|ocus y^e edge of <30v> a steele ruler accurately divided into equall parts (to measure y^e diameters or distances of starrs &c) w^ch should bee soe made y^t by a pinne or handle it may be placed i{illeg}|n| any posture & in any parte of y^e focus, w^thout otherwise altering y^e Telescope in observations.

Note that were not y^e glasses faulty they would not onely magnify objects but render vision more distinct; each of the penicilli passing through (perhaps but) the 10^th, 20^th or 100^th part{illeg}|e| of y^e pupill must bee more exactly refracted to one point {illeg} of y^e Tunica Retina yⁿ in ordinary visio in w^ch each of y^e penicill{illeg}|i| spreads over all the pupill.

☞ Note also that y^t y^e glasse a may be ground Parabolic \Hyperbolicall/ by y^e line cb, if it turne on y^e mandrill e whilst c{illeg}|b | turnes on y^e axis rd being inclined to it as was shewed before. If the edge $(cb)$ bee not durable enough, inough instead thereof use a long small cilinder: w^ch I conceive to bee the best way, of{illeg} all. For a Cilinder of all sollids is most easily made exact (being turned, as in the figure, by a gage untill its thicknesse bee every where equall). 2 the Cilinder may bee made {illeg} to slip up & downe & turne round whereby it will not onely grinde y^e glase crosse wise to take of all hubbes, but also y^e glasse & cilinder will grinde y^e one y^e other truer & truer. All y^e difficulty is in placing y^e axis rd perpendicular to the Mandrill ae \& vertex to vertex/, w^ch yet may bee done exactly severall ways. & untill yⁿ the glasse & Cilinder will not fit. & should y^e axis not intersect y^e glasse would bee still Hyperbolicall except a point at the vertex of it. The same instrument may also serve for severall glasses onely making df longer or shorter. Let the Cilinder han{g} over the glasse.

<31r>

To Grinde Sphæricall optick Glasses

If y^e glasse $(bc)$ is to bee ground sphærically hollow: naile a steele plate to y^e beame $(fg)$ , on y^e upper side: In w^ch make a center hole for y^e steele point $(f)$ of y^e shaft $(def)$ : to w^ch shaft fasten a plugg $(a)$ of stone or leade or leather &c: (w^th w^ch you intend to grinde y^e glasse $(bc)$ ): w^ch shaft & plugg being swung to & fro upon y^e center f will grind y^e glasse bc sphærically hollow.

The manner whereby glasses may bee ground sf|p|hærically convex may appeare by y^e annexed {illeg} figure (being y^e former way inverted). Also y^e plugg $(a)$ , in y^e $\begin{matrix} first \\ second \end{matrix}$ figure, is ground sphærically $\begin{matrix} convex. \\ concave. \end{matrix}$

But if this way bee not exact enough yet hereby may bee {grownd}{ground} plates of mettall well nigh sphæricall, And by those plates may bee ground glasses after y^e usual manner; If a circular hoope of steele \ $(abc)$ / bee put about y^e edge of y^e glasse $(d)$ to keepe it from grinding away at y^e edges faster yⁿ in y^e middle.

But the best way of all will bee to ti{illeg}|u|rne y^e glass circularly upon a mandrill whilest y^e plate is steadily rubbed upon it or else <31v> to turne y^e plate upon a mandrill whilest y^e glasse is rubbed upon it \or let sometimes y^e one, sometimes y^e other bee turned./: & by this meanes they will either of them weare the other to a truely sphericall forme. but however let there bee a hoope or of some mettall w^ch {illeg}|w|eares more difficultly then glasse to defend y^e glasse from wearing more at its edges then in y^e middle. Perhaps it may doe well \first/ to weare y^e plate sphæricall by y^e hoope alone w^thout the glasse.

The same meanes may bee used for gr{illeg}|i|nding plaine glasses.

Let not an object glasse bee ground sphærically convex on both sides, but sphaerically convex on one side & concave or plaine \plane or but a little convex ~/ on y^e other, & turne y^e convexest side towards y^e object.

<32r>

If the Glasses of a Telescope bee not truely ground, how to find where the fault is, & consequently to rectify it.

Take two plates abfh, dgkh of {illeg} wood or brasse i{illeg}|at| y^e midst of y^e sides of w^ch {boare} two very small holes \c & e/ (viz whose diameters are about y^e {illeg}|2|0^th or {illeg}|3|0^th pte of an inch, that they may transmit but soe much light as may serve to see {illeg}|y^e| edge of y^e sunne or a starre of y^e first magnitude)

Also make two other plates rs, & tv like y^e former but y^e holes in y^m must be as small as can bee (viz \about/ y^e 100^th pte of an inch in diameter or lesse). For y^e small end of y^e tube

Also make an\o/ther plate wzxy w^th a hole in \the midst of/ it about y^e 5^th or sixt pte of an inch in diameter (viz equall to y^e diameter of y^e pupill of y^e eye or eyeglasse). Make y^e like \plate A/ (but w^th a very small hole) for y^e eyeglaess {sic})

First cover y^e object glasse w^th y^e plates af, & gh distant about y^e y^e holes in y^m being distant about y^e sixt parte of an inch; & placed neare y^e center of y^e object glasse. Also cover y^e eye glasse w^th y^e plate A soe y^t its hole exactly respect y^e center of y^e eye glasse then turne y^e tube to a Starre w^ch will appeare like two starrs if y^e tube bee two {sic} long or short, w^ch bee shortned or lengthned untill there appeare but one, And then is y^e Tube of a good distance length for y^e vertices of y^e Glasses.

Secondly remove those plates, and instead thereof cover y^e object glasse w^th y^e plate wz, its hole exactly respecting y^e center of y^e glasse. (Fig 2)

<32v>

If y^e Glasses of a Telescope bee not truely ground The{illeg}|ir|e errors may bee may bee thus found.

Because an error is much more easily discernable in y^e object glasse yⁿ in y^e ey^e {sic} glasse let us first suppose y^e eye glasse to bee ground true towards its center, (tis exact enough if it be sphericall, & not Hyperbolicall), & so wee may find & rectifie y^e errors of y^e object glasse.

First make a thin plate $(A)$ of brasse & in the center of it a Small hole (whose diameter perhaps may bee about y^e 50^th or 100^dth parte of an inch. {illeg}|W|ith w^ch plate cover y^e eye glass y^e center of it respecting y^e center of y^e glasse.

Secondly make two other plates the one B w^th two holes \as/ neare to its edge as may bee their{e}{} distance being about y^e 5^t {sic} pte of an inch or lesse, & y^e other C w^th one hole close to y^e midst of its edge. Let y^e diameters of these 3 holes bee about a 20^th pte of an inch or lesse. And theire edges must bee true that they may slide one upon another, & yet not let y^e suns rays passe through, to w^ch purpose make y^m oblique. W^th these two plates cover y^e object glase {sic} (first stopping y^e hole of C \y^e holes of y^e other plate respecting y^e center of y^e glasse/ & looke at a stare (or y^e edge of y^e sunne &c) & if y^e object appeare double (like two starrs &c) make y^e Tube longer or shorter untill it appeare single. Then open y^e hole of C , & y^e plate B being fixed, slide y^e plate C up & downe still looking at y^e starre, When then appeares <33r> but one starre y^t part of y^e glasse under y^e hole of C is truely ground in respect of y^e 2 parts of y^e glasse under y^e two holes of B. But {no} when y^e starre appeares double. And y^e position of y^e starre caused by y^e hole of {illeg} C in repect {sic} of y^e starre caused by y^e holes of B, shews y^e error of y^e incli w^ch way y^e glasse under y^e hole of C is erroneously inclined; the distance of y^e two starres giving y^e quantity of y^t inclinati error.

Thus y^e errors of y^e of object glasse bein{g} found in every place of it they may bee all rectified, & found againe, & againe rectified, untill they almost or altogether vanish.

Then may y^e eye=glasse bee rectified much after y^e same manner, in every parte of it, & if it bee necessary y^e object glasse may bee aganie {sic} rectified & againe y^e eye=glasse untill y^e Teles{illeg}|c|ope bee as perfect as y^e workeman can make: Whome perhaps experience & other may teach by this & y^e former rules to make telescopes as perfect as {illeg} {illeg}men can \hope to/ make them.

These glasses may also bee rectified whilst on y^e Mandrill by observing y^e images mab|d|e by reflection from y^e vertex & all other ꝑts of y^e glasse w^t proportion they have one to another & how much they are longer yⁿ broader in one place then another. &c.

<33v>

$\begin{matrix} \begin{matrix} The sines measuring refractions are in \end{matrix} & \begin{matrix} Aere \\ 42 \end{matrix} & \begin{matrix} water \\ 56 \end{matrix} & \begin{matrix} Glasse \\ 65 \end{matrix} & \begin{matrix} christall \\ 70 \end{matrix} \\ \begin{matrix} The proportions of the motions of the \\ extreamely heterogeneous rays are in \end{matrix}} & 39,4 . 40,4 . & 70 \frac{3}{8} . 71 \frac{3}{8} . & 95 \frac{1}{10} . 96 \frac{1}{10} & 110 \frac{1}{3} . 111 \frac{1}{3} \\ \begin{matrix} \begin{matrix} The proportions of y^{e} sines of \\ refraction of the extreamely hetero- \\ geneous rays into aire out of \end{matrix} \\ Their common sine of incidence \\ Which substracted the difference is \end{matrix} & \begin{matrix} \begin{array}{l} c \end{array} \end{matrix} & \begin{matrix} \begin{matrix} 90 \frac{2}{3} . 91 \frac{2}{3} \end{matrix} \\ 68 \frac{1}{3} \\ 22 \frac{1}{3} . 23 \frac{1}{3} \end{matrix} & \begin{matrix} \begin{matrix} 68 . 69 \end{matrix} \\ 44 \frac{1}{4} \\ 23 \frac{3}{4} . 24 \frac{3}{4} \end{matrix} & \begin{matrix} \begin{matrix} 61 \frac{4}{5} . 62 \frac{4}{5} \end{matrix} \\ 36 \frac{4}{5} \\ 24 . 25 \end{matrix} \\ \begin{matrix} The like proportions for refrac- \\ tions made into water out of \end{matrix} & \begin{matrix} 275 \frac{4}{5} . 276 \frac{4}{5} \\ 238 \frac{2}{5} \\ 37 \frac{2}{5} . 38 \frac{2}{5} \end{matrix} & \begin{matrix} 196 \frac{1}{3} . 197 \frac{1}{3} \\ 157 \frac{4}{9} \\ 39 \frac{1}{9} . 40 \frac{1}{9} \end{matrix} \end{matrix}$

of the method of infinite series

I Newton

<35r>

Theoremata varia. Circa angulorum æqualitates.

si ang DAB = |&| DAE bisecentur a rectis FH et IG et ducatur quævis KLMN . Erit
1. $AK . AM ∷ KL . LM ∷$ \{illeg} $KN . MN$ / Euclid 6 3
2. $AK \times AM = AL q + KL \times LM =$ A {illeg}+ $KL \times LM - AL q$ . Scho{illeg}|{o}|te de {concis} /{æqu{is}}\
3. $AM + AK . PK ∷ AQ . AL$ posito $AP = AM$ .

Si in angulo quovis PAQ inseribantur æquales AB, BC, CD, DE, EF, FG, GH &c anguli BA{illeg} $P$ erit angulus CBQ dul|p|lus {illeg} \ DCP/ tripl, ED $P$ | $Q$ | quadr FEQ quint, GFQ sext, HGP sept. IHQ oct &c. Horu vero angulorum positi|o| ra{illeg}|di|o AB {illeg} sinus erunt Bβ , Cχ &c cosinus AB , Bχ , Cδ &c. Ergo si $AB = r$ , & $AB = x$ erit \ $AC = 2 x$ / $Aχ = \frac{2 x x}{r}$ . $Bχ = \frac{2 x x - 4 x}{r}$ . $AD = (2 Aχ - AB) = \frac{4 x x - r r}{r}$ . $Aδ =$ $\frac{4 x^{3} - r r x}{r r}$ &c

<79r>

To find the sume of y^e squares cu{bes} &c. of y^e rootes of an equation

If a , bg , c , d , e , f &c be the rootes of y^e equation $x^{6} + p x^{5} + q x^{4} + r x^{3} + s x x + t x + v = 0$ . yⁿ is {illeg} $a + b + c + d + e + f = p (= g)$ $a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} = p p - 2 q . (= p g - 2 q$ {illeg} $= h)$ $a^{3} + b^{3} + c^{3} + d^{3} + e^{3} + f^{3} = p^{3} - 3 p q + 3 r . (= p h - q g + 3 r = k)$ $a^{4} + b^{4}$ &c $= p^{4} - 4 p p q + 4 p r + 2 q q - 4 s . (= p k - q h + r g - 4 s = l)$ $a^{5}$ &c $= p^{5} - 5 p^{3} q + 5 p q q + 5 p p r - 5 p s - 5 q r + 5 t . (= p l - q k + r h - s g + 5 t$ {illeg} = { $m)$ } $a^{6}$ &c: $= p^{6} - 6 p^{4} q + 9 p p q q + 6 p^{3} r - 12 p q r - 6 p p s + 6 p t - 2 q^{3} + 3 r r + 6 q s - 6 v$ .

<80r>

$ag = a$ . $ab = x$ . $bh = \frac{d x}{c}$ . $bc = y$ . $bg = \sqrt{x x - a a}$ $gh = \sqrt{\frac{d d x x - e e x x + e e a a}{e e}}$ . $dg = b$ . $ce = \frac{y}{d x} \sqrt{d d x x - e e x x + e e a a} = fg$ . $cf = \frac{d x + e y}{d x} \sqrt{x x - a a}$ . $df = b - \frac{y}{d x} \sqrt{d d x x - e e x x + e e a a}$ . $z^{2} = {dc}^{2} = {\begin{matrix} x x - a a + \frac{2 e y x}{d} - \frac{2 e y a a}{d x} + b b + y y \\ \frac{- 2 b y}{d x} \sqrt{d d x x - e e x x + e e a a} \end{matrix}$ $\frac{2 b y}{d x} \sqrt{d d x x - e e x x + e e a a} = x x + y y - z z + b b - a a + \frac{2 e y x x - 2 e y a a}{d x}$ . $4 b b d d x x$ $\begin{matrix} d d x^{6} & + & 4 d e y x^{5} & + & 2 d d y y x^{4} & - & 4 d e a a y x^{3} & - & 4 b b d d y y x x & - & 4 d e y^{3} a a x & + & 4 e^{2} a^{4} y^{2} \\ - & 2 d d z z x^{4} & + & 4 d e y^{3} & + & 4 b b e e y y & + & 4 d e y z^{2} a^{2} \\ + & 2 d d b b & - & 4 d e y z z & - & 8 a a e e y y & - & 4 d e y b b a^{2} \\ - & 2 d d a a & + & 4 d e y b b & + & 2 d d y^{4} & + & 4 d e y a^{4} & - & 4 b b e^{2} a^{2} y^{2} \\ + & 4 e e y y & - & 4 d e y a a \end{matrix}$

Ad constructionem Canonis angularis.

$90^{gr}$ ${\frac{90}{5}}^{gr} = 18^{gr}$ . ${\frac{18}{5}}^{gr} = 3^{gr} + 36^{'}$ . Et ${\frac{60}{3}}^{gr} = 20^{gr}$ . ${\frac{20}{3}}^{gr} = 6^{gr} + 40^{'}$ $\frac{6^{gr} + 40^{'}}{2} = 3^{gr} + 20^{'}$ . $3^{gr} + 36^{'} - 3^{gr} - 20^{'} = 16^{'}$ . $\frac{16^{'}}{2} = 8^{'}$ . $\frac{8^{'}}{2} = 4^{'}$ . $\frac{4^{'}}{2} = 2^{'}$ $\frac{2^{'}}{2} = 1^{'}$ .

If $r =$ radius. Then
$\begin{matrix} \begin{matrix} y^{e} sine of \end{matrix} & \begin{matrix} 78^{degr} is, \frac{r \sqrt{5} - r + r \sqrt{30 + 6 \sqrt{5}}}{8} . \\ 66^{degr} is, \frac{r \sqrt{5} - r + r \sqrt{30 - 6 \sqrt{5}}}{8} . \\ 42^{degr} is, \frac{- \sqrt{5 r r} + r + \sqrt{30 r r + 6 r r \sqrt{5}}}{8} . \\ 6^{degr} is, \frac{\sqrt{30 r r - 6 r r \sqrt{5}} - \sqrt{5 r r} - r}{8} . \end{matrix} \end{matrix}$

<80v>

Suppose $gh = x$ . $nh = r$ . Then $\begin{matrix} gh & = & x . unisectio \\ ab \times r & = & 2 r r - x x . bisectio \\ h {}^{2}b \times r^{2} & = & 3 r r x - x^{3} . trisectio \\ a {}^{3}b \times r^{3} & = & 2 r^{4} - 4 r r x x + x^{4} . {quadrisec}^{o} . \\ h {}^{4}b \times r^{4} & = & 5 r^{4} x - 5 r r x^{3} + x^{5} . {quintusect}^{o} . \\ a {}^{5}b \times r^{5} & = & 2 r^{6} - 9 r^{4} x^{2} + 6 r r x^{4} - x^{6} . \\ hb \times r^{6} & = & 7 r^{6} x - 14 r^{4} x^{3} + 7 r r x^{5} - x^{7} . \\ ab \times r^{7} & = & 2 r^{8} - 16 r^{6} x^{2} + 20 r^{4} x^{4} - 8 r r x^{6} + x^{8} . \\ hb \times r^{8} & = & 9 r^{8} x - 30 r^{6} x^{3} + 27 r^{4} x^{5} - 9 r r x^{7} + x^{9} . \\ ab \times r^{9} & = & 2 r^{10} - 25 r^{8} x^{2} + 50 r^{6} x^{4} - 35 r^{4} x^{6} + 10 r r x^{8} - x^{10} . \\ hb \times r^{10} & = & 11 r^{10} x - 55 r^{8} x^{3} + 77 r^{6} x^{5} - 44 r^{4} x^{7} + 11 r r x^{9} - x^{11} . \end{matrix}$

< insertion from the top right of f 80v >

As on y^e other leafe excepting some signes here changed.

< text from f 80v resumes >

If $gh = x$ . $bh = y$ .Then $\begin{matrix} y = bh . duplicatio anguli hag \\ y y - x x = x \times h {}^{2}b . triplicatio anguli hag . \\ y^{3} - 2 x x y = x x \times h {}^{3}b . quadruplicatio. \\ y^{4} - 3 x x y y + x^{4} = x^{3} \times {}^{4}b h {. quint}^{o} \\ y^{5} - 4 x x y^{3} + 3 x^{4} y = x^{4} \times h {}^{5}b {. sext}^{o} \\ y^{6} - 5 x x y^{4} + 6 x^{4} y y - x^{6} = x^{5} \times hb {. sept}^{o} \\ y^{7} - 6 x x y^{5} + 10 x^{4} y^{3} - 4 x^{6} y = x^{6} \times hb {. oct}^{o} \\ y^{8} - 7 x x y^{6} + 15 x^{4} y^{4} - 10 x^{6} y y - x^{8} = x^{7} \times hb . nonc \\ y^{9} - 8 x x y^{7} + 21 x^{4} y^{5} - 20 x^{6} y^{3} + 5 x^{8} y = x^{8} \times hb . dec \\ y^{10} - 9 x x y^{8} + 28 x^{4} y^{6} - 35 x^{6} y^{4} + 15 x^{8} y^{2} - x^{10} = x^{9} \times hb . und, \\ y^{11} - 10 x x y^{9} + 36 x^{4} y^{7} - 56 x^{6} y^{5} + 35 x^{8} y^{3} - 6 x^{10} y = x^{10} \times hb . duod \end{matrix}$

<81r>

Of Angular sections

Suppose $ab = q$ . $\frac{ah}{2} = r$ . & $ag = x$ . & y^t y^e arches {illeg} hg, gb, bb are equall. By y^e following Equations an angle \bah/ may bee divided into any number of partes. $\begin{matrix} x = q . unisectio. \\ x^{2} - 2 r r = r q . bisectio. \\ x^{3} - 3 r r x = r r q . trisectio. \\ x^{4} - 4 r r x x + 2 r^{4} = r^{3} q . quadrisectio. \\ x^{5} - 5 r r x^{3} + 5 r^{4} x = r^{4} q . quintusectio. \\ x^{6} - 6 r r x^{4} + 9 r^{4} x x - 2 r^{6} = r^{5} q . sextusectio. \\ x^{7} - 7 r r x^{5} + 14 r^{4} x^{3} - 7 r^{6} x = r^{6} q . septusectio. \\ x^{8} - 8 r r x^{6} + 20 r^{4} x^{4} - 16 r^{6} x^{2} + 2 r^{8} = r^{7} q . \\ x^{9} - 9 r r x^{7} + 27 r^{4} x^{5} - 30 r^{6} x^{3} + 9 r^{8} x = r^{8} q . \\ x^{10} - 10 r r x^{8} + 35 r^{4} x^{6} - 50 r^{6} x^{4} + 25 r^{8} x^{2} - 2 r^{10} = r^{9} q . \\ x^{11} - 11 r r x^{9} + 44 r^{4} x^{7} - 77 r^{6} x^{5} + 55 r^{8} x^{3} - 11 r^{10} x = r^{10} q . \\ x^{12} - 12 r r x^{10} + 54 r^{4} x^{8} - 112 r^{6} x^{6} + 105 r^{8} x^{4} - 36 r^{10} x^{2} + 2 r^{12} = r^{11} q . \\ x^{13} - 13 r r x^{11} + 65 r^{4} x^{9} - 156 r^{6} x^{7} + 182 r^{8} x^{5} - 91 r^{10} x^{3} + 13 r^{12} x = r^{12} q . \\ x^{14} - 14 r r x^{12} + 77 r^{4} x^{10} - 210 r^{6} x^{8} + 294 r^{8} x^{6} - 196 r^{10} x^{4} + 49 r^{12} x^{2} - &c \\ x^{15} - 15 r r x^{13} + 90 r^{4} x^{11} - 275 r^{6} x^{9} + 450 r^{8} x^{7} - 318 r^{10} x^{5} + 140 r^{12} x^{3} - &c \\ x^{16} - 16 r r x^{14} + 104 r^{4} x^{12} - 352 r^{6} x^{10} + 660 r^{8} x^{8} - 672 r^{10} x^{6} + 336 r^{12} x^{4} - &c \\ x^{17} - 17 r r x^{15} + 119 r^{4} x^{13} - 442 r^{6} x^{11} + 935 r^{8} x^{9} - 1122 r^{10} x^{7} + 714 r^{12} x^{5} - &c \\ x^{18} - 18 r r x^{16} + 135 r^{4} x^{14} - 546 r^{6} x^{12} + 1287 r^{8} x^{10} - 1782 r^{10} x^{8} + 1386 r^{12} x^{6} - &c \\ x^{19} - 19 r r x^{17} + 152 r^{4} x^{15} - 665 r^{6} x^{13} + 1729 r^{8} x^{11} - 2717 r^{10} x^{9} + 2508 r^{12} x^{7} - &c \\ x^{20} - 20 r r x^{18} + 170 r^{4} x^{16} - 800 r^{6} x^{14} + 2275 r^{8} x^{12} - 4604 r^{10} x^{10} + 4290 r^{12} x^{8} - &c \end{matrix}$

This scheame is y^e forme\r/ inversed.

<81v>

Suppose y^e perifery {illeg} \ bgh / to bee a & y^e whole perifery to bee p. The line bh subtends these arches. a. $p - a$ . $p + a$ . $2 p - a$ . $2 p + a$ . $3 p - a$ . $3 p + a$ . $4 p - a$ . $4 p + a$ . $5 p - a$ . $5 p + a$ . $6 p - a$ . $6 p + a$ . &c: All w^ch are bisected, trisected, quadrisected, quintuse{illeg}|c|ted &c after same manner. |As for example|

The rootes of y^e equation $h {}^{2}b \times r r = 3 r r x - x^{3}$ . are 3. The first whereof subtends y^e arches $\frac{a}{3}$ . $\frac{3 p - a}{3}$ . $\frac{3 p + a}{3}$ . $\frac{6 p - a}{3}$ . $\frac{6 p + a}{3}$ . $\frac{9 p - a}{3}$ . $\frac{9 p + a}{3}$ &c. The second subtends y^e arches $\frac{p - a}{3}$ . $\frac{2 p + a}{3}$ . $\frac{4 p - a}{3}$ . $\frac{5 p + a}{3}$ . $\frac{7 p - a}{3}$ . &c. The 3^d $\frac{p + a}{3}$ . $\frac{2 p - a}{3}$ . $\frac{4 p + a}{3}$ . $\frac{5 p - a}{3}$ . $\frac{7 p + a}{3}$ &c.

Soe y^e rootes of y^e equation ~ $hb \times r^{4} = 5 r^{4} x - 5 r r x^{3} + x^{5}$ , doe y^e first subtend y^e arches $\frac{a}{5}$ . $\frac{5 p - a}{5}$ . $\frac{5 p + a}{5}$ &c: y^e 2^d $\frac{p - a}{5}$ . $\frac{4 p + a}{5}$ . $\frac{6 p - a}{5}$ . y^e 3^d $\frac{p + a}{5}$ . $\frac{4 p - a}{5}$ . $\frac{6 p + a}{5}$ &c. y^e 4^th $\frac{2 p - a}{5}$ . $\frac{3 p + a}{5}$ . &c y^e 5^t $\frac{2 p + a}{5}$ . $\frac{3 p - a}{5}$ . $\frac{7 p + a}{5}$ . &c.

Hence may appeare y^e reason of y^e number of rootes in these equations & y^t y^e points of y^e circuference {sic} \to w^ch they are extended/ æquidistant. & by y^e lower scheme may bee known w^ch rootes are affirmative & w^ch negative.

The numerall cöefficients of y^e afforesaid equations may bee deduced from this progression (if $∠ ∶ ∠ ∷ 1 ∶ n$ .) $1 \times \frac{− 0 + n \times - 1 + n}{1 \times 1 - n} \times \frac{n - 2 \times n - 3}{2 \times 2 - n} \times \frac{n - 4 \times n - 5}{3 \times 3 - n} \times \frac{n - 6 \times n - 7}{4 \times 4 - n} \times \frac{n - 8 \times n - 9}{5 \times 5 - n} \times \frac{n - 10 \times n - 11}{6 \times 6 - n}$ &c. As if x $n = 10$ . y^e ꝑression $1 \times - 10 \times \frac{- 7}{2} \times \frac{- 10}{7} \times \frac{- 1}{2} \times \frac{2}{25} \times 0$ . And y^e coefficients $1 . - 10 . + 35 . - 50 . + 25 . - 2$ .

<82r>

1663 _/4 January.

All y^e parallell lines w^ch can be understoode to bee drawne uppon any superficies are equiv{illeg}|a|lent to it, as all y^e lines drawne from $(ao)$ to $(co)$ may be used instead of y^e superficies $(aco .)$

If all y^e parallell lines drawne uppon any superficies be multiplied by another line they produce a Sollid like y^t w^ch relults {sic} from y^e superficies drawne into y^e lame {sic} line as if either al{illeg}|l| y^e lines in y^e superficies $(oac)$ or if y^e superficies oac be drawne into y^e line $(b)$ they both produce y^e same lollid {sic} $(d)$ whence All y^e parallell superficies w^ch {illeg} can bee understoode to bee in any sollid are equivalent to y^t Sollid. And If all y^e lines in any triangle, w^ch are parallell to one of y^e sides, be squ\a/red there results a Pyramid. if those in a square, there results a cube. If those in a crookelined figure there resuts {sic} a sollid w^th 4 sides {illeg} terminated & bended after according to y^e fasshion {sic} of y^e crookelined figure{.}

If each line in one superficies bee drawne into each correspondent line in another superficies as in aebk, & omnc if $ae \times dh$ . $bk \times cn$ . $qv \times wx$ . &c. they produce a sollid whos {sic} opposite sides are fashioned by one of y^e superfic as {illeg} figure|Sollid| fpsrg. where all y^e lines drawne from fr {illeg} to ps are equall to all equall to all the correspondent lines drawne from ow to mx. & those drawne from fg to fr are equall to y^e correspondent lines drawne from qz to vz.

<82v>

Theorema. 1

If in the Circle abcdeP there be inscribed any Poligon abcde w^th an odd number of sides, & from{illeg} any point in y^e circumference P there bee drawne lines Pe, Pa, Pb, Pc, Pd to e{illeg}|v|ery corner of y^e Pol{illeg}ygon: y^e summ of every other line {illeg} is equall to y^e summ of y^e rest, $Pa + Pb + Pc$ {illeg} $= Pd + Pe$ . & soe are their cubes ${Pa}^{3} + {Pb}^{3} + {Pc}^{3} = {Pd}^{3} + {Pe}^{3}$ . unless y^e figure be a Tri/gon\

Theor 2

If from{illeg} y^e points of y^e Polygon then {illeg}|b|ee drawne perpendicular ap, br, ct, ds, $e$ {illeg} $q$ to any Diameter pt: y^e summe of y^e Pe{illeg}|r|pendiculars on one side y^e Diameter is {equall}{equal} to their summe on y^e other $ap + br + ct = eq + ds$ . & soe is y^e summe of their cubes (unlesse wⁿ y^e figure is a Trigon), ${ap}^{3} + {br}^{3} + {ct}^{3} =$ ${eq}^{3} + {ds}^{3}$ . & of theire square cubes (except wⁿ y^e figure is a Trigon or Pentagon. &c.

Theor {illeg}|3|

If y^e 2 circles (fig 1 & 2) be equall w^th like Poligo{illeg}{ns} inscribed, & Pa in fig {illeg}|1| be assumed double to pa in fig 2. then are all y^e y^e other corresponding lines in fig 1 double to those in fig 2 viz $Pb = 2 rb$ , $Pc = 2 tc$ , $Pd = 2 sd$ , $Pe = 2 qe$ .

<83r>

To square y^e Parabola

In y^e Parabola cae suppose y^e Parameter $ab = r$ . $ad = y$ . $dc = x$ . & $r y = x x$ or $\frac{x x}{r} = y$ . Now suppose every y^e lines called x doe increase in arithmeticall proportion all y^e x's taken together y^e supp make y^e superficies dch w^ch is halfe a square let every line drawne from cd to hd be square & they produce {illeg}|a| Pyramid equall to every $x x = \frac{x^{3}}{3}$ . w^ch if divided by {illeg}y r {illeg} there remaines every $\frac{x^{3}}{3 r} =$ $= \frac{y x}{3}$ equall to every $\frac{x x}{r}$ {illeg}|e|quall to every $(y)$ or all y^e lines drawne from ag to accc equall to y^e superficies ag {illeg}| $c$ | equall to a 3^d pte of y^e superficies adcg & y^e superficie $acd = \frac{2 y x}{3}$ .

Otherwise. suppose $ce = b$ . $co = x$ . $to = y$ . & $r y = b x - x x$ y^e lines x increasing in arithmeticall proportion every x is equall to \4 times/ y^e superficies $cdh = \frac{b b}{2}$ w^ch drawne into b produceth y^e sollid $\frac{b^{3}}{2}$ but if every x be squarered {sic} {illeg}|t|hey poduce {sic} a {illeg}|p|yramid equall to $\frac{b^{3}}{3}$ . wherefore every $b x - x x = \frac{b^{3}}{6}$ equall to every $r y$ equall to y^e superfies|ci|es adce drawne into r & $\frac{b^{3}}{6 r} =$ to cade as before.

<83v>

* $pδ = pd = \frac{q + y}{5}$ & $\frac{q y + y y}{5} = x x$ ⊛

<84r>

To Square y^e Hyperbola

In y^e Hyperbola eqaw. suppose $ef = a$ . $fa = b$ . $ap = rq = y$ \{ $aλ = d$ }/ $pq = ar = x$ . $ad = q =$ \ $5$ / $oa =$ \ $5$ / $ac =$ \ $5$ / $d =$ | $r =$ |. & $da + ar ∶ ar ∷ ar ∶ rq$ . $x x = d y +$ {illeg} $y x$ . |In w^ch equation| Every x \taken together/ is equall to y^e triangle aβb \equall to $\frac{a a}{2}$ / & eve{illeg}|r|y $x x$ {illeg}|ta|k{illeg}en together is a pyramid $= \frac{a^{3}}{3}$ . Every y taken together is equall to y^e superficies $eba = mkt$ If yⁿ $gh = lm = ns = aλ = d$ . every $d y$ is equall to y^e solid nglmhs. If y^e angle mhk is a right one & if $mh = gl =$ $ba = ef = a$ {illeg} $f = kh$ , then all y^e lin that is if y^e triangle $mhk = abβ$ . every {illeg}|y | $x$ will be equall to y^e sollid mhstk Joyne these two sollids together as in $lmtng = \frac{a^{3}}{3}$ . *

⊛ Againe Suppose every x taken together to be equall to y^e superficies aef , {illeg} y^e line $qγ$ squared is $4 x x$ {illeg} every $4 x x$ composeth a Sollid like $(♊♉♌)$ a|n| quarter \eighth pte/ whereof (w^ch is equall to every $\frac{x x}{2}$ ) being like \ $♊♉♓o =$ / xyzv; xv will be equall to ♊ {illeg}|♓ | $= ef$ {illeg} $= a = st = xz = o♓ = hm$ . & $vz = ♊♓$ $= aβ$ {illeg}km . whence y^e {illeg} covexe {sic} superficies xyv of y^e figure xyvz will fitly joyne w^th y^e concave superficies mst of y^e figure shmkt . If every x is equall to y^e superficies aef , every y shall be equall to y^e triangle $afπ = \frac{b b}{2}$ . every $y y = \frac{b b b}{3}$ every $q y = \frac{q b b}{2}$ & therefor y^e Sollid $yxzv = \frac{b^{3}}{30} + \frac{q b b}{20} = \frac{2 b^{3} + 3 q b b}{60}$ . Joyne y^e Sollid shmkt to yxvz & there resulteth $shmztk =$ $= \frac{a a b}{2}$ from w^ch againe substract $xvzy = \frac{2 b^{3} + 3 q b b}{60}$ & there remaines y^e sollid $mhstk = \frac{30 a a b - 2 b^{3} - 3 q b b}{60}$ w^ch substract from y^e sollid $ntmlg = \frac{a^{3}}{3}$ & there remaines $nglmhs = \frac{20 a^{3} + 2 b^{3} + 3 q b b - 30 a a b}{60}$ b w^ch being divided by $θ = \sqrt{\frac{5 r r}{4}}$ . there remaines $\frac{40 a^{3} + 4 b^{3} + 6 q b b - 60 a a b}{r \sqrt{5}} =$ to y^e superficies abe

<85r>

The squareing of severall croked {sic} lines of y^e Seacond kind.

In a \any two/ crooked lines I call y^e Parameter or right side \of y^e greater./ $(r)$ . \but of y^e lesse $(s)$ / Transverse side $(q)$ . y^e right axis as cf $(x)$ \or $ef = v$ . / y Transverse axis as fe {illeg} y, \or/ fd z.

Suppose in{illeg} y^e Parab: $d d c$ : $ac = r$ . & in $e e c$ : $bc = s$ $r x = z z = {df}^{2}$ . $s x = y y = {fe}^{2}$ . $\sqrt{r x} - \sqrt{s x} = de =$ $= p$ . $r x = s x + p p + 2 p \sqrt{s x}$ . $r x - s x - p p = 2 p \sqrt{s x}$ $r r x x - 2 r s x x + s s x x - 2 p p r x - 2 p p s x + p^{4} = 4 p p s x$ . Or $p^{4} - 2 r x p p - 6 s x p p + r r x x - 2 r s x x + s s x x = 0$ . if $p = y$ . $x x = \frac{\begin{matrix} + 2 r y y \\ + 6 s y y \end{matrix} x - y^{4}}{r r - 2 r s + s s}$ . make $cf = a$ . $fd = b$ . $fe = c$ . $c e ef f = \frac{2 a c}{3}$ . & $c d def f = \frac{2 a b}{3}$ therefore $\frac{2 a b - 2 a c}{3} = c d de e$ y^e square of y^e crooked line $c d d$ (when y^e line $c e e$ is p|s|upposed to {sic} close w^th y^e line cf ) whose nature is exprest by y^e foregoing Equation.

2 $lk = b$ . $li = x$ . $qi = y$ . $in = z$ . {illeg} $+ b x$ \ $- x x$ / $= r y$ {illeg}− $b x$ \ $- x x$ / $= s z$ . $\frac{- x x + b x}{r} = y$ . $\frac{- x x + b x}{s} = z$ $qn = \frac{- s x x s + b s x + r x x r - b r x}{r s} = v =$ {illeg}y or, {illeg} \r/ $r x -$ {illeg} $s x x + b$ b\ $s$ / $x - b$ b\ $r$ / $x - r s y = 0$ . Or {illeg} $x x - b x - \frac{r s y}{r - s} = 0$

3 $tg = x$ . $dg = z$ . $gp = y$ . $r x - r z = {dp}^{2}$ . {illeg} $r x - r z + z z = y^{2}$ . $z z = r z - r x + y y$
$z = - \frac{r}{2} ⊻ \sqrt{\frac{1}{4} r r - r x + y y}$ . $r ∶ a ∷ \sqrt{x r - r z} ∶ z$ . $\frac{1}{2} r r ⊻ r \sqrt{\frac{1}{4} r r - r x + y y} = a \sqrt{r x - r z}$ . $\frac{1}{2} r^{4} + r r y y - r^{3} x$ {illeg} $- a a r x + r z a a = ⊻ \frac{r}{2} \sqrt{}$ $\frac{1}{2} r = ⊻ r \sqrt{\frac{1}{4} r r - r x + y y}$

<86r>

$\begin{matrix} \frac{1}{4} r^{6} & + & r^{4} y y & + & r r y^{4} & - & r^{5} x & = & 2 r^{3} y y x & + & r^{4} x x \\ - & r^{2} a a x & 2 r y y a a x & + & 2 r r a a x x \\ + & a^{4} x x \end{matrix}$ $r ∶ a ∷ r x - r z ∶ z z$ . $z z = a r x - a r z$ . $z z = - a z + a x$ . z {illeg} $= - \frac{1}{2} a + \sqrt{\frac{1}{4} a a + a x}$ . $a x - a z = r z - r x + y y$ Or $- y y + r x + a x = - \frac{1}{2} a a - \frac{1}{2} a r + \frac{a}{r} \sqrt{\frac{1}{4} a a + a x}$ . $\begin{matrix} 186 \\ 31 \end{matrix}$ $\begin{matrix} \begin{matrix} 354 \\ 59 \end{matrix} \\ \begin{matrix} 944 \end{matrix} \end{matrix}$

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

$17548875 . (5849625$ $\begin{matrix} 51) 358 (7,019608 . \\ \underline{357} - 58 . \\ \underline{100} \\ 490 \\ \underline{459} \\ 310 \\ \underline{306} \\ 400 \end{matrix}$ $\begin{matrix} 51) 197 (3,862745 \\ \underline{153} + 392 \\ 440 \\ \underline{408} \\ 320 \\ \underline{306} \\ 140 \\ \underline{102} \\ 380 \\ \underline{357} \\ 230 \\ \underline{204} \\ 26 \end{matrix}$ $\begin{matrix} \underline{93} \\ 53) 372 (7018868 \\ \underline{371} + 682 . \\ \underline{100} \\ 470 \\ \underline{424} \\ 460 \\ \underline{424} \\ 36 \\ \underline{318} \\ 42 \end{matrix}$ $\begin{matrix} 212) \underline{819} (3863207 \\ 636 70 \\ 1830 \\ \underline{1696} \\ 1340 \\ \underline{1272} \\ 680 \\ 636 \\ 44 \\ \underline{424} \\ 160 \end{matrix}$

<86v>

$\begin{matrix} \overset{●}{0} & \cdot & 0 & \cdot & 9 & \cdot & 7 & \cdot & 5 & \cdot & 4 & \cdot & 5 & \cdot & \overset{●}{5} & \cdot & \overset{●}{5} & \cdot & \overset{●}{9} & \cdot & \overset{●}{7} & \cdot & \overset{●}{7} & \cdot & \overset{●}{7} & \cdot & \cdot & \cdot & 3 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \overset{●}{0} & \cdot & \overset{●}{0} & \cdot & \overset{●}{0} & \cdot & \overset{●}{2} & \cdot & \overset{●}{0} & \cdot & \overset{●}{0} & \cdot & \overset{●}{2} & \cdot & \overset{●}{0} & \cdot & \overset{●}{0} & \cdot \end{matrix}$

<87r>

4 In y^e Hyperbola Parabola $cb = a$ . $be = x$ . $2 a a - 2 a x$ {illeg} $- a a + 2 a x - x x = {ed}^{2}$ $a a - x x = {ed}^{2} = y y$ . $\frac{a a - x x}{r} = y$ . $cp = cb$ $\sqrt{}$ $eb \times df = fg$ \{ $\times c$ }/ = {illeg} zc. $\frac{a a x - x^{3}}{r} + x x = zc$ $x^{3} - r x x - a a x + r c z = 0$ . Since all $eb \times df = \frac{1}{8}$ all ${co}^{2} = \frac{1}{4} ab \times ab \times r$ . $ab = b$ . all ${eb}^{2} = \frac{a^{3}}{3}$ therefore bgpf $\frac{1}{4} b b r + \frac{a^{3}}{3}$ .

<87v>

<88r>

$ab =$ {illeg}| $b$ | $e = y$ . $bd = x$ . $bq = dg = b$ . $nb = c$ . $\frac{y b x}{y c} = \frac{b x}{c} = ef$ Then shall $b q &c$ : be the axis of gravity in $f e b &c$ & bqgd.

<88v>

<89r>

In y^e 1^st figure.

$gc ∶ cd ∷ cfed ∶ ckhg = \frac{cd \times cfed}{gc}$ . $ac = gc$ . $x ∶ z ∷ z a ∶ x y$ . {illeg} $\frac{z z a}{x} = ckhg$ . {illeg} or $\frac{x x y}{z} = cdef$ . Suppose $cd ∶ ca ∷ ac ∶ bc ∷$ y^e swiftnesse of de ∶ to y^e swiftnesse of gh . $de \times its swiftnes ∶ gh \times its swiftness ∷ gc ∶ cd$ . $de \times cd ∶ gh \times ca ∷ de \times ac ∶ gh \times bc ∷ gc \times cd$ .

Fig 2d {sic}. 3^d.

$c$ d | θ | $∶ ca ∷ ac ∶ bc ∷ nm ∶ am ∷ swiftness θe ∶ swiftnesse gh$ . $de \times$ its {illeg}|s|wiftnesse $∶ gh \times$ it {sic} swifnes {sic} ∷ $c$ $d$ | $k$ | $\times de ∶ gh$ $c$ | × | $ac ∷ de \times ac ∶ gh \times bc$ $de \times$ $c$ | $k$ | $∶ gh \times ac ∷ de \times ac ∶ gh \times bc$ {illeg} \ ∷ / $de \times nm ∶ gh \times am ∷ gc ∶ cd$ . $de \times ck$ $de \times ck \times cd = gh \times ac \times gc$ . $de \times cd = gh \times bc$ . $de \times nm = gh \times gc$ .

Fig 4

$ck ∶ ca ∷$ motion of y^e point a from c ∶ motion of y^e point a from m ∷ $ck ∶ ca ∷$ increasing of $ac = gc$ ∶increasing of cd ∷motion of gh ∶ motion of de. &c as before.

These are to find such figures cghk, cfed, as doe equiponderate in respect of y^e axis acfk.

<89v>

<90r>

Reasonings concerning chance.

If by one of y^e equall chances a, I gaine p, by y^e chance the equall chances \ a , b , c , &c/ are such y^t one of y^m must necessarily happen, & y^t if one of y^e chances a happen I gaine p thereby, or q by one of y^e chances b , or r by one of y^e chances c . My chance or expectation is worth $\frac{p a + q b + r c}{a + b + c}$

1 If p is y^e number of chances by \one of/ w^ch I may gaine a, & q those by \one of/ w^ch I may gaine b, & r those by one of which I may gaine c; soe y^t those chances are \all/ equall & one of them must necessarily happen: My hopes or chance is worth $\frac{p a + q b + r c}{p + q + r} = A$ . |The same is true if p, q, r signify |an|y|y|^{e} proportion of chances for a, b, c.|

2. If I bargaine for more yⁿ one cha{illeg}c|n|ce (viz: y^t after I have taken y^e gaines by my first chance, from the stake $a + b + c$ ; I will venter another chance at y^e remaining stake &c) my second lott is worth $\frac{p a + q b + r c}{a + b + c \times p + q + r}$ {illeg} A $A \frac{- A A}{a + b + c} = A - \frac{A A}{a + b + c} = B$ . My third lot is worth $A \frac{- A A - A B}{a + b + c} = C$ . My Fourth lot is worth $A \frac{- A A - A B - A C}{a + b + c} = D$ . My Fift lot is worth $A \frac{- A A - A B - A C - A D}{a + b + c} = E$ . My sixt lot is worth $A - A \times \frac{A + B + C + D + E}{a + b + c}$ . &c

As if 6 me{illeg}|n| $(1.2.3.4.5.6 .)$ cast a die soe y^t he gaines a who throws a cise first: since there is but one point \chance/ to gaine a & 5 to gaine nothing at each cast, I make $b = 0 = c = r$ . $p = 1$ & $q = 5$ . Therefore by the <90v> The first mans lot is worth $\frac{a}{6}$ The 2^ds is worth $\frac{a}{6} - \frac{a}{36} = \frac{5 a}{36}$ . The thirds is worth $\frac{5 a}{36} - \frac{5 a}{216} = \frac{25 a}{216}$ . The 4^d|t|hs is $\frac{25 a}{216} - \frac{25 a}{1296} = \frac{125 a}{1296}$ The fifts lot is worth $\frac{125 a}{1296} - \frac{25 a}{7776} = \frac{625 a}{7776}$ . The Sixts lot is $\frac{625 a}{7776} - \frac{625 a}{46656} = \frac{3125 a}{46656}$ . &c. Soe y^t their lots are as $7776 : 6480 : 5400 : 4$ {illeg}| 5 | $00 :$ 4 | 3 | $950 : 3$ 3 | 1 | 25 .

Soe y^t if I cast a die two or more times tis $\dot{1}$ to 5 y^t I cast a c{illeg}|i|s{illeg}|e| at y^e first cast & 11 to 25 y^t I throw it at two casts, & 91 to 125 y^t I cast it at thrice, & 671 to 625 y^t I cast it once in 4 trialls, & 4651 to 3125 y^t I cast it once in 4|5| times. &c

3. If I bargaine to cast severall sorts of of {sic} lots successively at y^e same stake y^e valor of each lot is thus found viz: The first prop: gives y^e valor of y^e f{illeg}|ir|st lot; w^ch valor being destructed from y^e stake, y^e remainder is y^e stake of y^e 2^d lot w^ch therefore may bee also found by y^e first prop: &c.

As if {illeg}|I| gaine a by throwing 12 at y^e first cast, or 11 at y^e 2^d or 10 at y^e 3^d &c w^th two dice. Since at y^e first cast there is but one chance for a (viz 12 ) & 35 for nothing Therefore its valor is $\frac{a}{36}$ (by Prop 1). & y^e stake{illeg} for y^e 2^d cast is $a - \frac{a}{36} = \frac{35 a}{36}$ . Now since there is|ar|e two chances for i{illeg}|t| (viz: ⚅⚄ & ⚄⚅) at y^e 2^d cast & 34 for 0 at y^e 2^d cast therefore its valor is $\frac{2 \times 35 a}{36 \times 36} = \frac{35 a}{648}$ . as y^e stake for y^e 3^d chance lot is $\frac{595 a}{648}$ f{illeg}|o|r w^ch there are 3 chances (f{illeg}|vi|z ⚄⚄, ⚅⚃, ⚃⚅, {sic}) & 33 for nothing Therefore its valor is $\frac{595 a}{7776}$ .

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4 If I bargaine w^th one or two more to cast lots in order untill one of us by an assigned lott shall win y^e stake a: Since y^e chances may succede infinitly I onely consider y^e first revolution of them The valor of each mans \whole/ expectation being in such proportion one to another as y^e valors of their lots in one revolution. & y^e valors of each mans f{illeg}|ir|st lot being to y^e valor of his whole expectation as y^e summe of y^e valors of their f{illeg}|irs|t lots to y^e stake a.

As if I contend w^th another y^t who first throws 12 w^th 2 dice shall h{illeg}|a|ve a, I haveing y^e dice. {illeg}|M|y first lot is worth $\frac{a}{36}$ (by second prop 1), The 2^d \his first/ lot is worth $\frac{35 a}{36 \times 36}$ . And $\frac{a}{36} ∶ \frac{35 a}{36 \times 36} ∷ 36 ∶ 35 ∷$ $my expectation ∶ to his$ . for y^e two first lots make one revolution because {illeg} I have y^e same lot If I throw a 2^d time y^t I had at y^e first. Therefore $(36 + 35 = 71 ∶ a ∷ 36 ∶ \frac{36 a}{71})$ $\frac{36 a}{71}$ is my interest in y^e stake.

If o^r bargaine bee soe y^t there is some lott at y^e beginning of o^r play w^ch returnes not in y^e {illeg} after revolutions, detract y^e valor of those irregular lotts from y^e stake & y^e rest shall bee y^e stake of y^e regular lots w^ch follow & revolve successively. As if I contend w^th another y^t who first casts 11 must have a , onely I have {y^e} first cast for 12. My first lot is worth $\frac{a}{36}$ . & y^e stake for o^r after throws is $\frac{35 a}{36}$ . his firts lot being $\frac{35 a}{648}$ . & my next lot $\frac{595 a}{11664}$ . soe y^t his share in y^e stake $\frac{35 a}{36}$ is to mine as $\frac{35 a}{648} ∶ \frac{595 a}{11664} ∷ 18 ∶ 17$ . Soe y^t my share in it is $\frac{17 a}{36}$ . To w^ch adding {illeg} y^e valor of my f{illeg}|i|rst lot viz: $\frac{a}{36}$ , y^e summe is $\frac{18 a}{36} = \frac{a}{2}$ , my interest in y^e stake a at y^e begining.

5 If y^e Proportion of {illeg}|t|he chances for any stake bee irrationall {illeg}|t|he interest in the stake may bee found after y^e same manner. As if y^e Radij ab , ac , divide y^e \horizontall/ circle \ bcd / into two pts <91v> abec & abdc in such proportion as $\dot{1}$ |2 | to $\sqrt{5}$ . And if a ball falling perpendicularly upon y^e center a doth tumble into y^e portion abec I winn $(a)$ : {illeg}|b|ut if into y^e other portion, I win b . my hopes is worth $\frac{2 a + b \sqrt{5}}{2 + \sqrt{5}}$ .

Soe if a die bee not a Regular body but a Parallelipipi|e|don or otherwise unequall sided, it may bee found how much one side is cast is more easily gotten then another.

☞ 6 Soe y^t y^e facility of y^e chances & y^e stake belonging to each chance being knowne y^e worth of of {sic} the lott may bee ever found by y^e precedent precepts. And if they bee not both immediatly found known they must bee sought before y^e valor of y^e lott can bee found.

As if I want two games at Irish & my adversary three to win a , & I would know y^e value of my interest in y^e stake $(a .)$ my first lot {illeg} can gaine me nothing but y^e advantage of another lot, & therefore to know its vallue I must first find y^e value of y^t other lot &c. First therefore if wee each wanted one lot to win a o^r interest in it would bee equall viz my lot worth $\frac{a}{2}$ . \2^dly/ If I want thr{illeg} lo two {illeg} \one/ games & my adversary one|two|, {illeg} & I gaine y^e next game yⁿ I gaine a but if I loose it I onely gaine an{illeg} equall lot for a at y^e {illeg}|n|ext game w^ch is worth $\frac{1}{2} a$ , Therefore my interest in y^e stake is $\frac{a + \frac{1}{2} a}{2} = \frac{3 a}{4}$ . 3^dly If I want one game & my adversary {illeg}|t|hree & I gaine y^e next game I get a; but if I loose it, then I want one game & my adversary but two, y^t is I get $\frac{3 a}{4}$ : Therefore (there being one chance for a & one for $\frac{3 a}{4}$ ) my interest in y^e stake is $\frac{a + \frac{3 a}{4}}{2} = \frac{7 a}{8}$ . 4^thly If I want 2 games & my adversary 3; & I win I get $\frac{7 a}{8}$ . but if I loose I get $\frac{1}{2} a$ for o^r chances <92r> will then bee equall; Therefore my interest in y^e stake is $\frac{11 a}{16}$ . Soe if I want {illeg}| 1 | games & my adversary 4 my interest in a is $\frac{15 a}{16}$ . If I want two and hee 4 , it is $\frac{13 a}{16}$ . If I want 3 and hee 4 it is $\frac{21 a}{32}$ . If I 1 and hee 5 , it is: $\frac{31 a}{32}$ . If I 2 and hee 5 it is $\frac{57 a}{64}$ . If I 3 and hee 5 it is $\frac{99}{128} a$ . If I 4 and hee 5 , it is: $\frac{163}{256} a$ . |(|The like may bee done if 3 or more play together. |(|as if one wants one game, another 3 a third 4: Their lots are as $616 : 82 : 31$ . &c.|)| As also if their lots bee of divers sorts.|)|

By this meanes also some of y^e precedent questions may bee resolved. as if I have two throws for a cise to win a , w^th one die; If I have missed my first lot alredy {sic}, I have at my second cast five chances for nothing. & one for a . therefore y^t cast is worth $\frac{a}{6}$ . Soe y^t \in/ my first cast I had five chances for $\frac{1}{6} a$ & one for a , w^ch therefore is worth (w^th my 2^d cast) is worth $\frac{11}{36} a$ . That is tis {illeg} to 11 to 25 whith {sic} y^t I cast a cise at once in two throws. as before

By this meanes also my lot may bee known if I am to draw 4 cards of severall sorts out of 40 cards {illeg}| 1 | 0 of each s{illeg}i|or|t.

Or if out of two white & 3 black stones I am blindfold to chose a white & a black one.

<92v>

Symbol (equal subscript u) in text Equation

An equation given; & \if/ either \both/ x , & y , be of a /have\ divers quan dimensions, try if y^e roote \of one of y / may be extracted: {illeg} whereby y^e Symbol (equal subscript u) in text may often be diminished riduced to finer termes & sometimes to finer dimensions if it can {illeg} \{illeg}/ y^e line & {illeg} If a quantity wherein x | y | is not is divided by {illeg}| x | in y^e {illeg}q{illeg} {illeg}g line equall to x . y^t crooked cannot be squared.

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The line cdf is a Parab. \ $4$ / $ac =$ \ $2$ / $ad = r =$ \ $2$ / $na =$ \ $2$ /ge. $ce = x$ . $ef = y$ . $r x = y y$ . $g$ ^{illeg} $e ∶ ef ∷ \frac{1}{2} r ∶ \sqrt{r x} ∷ eo ∶ ap$ . $eo = ʒ$ . {illeg} $eo = z$ . $ap = a$ . $\frac{1}{2} r a = z \sqrt{r x}$ . $\frac{1}{4} r r a a = r z z x$ . $\frac{r a a}{4} = z z x$ . or supposeing $ea =$ x|y|. {illeg} $x = y + \frac{1}{4} r$ & $\frac{r a a}{4} = z z y +$ | $\frac{1}{4}$ | $z z r$ . {illeg} w^ch shews y^e nature of y^e crooked line po. now if $dt = ap$ . yⁿ $drst = eoap$ . for supposeing eo moves uniformely from ap , & rs moves from dt w^th motion decreaseing in y^e ꝑportio y^t y^e line eo doth shorten. Suppos $aq = ap = \frac{r}{2} = a$ & $eq =$ x | y |. $x = y - \frac{1}{4} r$ . then $\frac{r^{3}}{16} = z z y - \frac{1}{4} z z r$ . suppose $z = a + y$ . yⁿ $r^{3} = a a x + 2 a y x + y y x$ . Or $a a x + 2 a y x + y y x + \frac{1}{4} a a r + \frac{1}{8} a y r + \frac{1}{4} y y r = \frac{r^{3}}{16}$ . Or $a a x + 2 a y x + y y x - \frac{1}{4} a a r$ + $- \frac{1}{8} a y r - \frac{1}{4} y y r = \frac{r^{3}}{16}$ . Or suppose $z = y - a$ . yⁿ $\frac{r^{3}}{16} = a a x - 2 a y x + y y x$ . Or $a a x$ + / − \ $2 a y x + y y x + \frac{1}{4} a a r - \frac{1}{8} a y r + \frac{1}{4} y y r =$ | $\frac{r^{3}}{16}$ | Or $a a x - 2 a y x + y y x - \frac{1}{4} a a r + \frac{1}{8} a y r - \frac{1}{4} y y r = \frac{r^{3}}{16}$ . Or, if $x = a -$ y | r |. $\frac{r^{3}}{16} = z z a - z z$ y | x |. &
$a^{3} +$ \ $2$ / $a a y + a y y$ {illeg} − {illeg} $a a x - 2 a y x -$ a |y | $y x = \frac{r^{3}}{16}$ . Or $a^{3} - 2 a a y + a a y - a a x + 2 a y x - y y x = \frac{r^{3}}{16}$ . ${oƅ}^{2} =$ \ $2$ / $z^{2}$ . $mp =$ {illeg}y x |ʒ |. $mq = a +$ x | ʒ | $= mƅ$ . $mo = y = a +$ x | ʒ | $- \sqrt{2 z^{2}}$ {illeg} $pq (= a) ∶ aq (= \frac{1}{2} r) ∷ ƅq (= x + z) ∶ mƅ = y + \sqrt{2 z z}$ $a y + a \sqrt{2 z z} = \frac{1}{2} r x + \frac{1}{2} r z$ . $\frac{2 a y + 2 z \sqrt{2 a a}}{r} - \frac{1}{2} z = x =$ $\frac{r^{3} + 4 z z r}{16 z z}$ . $32 z z a y + 32 z^{3} a \sqrt{2} - 8 z^{3} r - r^{4} - 4 z z r r = 0$ $mv = ξ = a + ʒ -$ r $z \sqrt{2}$ . $\frac{ξ - a - ʒ}{\sqrt{2}} = z$ . $\frac{ξ^{2} - 2 ξ a - 2 ξ ʒ + a a + 2 a ʒ + ʒ^{2}}{2}$ $= z z$ . $\frac{ξ^{3} - 3 ξ^{2} a - 3 ξ^{2} ʒ + 3 a a ξ + 6 a ʒ ξ + 3 ʒ ʒ ξ - a^{3} - 3 a a ʒ - 3 a ʒ ʒ - ʒ^{3}}{\sqrt{8}} = z^{3}$ $a \sqrt{2} +$ $32 \sqrt{2 a a} - 8 r = c$ $\begin{matrix} c z^{3} + 32 a ξ z z - r^{4} \\ - 4 r r \end{matrix} = 0$ <93v> <94r> $\begin{matrix} \begin{matrix} c ξ^{3} & - & 3 c a ξ^{2} & + & 3 c a a ξ & - & c a ʒ \\ + & 32 a ξ^{3} & - & 3 c ʒ ξ^{2} & + & 6 c a ʒ ξ & - & 3 c a^{2} ʒ \\ + & 3 c ʒ ʒ ξ & - & 3 c a ʒ^{2} \\ - & 64 a a ξ^{2} & + & 32 a^{3} ξ & - & c ʒ^{3} \\ - & 64 a ʒ ξ^{2} & + & 64 a a ʒ ξ & - & 4 r r a a \\ - & 4 r r ξ^{2} & + & 32 a ʒ ʒ ξ & - & 8 r r a ʒ \\ + & 8 r r a ξ & - & 4 r r ʒ ʒ \\ + & 8 r r ʒ ξ & - & r^{4} \end{matrix} & \begin{matrix} = & 0 \end{matrix} \end{matrix}$ Or
$\begin{matrix} \begin{matrix} ξ^{3} & - & \frac{d}{e} ʒ ξ^{2} & + & g g ξ & - & f ʒ \\ - & d & + & h ʒ & - & ʒ m^{2} \\ + & \frac{i}{g} ʒ ʒ & - & n ʒ^{2} \\ - & \frac{o}{n} ʒ^{3} \end{matrix} & \begin{matrix} = & 0 \end{matrix} \end{matrix}$ .

Let $ed = a$ . $ae = x$ $ab = y$ a {illeg} y {illeg}/{illeg}\ $se = v$ . $sb = s$ . $s s = y^{2} + v v + x x - 2 v x$ . $y^{3} = - v v y + s s y + 2 v x y - x x y = a^{2}$ {illeg} a {illeg}²{illeg} v {illeg} $s s$ {illeg} $x - v$ ^{illeg} $- v^{4} x^{2}$ $a a x + a^{3} = y y x$ $y^{2} = s s - v v + 2 v x - x x$ $a a x + a^{3} = s s x$ + / − \ {illeg} $v v x + 2 v x^{2} - x^{3}$ $\begin{matrix} x^{3} & - & 2 v x^{2} & \begin{matrix} + & v v \\ - & s s \\ + & a a \end{matrix} x & + & a^{3} & = & 0 \\ 2 & 1 & 0 & - & 1 \end{matrix}$ $2 x^{3} - 2 v x^{2}$ + $- a^{3} = 0$ . $\frac{2 x^{3} - a^{3}}{2 x x} = v$

$a a x = y y x + y y a$ . $y^{2}$ + $= s s - v v + 2 v x$ {illeg} $- x x$ $a a x = s s x - v v x + 2 v x x - x^{3} + s s a + 2 a v x$ {illeg} $- a x x - a v v$ . $\begin{matrix} x^{3} & \begin{matrix} - & 2 v \\ + & a \end{matrix} x^{2} & \begin{matrix} - & s s \\ + & v v \\ - & 2 a v \\ + & a a \end{matrix} x & \begin{matrix} - & s s a \\ + & a v v \end{matrix} & = & 0 \end{matrix}$ . $\begin{matrix} x^{2} - 2 e x + e e \times x + f \\ \begin{matrix} x^{3} & \begin{matrix} - & 2 e \\ + & f \end{matrix} x^{2} & \begin{matrix} + & e e f x \\ + & 2 e e x \end{matrix} & + & e e f & = & 0 \end{matrix} \end{matrix}$ $- 2 v + a = f - 2 e$ . $f = a + 2 e$ {illeg} $- 2 v$ . $e e f = a v v - a s s$ $\frac{- e e a - 2 e^{3} + 2 e e v + a v v}{a} = s s$ . $\begin{matrix} v v x & - & 2 v v x & + & x^{3} \\ v v a & - & 2 v a x & + & a x^{2} \\ + & a a x \end{matrix} = \begin{matrix} a \\ x \end{matrix}} \frac{2 e e v + a a v - 2 e^{3} - e e a}{a}$

<94v>

To square those lines in w^ch is y onely

If y is in but one terme onely of y^e Equation (as $x x = a y$ . or, $a^{3} = x x y$ ) resolve y^e Eq: into y^e proport $y ∶ a$ (as $y ∶ a ∷ x x ∶ a a$ . or, $y ∶ a ∷ a a ∶ x x$ .) If y^e line hath Assymptotes
$x^{3} = a a y$ . $v = \frac{3 x^{5}}{a^{4}} + x$ .

<95r>

$\begin{matrix} \begin{matrix} a \\ x \\ - & a \\ - & x \end{matrix} & \begin{matrix} v v \end{matrix} & \begin{array}{l} - & 2 v {\dot{x}}^{2} \\ - & 2 v a x \\ - & 2 v e \dot{e} \\ - & \frac{2 x}{a} e e v \end{array} & \begin{array}{l} + & {\dot{x}}^{3} \\ + & a {\dot{x}}^{2} \\ + & a \dot{a} x \\ + & 2 {\dot{e}}^{3} \\ + & e \dot{e} a \\ + & \frac{2 x}{a} {\dot{e}}^{3} \\ + & x e \dot{e} \end{array} & \begin{matrix} = & 0 \end{matrix} \end{matrix}$ $\begin{matrix} v = \frac{4 x^{3} + 2 a x x + a \dot{a} x + 2 \frac{x^{4}}{a}}{4 x x + 2 a x + 2 \frac{x^{3}}{a}} \\ v = \frac{4 a x^{2} + 2 a a x + a^{3} + 2 x^{3}}{4 a x + 2 a a + 2 x x} \\ sa = \frac{\begin{matrix} 4 a x^{2} + 2 x^{3} + 2 a a x + a^{3} - \\ - 4 a x^{2} - 2 x^{3} - 2 a a x \end{matrix}}{4 a x + 2 a a + 2 x x} \\ sa = \frac{a^{3}}{2 x^{2} + 4 a x + 2 a a} \end{matrix}$ $\frac{2 a^{4} x}{x + a} ∶ \frac{2 a^{6}}{4 x x + 8 a x + 4 a a} ∶$ {illeg} $2 a a x ∶ \frac{2 a^{6}}{8 x^{3} + 24 a x + 24 a a x + 8 a^{3}} ∶$ \ $2$ / $x^{3} + 2 a x^{2}$ ad. $\frac{2 a^{4} x^{2} + 2 a^{5} x}{8 x^{3} &}$ = $\frac{a^{4} x}{4 x x + 8 a x + 4 a a}$ $a \sqrt{a a}$ $x = \frac{y y}{a a - y y}$ $x^{2} = \frac{y^{4} a^{2}}{a^{4} + 2 a^{2} y^{2} + y^{4}}$ $\frac{a^{5} y y}{\begin{matrix} 4 a a x^{2} + 8 a^{3} x + 4 a^{4} \\ - 4 y y - 8 y y a - 4 y y a a \end{matrix}}$ $\begin{matrix} 2 x^{3} + 4 a x^{2} + 2 a a x - a^{3} = 0 \\ oe = p \end{matrix}$ $a y y$ $\frac{\sqrt{a a x}}{a + x} ∶ \frac{a^{3}}{2 x^{2} + 4 a x + 2 a a} ∷ p ∶ z$ $\frac{z^{2} a a x}{a + x} = \frac{a^{6} p p}{\begin{matrix} 4 x^{4} & + & 16 a x^{3} & + & 8 a a & x x & + & 16 a^{3} x & + & 4 a^{4} \\ + & 16 a a \end{matrix}}$ $a^{5} p p + a^{4} x p p = 4 z z x^{5} + 16 z z a x^{4} + 24 a a z z x^{3} + 16 z z a^{3} x^{2} + 4 a^{4} z^{2} x$ divided by $x + a$ it produceth. $4 z z x^{4} + 12 z z a x^{3} + 12 z z a a x^{2} + 4 z z a^{3} x$ + / − \ $a^{4} p p = 0$

<95v>

By y^e Squares of y^e simplest lines to square lines more compound. 1^s {sic} those whein y.

find y^e valor of y. If y^e number of y^e termes in y^e denom: thereof be neither $1.3.6.10.15.21.28 .$ &c. y^e line cannot be squared If it have but one terme tis squared by finding y^e square of \{illeg} each/ particular terme in y^e valor of y & yⁿ adding all those squares together. Example 1^st. $3 x^{4} + a^{4} = y a x x$ . & $y = \frac{3 x^{4} + a^{4}}{a x x}$ . Then makeing y equall to each particular terme. $\frac{3 x x}{a} = y$ . $\frac{a^{3}}{x x} = y$ or $3 x x = a y$ whose □ is $\frac{x^{3}}{a}$ . & $a^{3} = x x y$ . whose □ is $\frac{a^{3}}{x}$ Ad {sic} these 2 □s together & they (viz: $\frac{x^{4} + a^{4}}{a x}$ ) are the □ of y^e line $3 x^{4} + a^{4} = a y x x$ . Againe $2 a^{7}$ + \ − / $2$ a | b | $x^{6} + x^{7} = a^{3} x^{3} y$ . Or $y = \frac{2 a^{7} - 2 b x^{6} + x^{7}}{a^{3} x^{3}}$ . yⁿ disjoynting y^e valor of y. $y = \frac{2 a^{4}}{x^{3}}$ . $y = \frac{x^{4}}{a^{3}}$ . $y = \frac{2 b x^{3}}{a a a}$ Or $x^{3} y = 2 a^{4}$ , whose □ is $\frac{a^{4}}{x x}$ . $y a^{3} = x^{4}$ , whose □ $\frac{x^{5}}{5 a^{3}}$ . $y$ {illeg}| $a^{3}$ | $= - 2 b x^{3}$ , whose □ $- \frac{x^{4} b}{2 a^{3}}$ . which 3 □s (viz $\frac{10 a^{7} + 2 x^{7} - 5 b x^{6}}{10 a^{3} x x}$ ) is y^e □ {illeg} taken together are y^e square sought for. And these lines may bee ever squared unless in y^e valor of y there bee found $\frac{a a}{x}$ , $\frac{a b}{x}$ , $\frac{c c + d e}{x}$ , &c. for y^e Squareing of y^t line depends on y^e squareing of y^e Hyperbola. As in y^e line {illeg}| x | $x x y =$ {illeg} $x x$ / $x^{4}$ \ $+ a^{3} x + a^{4}$ .

<96r>

$a a x = y^{3}$ . $cd = a$ . $ce = x$ . $eh = y$ . $oc = v$ . $ok = s$ . $s s = y^{2} + x^{2} - 2 v x + v^{2}$ . $x^{2} = 2 v x + s s - y^{2} - v^{2}$ . $x = v - \sqrt{+ s s - y^{2}}$ . $a a v - a a \sqrt{s s - y^{2}} = y^{3}$ . $\begin{matrix} a^{4} v^{2} & - & 2 a a v y^{3} & + & y^{6} & - & a^{4} s s & + & a^{4} y^{2} & = & 0 \\ 0 & 3 & 6 & 0 & 2 \end{matrix}$ $6 y^{6} - 6 a a v y^{3} +$ \ $2$ / $a^{4} y y = 0$ $v = \frac{a^{4} + 3 y^{4} - 3 y^{4}}{3 a a y}$ . $y ∶ a^{4} + 3 y^{4} 3 a a y ∷ \frac{y^{3}}{a a} ∶ \frac{a^{4} y^{3} + 3 y^{7}}{3 a^{4} y y} = \frac{a^{4} y + 3 y^{5}}{3 a^{4}}$ $v - x = a^{4} + 3 y^{4} - 3 y^{4} 3 a a y$ $v - x = \frac{a^{4}}{3 a a y} = \frac{a a}{3 y}$ . $y ∶ \frac{a a}{3 y} ∷ \frac{y^{3}}{a a} ∶ \frac{a a y^{3}}{3 a a y y} = \frac{y}{3}$ . make $md = dv$ that is $a a = 3 y y$ . & $\frac{a}{\sqrt{3}} = y = md = mv$ . $x = dc = \frac{a}{\sqrt{27}} = x$ $ds = dc = \sqrt{\frac{a a}{27}}$ $\sqrt{c a a y} ∶ \frac{a a}{3 y} ∷ \sqrt{\frac{a a}{27}} ∶ z = en$ $a a x ∶ \frac{a^{6}}{3 a a x} ∷ \frac{a^{3}}{27 \sqrt{27}} ∶ z^{3}$ $a a x^{2} z^{3} = \frac{a^{7}}{81 \sqrt{27}}$ $x x z^{3} = \frac{a^{5}}{81 \sqrt{27}}$ w^ch expresseth y^e nature of y^e crooked line ns . & $vl = ds$ . $vlfr = dens$ .

2dly. I{illeg}|f| it have 2|3| termes. bee $6, 10, 15, 21,$ termes See if it may be reduced to ${\begin{matrix} one or \\ fewer \end{matrix}}$ dimensions by addin{illeg}|g| or subtracting a{illeg} knowne quantity to or from x . Example. $2 b a x + a x x = b b y + 2 b x y + x x y$ . w^ch (makeing $x + b = z$ ) i{illeg}|s| thus red{illeg}|u|ced
$z z y =$ {illeg} − {illeg} $b b a + a z z$ . Or $\frac{- b b a + a z z}{z z} = y$

<97r>

$a a x + b b y = y^{3}$ . $x = v - \sqrt{s s - y y}$ . $a a v - a a \sqrt{s s - y^{2}} + b b y =$ \ $y^{3}$ / $a a v + b b y - y^{3} = a a \sqrt{s s - y^{2}}$ .
$\begin{matrix} a^{4} v^{2} & + & 2 a a v b b y & - & 2 a a v y^{3} & + & b^{4} y^{2} & - & 2 b b y^{4} & + & y^{6} \\ - a^{4} s s & + & a^{4} y^{2} \\ 0 & 1 & 3 & 2 & 4 & 6 \end{matrix}$ $\frac{6 y^{5} - 8 b b y^{3} + 2 b^{4} y + 2 a^{4} y}{- 2 a a b b + 6 a a y y} = v$ $\begin{matrix} \begin{matrix} v - x & = \end{matrix} & \begin{matrix} - & 2 b^{4} y & + & 6 b b y^{3} & - & 6 y^{5} \\ + & 2 b^{4} y & + & 2 b b y^{3} & + & 6 y^{5} \\ + & 2 a^{4} y & - & 8 b b y^{3} \end{matrix} \end{matrix}$ $v - x = \frac{2 a^{4} y}{6 a a y^{2} - 2 a a b b}$ $v - x = \frac{a a y}{3 y^{2} - b b}$ . $\frac{- b b y +}{a a} | \frac{y}{} ∶ \frac{a a y}{3 y^{2} - b b} ∶ \frac{y^{3} - b b y}{a a} ∶$ & ad. $\frac{a a y^{4} - a a b b y y}{3 a a y^{3} - a a b b y} ∶ \frac{y^{3} - b b y}{3 y^{2} - b b}$ .

{illeg} $\frac{y}{3 y^{2} - b b} ∷ b b$ {illeg} y $\frac{a a y}{3 y^{2} - b b} = \frac{y^{3} - b b y}{a a .}$ $a^{4} = 3 y^{4} - 4 b b y y + b^{4}$
{illeg} $y^{4} = \frac{4 b b y y - b^{4} + a^{4}}{3}$ $y = \sqrt{\frac{b b}{3} \sqrt{{4b}^{4}}}$ / $a = b$ \ $y^{2} = \frac{4 b b}{3}$ . $y = \frac{2 b}{\sqrt{3}} = dm = dv$ . $\frac{8 b^{3}}{3 \sqrt{3}}$ {illeg} $- \frac{2 b b b}{\sqrt{3}} - a a x$ . $\frac{2 b}{3 \sqrt{3}} = dc = ds$ . $y ∶ \frac{a a y}{3 y^{2} - b b} ∷ \frac{2 b}{3 \sqrt{3}} ∶ z$ . $y z = \frac{2 a a b y}{9 y^{2} \sqrt{3} - 3 b b \sqrt{3}}$ $9 y y z - 3 b b$ {illeg} $z = \frac{2 a a b}{\sqrt{3}}$ An Equation expressing y^e nature of y^e line ns .

<98r>

$a a x + b y x = y^{3}$ . $x = v - \sqrt{s s - y y}$ .
$a a v + b y v - y^{3} = \begin{matrix} a a \\ b y \end{matrix} \sqrt{s s - y y}$
$\begin{matrix} a^{4} v^{2} & + & 2 a a v^{2} b y & - & 2 a a v y^{3} & + & b b v v y^{2} & - & 2 b v y^{4} & + & y^{6} \\ - & a^{4} s s & + & a^{4} y y & + & b b \\ - & b b s s \\ - 2 & - 1 & + 1 & 0 & 2 & 4 \\ 0 & 1 & 3 & 2 & 4 & 6 \end{matrix}$ $s s = \frac{2 a^{4} v v + 2 a a v v b y + a a v y^{3} + 4 b v y^{4} - 2 b b y^{4} - 4 y^{6}}{2 a^{4}}$ $\begin{matrix} 12 a^{4} y^{6} & - & 16 a^{4} b v y^{4} & + & 4 a^{4} b b y^{4} & + & 4 b b a^{4} v v y^{4} & - & 12 a^{6} v y^{3} & + & 4 a^{6} v v b y & - \\ + & 4 a^{8} y y \end{matrix}$ $\begin{array}{l} \begin{matrix} - 12 a^{4} y^{6} & - & 16 a^{4} b v y^{4} & + & 4 a^{6} v v b y \\ - & 8 b^{3} v y^{6} & + & 8 a^{4} b b y^{4} & - & 12 a^{6} v y^{3} & + & 4 a^{8} y y \\ + & 4 b^{4} y^{6} & - & 2 a a b b v y^{5} & - & 4 a a v v b^{3} \end{matrix} & } & 0 \\ \begin{matrix} + & \underline{8 y^{8} b b} \end{matrix} \\ \begin{array}{l} \begin{array}{l} + & 4 a^{6} b y \\ + & 4 a a b^{3} y^{3} \end{array} & {\begin{matrix} v v & = \end{matrix} & \begin{matrix} \begin{matrix} 12 a^{6} y^{3} a v & - & 8 a^{4} b b y^{4} \\ 2 a a b^{2} y^{5} & - & 8 y^{8} b b \\ 16 a^{4} b y^{4} & - & 4 b^{4} y^{6} \\ 8 b^{3} y^{6} & - & 12 a^{4} y^{6} \end{matrix} \\ \begin{matrix} 4 a^{6} b y & - & 4 a a b^{3} y^{3} \end{matrix} \end{matrix} \end{array} \end{array}$ $\begin{matrix} \begin{matrix} \begin{matrix} v & = \end{matrix} & \begin{matrix} \begin{array}{l} 6 a^{6} y^{2} \\ + & a^{2} b^{2} y^{4} \\ + & 8 a^{4} b y^{3} \\ + & 4 b^{3} y^{5} \end{array} \\ \begin{matrix} 4 a^{6} b & - & 4 a a b^{3} y^{2} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} ♉ & \sqrt{\begin{matrix} \begin{matrix} \begin{matrix} 36 a^{12} y^{4} & + & 8 a a b^{5} y^{9} \\ + & 12 a^{8} b^{2} y^{6} & + & 64 a^{8} b^{8} y^{6} \\ + & 16 a^{10} b y^{5} & + & 16 b b y^{10} \\ + & 24 a^{8} b^{3} y^{7} \\ + & a^{4} b^{4} y^{8} \\ b^{5} \end{matrix} \\ \begin{matrix} 16 a^{12} b^{2} - 32 a^{8} b^{4} y^{2} \\ + 16 a^{4} b^{6} y^{4} \end{matrix} \end{matrix} & \begin{matrix} \end{matrix} & \begin{matrix} \begin{matrix} - & 8 a^{4} b b y^{3} \\ - & 8 y^{7} b b \\ - & 4 b^{4} y^{5} \\ + & 12 a^{4} y^{5} \end{matrix} \\ \begin{array}{l} + & 4 a^{6} b \\ - & 4 a a b^{3} y^{2} \end{array} \end{matrix} \end{matrix}} \end{matrix} \end{matrix}$

<99r>

$a^{3} x = y^{4}$ . $x = v - \sqrt{s s - y y}$ . $a^{3} v - y^{4} = a^{3} \sqrt{s s - y y}$ $\begin{matrix} a b v^{2} & - & 2 a^{3} v y^{4} & + & y^{8} & + & a^{6} y y \\ - & a b s s \\ 0 & 4 & 8 & 2 \end{matrix}$ . $v = \frac{4 y^{6} + a^{6}}{4 a^{3} y^{2}}$ $v - x = \frac{4 y^{6} + a^{6} - 4 y^{6}}{4 a^{3} y^{2}} = \frac{a^{3}}{4 y^{2}}$ .
$y \cdot \frac{a^{3}}{4 y^{2}} ∷ \frac{y^{4}}{a^{3}} ∶ \frac{a^{3} y^{4}}{4 a^{3} y^{3}} = \frac{y}{4}$

<100r>

$a a x - a a y = y^{3}$ . $a a v - a a y - y^{3} = a a \sqrt{s s - y y}$ $\begin{matrix} a^{4} v^{2} & - & 2 a^{4} v y & - & 2 a a v y^{3} & + & 2 a^{4} y^{2} & + & 2 a a y^{4} & + & y^{6} \\ - & a^{4} s s \\ 0 & 1 & 3 & 2 & 4 & 6 \end{matrix}$ $v = \frac{3 y^{5} + 4 a a y^{3} + 2 a^{4} y}{a^{4} + 3 a a y y}$
$v - x = \frac{3 \dot{y} 5 + 4 \dot{a a} y^{3} + 2 a^{4} y - \dot{a a} y^{3} - 3 {\dot{y}}^{5} - {\dot{y a}}^{4} - 3 \dot{a a} y^{3}}{a^{4} + 3 a a y y}$ $v - x = \frac{a a y}{a a + 3 y y}$ . $y ∶ \frac{a a y}{a^{2} + 3 y^{2}} ∷ \frac{a^{2} y + y^{3}}{a a} ∶ \frac{a^{4} y^{2} + a^{2} y^{4}}{a^{4} y + 3 a^{2} y^{3}}$ $= \frac{a a y + y^{3}}{a a + 3 y y}$ $\frac{y^{3} + a a y}{a a} = \frac{a a y}{a a + 3 y y}$ $y y a a + 3 y^{4} +$ \ $3$ / $a a y y + a^{4} = y$ {illeg} $y y$ {illeg} $a a y$

<101r>

$a a x = b y^{2} + y^{3}$ . $a a v - b y^{2} - y^{3} = a a \sqrt{s s - y y}$ $\begin{matrix} a^{4} v^{2} & - & 2 a a b y y v & - & 2 a a v y^{3} & + & b v y^{4} & + & 2 b y^{5} & + & y^{6} \\ - & a^{4} s s & + & a^{4} \\ 0 & 2 & 3 & 4 & 5 & 6 \end{matrix}$ $v = \frac{3 y^{4} + 5 b y^{3} + 2 b b y^{2} + a^{4} y y}{2 a a b + 3 a a y}$ $v - x = \frac{3 {\dot{y}}^{4} + 5 b {\dot{y}}^{3} + 2 b b {\dot{y}}^{2} - 2 b b y y - 3 b {\dot{y}}^{3} - 2 b y^{3} - 3 {\dot{y}}^{4} + a^{4}}{2 a a b + 3 a a y}$ {illeg} $v - x = \frac{a^{4}}{2 a a b + 3 a a y}$ $y ∶ \frac{b y^{3} + b b y^{2}}{a a b + 3 a a y} ∷ \frac{b y^{2} + y^{3}}{a a} ∶ ad$ $\frac{2 b b y^{4} + b y^{5} + b^{3} y^{3}}{a^{4} + 3 a a y} ∶$ $a = b$ . $a^{3} + 3 a a y = b y y + b b y$ $a a + 3 a y - y y - a y = 0$ $y y = 2 a y + a a$ $y = a + \sqrt{a a + a a}$ . $y = a + a \sqrt{2} = dm = vd$ . $\frac{\begin{matrix} a^{3} & + & 3 a^{3} \sqrt{2} & + & 6 a^{3} & + & 2 a^{3} \sqrt{2} \\ a^{3} & + & 2 a^{3} \sqrt{2} & + & 2 a^{3} \end{matrix}}{a a} = 10 a + 7 a \sqrt{2} = dc$ $y ∶ \frac{y^{3} + b y^{2}}{b b + 3 b y} ∷ 10 b + 7 b \sqrt{2} ∶ z$ . $y z = \frac{\begin{matrix} 10 y^{3} & + & 10 b y^{2} & + & 7 y^{3} \sqrt{2} \\ + & 7 b y^{2} \sqrt{2} \end{matrix}}{b + 3 y}$ $b z + 3 y z = 10 y y + 10 b y + 7 y y \sqrt{2} + 7 b y \sqrt{2}$ .

<102r>

$a x y = y^{3} + b^{3}$ . $a y v - y^{3} - b^{3} = a y \sqrt{s s - y y}$ . $\begin{matrix} a a v v y^{2} & - & 2 a v y^{4} & - & 2 a v b^{3} y & + & y^{6} & + & 2 b^{3} y^{3} & + & b^{6} \\ - & a a s s & + & a a \\ 0 & 2 & - 1 & 4 & 1 & - 2 \end{matrix}$ $v = \frac{4 y^{6} + 2 b^{3} y^{3} + 2 a a y^{4} - 2 b^{6}}{4 a y^{4} - 2 a b^{3} y}$ $v - x = \frac{\begin{matrix} 4 {\dot{y}}^{6} & + & 2 b^{3} {\dot{y}}^{3} & + & 2 a a y^{4} & - & 2 {\dot{b}}^{6} & - & 4 {\dot{y}}^{6} & + & 2 b^{3} {\dot{y}}^{3} & - & 4 b^{3} {\dot{y}}^{3} \\ + & 2 {\dot{b}}^{6} \end{matrix}}{4 a y^{4} - 2 a b^{3} y}$ $v - x = \frac{a y^{3}}{2 y^{3} - b^{3}}$ . $y ∶ \frac{a y^{3}}{2 y^{3} - b^{3}} ∶ \frac{y^{3} + b^{3}}{a y} ∶ \frac{y^{4} + b^{4}}{2 y^{3} - b^{3}}$

<102v>

<103r>

$a^{3} = x y y$ . $a = dg = dp$ . $go = x$ . $oa = y$ . $cg = v$ . $ca = s$ . $s s = y^{2} + x x - 2 v x + v^{2}$ . $\begin{matrix} a^{3} & - & x s s & + & x^{3} & - & 2 v x^{2} & + & v v x = 0 \\ - 1 & 0 & 2 & 1 & 0 \end{matrix}$ $\frac{2 x^{3} - a^{3}}{2 x^{2}} = v$ . $\sqrt{}$ $\frac{a^{3}}{x} ∶ \frac{4 x^{6} - 4 a^{3} x^{3} + 4 a^{6}}{4 x^{4}} ∷ x$ . { Ad } $\frac{4 x^{6} - 4 a^{3} x^{3} + a^{6}}{4 x x a^{3}}$ . $v - x =$ $x - v = \frac{a^{3}}{2 x^{2}}$ . $\sqrt{\frac{a^{3}}{x}} ∶ \frac{a^{3}}{2 x x} ∷ p ∶ z$ $\frac{z z a^{3}}{x} = \frac{a^{6} p p}{4 x^{4}}$ $4 x^{3} z z = a^{3} p p$ $go = y$ . $oa = x$ . &c: $a^{3} = x y y$ . x |s| $s = x x + y y + v v - 2 v y$ $x x = s s - y y + 2 v y + v v$ . $\begin{matrix} a^{6} & - & s s y^{4} & + & y^{6} & - & 2 v y^{5} & - & v v y^{4} & = & 0 \\ - 4 & 0 & 2 & 1 & 0 \end{matrix}$ $v = \frac{2 y^{6} - 4 a^{6}}{2 y^{5}}$ $y - v = \frac{2 a^{6}}{y^{5}}$ $y y = s s - x x + v x$ $v v$ $\frac{a^{3}}{y^{2}} ∶ \frac{2 a^{6}}{y^{5}} ∷ p ∶ z$ $z = \frac{2 a^{3} p}{y y y}$ . $z y^{3} = 2 a^{3} p$ . w^ch shewes y^e nature of another crooked line y^t may be squared.

<104r>

$a x x = y^{3}$ . $x = v - \sqrt{s s - y y}$ . $x^{2} = v^{2} - 2 v \sqrt{s s - y^{2}} + s s - y^{2}$ . $a v^{2} + a s s - a y^{2} - y^{3} = 2$ / a \ $v \sqrt{s s - y y}$ .
$\begin{matrix} a^{2} v^{4} & - & 2 a v^{2} y^{3} & + & a a y^{4} & - & 2 a y^{5} & + & y^{6} \\ + & a a s^{4} & - & 2 a a s s y^{2} & - & 2 a s s y^{3} \\ - & 2 a^{2} v^{4} s s \\ + & 2 v v a a y^{2} \\ + 3 & 1 & - 0 & - 1 & - 2 & - 3 \end{matrix}$ $\begin{matrix} 3 a^{2} s^{4} & - & 2 a^{4} y^{2} s s & + & 3 a^{2} v^{4} & &c & + & 2 v^{2} a^{2} y^{2} & - & a^{2} y^{4} & + & 4 a y^{5} & - & 3 y^{6} \\ - & 6 a a v v \end{matrix}$ $s s = \begin{matrix} \frac{2 y y}{3} \\ + \frac{6 v v}{3} \end{matrix} + \sqrt{\frac{4 y^{4}}{9} \frac{- v^{4} + y^{4}}{3} \frac{+ y^{6} - 2 a y^{5}}{3 a a}}$ $s s = \frac{2 y y + 6 v v}{3} - \sqrt{\frac{\begin{matrix} 4 {\dot{y}}^{4} a^{2} + 24 a a y \dot{y} v v + 36 a a v^{4} \\ - 9 a a v^{4} - 6 a a y \dot{y} v v - 12 a {\dot{y}}^{5} \\ + 3 a a {\dot{y}}^{4} + 9 {\dot{y}}^{6} \end{matrix}}{9 a a}}$ $s s = \frac{2 y^{2} + 6 v^{2}}{3} - \sqrt{\frac{\begin{matrix} 9 y^{6} - 12 a y^{5} + 7 a a y^{4} + 18 a a v v y^{2} \\ + 27 a a v^{4} \end{matrix}}{9 a a}}$

<104v>

$a = b - A = C - B = D - d = e - D = F - E = G - f = 1,117313$ . $A - a = d - C = ff - F = 0,921787$ . $B - b = E - e = 0,706724$ . $A = b - a = d - B = D - C = f - E = G - F = 2,039100$ . $B - A = C - b = E - D = F - e = 1,824037$ . $e - d = aa - f = 2,234626$ . $b = D - B = e - C = G - E = aa - F = AA - f = 3,156413$ . $C - A = E - d = F - D = 2,941350$ . $B - a = d - b = f - e = 2,745824$ $B = C - a = d - A = D - b = E - C = f - D = G - e = 3,863137$ . $e - B = aa - E = bb - f = 4,273726$ . $F - d = 4,058663$ . $A^{2} - F = 4,078200$ . $C = D - A = e - b = E - B = F - C = f - d = G - D = aa - e = B^{2} - f = 4,980450$ $A^{2} - E = bb - F = 5,195513$ . $d - a = 4,784924$ . $d = D - a = f - C = A^{2} - e = B^{2} - F = 5,902237$ . $bb - e = 6,312826$ . $e - A = F - B = G - d = aa - D = C^{2} - f$ / $= 6,097763$ . \ $E - b = 5,687174$ . $D = e - a = f - B = G - C = A^{2} - D = bb - e = B^{2} - E = C^{2} - F = dd - f = 7,01955$ $E - A = F - b = 6,804487$ . $aa - d = 7,215076$ . $e = G - B = aa - C = A^{2} - d = bb - D = C^{2} - E = D^{2} - f = 8,136863$ $E - a = f - b = B^{2} - e = 7,726274$ . $F - A = 7,921800$ . $dd - F = 7,941337$ . $E =$ $F - A$ $F - a = f - A = G - b = B^{2} - D = C^{2} - e = 8,843587$ . $A^{2} - d = dd - E = D^{2} - F = 9,058650$ . $aa - B = bb - d = ee - f = 9,254176$ . $F = G - A = aa - b = B^{2} - d = C^{2} - D = E^{2} - f = 9,960900$ . $A^{2} - B = bb - C = D^{2} - E = ee - F = 10,175963$ . $f - a = dd - e = 9,765374$ . $f = G - a = A^{2} - b = B^{2} - C = dd - D = D^{2} - e = E^{2} - F = 10,882687$ . $aa - A = C^{2} - d = F^{2} - f = 11,078213$ . $B^{2} - C = ee - E = 11,293276$ .

This table shews y^e distance of any two notes As y^e distance of C & E is B , or a 3^d, or 3,863137 halfe notes. Of B & E tis a 4^th, or 4,98045 halfe notes. of B & F tis 6,097763 halfe notes, or {illeg} greater yⁿ a 5^t ♭, by 0,095526 halfe notes &c.

<105r>

$a a = x y$ . $eg = eh = a$ . $ga = x$ . $ad = y$ . {illeg}| $d$ | $c = s$ . $cg = v$ $y^{2} = s s - x x + 2 v x - v v$ . $\begin{matrix} - & a^{4} & + & x^{2} s^{2} & - & x^{4} & + & 2 v x^{3} & = & 0 \\ - & x^{2} v^{2} \\ + 2 & 0 & - 2 & - 1 \end{matrix}$ $\frac{2 x^{4} - 2 a^{4}}{2 x^{3}} = v$ $x - v = \frac{a^{4}}{x^{3}}$ . $\frac{a a}{x} ∶ \frac{a^{4}}{x^{3}} ∷ p ∶ z$ . $z x x = a^{2} p$ .

$8^{th} - 5^{t} = 4^{th} = G - D = C = 6^{t} - 3^{d} = E - B$
$5^{t} - 4^{th} = 5^{t} + 5^{t} - 8^{th} = 2^{d} = A$ .
$4^{th} + 4^{th} = 8^{th} + 8^{th} - 5^{t} - 5^{t} = 7^{th} = F$ .
$4^{th} - 3^{d} = 2^{d} ♭ = a$
$8^{th} - 3^{d} = 6^{t} ♭ = e$
$4^{th} + 3^{d} = 6^{t} = E$
$5 - 3^{d} = 3^{d} ♭ = 8^{th} - 6^{t} = b$
$3^{d} + 5^{t} = 7^{th} ♯ = f$
{illeg} $7^{th} ♯ - 4^{th} = 2^{d} + 3^{d} = 5^{t} ♭ = d = 3^{d} + 5^{t} - 4^{th} = - 2^{d} ♭ + 5^{t}$ .

By y^e helpe of concordant notes all y^e notes in y^e Gam ut may bee thus tuned viz:

First tune y^e 8^ths, $G, G^{2}, G^{3}, G^{4}$ &c.

Seacondly tune 5^ts to y^m both above y^m $D, D^{2}, D^{3}, D^{4} .$ & below them ${}^{2}C, C, C^{2}, C^{3}$ .

Thirdly tune 3^ds to y^m both above y^m $B, B^{2}, B^{3}, B^{4},$ & below them ${}^{2}E ♭, E ♭, E^{2} ♭, E^{3} ♭$ .

Fourthly from each $B, B^{2}, B^{3}, B^{4}$ rise a fift for $F ♯, F^{2} ♯, F^{3} ♯, F^{4} ♯$ & fall a 5^t for ${}^{2}E, E, E^{2}, E^{3}$ .

Fiftly from ${}^{2}E ♭, E ♭, E^{2} ♭, E^{3} ♭$ rise a fift for $B ♭$ {illeg} $B^{2} ♭, B^{3} ♭, B^{4} ♭ .$ & fall a fift for ${}^{2}A ♭, A ♭, A^{2} ♭, A^{3} ♭$ .

Sixtly from $D,$ ris| $D^{2}$ | $, D^{3}, D^{4} .$ rise a fift for $A^{2}, A^{3}, A^{4}$ $A^{5} .$ & from ${}^{2}C, C, C^{2}, C^{3}$ fall a fift for ${}^{3}F, {}^{2}F, F, F^{2}$ .

Seaventhly from each $F ♯, F^{2} ♯, F^{3} ♯, F^{4} ♯ .$ rise a fift for $D^{2} ♭, D^{3} ♭, D^{4} ♭, D^{5} ♭$ . The rest as $A, D ♭$ are supplyed by 8^ths viz to $A^{2}, D^{2} ♭$ &c.

<105v>

November 20. 1665.

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{1}{2} & 360 \end{matrix} & 2,55630247 & 2,5563025 & 360,00000,0 & 12,00000 . & G \\ \begin{matrix} \frac{8}{15} & 384 \end{matrix} & 2,58433118 & 2,5813883 & 381,40667,8 . & 10,88268,7 & f \\ \begin{matrix} \frac{9}{16} & 405 \end{matrix} & 2,60745497 & 2,6064742 & 404,08640,6 . & 9,96090,0 & F \\ \begin{matrix} \frac{3}{5} & 432 \end{matrix} & 2,63548369 & 2,6315600 & 428,11458,1 . & 9,84358,7 . & E \\ \begin{matrix} \frac{5}{8} & 450 \end{matrix} & 2,65321247 & 2,6566458 & 453,57157,8 . & 8,13686,3 . & e \\ \begin{matrix} \frac{2}{3} & 480 \end{matrix} & 2,68124123 & 2,6817317 & 480,54236,7 . & 7,01955,0 & D \\ \begin{matrix} \frac{32}{45} & 512 \end{matrix} & 2,70926992 & 2,7068175 & 509,11688,24543 . & 5,90223,7 . & d \\ \begin{matrix} \frac{3}{4} & 540 \end{matrix} & 2,73239371 & 2,7319033 . & 539,39055,9 . & 4,98045,0 . & C \\ \begin{matrix} \frac{4}{5} & 576 \end{matrix} & 2,76042244 & 2,7569892 & 571,46447,4 . & 3,86313,7 . & B \\ \begin{matrix} \frac{5}{6} & 600 \end{matrix} & 2,77815121 & 2,7820750 & 605,44546,7 . & 3,15641,3 . & b \\ \begin{matrix} \frac{8}{9} & 640 \end{matrix} & 2,80617993 & 2,8071608 & 641,44697,3 . & 2,03910,0 . & A \\ \begin{matrix} \frac{15}{16} & 675 \end{matrix} & 2,82930373 & 2,8322467 & 679,589514,9 . & 1,11731,3 . & a \\ \begin{matrix} 1 & 720 \end{matrix} & 2,857332447 . & 2,8573325 & 720,00000,0 . & 0,00000,0 . & G \end{matrix} \\ \begin{matrix} \begin{matrix} How y^{e} string 1 \\ or 720 is to bee di= \\ =vided y^{t} it may sound \\ all y^{e} musicall notes \\ & halfe notes in an \\ eight \end{matrix} & \begin{matrix} The proportion w^{ch} \\ those musicall \\ notes & \frac{1}{2} notes beare \\ y^{e} one to y^{e} other (viz \\ y^{e} logarithmes of y^{e} \\ string sounding them) \end{matrix} & \begin{matrix} Twelve exact or \\ equidistant \frac{1}{2} notes \\ (or y^{e} logarithmes of a \\ cord divided into 12 geome \\ tricall partes) y^{e} dist \bar{a} ce \\ of each \frac{1}{2} note being \\ 0,025085833333 &c. \\ A just note being \\ 0,050171666666 &c. \end{matrix} & \begin{matrix} A string (720) divided \\ into 12 (geometrically \\ progressionall) parts, y^{t} \\ it may sound y^{e} 12 \\ exact \frac{1}{2} notes in an \\ eight \end{matrix} & \begin{matrix} The proportion of all \\ y^{e} 12 musicall \frac{1}{2} notes \\ in a eight; An exact \\ halfe note being a \\ unite. \end{matrix} \end{matrix} \end{matrix}$

<106r>

$a^{4}$ {illeg} $x x y y$ . $\begin{matrix} a^{4} & + & s s x^{2} & + & v x^{3} & - & x^{4} & = & 0 \\ - & v v \\ 2 & 0 & - 1 & - 2 \end{matrix}$ $\frac{x^{4} - a^{4}}{x^{3}} = v$ $x - v = \frac{a^{4}}{x^{3}}$ \ 2,635483 74 |69 | / $a^{4} = x y^{3}$ . $ad = x$ . $\begin{matrix} - & a^{8} & + & y^{6} s s & - & y^{8} & + & 2 v y^{7} & - & v v y^{6} & = & 0 \\ + 6 & 0 & - 2 & - 1 & 0 \end{matrix}$ $\frac{2 y^{8} - 6 a^{8}}{2 y^{7}} = v$ . $v - y = \frac{3 a^{8}}{2 y^{7}}$ . {illeg} $\frac{a^{4}}{y^{3}} ∷ \frac{3 a^{8}}{2 y^{7}} ∷ p z$ $\frac{z a^{4}}{y^{3}} = \frac{3 a^{8} p}{3 y^{7}}$ . \{illeg}/ $z y^{4} = \frac{3 a^{4} p}{2}$

$0,921787 = A - a = d - C = f - F$ . $2,039100 = A = b - A = d - B = D - C = f - E = b - F$ $0,706724 = B - b = E - e$ . $F - e = 1,824037 = B - A = C - b = E - D$ . $e - d = 2,234626$ / $aa - f = 2,234626$ \ $2,745824 = B - a = d - b =$ F $f - e$ . $aa - E =$ \ $bb - f =$ / $= 4,273726 = e - B$ . $F - d = 4,058663$ $2941350 = C - A = E - d = F - D$ . $4,078200 = A^{2} - F$ $3,156413 = b = D - B = e - C = G - E = aa - F = AA - f$ . $bb - e = 6312826$ $3863137 = B = C - a = d - A = D - b = E - C = f - D = G - e$ $4980450 = C = D - A = e - b = E - B = F - C = f - d = G - D = aa - e = B^{2} - f$ $4784924 = d - a$ $5,195513 = A^{2} - E = bb - F$ $5,902237 = d = D - a = f - C = A^{2} - e = B^{2} - F$ = . $7,9218 = F - A$ $6,097763 = e - A = F - B = G - d = aa - D = C^{2} - f$ $7941337 = dd - F$ $5,687174 = E - b$ $6,804487 = E - A = F - b$ $7,019550 = D = e - a = f - B = G - C = A^{2} - D = bb - e = B^{2} - E = C^{2} - F = dd - f$ $5,215076 = aa - d$ . $7,726274 = E - a = f - b = B^{2} - e$ $8,136863 = e = G - B = a^{2} - C = A^{2} - d = bb - D = C^{2} - E = D^{2} -$ Ff $9,254176 = aa - B = bb - d = ee - f$ . $D^{2} - F = 9,058650 A^{2} - d = dd - E$ . $9,765374 = f - a = dd - e$ $10,175963 = A^{2} - B = bb - C = D^{2} - E = ee - F$ $11,078213 = aa - A = C^{2} - d = F^{2} - f$ $11,293276 = B^{2} - C = ee - E$ perhaps $d = E - b$ is better yⁿ $d = 3^{d} + 5^{t} - 4^{th}$ . $\begin{matrix} \begin{matrix} 0 & 1,1 & 2 & 3,1 & 3,9 & 5 & 5,9 & 7 & 8,1 & 8,9 & 10 & 109 & 12 \\ G & a & A & b & B & C & d & D & e & E & F & f & G \\ 0 & . & 1,2 & . & 2 & . & 3,2 & . & 3,8 & . & 5 & . & 5,8 & . & 7 & . & 8,2 & . & 8,8 & . & 10 & . & 108 & . & 12 \\ 0 & . & 6 & . & 10 & . & 16 & . & 19 & . & 25 & . & 29 & . & 35 & . & 41 & . & 44 & . & 50 & . & 54 & . & 60 & . \end{matrix} \end{matrix}$

<106v>

$\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{array}{l} \frac{64}{45} = 14222 &c \end{array} & \begin{array}{l} 2,00000 & g & 2 = 2,0000 . \\ 1,88774 & \frac{15}{8} = 1,8750 . \\ 1,7818 & a & \frac{16}{9} = 1,777777 &c \\ 1,6818 & \frac{5}{3} = 1,66666 &c \\ 1,5874 & b & \frac{8}{5} = 1,6000 \\ 1,4983 & c & \frac{3}{2} = 1,5000 \\ 1,4142136 & \frac{10}{7} = 1,428571428571 &c \\ 1,3349 & d & \frac{4}{3} = 1,33333 &c \\ 1,2599 & \frac{5}{4} = 1,2500 . \\ 1,18920 & e & \frac{6}{5} = 1,2000 . \\ 1,12245 & f & \frac{9}{8} = 1,1250 . \\ 1,05946 & \frac{16}{15} = 1,066666 &c \\ 1,00000 & g & 1 = 1,0000 . \\ \begin{matrix} A string divided \\ in a geometricall progres \\ =sion \end{matrix} & \begin{matrix} A string divided by \\ a musicall progre \bar{ssi} \end{matrix} \end{array} \end{matrix} \end{matrix} & \begin{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} By this table it may appeare that a \\ \begin{array}{l} \begin{array}{l} Second minor \\ Second \underline{} \\ Third minor \end{array}} is higher by y^{e} {\begin{matrix} 17^{th} + \\ 51^{th} + \\ 13^{th} - \end{matrix} \\ \begin{array}{l} Third \underline{} \\ Fourth \underline{} \end{array}} is lower by y^{e} {\begin{matrix} 15^{th} - \\ 102^{th} \end{matrix} \\ Fift {min}^{e} is \begin{array}{l} lower by y^{e} 20^{th} \frac{1}{2} \\ or \\ higher by y^{e} 20^{th} \frac{1}{2} \end{array} \\ \begin{matrix} Fift \underline{} \\ Sixt minor \end{matrix}} is higher by y^{e} {\begin{matrix} 102^{th} \\ 15^{th} \end{matrix} \\ \begin{array}{l} Sixt \\ Seaventh \\ Eight minor \end{array}} is lower by y^{e} {\begin{matrix} 13^{th} \\ 51^{th} \\ 17^{th} \end{matrix} \end{array}} \begin{matrix} parte of a note then it would \\ bee were y^{e} musicall chord \\ divided in geometricall \\ progression. \end{matrix} {\begin{cases} \frac{1}{2} notes, notes \\ = 1,058522 . \\ = 2,02 . \\ = 3,07 \\ = 3,93 \\ 4,99 . \\ 5,95 . \\ 6,05 . \end{cases} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} 1 & , & \begin{matrix} \frac{15}{16} & , \\ \frac{24}{25} & , \\ \frac{20}{21} & , \end{matrix} & \begin{matrix} \frac{8}{9} & , \\ \frac{7}{8} & , \\ \frac{9}{10} \end{matrix} & \frac{5}{6} & , & \frac{4}{5} & , & \frac{3}{4} & , & \begin{matrix} \frac{18}{25} \\ \frac{5}{7} \\ \frac{32}{45} & , \\ \frac{45}{64} & , \\ \frac{7}{10} \\ \frac{25}{36} \end{matrix} & \frac{2}{3} & , & \frac{5}{8} & , & \frac{3}{5} & , & \begin{matrix} \frac{9}{16} & , \\ \frac{4}{7} & , \\ \frac{5}{9} & , \end{matrix} & \begin{matrix} \frac{8}{15} & , \\ \frac{25}{48} \\ \frac{21}{40} \end{matrix} & \frac{1}{2} \end{matrix} \\ \begin{matrix} \frac{1}{2} & , & \frac{15}{32} & , & \frac{4}{9} & , & \frac{5}{12} & , & \frac{2}{5} & , & \frac{3}{8} & , & \frac{16}{45} & , & \frac{1}{3} & , & \frac{5}{16} & , & \frac{3}{10} & , & \frac{9}{32} & , & \frac{4}{15} & , & \frac{1}{4} \\ \frac{1}{4} & , & \frac{15}{64} & , & \frac{2}{9} & , & \frac{5}{24} & , & \frac{1}{5} & , & \frac{3}{16} & , & \frac{8}{45} & , & \frac{1}{6} & , & \frac{5}{32} & , & \frac{3}{20} & , & \frac{9}{64} & , & \frac{2}{15} & , & \frac{1}{8} \\ \frac{1}{7} \end{matrix} \end{matrix} & \begin{array}{c} A 3^{d} {min}^{e} is higher y^{¯} \\ just, by y^{e} {\frac{5}{64}}_{+}^{th} p^{te} of \\ a note \\ A Fift minor is \\ h y^{n} just by \\ y^{e} {\frac{2}{41}}^{th} p^{te} of a \\ note \end{array} \end{matrix} \end{matrix} \end{matrix}$

<107r>

$a^{5} = x x y^{3}$

By this table may bee knowne y^e distance of any t{illeg}|wo| notes whither a {trew} second of y^e lesse, second, third f y^e lesse, a third fourth &c: As to know y^e distance twixt A re & B sol re I follow y^e pricked stroke from A to D or from D to A where I find it crossed by a black crooked line & {illeg}o \against it/, a Fourth written, therefore I conclude A re & D la sol distant a true fourth.

<107v>

And Thus to find y^e distance of B mi & D la sol re I follow y^e prick line from y^e top B to y^e right hand side thence to y^e bottom B thence towards y^e left hand side untill I come {over} D. Or (w^ch is y^e same) I follow y^e prick{t}{{illeg}} line from y^e top D to {illeg}|th|e left hand side thence to y^e bottom D, thence toward y^e right hand side untill I come just over B, where{illeg} I find y^e pricked line to be crossed by a Symbol (roundAngle) in text stroke & against it to bee written on y^e upper line a tenth , on the lower a ninth therefore tis a tenth minor exactly. But if it

<108r>

$a^{3} = x x y$ . $x = v + \sqrt{s s - y y}$ . $a^{3}$ {illeg} $v y y = y y \sqrt{s s - y y}$ $\begin{matrix} a^{6} & - & 2 a^{3} v y y & + & v v y^{4} & + & y^{6} & = & 0 \\ - & s s \\ - 4 & - 2 & 0 & 2 \end{matrix}$ {illeg} $\frac{- 2 y^{6} + 4 a^{6}}{2 a^{3} y^{2}}$ . $\frac{a^{3} -}{y^{2}}$ $x - v = \frac{2 y^{6} - 2 a^{6}}{2 a^{3} y^{2}}$ . $2 a^{3} y^{3} ∶ 2 y^{6} - 2 a^{6} ∶ p ∶ z$ $2 a^{3} y^{3} z = 2 y^{6} p - 2 a^{6} p$ {illeg} $p y^{6} - a^{3} y^{3} - a^{6} p$

$\begin{matrix} 0 & . & 5 & . & 9 & . & 14 & . & 16 & . & 21 & . & 25 & . & 30 & . & 35 & . & 37 & . & 42 & . & 46 & . & 51 & . \\ 0 & . & 5 & . & 9 & . & 14 & . & 17 & . & 22 & . & 26 & . & 31 & . & 36 & . & 39 & . & 44 & . & 48 & . & 53 & . \\ g & . & a & . & A & . & b & . & B & . & C & . & d & . & D & . & e & . & E & . & F & . & f & . & G & . \\ 0 & . & 4 & . & 7 & . & 11 & . & 13 & . & 17 & . & 20 & . & 24 & . & 28 & . & 30 & . & 34 & . & 37 & . & 41 & . \\ 0 & . & 3 & . & 5 & . & 8 & . & 9 & . & 12 & . & 14 & . & 17 & . & 20 & . & 21 & . & 24 & . & 26 & . & 29 & . \\ 0 & . & 4 & . & 6 & . & 10 & . & 11 & . & 15 & . & 17 & . & 21 & . & 25 & . & 26 & . & 30 & . & 32 & . & 36 & . \\ 0 & . & 11 & . & 20 & . & 31 & . & 39 & . & 50 & . & 59 & . & 70 & . & 81 & . & 89 & . & 100 & . & 109 & . & 120 & . \\ 0 & . & 4 & . & 2 & . & 6 & . & 1 & . & 5 & . & 3 & . & 7 & . & 11 & . & 6 & . & 10 & . & 8 & . & 12 \\ 0 & . & 2 & . & 5 & . & 7 & . & 8 & . & 10 & . & 13 & . & 15 & . & 17 & . & 18 & . & 20 & . & 23 & . & 25 & . \\ 0 & . & 1 & . & 4 & . & 5 & . & 7 & . & 8 & . & 11 & . & 12 & . & 13 & . & 15 & . & 16 & . & 19 & . & 20 \end{matrix}$

<108v>

$\begin{matrix} 0 & . & a & . & . & b & . & b + a & . \\ 0 & . & a & . & a + b & . & a + b + c & . & 2 a + b + c & . & 2 a + 2 b + c & . \end{matrix}$

$\begin{array}{l} \begin{matrix} The proportion of their & \begin{matrix} How they may \\ bee otherwise \\ distinguished by \\ figures \end{matrix} \end{matrix} \\ \begin{matrix} The notes & \begin{matrix} sounds \end{matrix} & or thus & or thus & thus & thus & \begin{matrix} or \\ thus \end{matrix} & or thus & or thus \\ G & 53 & 612 & 59 & 100 & 100 & 36 & 29 & 12 & 12 . \\ f & 48 & 555 & 90 & 89 & 32 & 26 & 10 \frac{2}{3} & 10,9 \\ F & 44 & 508 & 82 & 30 & 24 & 10 & 10 \\ E & 39 & 451 & 72 & 26 & 21 & 8 \frac{2}{3} & 8,9 \\ e & 36 & 415 & 40 & 69 & 21 & 17 & 7 & 7 \\ D & 31 & 358 & 59 & 25 & 20 & 8 \frac{1}{3} & 8,1 \\ d & 26 & 301 & 49 & 48 & 17 & 14 & 5 \frac{2}{3} & 5,9 \\ C & 22 & 254 & 41 & 41 & 15 & 12 & 5 & 5 \\ B & 17 & 197 & 19 & 31 & 30 & 11 & 9 & 3 \frac{2}{3} & 3,9 \\ b & 14 & 161 & 28 & 29 & 10 & 8 & 3 \frac{1}{3} & 3,1 \\ A & 9 & 104 & 18 & 18 & 6 & 5 & 2 & 2 \\ a & 5 & 57 & 10 & 11 & 4 & 3 & 1 \frac{1}{3} & 1,1 \\ G & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \end{array}$

<109r>

$ao = a = ad$ . $dc =$ {illeg}p . $ai = x$ . $oi = y$ . $a a - x x = y^{2}$ $in = z$ . $x x ∶ a a - x x ∷ p p ∶ z z$ . $z z x x = a a p p - p^{2} x^{2}$ . $id = x$ . $oi = 2 a x - x x$ . $a a - 2 a x +$ $a a - 2 a x + x^{2} ∶ 2 a x - x^{2} ∷ p p ∶ z z$ $\begin{matrix} z z x^{2} & - & 2 a z z x & + & a a z z & = & 0 \\ p p & - & 2 a p p \end{matrix}$ .

The 10 \3/ meanes are best there being an imperfect 5^t in y^e {illeg}|ou|tward extreame & a trit{illeg}|o|nus in y^e inmost. Modi
$1.6.4.3.2.5 . / 10.7.2.5 .$ $. 6.8.4 .$ $6.11.8 .$ $3.10.2 .$ $3.9.10.12.7 .$ $1.6.11.8.4.3 : 9 : 10 : 12.7.2.5 .$

<109v>

<110r>

In y^e Para|Hyper|bola dm . suppose $ak = a = kh$ $ao = x$ . $od = y$ . $dc =$ {illeg} a secant. $ca =$ {illeg}|v | $og = z$ . $x y = a a$ . {illeg}| y | $y = s s - x x + 2 v x - v v$
$\begin{matrix} \begin{matrix} - & a^{4} & + & x x s s & - & x^{4} & + & 2 v x^{3} & = & 0 \\ - & v v x x \\ + & 2 & 0 & - & 2 & - & 1 \end{matrix} & \begin{array}{l} double route equall \\ \frac{x^{4} - a^{4}}{x^{3}} = v \end{array} \end{matrix}$ . $x - v = \frac{a^{4}}{x^{3}} = oc$ . $od ∶ oc ∷ kh ∶ og$ . $\frac{a a}{x} ∶ \frac{a^{4}}{x^{3}} ∷ a ∶ z$ . $\frac{a a z}{x} = \frac{a^{5}}{x^{3}}$ . $z x x = a^{3}$ . w^ch equation continues y^e nature of y^e crooked line gh. Now supposeng {sic} y^e line og always moves over y^e same superficies in y^e same {illeg}|t|ime, it will increase in {illeg} motion from kh in y^e same proportion y^t it decreaseth in lenght \&/ y^e line ne will move uniformely through from |(| $mq$ |)|, soe y^t y^e space $mqen = gokh$ . suppose $ok = a$ . $ao = 2 a$ . $od = \frac{a}{2} = nm$ . & $mqen = \frac{1}{2} a a = ogkh$ .

Modi $\begin{matrix} \begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\ 5 & 5 & 5 & 5 & 5 & 5 & 5 \\ 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ 7 & 7 & 7 & 7 & 7 & 7 & 7 \\ 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8 \\ 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 \\ 10 & 10 & 10 & 10 & 10 & 10 & 10 \\ 11 & 11 & 11 & 11 & 11 & 11 & 11 & 11 \\ 12 & 12 & 12 & 12 & 12 & 12 & 12 \end{matrix} \end{matrix}$ $1.6 .$

<110v>

In y^e order of y^e musicall {illeg}|t|ones y^e 2 halfe notes may not {illeg} \be/ together 1^st because every note would yⁿ bee distant 3 tones from some other w^ch is most ungratefull 2^dly whole notes ought to bee interposed to moderate their harsh{illeg}|nes|se. 3^dly since {illeg}|th|eir {sic} must bee a 5^t to y^e gr{illeg}|ou|nd: these $\frac{1}{2}$ notes must bee next either next y^e ground or its 5^t w^ch would make y^m harsh & y^t they \wee/ could not gradually passe to or from them.

Neither ought they to be distant but one tone for y^e 2^d reason {afforesd} & because they will bee more consonant by y^e absense of more 2|3| tons {sic} &c if they be distant 2 tones yet perhaps they may not bee wholly uselesse. S{illeg}|ee| {illeg} y^e last modes.

A catalogue of y^e 12 Musicall modes in theire order of gratefulnesse. $\begin{matrix} \begin{array}{l} \begin{matrix} 1 & G & . & a & . & b & c & . & d & . & e & f & . & g \\ 3 & c & . & d & . & e & f & . & g & . & a & . & b & c \\ 2 & d & . & e & f & . & g & . & a & . & b & c & . & d \\ 4 & a & . & b & c & . & d & . & e & f & . & g & . & a \\ 5 & e & f & . & g & . & a & . & b & c & . & d & . & e \\ 6 & f & . & g & . & a & . & b & c & . & d & . & e & f \\ b & c & . & d & . & e & f & . & g & . & a & . & b \\ . & b & c & . & d & . & e & f & . & g & . & a & . \\ . & a & . & b & c & . & d & . & e & f & . & g & . \\ . & d & . & e & f & . & g & . & a & . & b & c & . \\ . & g & . & a & . & b & c & . & d & . & e & f & . \\ . & e & f & . & g & . & a & . & b & c & . & d & . \end{matrix} \\ \begin{matrix} 0 & . & 1 & . & 2 & . & 3 & . & 4 & . & 5 & . & 6 & . & 7 & . & 8 & . & 9 & . & 10 & . & 11 & . & 12 \end{matrix} \\ \begin{matrix} 2 & G & . & . & a & . & b & c & . & d & e & . & f & . & g \\ 1 & c & . & d & e & . & f & . & g & . & a & . & b & c \\ 3 & d & e & . & f & . & g & . & a & . & b & c & . & d \end{matrix} \end{array} & \begin{array}{l} \begin{matrix} 0 & . & 9 & . & 17 & . & 22 & . & 31 & . & 39 & . & 44 & . & 53 & . \\ 0 & . & 9 & . & 17 & . & 22 & . & 31 & . & 40 & . & 48 & . & 53 & . \\ 0 & . & 9 & . & 14 & . & 22 & . & 31 & . & 40 & . & 45 & . & 53 & . \\ 0 & . & 9 & . & 14 & . & 22 & . & 31 & . & 36 & . & 45 & . & 53 \\ 0 & . & 5 & . & 14 & . & 22 & . & 31 & . & 36 & . & 45 & . & 53 & . \\ 0 & 9 & 17 & . & 26 & . & 31 & . & 40 & . & 48 & . & 53 \end{matrix} \\ \begin{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} G & . & a & . & b & . & cde & . & f & . & g \\ de & . & f & . & g & . & a & . & b & . & cd \end{matrix} \\ \begin{matrix} f & . & g & . & a & . & bc & . & de & . & f & . \end{matrix} \end{matrix} \end{array} \end{matrix}$

<111r>

suppose y^e line {illeg}|l|ast found to be md . $mk = kh = a = ka$ $ao = x$ . $od = y$ . $dc = s$ . $ca = v$ . $y x x = a^{3}$ . $y y = s s - v v + 2 v x - x x$ . $\begin{matrix} s s x^{4} & + & 2 v x^{5} & - & x^{6} & - & a^{6} & = & 0 \\ - & v v \\ 0 & - 1 & - 2 & + 4 \end{matrix}$ $v = \frac{x^{6} - 2 a^{6}}{x^{5}}$ . $x - v = \frac{2 a^{6}}{x^{5}}$ . to find at w^t point $do = oc$ : $\frac{a^{3}}{x x} = \frac{2 a^{6}}{x^{5}}$ . $x^{3} = 2 a^{3}$ . {illeg} $x = a \sqrt{c: 2} = af$ . $mq = fh = a$ . $og = z$ . | $od ∶ oc ∷ fh ∶ og$ . | $\frac{a^{3}}{x x} ∶ \frac{2 a^{6}}{x^{5}} ∷ a ∶ z$ . $a^{3} z x^{5} = 2 a^{7} x x$ . $z x^{3} =$ {illeg} \ 2 / $a^{4}$ . w^ch shews y^e nature of y^e line $(gh)$ . & $mneq = gofh$ or $nbpe = gokl$ . suppose $ko = ka = a$ . $oa = 2 a =$ {illeg} x $od = \frac{a^{3}}{x x} = \frac{a^{3}}{4 a a} = \frac{a}{4}$ . $bn = \frac{3 a}{4}$ . $bpen = \frac{3 a a}{4} = gokl$ .

Suppose {illeg}| $k$ | $l =$ $ka = a = kb$ . $af = x -$ {illeg} $\sqrt{}$ {illeg} $x - v = \frac{2 a^{6}}{x 5} = \frac{2 a^{6}}{a^{5} \sqrt{c: 3}} = \frac{2 a}{\sqrt{c: 3}}$ $\frac{a}{\sqrt{c: 4}} ∶$ $\frac{a}{\sqrt{c: 4}} ∶ \frac{^{3}}{a}$ Suppose $kl = ka = kb = a$ $ak = x = a$ . $bk = y = a$ . $bs = s$ . $as = v$ . $y y = s s - v v - 2 v x - x x$ . $y y x x x x = a^{6}$ . $\begin{matrix} s s x^{4} & - & 2 v x^{5} & - & x^{6} & - & a^{6} \\ - & v v \\ 0 & - & 1 & - & 2 & + & 4 \end{matrix}$ . $\frac{- 2 x^{6} + 4 a^{6}}{2 x^{5}} = v$ $v + x = \frac{2 a^{6}}{x^{5}} = ks = 2 a$ . $ks ∶ bk ∷ kl ∶ fh$ . $2 a ∶ a ∷ a ∶ \frac{a}{2}$ . $fh = \frac{a}{2} = ne = mq = rp$ . $mn = \frac{3 a}{4} = qe$ $m$ {illeg}= $\frac{3 a a}{8} = mneq = lkog$ . $ao = 2$ {illeg}|a| $= x$ . $\frac{a^{3}}{4 a a} = do$ $do = \frac{a}{4}$ . $oc = \frac{2 a^{6}}{32 a^{5}} = \frac{a}{16}$ . $\frac{a}{4} ∶ \frac{a}{16}$ $∷ \frac{a}{2} ∶ og = \frac{a}{8}$ $\frac{2 a^{6}}{x^{5}} ∷ \frac{a^{3}}{x x} ∶ z ∷ \frac{a}{2}$ . $a^{4} = z x^{3}$ .

<111v>

$a, b, c = \frac{1}{2}$ tone max; m{illeg}|ed|î: minimꝰ. $a + b =$ {illeg} $a + c =$ {illeg} $a + a = d$ {illeg}, $a + b = e$ , $a + c = f ~$ tone {maj me:mi.} $a + a + b = a + e = 3^{d} ♭$
$a + a + b + c = e + f = 3^{d} ♯$ .
$3 a + b + c = a + e + f = f + 3^{d} ♭ = 4^{th}$

$r . s . t . r . s . t . r : r . s t$ $2 r + s + t = 5^{t}$ . $\begin{array}{l} 3^{d} maj = r + s . r + t = 3^{d} mi \\ 6^{t} min = 2 r + s + 2 t \\ 6^{th} maj = 2 r + 2 s + t \\ 4^{th} = r + s + t \end{array}$ $\begin{matrix} s & . & r & . & t & . & r & . & s & . & t & . & r & , & s & . & r & . & t & . & r & . & s & . & t & . & r & . \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 5^{ts} \\ 1 & 2 & 3 & 4 & 5 \end{matrix}$ hath 8 5^ts

$\begin{matrix} \begin{matrix} \begin{matrix} r & . & s & . & t & . & r & . & s & . & t & . & r & , & r s t r s t r . \\ 1 & 2 & 3 & 4 & 5 & fifts \\ 1 & 2 & third {maj}^{s} & 2 \\ 1 & 2 & third min. \\ 1 & 2 & . & 6^{ts} maj \\ 1 & 1 & 2 & sixt minors \\ 1 & 2 & 3 & 4 & 5 & forths \end{matrix} \\ \begin{matrix} s & r & t & r & s & t & r & , & s . r . t . r s t \\ 1 & 2 & 3 & 4 & 5 & fourths & 3^{d} mode \\ 1 & 2 & 3 & 4 & 5 & fifts \\ 1 & 2 & 3 & 3^{d} maj. & 1 \\ 1 & 2 & 3 & third min \\ 1 & 2 & 3 & Sixt maj. \\ 1 & 2 & 3 & Sixt min. \end{matrix} \\ \begin{matrix} r & s & t & r & r & t & s & , & r s t r r t s . \\ 1 & 2 & 3 & 4 & fourths \\ 1 & 2 & third maj. & 4 \\ 1 & 2 & third minor \end{matrix} \\ \begin{matrix} s & r & t & r & r & t & s & , & s r t . \\ 1 & 2 & fourths & 5 \\ 1 & 3^{d} maj \\ 1 & 2 & 3 & 3^{d} min. \end{matrix} \\ \begin{matrix} r & . & r & . & t & . & s & s & t & r & , & r r t s \\ 1 & 2 & fourths & 6 \\ 0 & 3^{d} maj \\ 1 & 2 & 3^{d} min \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} r & r & t & s & r & t & s & , & r r t s \\ 1 & 2 & 3 & 4 & 5 & fourths \\ 1 & 2 & third maj. & 3 \\ 1 & 2 & third min. \end{matrix} \end{matrix} \\ \begin{matrix} 6^{t} . mode \\ r & s & r & t & s & r & t & , & r s \\ 1 & 2 & 3 & 4 & 5 & . & 4^{ths} \\ 1 & 2 & 3 & 3^{d} maj \\ 1 & . & 2 & . & 3 & . & 3^{d} min \end{matrix} \\ \begin{matrix} s & r & r & t & r & s & t & , & s \end{matrix} \\ \begin{matrix} ac & , & ab & , & a & , & ab & , & ac & , & a & , & ab & : & ac & , & ab \\ ac & , & ba & , & aaa & ba \\ ab & . & ac & . & aab & : & a & c & a & b & a & . & a \\ . & . & . & b & . & c & . & b & . & c & . & b \\ a & b & a & a & c & a & b & a & a & c & a & b & , & a \\ a & b & a & c & a & b & a & a & c & a & b & a \end{matrix} \\ \begin{matrix} 9 \times 4^{ths} & . & 7 \times 3^{ds} ♯ & . & 6 3^{ds} ♭ & . \end{matrix} \end{matrix} \end{matrix}$

<112r>

Suppose againe y^e last line whose nature is comprisd {sic} in this equation {illeg}|y | $x^{3} =$ {illeg} $a^{4}$ . $ak = bk = lk = a$ $ao = x$ . $ac = v$ . $do = y$ . $dc = s$ . $og = z$ . $\begin{matrix} s s x^{6} & + & 2 v x^{7} & - & x^{8} & - & a^{8} & = & 0 \\ - & v v \\ 0 & - 1 & - 2 & + 6 \end{matrix}$ $v = \frac{2 x^{8} - 6 a^{8}}{2 x^{7}}$ $x - v = \frac{3 a^{8}}{x^{7}}$ . to find where $do = dc$
$\frac{a^{4}}{x^{3}} = \frac{3 a^{8}}{x^{7}}$ {illeg} $a^{4} x^{4} =$ 6|3| $a^{8}$ . $x^{4} =$ 6|3| $a^{4}$ . \{illeg}/ $x = \sqrt{qq: 3}$ $af = a \sqrt{qq: 3 :}$ $bk = y = a$ . \ $bs = s$ $as = v$ / $ak = x = a$ $x + v = \frac{3 a^{8}}{x^{7}} =$ $x + v =$ {illeg} $3 a$ . {illeg} $3 a ∶ a ∷ ki (= a) ∶ fh = \frac{a}{3} = mq = ne$ $oa = x =$ \ $2$ / a. $\frac{a^{4}}{x^{3}} = do = y = \frac{a^{4}}{8 a^{3}} = \frac{a}{8} = do$ . $mn = \frac{7 a}{8} = qe$ . $mqen = \frac{7 a a}{24} = lkog = \frac{7 a a}{24}$ .

$\begin{matrix} \begin{matrix} \begin{matrix} r s t r s t r; r s \\ 2 \times 3^{d} ♭ . 2 \times 3^{d} ♯ . \\ 5 \times 4^{th} . \\ no 4^{th} \\ 1^{st} mode 2^{d} mode. 3^{d} . \end{matrix} & \begin{matrix} s r t r s t r; s r \\ 3 \times 3^{d} ♭ . 3 \times 3^{d} ♯ . \\ 5 \times 4^{th} . \\ 6^{t}, 4^{th} & 5^{t} \bar{m} ode \\ 3^{d} \bar{m} ode. 1^{st} \bar{m} ode. no 2^{d} \end{matrix} & \begin{matrix} r s t s r t r, r s \\ 2 \times 3^{d} ♭ . 2 \times 3^{d} ♯ . \\ 3 \times 4^{th} . \end{matrix} & \begin{matrix} s r t s r t r, s r \\ 3 \times 3^{d} ♭ . 3 \times 3^{d} ♯ . \\ 5 \times 4^{th} . no 1^{st} \\ 5^{t} . 4^{th} \bar{m} ode. 3^{d} mode \\ 6^{t} \bar{m} ode. 2^{d} \bar{m} ode \end{matrix} \\ \begin{matrix} r r t s s t r; r r \\ 2 \times 3^{d} ♭ . 0 \times 3^{d} ♯ . \\ 2 \times 4^{th} . \end{matrix} & \begin{matrix} s s t r r t r; s s \\ 3 \times 3^{d} ♭ . 1 \times 3^{d} ♯ . \\ 2 \times 4^{th} . \end{matrix} & \begin{matrix} r s t r r t s; r s \\ 2 \times 3^{d} ♭ . 2 \times 3^{d} ♯ . \\ 2 \times 4^{th} . \\ 2^{d} mod. \end{matrix} & \begin{matrix} s r t r r t s; s r . \\ 3 \times 3^{d} ♭ . 1 \times 3^{d} ♯ . \\ 2 \times 4^{ths} . \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} r r t r s t s; r r . \\ 2 \times 3^{d} ♭ . 2 \times 3^{d} ♯ . \\ 3 \times 5^{t} . \end{matrix} & \begin{matrix} r r t s r t s, r r \\ 2 \times 3^{d} ♭ . 2 \times 3^{d} ♯ . \\ 5 \times 4^{ths} . \\ 2^{d} . \begin{matrix} noe 3^{d} . \end{matrix} \\ 4^{th} mode. 6^{t} mode \end{matrix} \end{matrix} \end{matrix}$

<112v>

$\begin{matrix} \begin{matrix} g & a & b & c & d & e & f & g \end{matrix} \\ \begin{matrix} e & d & e & f & g & a & b & c \\ f & g & a & b & c & d & e & f \\ g & a \end{matrix} \end{matrix}$
$\begin{matrix} \begin{matrix} 1 . & e & f & g & a & b & c & d & e & 5 \\ 2 . & a & b & c & d & e & f & g & a & 4 \\ 3 . & d & e & f & g & a & b & c & d & 2 \\ 4 . & g & a & b & c & d & e & f & g & 1 \\ 5 . & c & d & e & f & g & a & b & c & 3 \\ 6 . & f & g & a & b & c & d & e & f & 6 \end{matrix} \end{matrix}$

$r = ton. maj$ . $s = ton min$ . $t = semit maj$ . $v = semit min$ .

1^st. $gd = cg = da = 5^{t} = 2 r + s + t$ . g{illeg} $dg = gc = ad = r + s + t$ . $cd = ga = r$ . $ac = s + t$ . $ca = 3 r + s + t$ . $ab = s$ . $bc = t$ . $dc = 2 r + 2 s + 2 t$ . $de =$ ton s. $ef = t$ . per sup. et fg $\begin{matrix} \begin{matrix} 3 \\ 4 \\ 5 \end{matrix} & \begin{matrix} s & t & r & r & s & t & r & ♭ \\ r & s & t & r & s & t & r \\ r & s & t & r & r & s & t & ♯ \end{matrix} \end{matrix}$ Modus / Symbol (largeAsteriskEightBars) in text \ harum \vocum/ optimus respectu fundamenti.

2^d. $gd = da = ae = 5^{tæ} =$ \ $2$ / $r + s + t$ . $dg = ad = ea = r + s + t$ . $ga = da - dg =$ $de = ae - ad = r$ . $ge = 3 r + s + t = gd + de$ . $eg = s + t$ . $ef = t$ . $fg = s$ . And if $ab = s$ . $bc = t$ . Then $cd = r$ & y^e voyces in respect of their base /{ground}\ are \best/ $\begin{matrix} \begin{matrix} 2 \\ 3 \\ 4 \end{matrix} & \begin{matrix} s & t & r & r & t & s & r & . & ♭ \\ r & t & s & r & s & t & r & . \\ r & s & t & r & r & t & s & . & ♯ \end{matrix} \end{matrix}$ Symbol (largeAsteriskSixBars) in text

3^d. $fc = cg = gd = 2 r + s + t$ . $cf = gc = dg = r + s + t$ . $fg = cg - cf = cd = gd - gc = r$ . $df = cf - cd = s + t$ . $de = s$ . $ef = t$ . {illeg} & if $ga = s$ , yⁿ $ab = r$ . $bc = t$ whence $\begin{matrix} \begin{matrix} 4 . \\ 5 . \\ 6 . \end{matrix} & \begin{matrix} s & r & t & r & s & t & r & ♭ \\ r & s & t & r & s & r & t \\ r & s & r & t & r & s & t & ♯ \end{matrix} \end{matrix}$ Symbol (largeAsteriskEightBars) in text $\begin{matrix} Or if ga = r . y^{n} ab = s \\ & bc = t whence \end{matrix}$ $\begin{matrix} \begin{matrix} 4 \\ 5 \\ 6 \end{matrix} & \begin{matrix} r & s & t & r & s & t & r & . \\ r & s & t & r & r & s & t & . \\ r & r & s & t & r & s & t & . \end{matrix} \end{matrix}$ These differ in y^t this hath but 2 exact $2^{ds}$ & this but 2 exact /thirds\

If in y^e 1^st case $de = r$ . yⁿ $\begin{matrix} \begin{matrix} 3 . \\ 4 . \\ 5 . \end{matrix} & \begin{matrix} r & t & s & r & s & t & r . & ♭ \\ r & s & t & r & r & s & t . \\ r & r & s & t & r & s & t . & ♯ \end{matrix} \end{matrix}$ / Symbol (largeAsteriskSixBars) in text \ If in y^e 2^d case $ab = r$ then $\begin{matrix} \begin{matrix} 2 . \\ 3 . \\ 4 . \end{matrix} & \begin{matrix} r & t & s & r & t & s & r . & ♭ \\ r & t & s & r & r & t & s . \\ r & r & t & s & r & t & s . & ♯ \end{matrix} \end{matrix}$ Symbol (largeAsteriskEightBars) in text

$\begin{matrix} \begin{matrix} \begin{matrix} 2 & s & t & r & r & t & s & r . \\ 3 & r & t & s & r & s & t & r . \\ 4 & r & s & t & r & r & t & s . \\ 5 & r & r & t & s & r & s & t . \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 1 . & t & s & r & r & t & s & r . \\ 2 . & r & t & s & r & t & s & r . \\ 3 . & r & t & s & r & r & t & s . \\ 4 . & r & r & t & s & r & t & s . \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} 3 . & s & t & r & r & s & t & r . \\ 4 . & r & s & t & r & s & t & r . \\ 5 . & r & s & t & r & r & s & t . \\ 6 . & r & r & s & t & r & s & t . \end{matrix} \end{matrix} \end{matrix}$

<113r>

Likewise supposing y^e line $y x^{4} = a^{5}$ . $x - v = \frac{4 a^{10}}{x^{9}} = oc$ . $af =$ \ $a$ / $\sqrt{qc: 4}$ . $ka (= x) + v = 4 a$ $fh = \frac{a}{4}$ . $do = \frac{a}{16}$ . $mq = ne = \frac{a}{4}$ . $\frac{15 a}{16}$ $\frac{15 a a}{64}$ . &c whence supposeing x to be a line increasing in arithmeticall proportion from y^e quantity of y^e line $(a)$ untill it be as long as b. y^e superfices resulting out of $\frac{a^{3}}{x x} . \frac{a^{4}}{x^{3}}$ &c is found {illeg} as follows.
$\frac{a^{3}}{x x} = a a - \frac{a^{3}}{b}$ . $\frac{a^{6}}{x^{5}} = \frac{a a}{4} - \frac{a^{6}}{4 b^{4}}$
$\frac{a^{4}}{x^{3}} = \frac{a a}{2} - \frac{a^{4}}{2 b b}$ . $\frac{a^{7}}{x^{6}} = \frac{a a}{5} - \frac{a^{7}}{5 b^{5}}$ .
$\frac{a^{5}}{x^{4}} = \frac{a a}{3} - \frac{a^{5}}{3 b^{3}}$ . $\frac{a^{8}}{x^{7}} = \frac{a a}{6} - \frac{a^{8}}{6 b^{6}}$ . &c

$\begin{matrix} 1 & t & r & s & r & t & s & r \\ 2 & r & t & s & r & t & r & s \\ 3 & r & t & r & s & r & t & s \end{matrix}$

1 Of y^e Key or Ground sound. 2^dly, Of its $8^{ths}$ . 3^dly, of their divisions into $5^{ts}$ & $4^{ths}$ $6^{ts}$ & $3^{ds}$ , illustrated by y^e division of a corde. 4^thly, {illeg}|T|he order of y^e con{illeg}|c|ords in respect of gratefulnes deduced thence & from other considerations. 5^thy {sic} y^e degrees deduced thence & of the{illeg} proportion of y^e concords & degrees i.e. y^e logarrithmes of their strings. 6 Of y^e various ordering of y^e degrees \& distance of y^e halfe notes/ , y^e keys fift being onely stable 7 Of y^e moodes ariseing thence & their dignity; explained by one line, $o . p .$ $qr . s . tv . o . p . qr . s . tv . o . p .$ &c. 8^thly, How y^e tones major & minor are best ordered in every Moode. 9^thly of passing from one moode to another explained by 3 lines $\begin{matrix} g & . & a & . & b c & . & d & . & e f & . & g \\ c & . & d & . & e f & . & g & . & a & . & b c \\ f & . & g & . & a & . & b c & . & d & . & e f \end{matrix}$ 10 How y^e notes major and minor to be
ordered for y^t purpose.

<114v>

<115r>

in y^e Par|Hyp|a {sic}\r/b: $ed = q$ . $be = x$ . $ab = y$ . $\begin{matrix} r x & + & \frac{r}{q} x x & = & y y & = & s s & - & x x^{2} & + & 2 v x & - & v v \\ 1 & 2 & 0 & - 2 & - & 0 \end{matrix}$ $\frac{r}{2}$ {illeg}+ {illeg} $\frac{r}{q}$ {illeg}| x |{illeg}+ {illeg}x{illeg} $= v$ {illeg} $\frac{r}{2} + \frac{r}{q} x = v - x$ {illeg}|r | $x + \frac{r}{q} x x ∶ \frac{r r}{4} + \frac{r r x x}{q q} + \frac{r r x}{q} ∷ p ∶ z$ . $r x z + \frac{r}{q} x x z - \frac{r r p}{4} - \frac{p r r x x}{q q} - \frac{r r x p}{q}$ $z z = \frac{p^{2} q q r r + 4 r r x x p^{2} + 4 q r r x x p}{4 q q r x + 4 q r x x}$ $z z = \frac{b b b + 4 c x x + 4 d x b}{q x + x x}$

<115v>

<116r>

$pd = x$ . $db = y$ . $a$ {illeg}|y| $y = x^{3}$ . $ap =$ {illeg}|v|. $ab = s$ . $op = og = b$ . $\begin{matrix} a s s & - & a v v & + & 2 a v x & - & a x x & - & x^{3} & = & 0 \\ 0 & 0 & 1 & 2 & 3 \end{matrix}$ $\frac{2 a x x + 3 x^{3}}{2 a x} = v$ $v - x = \frac{3 x x}{2 a} = ad$ : ${ad}^{2} = \frac{9 x^{4}}{4 a a}$ . $\frac{x^{3}}{a} ∶ \frac{9 x^{4}}{4 a^{2}} ∷ b b ∶ z z$ . $\frac{z^{2} x^{3}}{a} = \frac{9 x^{4} b}{4 a a}$ . $4 a z^{2} = 9 b^{2} x$ .

<120r>

A Method whereby to square those crooked lines w^ch may be squared.

That a line may be squared Geometrically tis required may be y^t its area may be expressed in generall by some equation in w^ch there is an unknowne quantity, so y^t this quantity being determined y^e area thereof {illeg}(comprehended by y^e cr|o|oked line, y^e two lines (whose intersections describe y^e crooked line), & another streight line{illeg} w^ch guides these two lines in theire motion wⁿ they describe it{illeg}) \to w^ch all y^e points in y^e crooked line are referred)/ is limited & {by} may bee {illeg}{b}|f|ound by y^e same equation. Also {every} every such equation must be of two dimension\s/ because it expresseth y^e quanti{f}|t|y of a superficies.

That an super equation expresse y^e area of a straight /crooked\ line tis required y^t y^e superficie{s} increase in |an|{illeg} |un|equall proportion, y^t when y^e line (considered as unknowne) increaseth {equall} in arithmeticall proportion, wherefore (suppo\s/ ing x always to signifie y^e unknowne quantity: a, b, c, &c; {illeg}|t|o signifie y^e quanitys {sic} gi{illeg}|v|en) ax, or xx either alone or ade|d|ed to any other supperficiess, serve not to find y^e area of any crooked line w^ch may not be found w^th out y^m. {How}

<120v>

<121r>

Prop:

Haveing an equation of 2 dimensions to find w^t crooke line it is {w^ch} whose are{illeg}|a| it doth expresse, suppose y^e equation is $\frac{x^{3}}{a}$ . nameing y^e quantitys; a = dh = kl. {illeg} bg = y. db = mk = x = gp. y^e{illeg} superficies $dbg = \frac{x^{3}}{a}$ supose y^e square dkhl {illeg}is equall to y^e superficies gbd; yⁿ $dk = z = bm = lh = \frac{x^{3}}{aa}$ , & $aaz = x^{3}$ . w^ch is an equation expressing y^e nature of y^e line fmd.

I|N|ext making nm={illeg}|s| \a line w^ch cutteth {f} dmf at right ang angles/. nd=v. $\begin{matrix} \begin{matrix} s s & - & v v & + & 2 v x & - & v v & = & \frac{x^{6}}{a^{4}} & = \\ 0 & 0 . & 1 . & 2 . & 6 . \end{matrix} & \begin{array}{l} mb squared. w^{ch} is an \\ equation haveing 2 equall \\ rootes & therefore multiplyed \\ according to Huddenius his \\ method, produceth another. \end{array} \end{matrix}$ $2 v x = 2 x x +$ \6/ $\frac{x^{6}}{a^{4}}$ $v = x + \frac{3 x^{5}}{a^{4}}$ . & $nb = v - x = \frac{3 x^{5}}{a^{4}}$ . {illeg}|N|ow supposeing, mb:b{illeg}|n|::dh:bg. that is that is, $\frac{3 x^{5}}{a^{4}} ∶ \frac{x^{3}}{a a} ∷ y ∶ a$ . | ${\begin{matrix} 3 & x & x & = & a & y & . \\ 3 & x & x & = & a^{2} & y & . \end{matrix}$ | Which is y^e nature of y^e line dgw & y^e area $dbg = dklh = \frac{x^{3}}{a}$ , makeing db=x. dh={illeg}|a|. or. $diw = deoh = \frac{x^{3}}{a}$ , determing {sic} ({illeg}|d|i) to be ({illeg}|x|). &c

The Demonstration whereof is as followeth

Suppose ω♊♌, ♌mz, zfv; &c are tangents of {illeg}|y|^e line dmf. & from theire intersections z, ♌, v, draw va, zq. ♌s. ωx, parallel & fr{illeg}|o|m theire touch points draw fw, mg, ♊ξ. all parallell to {mg} kp. also from y^e same point of intersection draw vσ, {mf} zλ, ♌{illeg},|ν|. ωζ {sic}.

<122r>

And nb:bm::bm:bt::bg:kl::B{m}:B♌. And mb:nb::bt:bm::kl|♌|B:Bm::kl:bg. wherefore ♌B×bg=Bm×kl. {illeg}|th|at is y^e rectangle klνμ=bρsg. And. πρs♉=θλνμ. in like manner it may be demonstrated y^t ;aqπν {sic}=θλσρ, & ρωxy=μdν{illeg}|h|. &c so y^t y^e rectangle ρshd is equall to any number of such like squares inscribed {twixt} y^e line ny & y^e point d, w^ch squares if they bee infinite in number, they will bee equall to y^e superficies dnywgξ.

This being demonstrated tha{illeg}|t| I may shunne confusion in {y^e} squareing y^e {illeg}|l|ines of every sort {illeg}|I| shall use this method to /in.\ distinguish|ing| y^m. viz: first such lines whose area is exprest by equations in w^ch y^e unknowne quantity is numerator, & y^t 1^st all y^e sines being affirmative, 2^dly mixed.

|2dly| {L}>|l|ines whose area is exprest by quantitys in w^ch y^e unknowne quantity is divisor, & those 1^st under affirmative sines, 2^d under mixt one's |3| lines squared by equations mixt of y^e 2 former kinds, whose quantitys are all 1^s affirmative 2dly mixt.

<124r>

The squareing of those lines whose area is exprest by affirmative quantitys in which y^e unknowne quantity is {n}umeral{\e/}

$\begin{matrix} \begin{matrix} The equations expressing \\ the nature of y^{e} lines. \end{matrix} & Theire square. \\ 3 x x = a y . & Parab: & \frac{x^{3}}{a} . \\ 4 x^{3} = a a y . & ----- & \frac{x^{4}}{a a} \\ 5 x^{4} = y a^{3} & ----- & \frac{x^{5}}{a^{3}} \\ 6 x^{5} = y a^{4} & ----- & \frac{x^{6}}{a^{4}} \\ 7 x^{6} = y a^{5} & ----- & \frac{x^{7}}{a^{5}} . \end{matrix}$ & soe infini{illeg}|t|ely. $\begin{matrix} 4 x^{4} + 7 b x^{3} + 3 b b x x - b a x y - b b a y = 0 . & \frac{x^{3}}{a} + \frac{x^{4}}{a b} \\ \begin{matrix} 5 x^{6} + 8 b b x^{4} + 3 b^{4} x^{2} - a b^{4} y = 0 . \\ - a b b y x^{2} \end{matrix} & \frac{x^{3}}{a} + \frac{x^{5}}{a b b} \end{matrix}$

$\begin{matrix} 4 x^{3} + 3 b x^{2} = b a y & \frac{x^{3}}{a} + \frac{x^{4}}{a b} \\ 5 x^{4} + 3 b^{2} x^{2} = b b a y & \frac{x^{3}}{a} + \frac{x^{5}}{a b b} \\ 6 x^{5} + 3 b^{3} x^{2} = b^{3} a y & \frac{x^{3}}{a} + \frac{x^{6}}{a b^{3}} \\ 7 x^{6} + 3 b^{4} x^{2} = b^{4} a y & \frac{x^{3}}{a} + \frac{x^{7}}{a b^{4}} \end{matrix}$

Soe y^t y^e nature of every crooked line, whose {illeg}|a|rea is compounded of y^e area of 2 or more of y^e former lines, {illeg}|o|r of y^e difference of \y^e area of/ 2 or more of y^e former lines, is exprest by an equation compounded or 2 or more of y^e equations expresing y^e nature of {illeg}|th|ose lines.

$\begin{matrix} 4 x^{3} - 3 b x^{2} = b a y & \frac{x^{4} - x^{3} b}{a b} \\ 5 x^{4} - 3 b b x^{2} = b b a y & \frac{x^{5} - x^{3} b b}{a b b} \\ 6 x^{5} - 3 b^{3} x^{2} = b^{3} a y & \frac{x^{6}}{a b^{3}} - \frac{x^{3}}{a} . \end{matrix}$

<124v>

<125r>

The squareing those lines whose area is exprest by an equation in w^ch y^e unknown quantity is denominator.

$\begin{matrix} \begin{matrix} The Equations expressing \\ y^{e} nature of y^{e} lines. \end{matrix} & \begin{matrix} The square thereof when \end{matrix} \\ x x y = a^{3} & -------- & \frac{a^{3}}{x} \\ x^{3} y = 2 a^{4} & -------- & \frac{a^{4}}{x x} \\ x^{4} y = 3 a^{5} & -------- & \frac{a^{5}}{x^{3}} \\ x^{5} y = 4 a^{6} & -------- & \frac{a^{6}}{x^{4}} \\ x^{6} y = 5 a^{7} & -------- & \frac{a^{7}}{x^{5}} . \end{matrix}$

$\begin{matrix} 4 y y = 9 a^{8} x & Parab. --- & x \sqrt{a x} \\ 4 a y y = 25 x^{3} & -------- & \frac{x x}{a} \sqrt{a x} \\ 4 a^{3} y y = 49 x^{5} & -------- & \frac{x^{3}}{a a} \sqrt{a x} \\ 4 a^{5} y y = 81 x^{7} & -------- & \frac{x^{4}}{a^{3}} \sqrt{a x} \\ 4 a^{7} y y = 121 x^{9} & -------- & \frac{x^{5}}{a^{4}} \sqrt{a x} \end{matrix}$

$\begin{matrix} 4 x y y = a^{3} & -------- & a \sqrt{a x} \\ 4 x^{3} y^{2} = a^{5} & -------- & \frac{a a}{x} \sqrt{a x} \\ 4 x^{5} y^{2} = 9 a^{7} & -------- & \frac{a^{3}}{x x} \sqrt{a x} \\ 4 x^{7} y^{2} = 25 a^{9} & -------- & \frac{a^{4}}{x^{3}} \sqrt{a x} \\ 4 x^{9} y^{2} = 49 a^{11} & -------- & \frac{a^{5}}{x^{4}} \sqrt{a x} \end{matrix}$

<126r>

$\begin{array}{l} 9 a x^{2} + 12 a a x + 4 a^{3} = 4 y y x + 4 a y y . & x \sqrt{a x + a a} \\ 25 x^{4} + 40 a x^{3} + 16 a a x^{2} = 4 a x y^{2} + 4 a a y^{2} . & \frac{x^{2}}{a} \sqrt{a x + a a} \\ 49 x^{6} + 84 a x^{5} + 36 a^{2} x^{4} = 4 a^{3} x y^{2} + 4 a^{4} y^{2} . & \frac{x^{3}}{a a} \sqrt{a x + a a} \\ 81 x^{8} + 144 a x^{7} + 64 a a x^{6} = 4 a^{5} x y^{2} + 4 a^{6} y^{2} . & \frac{x^{4}}{a^{3}} \sqrt{a x + a a} \end{array}$

$\begin{array}{l} a^{3} = 4 x y^{2} + 4 a y y . & a \sqrt{a x + a a} \\ a^{5} x^{2} + 4 a^{6} x + 4 a^{7} = 4 x^{5} y y + 4 a x^{4} y^{2} . & \frac{a a}{x} \sqrt{a x + a a} \\ 9 a^{7} x^{2} + 24 a^{8} x + 16 a^{9} = 4 x^{7} y^{2} + 4 a x^{6} y^{2} . & \frac{a^{3}}{x^{2}} \sqrt{a x + a a} \\ 25 a^{9} x^{2} + 60 a^{10} x + 36 a^{11} = 4 x^{9} y^{2} + 4 a x^{8} y^{2} . & \frac{a^{4}}{x^{3}} \sqrt{a x + a a} \end{array}$

$\begin{array}{l} 9 a x^{2} - 12 a a x + 4 a^{3} = 4 x y^{2} - 4 a y y . & x \sqrt{a x - a a} \\ 25 x^{4} - 40 a x^{3} + 16 a a x^{2} = 4 a x y^{2} - 4 a a y y . & \frac{x^{2}}{a} \sqrt{a x - a a} \end{array}$

$\begin{array}{l} a^{3} = 4 x y^{2} - 4 a y^{2} . & a \sqrt{a x - a a} \\ a^{5} x^{2} - 4 a^{6} x + 4 a^{7} = 4 x^{5} y y - 4 a x^{4} y y . & \frac{a a}{x} \sqrt{a x - a a} \end{array}$

$\begin{array}{l} 9 a x^{2} - 12 a a x + 4 a^{3} = 4 a y y - 4 x y y . & x \sqrt{a a - a x} \\ 25 x^{4} - 40 a x^{3} + 16 a a x^{2} = 4 a^{2} y y - 4 a x y y . & \frac{x^{2}}{a} \sqrt{a a - a x} \end{array}$

$\begin{array}{l} a^{3} = 4 a y^{2} - 4 x y^{2} . & a \sqrt{a a - a x} \\ a^{5} x^{2} - 4 a^{6} x + 4 a^{7} = 4 a x^{4} y y - 4 x^{5} y y . & \frac{a a}{x} \sqrt{a a - a x} . \end{array}$

Note y^t the lines whose nature is expres{illeg}|t| by y^e {2}|4| latter sorts of equations, are y^e same w^th the lines of y^e 2 former sorts. D{illeg}|o|ubtfull.

<127r>

$\begin{array}{l} 9 a x x + 24 a x x + 16 x^{3} = 4 x y y + 4 a y y . & x \sqrt{a x + x x} \\ 25 a a x^{3} + 60 a x^{4} + 36 x^{5} = 4 a a x y y + 4 a^{3} y y . & \frac{x x}{a} \sqrt{a x + x x} \\ 49 a a x^{5} + 112 a x^{6} + 64 x^{7} = 4 a^{4} x y y + 4 a^{5} y y = & \frac{x^{3}}{a a} \sqrt{a x + x x} \\ 81 a a x^{7} + 180 a x^{8} + 100 x^{9} = 4 a^{6} x y y + 4 a^{7} y y = & \frac{x^{4}}{a^{3}} \sqrt{a x + x x} \end{array}$

$\begin{array}{l} a^{4} + 4 a^{3} x + 4 a a x x = 4 a x y y + 4 x x y y . & a \sqrt{a x + x x} . \\ a^{6} = 4 a x^{3} y y + 4 x^{4} y y . & \frac{a a}{x} \sqrt{a x + x^{2}} . \\ 9 a^{8} + 12 a^{7} x + 4 a^{6} x^{2} = 4 a^{} x^{5} y y + 4 x^{6} y^{2} . & \frac{a^{3}}{x x} \sqrt{a x + x x} . \\ 25 a^{10} + 40 a^{9} x + 16 a^{8} x^{2} = 4 a^{} x^{7} y y + 4 x^{8} y y . & \frac{a^{4}}{x x} \sqrt{a x + x x} . \end{array}$

$\begin{array}{l} 9 a a x - 24 a x x + 16 x^{3} = 4 a y y - 4 x y y . & x \sqrt{a x - x x} . \end{array}$

<128r>

$\begin{array}{l} 4 x^{4} + 4 a a x x + a^{4} = x x y y + a a y y & x \sqrt{a a + x x} \\ 9 x^{6} + 12 a a x^{4} + 4 a^{4} x^{2} = a a x x y^{2} + a^{4} y^{2} . & \frac{x x}{a} \sqrt{a a + x x} \\ 16 x^{8} + 24 a a x^{6} + 9 a^{4} x^{4} = a^{4} x^{2} y^{2} + a^{6} y^{2} . & \frac{x^{3}}{a^{2}} \sqrt{a a + x x} \\ 25 x^{10} + 40 a a x^{8} + 16 a^{4} x^{6} = a^{6} x^{2} y^{2} + a^{8} y^{2} . & \frac{x^{4}}{a^{3}} \sqrt{a a + x x} . \end{array}$

$\begin{array}{l} a a x x = a a y y + x x y y . & a \sqrt{a a + x x} \\ a^{8} = a a x^{4} y y + x^{6} y y & \frac{a a}{x} \sqrt{a a + x x} \\ 4 a^{10} + 4 a^{8} x x + a^{6} x^{4} = a a x^{6} y y + x^{8} y y . & \frac{a^{3}}{x x} \sqrt{a a + x x} \\ 9 a^{12} + 12 a^{10} x x + 4 a^{8} x^{4} = a a x^{8} y^{2} + x^{10} y y . & \frac{a^{4}}{x^{3}} \sqrt{a a + x x} . \end{array}$

$\begin{array}{l} 4 x^{4} - 4 a a x x + a^{4} = a a y y - x x y y . & x \sqrt{a a - x x} \end{array}$

$\begin{array}{l} a a x x = a a y y - x x y y . & a \sqrt{a a - x x} . \\ a^{8} = a a x^{4} y y - x^{6} y y & \frac{a a \sqrt{a a - x x}}{x} \\ a \sqrt{\frac{a^{3} - a x x}{x}} . \end{array}$

<129r>

$\begin{array}{l} \begin{array}{r} a^{4} + 3 a^{2} b x + 4 a a x^{2} + \frac{9}{4} b b x^{2} + 6 b x^{3} + 4 x^{4} = a^{2} y^{2} . \\ + b x \\ + x x \end{array} & x \sqrt{a a + b x + x x} \\ a a b b + 4 a a b x + 4 a a x x = 4 c c y y + 4 b x y^{2} + 4 x x y^{2} & a \sqrt{c c + b x + x x} \\ \sqrt{a^{3} x + x^{4}} \\ \sqrt{a^{4} + a x^{3}} \\ 9 a x^{4} + 6 a^{3} x^{2} + a^{5} = 4 a a x y y + 4 x^{3} y y & \sqrt{a^{3} x + a x^{3}} . \\ \sqrt{a^{4} + x^{4}} \end{array}$

<130r>

$\frac{a^{3}}{a + x}$
$\frac{a^{2} x}{a + x}$
$\frac{a x^{2}}{a + x}$
$\frac{x^{3}}{a + x}$

$\frac{a^{3}}{a - x}$

$\frac{a^{3}}{x - a}$

<131r>

$x^{4} + 3 a x^{3} + a x^{3} + 3 a^{2} x^{2} + a^{3} x a^{4} = a^{3} y + 2 a^{2} x y + a x^{} y$ .

$\begin{matrix} a^{3} = b b y + 2 b x y + x x y & ----- & \frac{a^{3}}{b + x} \\ 2 b^{4} x + b^{4} a = x^{4} y + 2 a x^{3} y + a a x x y & ----- & \frac{b^{4}}{a x + x^{2}} \\ 3 b^{5} x + 2 b^{5} a = x^{5} y + 2 a x^{4} y + a^{2} x^{3} y & ----- & \frac{b^{5}}{a^{} x^{2} + x^{3}} \\ 4 b^{6} x + 3 b^{6} a = x^{6} y + 2 a x^{5} y + a^{2} x^{4} y & ----- & \frac{b^{6}}{a x^{3} + x^{4}} \\ \frac{x^{3}}{a + x} \\ \frac{x^{4}}{a a + a x} \\ \frac{x^{5}}{a^{3} + x a^{2}} \\ \frac{x^{6}}{a^{4} + x a^{3}} \\ a a b = b b y + 2 b x y + x x y . & ----- & \frac{a^{2} x}{b + x} \\ 2 b a x + a x x = b b y + 2 b x y + x x y . & ----- & \frac{a x^{2}}{b + x} \end{matrix}$

<132r>

$\begin{matrix} 2 a^{4} x^{} = b^{4} y + 2 b b x x y + x^{4} y & \frac{a^{4}}{b b + x^{2}} \\ \frac{a^{5}}{b b x + x^{3}} \\ \frac{a^{6}}{b b x^{2} + x^{4}} \\ \frac{a^{7}}{b b x^{3} + x^{5}} \\ \frac{x^{4}}{b b + x^{2}} \\ \frac{x^{5}}{b^{3} + x^{2} b} \\ \frac{x^{6}}{b^{4} + b b x^{2}} \\ \frac{x^{7}}{b^{5} + b b x^{2}} \\ \frac{a^{3} x}{b b + x^{2}} \\ \frac{a a x x}{b b + x x} \\ \frac{a x^{3}}{b b + x x} \end{matrix}$

<132v>

$\begin{matrix} 27 x^{2} y^{3} = a^{5} & ———— & a \sqrt{c ∶ a a x} . \\ 27 x^{5} y^{3} = 8 a^{8} & ———— & \frac{a a}{x} \sqrt{c ∶ a a x} \\ 27 x^{8} y^{3} = 125 a^{11} & ———— & \frac{a^{3}}{x^{2}} \sqrt{c ∶ a a x} \\ 27 x^{11} y^{3} = 512 a^{14} & ———— & \frac{a^{4}}{x^{3}} \sqrt{c ∶ a a x} \\ 27 x y^{3} = 8 a^{4} & ———— & a \sqrt{c ∶ a x x} . \\ 27 x^{4} y^{3} = a^{7} & ———— & \frac{a a}{x} \sqrt{c ∶ a x x} . \\ 27 x^{7} y^{3} = 64 a^{10} & ———— & \frac{a^{3}}{x x} \sqrt{c ∶ a x x} . \\ 27 x^{10} y^{3} = 343 a^{13} & ———— & \frac{a^{4}}{x^{3}} \sqrt{c ∶ a x x} . \\ 27 y^{3} = 64 a a x & ———— & x \sqrt{c ∶ a a x} . \\ 27 a y^{3} = 343 x^{4} & ———— & \frac{x x}{a} \sqrt{c ∶ a a x} . \\ 27 a^{4} y^{3} = 1000 x^{7} & ———— & \frac{x^{3}}{a a} \sqrt{c ∶ a a x} . \\ 27 y^{3} = 125 a x^{2} & ———— & x \sqrt{c ∶ a x x} \\ 27 a^{2} y^{3} = 512 x^{5} & ———— & \frac{x x}{a} \sqrt{c ∶ a x x} \\ 27 a^{5} y^{3} = 1331 x^{8} & ———— & \frac{x^{3}}{a} \sqrt{c ∶ a x x} . \end{matrix}$

<133r>

$\frac{a^{5}}{b^{3} + x^{3}}$ $\frac{a^{6}}{b^{3} x + x^{4}}$ $\frac{a^{7}}{b^{3} x^{2} + x^{5}}$ $\frac{a^{8}}{b^{3} x^{3} + x^{6}}$ $\frac{x^{5}}{b^{3} + x^{3}}$ $\frac{x^{6}}{b^{4} + x^{3} b}$ $\frac{x^{7}}{b^{5} + b b x^{3}}$ $\frac{x^{8}}{b^{6} + b^{3} x^{3}}$ $\frac{a^{4} x}{b^{3} + x^{3}}$ $\frac{a^{3} x^{2}}{b^{3} + x^{3}}$ $\frac{a^{2} x^{3}}{b^{3} + x^{3}}$ $\frac{a x^{4}}{b^{3} + x^{3}}$

<133v>

$\begin{matrix} 256 x^{3} y^{4} = a^{7} . & ———— & a \sqrt{q q ∶ a a a x} . \\ 256 x^{7} y^{4} = 81 a^{11} . & ———— & \frac{a a}{x} \sqrt{q q ∶ a a a x} . \\ 256 x^{11} y^{4} = 2401 a^{15} & ———— & a a a \sqrt{q q ∶ a a a x} . \\ 256 x y^{4} = 81 a^{5} & ———— & a \sqrt{q q ∶ a x^{3}} \\ 256 x^{5} y^{4} = a^{9} & ———— & \frac{a a}{x} \sqrt{q q ∶ a x^{3}} \\ 256 x^{9} y^{4} = 625 a^{13} & ———— & \frac{a^{3}}{x x} \sqrt{q q ∶ a x^{3}} \\ 256 x^{13} y^{4} = 6561 a^{17} & ———— & \frac{a^{4}}{x^{3}} \sqrt{q q ∶ a x^{3}} \end{matrix}$

<134v>

<135r>

A Method whereby to square such crooked lines as may be squared.

If y^e crooked lines σha \& aoθ/ are of such a nature that (supposeing [gh] parallell to [qa], & [bh] perpendic: to σha & [an] a given line) gh∶bg∷an∶ge. Then y^e area [age]=[qlna] y^e rectangle made by [an] & [gh].

Demonstration.

Suppose σi, id, de, &c; are tangents of σha, from whose intersections or ends are drawne, ec, dv|f|, iz, σ{illeg}|w|, &{c}{illeg} & from whose touch points are drawne βθ, ho, λμ, &c: all parallel to av. From y^e said intersections draw sw, ik, {illeg}|d|m, es, &c. paralle{illeg}|l| {illeg}|t|o bn. {illeg}|Sin|ce gh∶bg∷{illeg}|p|d∶ip∷an∶ge. pd×ge=iν×an. that is □pkmt=□uτfs. by y^e same reason {illeg}|t|mso= =τνcy; & vpkw=ζuzx &c: Thus also it may be prooved y^t y^e ▭vwna{illeg} is equall to any{illeg}|y| number of such li{illeg}|k|e ▭s w^ch> inscribed twixt y^e lin{illeg}|e| ζω & y^e point i|a|, w^ch if they be infinite are equall t{illeg}|o| superficies ζaω=vwna .also {illeg}gπμo. {illeg}=ql♊♌. &c.

Prop 1

To find y^e line whose area is exprest by any given equac|t|ion. Suppose y^e equatiō is $\frac{x^{3}}{a}$ . {is} nameing y^e quantitys a=an. x=ag. $\frac{x^{3}}{a} = qlna = goa$ $gh = qa = \frac{x^{3}}{a a}$ . bh=s. kge. ba=v. $\begin{matrix} s s & - & v v & + & 2 v x & - & x x & = & \frac{x^{6}}{a^{4}} \\ 0 & 0 & 1 & 2 & 6 \end{matrix}$ |{illeg} equaco hath 2 equall rootes & is therefore multiplied according to Huddenius his Meth{illeg}| $v x = x^{2} + 3 \frac{x^{6}}{a^{4}} . gb = v - x = \frac{3 x^{5}}{a^{4}}$ . Wherefore if $\frac{x^{3}}{a a} ∶ \frac{3 x^{5}}{a^{4}} ∷ a ∶ \frac{3 x x}{a} = ge$ . therefore aoω is a Para{illeg}|b|: & $age = \frac{x^{3}}{a} = qlna$

<135v>

Also if y^e Equation be $\frac{a^{3}}{x}$ . Then makeing $a = an . x = ag . \frac{a^{3}}{x} = qlna = gea. qa = \frac{a a}{x} = gh$ . $bh = s . ba = v$ . $\begin{matrix} \begin{matrix} s s & - & v v & + & 2 v x & - & x x & = & \frac{a^{4}}{x} . \\ 0 & 0 & - 1 & - 2 & + 2 \\ - 2 v x & - & + 2 x x & = & \frac{2 a^{4}}{x x} . \end{matrix} \\ \begin{matrix} v = x - \frac{a^{4}}{x^{3}} . \end{matrix} \end{matrix}$ . |w^ch multiplied by Huddenius his method by reasō of z equall rootes.| $x - v = gb = \frac{a^{4}}{x^{3}}$ . Lastly, $\frac{a a}{x} ∶ \frac{a^{4}}{x^{3}} ∷ a ∶ \frac{a^{3}}{x x} ∶ \frac{a^{3}}{x x} = ge = y . & a^{3} = x x y$ . w^ch last equation expresseth y^e nature of y^e line aθo, whose surface $aeg = qlan = \frac{a^{3}}{x}$ .

Note y^t line I call y^t line [x] to w^ch both y^e lines \σha & aoω/ have respect & increaseth & dimi{n}isheth w^thout moveing over any superficies as π{a}|α|, ga, &c. but y^t line to w^ch but one line h{illeg}|a|th respect I call [y as go, πμ: or [z] as gh, πλ, &c.

If $a x^{m} = b y^{n}$ .(m & n being numbers y^t signifie y^e dimensions of x & y), then $\frac{n x y}{n + m} = ago$ , y^e area of y^e line {illeg}|aμo|. And if $a = b \times x^{m} \times y^{n}$ . $y^{n}$ is $\frac{n x y}{x - m} = ago$ . y^e area of y^t line.

<136r>

The squareing of y^e simplest lines in w^ch y is but of one dimension.

$\begin{matrix} \begin{matrix} Equations expressing \\ y^{e} nature of y^{e} lines. & Theire squares. \\ 3 x x = a y . & Parab: —— & \frac{x^{3}}{a} . \\ 4 x^{3} = a a y & ————— & \frac{x^{4}}{a a} . \\ 5 x^{4} = y a^{3} & ————— & \frac{x^{5}}{a^{3}} . \\ 6 x^{5} = y a^{4} & ————— & \frac{x^{6}}{a^{4}} . &c. \end{matrix} & \begin{matrix} Lines. & □ . \\ x x y = a^{3} & ————— & \frac{a^{3}}{x} . \\ x^{3} y = 2 a^{4} & ————— & \frac{a^{4}}{x x} . \\ x^{4} y = 3 a^{5} & ————— & \frac{a^{5}}{x^{3}} . \\ x^{5} y = 4 a^{6} & ————— & \frac{a^{6}}{x^{7}} . &c \end{matrix} \end{matrix}$

The square of y^e simplest lines in w^ch y is of 2 dimensions.

$\begin{matrix} \begin{matrix} \begin{matrix} The lines \\ squared. & Theire S quares. \\ 4 y y = 9 a x . & Parab: —— & x \sqrt{a x} . \\ 4 a y y = 25 x^{3} & ————— & \frac{x x}{a} \sqrt{a x} . \\ 4 a^{3} y y = 49 x^{5} & ————— & \frac{x^{3}}{a a} \sqrt{a x} . \\ 4 a^{5} y y = 81 x^{7} & ————— & \frac{x^{4}}{a^{3}} \sqrt{a x} . \end{matrix} & \begin{matrix} Lines squared. & □ s. \\ 4 x y y = a^{3} & ————— & a \sqrt{a x} . \\ 4 x^{3} y y = a^{5} & ————— & \frac{a a}{x} \sqrt{a x} . \\ 4 x^{5} y y = 9 a^{7} & ————— & \frac{a^{3}}{x x} \sqrt{a x} . \\ 4 x^{7} y y = 25 a^{9} . & ————— & \frac{a^{4}}{x} \sqrt{a x} &c \end{matrix} \end{matrix} \\ &c. \end{matrix}$

The square of those $\sqrt{lines}$ where y is of 3 dimensions onely.

$\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} The lines squared. & Their squares \\ 27 x y^{3} = 8 a^{4} & ————— & a \sqrt{c ∶ a x x} . \\ 27 x^{4} y^{3} = a^{7} & ————— & \frac{a a}{x} \sqrt{c ∶ a x x} . \\ 27 x^{7} y^{3} = 64 a^{10} & ————— & \frac{a^{3}}{x x} \sqrt{c ∶ a x x} . \\ 27 x^{10} y^{3} = 343 a^{13} & ————— & \frac{a^{4}}{x^{3}} \sqrt{c ∶ a x x} . \\ &c \end{matrix} & \begin{matrix} Lines squared. & Theire squares \\ 27 y^{3} = 64 a a x & ————— & x \sqrt{c ∶ a a x} . \\ 27 a y^{3} = 343 x^{4} & ————— & \frac{x x}{a} \sqrt{c ∶ a a x} . \\ 27 a^{4} y^{3} = 1000 x^{7} & ————— & \frac{x^{3}}{a a} \sqrt{c ∶ a a x} . \\ 27 a^{7} y^{3} = 2197 x^{10} . & ————— & \frac{x^{4}}{a^{3}} \sqrt{c ∶ a a x} \\ &c \end{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} 27 x x y^{3} = a^{5} & ————— & a \sqrt{c ∶ a a x} . \\ 27 x^{5} y^{3} = 8 a^{8} & ————— & \frac{a a}{x} \sqrt{c ∶ a a x} . \\ 27 x^{8} y^{3} = 125 a^{11} & ————— & \frac{a^{3}}{x x} \sqrt{c ∶ a a x} . \\ 27 x^{11} y^{3} = 512 a^{14} & ————— & \frac{a^{4}}{x^{3}} \sqrt{c ∶ a a x} . \\ &c \end{matrix} & \begin{matrix} 27 y^{3} = 125 a x^{2} & ————— & x \sqrt{c ∶ a x x} . \\ 27 a a y^{3} = 512 x^{5} & ————— & \frac{x x}{a} \sqrt{c ∶ a a x} . \\ 27 a^{5} y^{3} = 1331 x^{8} & ————— & \frac{x^{3}}{a a} \sqrt{c ∶ a x x} . \\ 27 a^{8} y^{3} = 2744 x^{11} & ————— & \frac{x^{4}}{a^{3}} \sqrt{c ∶ a x x} \\ &c \end{matrix} \end{matrix} \end{matrix} \end{matrix}$

<138r>

Of Musick.

1. First some \one/ sound must bee pitched upon, to w^ch all y^e musick must bee more especially refered yⁿ to any other sound, (as number to an unit) let this sound be called y^e Cliffe or Key of y^e song.

2. Then consider {illeg} y^e sound w^ch is one or two or thre 8^ths above or below y^t key (for Musick seldome takes a larger compasse yⁿ 3 8^ths) The cheife of {illeg}|w|^ch is y^e 8^th next above y^e Key. 3.Each of these Eights are alike divided into ꝑ{illeg}|t|s, t|f|or y^e ꝑ{illeg}|t|s of y^e higher eight are {of}an Eight abo{illeg}|v|e their correspondent ꝑts {illeg}|o|f y^e lower eight. so y^t y^e ꝑts of one Eight knowne give all y^e rest, y^e other Eights being but a repetition of y^t. {illeg}|in| {illeg}|{a}|more base or treble sound. (Hence s{illeg}|o|me call an 8^th y^e largest consonant.) Let this division of an 8^th bee called y^e Mode of y^e song.

4. This Eight i{illeg}|s| first divided into a 5^t & 4^th, y^e fift being next above y^e Key; to w^ch it{illeg} adds so much sweetnesse y^t should this fift bee omitted in any song, y^e Key would {illeg}|i|mparte its name & nature {illeg}|t|o some sound w^{illeg}|ch| ha{illeg}|th| a fift above it. And since all ha{illeg}|r|mony w^{illeg}|t|hout a fi{illeg}|f|t is flat, therefore the key must necessarily have a fift above i{illeg}|t|. ✝ < insertion from f 137v > ✝ here annex a discourse of y^e motion of strings sounding an 8^t 5^t & 4^th & of y^e Logarithmes of those strings, or distances of y^e notes.

< text from f 138r resumes >

5. An 8^th is next divided into a third major & 6^t minor, & lastly into a 3^d minor & 6^t major. * < insertion from f 137v > * these are all y^e concords conteined in an Eight. Hereto annex a discour{illeg}|se| of y^e 3^ds & 6^ts

The notes in order of concordance

Eight. 5^t. 3^d maj. 4^th. 6^t maj. 3^d min. 6^t min. 2^d maj. 7^th maj 7^th min. 2^d min. 5^t min. < text from f 138r resumes > But as too suddaine a change from lesse to greater light offfends y^e eye by reason y^t, y^e \spirits rarified by the/ augmented motion of y^e light too violently stretch y^e optick nerve: soe y^e suddaine passing f{illeg}|r|om grave to acute sounds is not so pleasant as if it were done by degrees, for there \because/ of two {sic} greate a change of motion made \thereby/ in the auditory spirits ♀.|✝| < insertion from f 137v > ✝{illeg}|A|nd as a man \suddainely/ com̄e|i|ng from greater to lesse light, cannot discerne objects thereby so well, as when if he came to it by degrees or as when hee hath staid some whi{illeg}|l|e in y^e lesser light (by reason y^t y^e motion of y^e spirits in y^e optick nerve caused by y^e greater \light/, doth, untill it bee allayed; {illeg} disturbe \& as it were drowne/ y^e motion motion of y^e weaker light) soe if {illeg}|t|he slower motion of y^e lower sound immediately succede y^e m{illeg}|uch| more smart motion of y^e higher its impression {illeg} \on y^e auditory spirits — being then/ less perceptible, y^e consonance \lower sound/ must bee {more} less pleasant y^t {had} if moved y^e step had beene graduated, thus that coldness | warmth is \{illeg}/ pleasant to a hot | cold man which is imperceptible to y^e same man cold | hot. Thus a little {illeg}|h|eate is at y^e first un\ least /perceptible to one newly come from a greater. Coroll: 1. The distance of sounds adds to y^e imperfection of their concordance. Cor: 2: Tis better to descend yⁿ ascend by {illeg}|le|apes y^e first makeing y^e highest sound harsher, y^e seacond makeing y^e lower \one{illeg}|ly|/ lesse perceptible. < text from f 138r resumes > Which graduation may be thus don.

{illeg}|6|. The prime ꝑts of an 8^th are a 5^t & 4^th: of a 4^th fift are a 3^d major & 3^d minor: w^ch two consist y^e first of a tone major & tone minor, y^e 2^d of a tone major {illeg}|&| semitone. A 4^th consists of a tone major, minor {illeg}|&| semitone. Soe y^t an eight consists of th{illeg}|r|e <139r> tone majors, 2 tone minors, & 2 semitones. [The tones might be againe divided into $\frac{1}{2}$ tones & $\frac{1}{4}$ tones, but they would bee of noe use for tones $\frac{1}{2}$ tones & $\frac{1}{4}$ tones \being discor{illeg}|d|s/ can onely serve to move {illeg}|b|y from concord to concord w^ch if done by $\frac{1}{2}$ tones & $\frac{1}{4}$ tones y^e number of discords twixt each concord{illeg} would much m{illeg}|o|re bee harsh yⁿ y^e concord would {illeg}|b|ee pleasan{illeg}|t|, besides $\frac{1}{2}$ tones & $\frac{1}{4}$ tones are harshe{illeg}|r|{illeg} discords by far yⁿ tones, & experience speakes y^t an 8^th run over by $\frac{1}{2}$ notes is unpleasant]. {y}|Y|et perhaps $\frac{1}{2}$ or $\frac{1}{4}$ notes passed over very hastily w^th a larger stay upon y^e concords twixt w^ch they are, might bee delightfull. {illeg}|But| since they are such discords, inserted as 'twere by accident /{illeg}\ {illeg}|o|nely to graduate concords, & soe quickly slipt o{illeg}|v|er, y^e sen{illeg}|c|e cannot perceive any error or exactnesse in y^m, & therefore \bee they usefull yet/ to treate of y^m would be lost labor]

7. The degrees (viz 2 tone majors, a tone minor & semitone in {illeg}|y|^e{illeg} 5^t & a tone major, a tone minor & semitone in a 4^th) are \12/ severall ways ordered in y^e 8^th w^ch orders are called m|M|odes, generally, because they see \much/ limit y^e tune as to prevent \partes of y^e tune from/ discords sounds w^th y^e key, w^th its 5^t {one} w^th one another, \of one w^th another/ particularly bes|c|ause tunes framed by divers of them differ in their aires or Modes.

8. These modes are 3 fold, {illeg}|v|iz: 6 in w^ch y^e $\frac{1}{2}$ notes are distant 2 tones: foure in w^ch they are distant one tone: & 2 in w^ch they are together. The last two are of small or noe use, because every note \sound/ is distant 3 tones from some other \excepting y^t there are but 2 {illeg}|f|ifts/. Also thos $\frac{1}{2}$ notes are two harsh to come together much more to bee annext to y^e Key or its fift. Neither is the seacond sort very useful for one of y^e $\frac{1}{2}$ notes are annexed {illeg}|ei|ther to the Key or its 5^t or 8^t, also 4 of its ꝑts \sounds/ are distant 3 notes \& but 4 of them are distant a fift/ from some other: whereas there are but 2 in {illeg}|t|hose of y^e first sort \distant {those} notes & six of them distant fifts from other sounds./; the harshnes of y^e $\frac{1}{2}$ notes being there also more moderated by their distance. And therefore y^e first 6 are yet in use.

<139v>

9. The following table may expresse y^e 12 Modes in their order of Elegancy. In w^ch y^e tone major & minor are not distinguished, their difference being too little to make new modes by their \order/ changed, though thereby th{illeg}|e|y may add much grace or harshnesse to any particular mode.

$\begin{matrix} \begin{array}{r} y^{e} key & 2^{d} & 3^{d} minor & 3^{d} major & 4^{th} & Tritonus & 5^{t} & 6^{t} minor & 6^{t} major & 7^{th} & 8^{th} \end{array} \\ \begin{matrix} \begin{array}{c} \begin{matrix} 4 . \\ 3 . \\ 5 \\ 2 \\ 1 \\ 6 \end{matrix} & \begin{matrix} O & . & p & . & q & r & . & s & . & t & v & . & o \\ s & . & t & v & . & o & . & p & . & q & r & . & s \\ r & . & s & . & t & v & . & o & . & p & . & q & r \\ p & . & q & r & . & s & . & t & v & . & o & . & p \\ t & v & . & o & . & p & . & q & r & . & s & . & t \\ v & . & o & . & p & . & q & r & . & s & . & t & v \end{matrix} & \begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{matrix} \end{array} \\ \begin{array}{r} \begin{matrix} o & . & p & . & q & r & . & s & x & . & v & . & o \\ r & . & s & x & . & v & . & o & . & p & . & q & r \\ s & x & . & v & . & o & . & p & . & q & r & . & s \\ v & . & o & . & p & . & q & r & . & s & x & . & v \end{matrix} & \begin{matrix} 7 \\ 8 \\ 9 \\ 10 \end{matrix} \\ \begin{matrix} o & . & p & . & q & . & y & s & x & . & v & . & o \\ s & x & . & v & . & o & . & p & . & q & . & y & s \end{matrix} & \begin{matrix} 11 \\ 12 \end{matrix} \end{array} \end{matrix} \end{matrix}$

This order may be thus evinced. The first Mode excells y^e 2^d, by reason of y^e $\frac{1}{2}$ Note's being more equally \{distantly}/ /convent\ placed twixt y^e Key & its fift, {it}|it| lesse detracting from y^e fift {illeg} \because/ of its greater distance from it. Also y^e key hath its 3^d major & y^e fift its 3^d minor in y^e i|1|^st mode, but contrarily in y^e 2^d mode y^e key hath its 3^d minor & y^e 5^t its 3^d major. The 3^d {m} sweetness of y^e key in y^e 3^d mode is still more diminished by haveing y^e $\frac{1}{2}$ n{illeg}|o|te imediate{illeg}|e|ly below it & its 8^ts. The 4^th Mode succedes as partakeing of y^e 3^ds def{f}ect; y^e sweetnesse of its key's 5^t, & consequently of its key, being \also/ diminished by y^e $\frac{1}{2}$ note {a}|i|mmediately above it. The 5^t mode succeds because to y^e imperfections of y^e 4^th this is added y^t its first $\frac{1}{2}$ note is next above y^e key \& its fifts have tritones/. The 6^t mode is yet more unpleasant <141r> for both y^e key, its 5^ts, & eights have a $\frac{1}{2}$ note next below them: Al{illeg}|s|o y^e key & its eights have tritones above & below them. Other reasons might {illeg}|b|ee added for this order, & also for y^e order of y^e six{illeg}|t| last modes; & it might \perhaps bee/ shown y^t y^e 7^th mode may bee as usefull as y^e Sixt, but that would {illeg}|b|ee tedious. Note, y^t sometime a note is put out of its place for some particular reason (as to prevent a greater discord &c) but y^t seemes soe rare & acci{illeg}|d|entall to y^e song{illeg} as not to change its aire or constitute a new mode.

10. The tones major & minor may bee six severall ways ordered in each mode but y^e best way generally is to make y^e first \seacond/ third & fift a{illeg} tone major, the \first/ 2^d & 4^th a tone minor; especially in a solitary voyce. \{illeg} & but 10 severall ways in all y^e six first modes./ |.| the first is by makeing y^e distances, pq, rs, vo, to bee a {illeg} tone major/s\ op, & st, to bee tone minors. In this order there are five 5^ts{illeg}, {illeg} 4^th 3 third majors, & 3 third minors \in an 8^th/. Thus is y^e 3^d 5^t & first mode best ordered, & thus may y^e 4^th & 6^t moode bee ordered but not y^e 2^d \well/ for its \keys/ fift will |yⁿ|bee {illeg}|oo| {illeg}|fla|t. The 2^d way is by putting y^e {illeg}|t|one minor twix{illeg}|t|, o & p, r & s. This order makes also 5 fif{illeg}|t|{illeg}s, thre 3^d {illeg} majors & 3 3^d minors, in each 8^th. And by it \thus/ may y^e 4^th, 6^t, & 2^d Moode bee \best/ ordered; the 3^d & 5^t moode may bee {illeg} also ordered thus, but not y^e first \not well/, for y^e Keys 5^t will |yⁿ| bee too flatt. The 3^d way is by putting y^e {illeg} \minor/ note in v|b|etwixt r & s, v & o. & thus each 8^th will have five fifts, 2 third majors & 2 minor thirds. The 4^th 6^t & 2^d moode may v|b|ee well thus ordered y^e {3}|1|^d|s|t & 5^t not so well & y^e 3^d {illeg}|w|orst of all. The 4^th Order is by putting y^e minor tone {illeg}|t|wi{illeg}|x|t p & q, s & t & thus each 8^th hath 5 5|f|ifts, 2 minor 3^ds, & 2^d moode bee ordered well, v|b|ut y^e 6^t & 4^th moode not well. The other six orders are l{illeg}|e|sse convenient to y^e Moodes. Note y^t, In every 8^th there are 6 5^ts, 3 major thirds & 4 minor thirds whereof one or more of them are mad{e} too flat {illeg}|o|r sharpe {illeg}|b|y about y^e 10^th ꝑte of a note, {illeg}|b|ut in this computation I onely reckon y^e exact concords <141v> Esteeming y^t order most \more/ perfect whose sounds agree in {y^e} most \more of y^e/ exa{illeg}|c|t concords. Note also y^t every order Eight hath soe many exact 4^ths, 6^t minors & third majors as it hath 5^ts, 6|3|^t|d| majors & 3^d minors their comple^nts to an 8^th.

{illeg}|11| |12|. It m{illeg}|ay| bee required sometimes {illeg}|t|o raise or let fall y^e voyce in singing w^ch is best done by r{illeg}i|a|sising or depressing y^e {illeg}|ke|y of {illeg}|y|^{illeg}|e| song a fif{illeg}|t|, (if an 8^{illeg}|t| be too greate), for y^t will v|b|ee consonant w^th y^e former sound w^ch is now become (for y^e present) gratefull to y^e eare. Also instruments are usually tuned one a fif{illeg}|t| abov{illeg}|e| another \if the keys of severall parts being a fift one above another/; & a tune might bee pricked for too high a voy{illeg}|c|e in one ꝑte of y^e Gamut & {illeg}|t|oo {illeg}|b|ase a voyce if removed an 8^th lower. {illeg}|H|ence ariseth a comparison of y^e same moode w^th it selfe placed a fift higher. The precedent scheme may serve {illeg}|t|o represent any of y^e six modes set \repeated/ six times w^th y^e distance of a fif{illeg}|t| twixt each, according {illeg}|t|o y^e order of y^e left hand figures. T|B|ut they cannot bee soe repeated more yⁿ 3 times, unlesse {illeg}|w|ith |more| discord yⁿ harmony.

$\begin{matrix} \begin{matrix} 1 & . & o & . & p & . & q & r . & s & . & t & v . & o & . & p & . & q & r . & s & . & t & v . & o & . & p & . & q & r . & s & . \\ 2 & . & r & . & s & . & t & v . & o & . & p & . q & r & . & s & . & t & v . & o & . & p & . q & r & . & s & . & t & v . & o & . \\ 3 & . & v & . & o & . & p & . q & r & . & s & . t & v & . & o & . & p & . q & r & . & s & . t & v & . & o & . & p & . q & r & . \\ ︸ & ︸ \\ | & | & | & | & | & | & | & | & | & | \end{matrix} \\ \begin{matrix} f & G & , & a & , b, & c & , & d & , e, & f & , & g & , & a & b & c & d & e & f & g & a & b & c \\ 3 \end{matrix} \end{matrix}$ Symbol (bass clef) in text Symbol (musical symbol) in text Symbol (g symbol) in text

Any of y^e 6 Moodes \with its eights/ may bee represented by any of t{illeg}|he|se 3 orders of letters for y^e key being o they re present y^e first Moode, & y^e second it being, s, & y^e 3^d if it be r &c: Also y^e first ranke being lowest y^e 2^d a fif{illeg}|t| above {illeg}|it| & y^e 3^d a fift above y^t,|this|{w^h} scheame may represent {illeg}|an|y of y^e Modes w^th y^e same mode one or 2 fifts above or below it.

<142r>

|12| 11. These degrees have of old beene e{illeg}|x|pressed by |y^e| Six notes, vt, re, mi, fa, sol, la, the 7^th note being omitted as be{illeg}|i|ng a discord to y^e key in y^e first moode. But of late y^e usuall notes are sol, {illeg}|l|a, mi, fa, sol, la, fa, hitherto expressed by y^e letters o, p. q. r. s.t. v. Tis generally best (by se{illeg}|e| 10) to make from the distance from sol to {illeg}|l|a, to be a minor tone, from la to mi & fa to sol a major tone, & a semitone from la to fa /& mi to fa\. Onely in y^e 2^d Mode make sol & la /{illeg}\ {illeg}|&| mi to bee distant a major tone, fa to be dis a minor tone from sol els y^e fift to y^e key will bee too flat. Or thus if y^e key bee f, a, b, or c make y^e distances twixt g & a, c & d to bee a minor tone if y^e key bee d or e make y^e distances from f to g & c to d a minor tone, but if it be {illeg}|g| make {a−}g a−g=d−c=g−f.

<143r>

13. Tis usuall to passe from{illeg} one moode to another in y^e midst of a song w^ch how & to w^t moode it may be done will appeare by y^e precedent scheme. For the 3 rankes may si{illeg}|g|nifie any {of y^e} three k Moodes w^ch have one common key, as F is the key of y^e first third & sixt Moode, {illeg}|G| y^e key of y^e first 2^d & 4^th mood &c: And wee may passe from any of those Moodes to another w^ch in y^t scheme have y^e same key. But this transition is better done from one key to y^e key next it, yⁿ to y^e remoter key. Neither may it bee done twixt any other Moodes {illeg} as twixt y^e fi{illeg}|r|st & fift \or 3^d & 4^th/ by reason of their great difference, w^ch would soe change y^e aire of y^e song as to make y^e ꝑts of it rather seeme divers songs.

14. It may app{illeg}|1|0) y^t if the key bee f, or b, or e, y^e the {tra} transition may be best done y^e moode \degrees of y^e Moode/ being ordered y^e first way. If y^e key bee {illeg}|a| or d y^e 2^d order is best. If y^e key bee g y^e 3^d order is best, & the fourth y^e key being c. But in generall, if y^e degrees bee ordered y^e {illeg}|4|^d|t|h way in y^e 2^d Moode & y^e 1^st way in all y^e rest, this transition will \may/ bee well {illeg}|d|one|.| {illeg}

15. from y^e consideration of pa{illeg}|s|sing from one moode to another in y^e same song \two other moodes may bee usefull/ y^e one whereof wan{illeg}|t|s y^e {illeg}|k|ey y^e other its fift, but these defects are ꝑly supplyed by their use in y^e former {illeg} ꝑte of y^e song, whereby such {an} \the {eare} retaining the/ imp{illeg}{illeg}|r|ession \of their {sweetnes}/ may be made in y^e song as thereby eare y^t made \the eares retaining the impression of/ their sweetness made {illeg}|b|y y^e former ꝑte of y^e song. q is y^e key of one moode & v y^e key's 5^t in |y^e| other moode.

<147r>

A Method whereby to find y^e areas of Those Lines w^ch can bee squared.

Prop: 1^st. If ab=x{illeg} \{illeg}/ ⊥y=be. cb=z. bd=v{illeg} secant=cd. m & n are numbers expressing y^e dimensions of x, y, or z. a, b, c, d,|&c:|are knowne quantitys, & $\frac{a x^{\frac{m}{n}}}{b} = z$ . yⁿ $\frac{m a z x^{\frac{m}{n}}}{n b x} =$ $= \frac{m a z x^{\frac{m - n}{n}}}{n b} = v$ . And in generall what ever y^e relation t{illeg}|w|ixt x & z bee, make all y^e termes equall to nothing, multiply each terme by so ma{illeg}|n|y times zz as x hath dimensions in y^t terme, for a Numerator: yⁿ multiply each {illeg}|t|erme by soe ma{illeg}|n|y times −x as z hath dimensions in y^t terme f{illeg}|o|r a denominator in y^e valor of v.

Prop: 2^d. If hi=r. & rv=zy. yⁿ hi & be describe equall) spaces hi{illeg}|g|k, {illeg}|o|r hiak & abef. that is abef=aik{h}

Prop: 3^d. If $a^{n} x^{m} = b^{n} y^{m}$ . Or $\frac{a x^{\frac{m}{n}}}{b} = y$ . yⁿ is $\frac{n x y}{n + m} =$ $\frac{n \times a \times x^{\frac{m + n}{n}}}{n b + m b} = abef$ y^e area of y^e line aef. And if $\frac{a}{b x^{\frac{m}{n}}} = y$ : yⁿ is $\frac{n x y}{n - m} = \frac{n a}{\bar{n - m} \times b x^{\frac{- n + m}{n}}} = abef = \frac{n a}{\bar{n - m} \times b x^{\frac{m - n}{n}}}$

Demonst{illeg}|ra|con.

For $\frac{a x^{\frac{m}{n}}}{b} = (sup) y$ . Therefore Suppose akhi is a paralllelogram {sic} & equall to $\frac{n a x^{\frac{m + n}{n}}}{n b + m b}$ . yⁿ is $\frac{n a x^{\frac{m + n}{n}}}{n b r + m b r} = ai = z$ . & (prop i) $\frac{a z^{2} x^{\frac{m + n}{n}}}{b r x z} = \frac{a z x^{\frac{m}{n}}}{b r} = v$ . & (prop 2^d) rv=zy. y^t is $\frac{a x^{\frac{m}{n}}}{b} = y$ .

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Prop: 4^th. If $y = a x^{m} + b x^{n}$ . yⁿ is $\frac{a x^{m + 1}}{m + 1} + \frac{b x^{n + 1}}{n + 1} = abef$

And in generall if y^e valor of y consists of severall termes so y^t x hath more y^e one \is not of divers/ dimensions in y^e denominator /{illeg}\ of any \no/ terme, are is not of divers dimensions in y^e denominator of any terme, yⁿ multiply each terme by x & divide it by y^e number of y^e dimensions of x, all those produc{illeg}|t|s shall bee y^e area{illeg} of y^e given line: supposeing al{illeg}|s|o y^t either none or all {illeg}|y|^e signes of those t{illeg}ermes are changed by this operation. For i{illeg}|f| some bee changed & others bee not y^e area is infinite \they proeed {sic} divers ways & joyne no{illeg}|t|/, & yⁿ y^e quantitys {illeg} y or x must be increased or diminished or otherwise altered.

The reason of this is|p|rop: is, y^t y^e {illeg}|a|rea described by {illeg}|y| is also described by {illeg}|it|s parts y^t is by y^e termes of its va{illeg}|l|or, & w^t areas those termes describe appeares by prop 3^d.

Prop 5^t. The progressions in this Table may bee designed by these g{illeg}|e|omet: lines. Whereby also any intermediate termes may bee found.

$\begin{matrix} \begin{matrix} \begin{matrix} a & b & b & b & b & b \end{matrix} \\ \begin{array}{r} 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . & 1 & . \\ - 2 & . & - 1 & . & 0 & . & 1 & . & 2 & . & 3 & . & 4 & . & 5 & . & 6 & . & 7 & . & 8 & . & 9 & . \\ 3 & . & 1 & . & 0 & . & 0 & . & 1 & . & 3 & . & 6 & . & 10 & . & 15 & . & 21 & . & 28 & . & 36 & . \\ - 4 & . & - 1 & . & 0 & . & 0 & . & 0 & . & 1 & . & 4 & . & 10 & . & 20 & . & 35 & . & 56 & . & 84 & . \\ 5 & . & 1 & . & 0 & . & 0 & . & 0 & . & 0 & . & 1 & . & 5 & . & 15 & . & 35 & . & 70 & . & 126 & . \\ - 6 & . & - 1 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 1 & . & 6 & . & 21 & . & 56 & . & 126 & . \\ 7 & . & 1 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 1 & . & 7 & . & 28 & . & 84 & . \\ - 8 & . & - 1 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 0 & . & 1 & . & 8 & . & 36 & . \end{array} \end{matrix} & \begin{array}{l} \sim & 1 = y . \\ \sim & x = y . \\ \sim & x x - x = 2 y . \\ \sim & x^{3} - 3 x x + 2 x = 6 y . \\ \sim & x^{4} - 6 x^{3} + 11 x x - 6 x = 24 y . \\ \sim & x^{5} - 10 x^{4} + 35 x^{3} - 50 x^{2} + 24 x = 120 y . \\ \sim & x^{6} - 15 x^{5} + 85 x^{4} - 225 x^{3} + 274 x x - 120 x = 720 y . \\ \sim & x^{7} - 21 x^{6} + 175 x^{5} - 735 x^{4} &c = 5040 y . \end{array} \end{matrix}$

The distance of y^e terme b from y^e terme a being called x. y & the quantity of y^t termes being y. & each terme being distant an unit from y^e next. The nature of w^ch table is such y^t y^e sum̄e of any figure & y^e figure above it is equall to y^e figure after it. & the nature of y^e lines are such y^t y^e precedent \any/ figure; multiplyed by y^e dimen number of dimensions of x in y^e first terme, being {illeg}|s|ubstracted from y^e figure following |it|, is equall to y^e figure under y^t following figure. And y^t y^e numbers of y may be deduced hence 1×2×3×4×5×6×7 &c.

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Prop 6^t. If $\frac{m}{n} = x$ . This Progression $\frac{n \times m \times \bar{m - n} \times \bar{m - 2 n} \times \bar{m - 3 n} \times \bar{m - 4 n} \times \bar{m - 5 n}}{n \times n \times 2 n \times 3 n \times 4 n \times 5 n \times 6 n}$ &c gives all y^e quantitys downward, in y^e preceding {illeg}|t|able. As if m=3. n=1. {illeg} the {illeg}|qu|antitys downward are $\frac{1}{1} . \frac{m}{n} . \frac{m \times m - n}{n \times 2 n}$ . $\frac{m \times \bar{m - n} \times \bar{m - 2 n}}{n \times 2 n \times 3 n} . \frac{m \times \bar{m - n} \times \bar{m - 2 n} \times \bar{m - 3 n}}{n \times 2 n \times 3 n \times 4 n}$ . &c y^t is 1. 3. 3. 1. 0. &c. So if $\frac{m}{n} = \frac{1}{2} = x$ . y^t $1 . \frac{1}{2} . \frac{-1}{8} . \frac{1}{16} . \frac{-5}{128} . \frac{7}{256}$ . &c. are y^e terms downward.

Prop {illeg}|7|^th. ${\bar{a + b}}}^{\frac{m}{n}} = a^{\frac{m}{n}} + \frac{m}{n} \times \frac{b}{a} \times a^{\frac{m}{n}} + \frac{m}{n} \times \frac{m - n}{2 n} \times \frac{b b}{a a} \times a^{\frac{m}{n}}$ .{As} $+ \frac{m}{n} \times \frac{m - n}{2 n} \times \frac{m - 2 n}{3 n} \times \frac{b^{3}}{a^{3}} \times a^{\frac{m}{n}}$ . &c As may bee deduced from $a^{\frac{m}{n}} \times \frac{m b}{n a} \times \frac{\bar{m - n} \times b}{2 n a} \times \frac{\bar{m - 2 n} \times b}{3 n a} \times \frac{\bar{m - 3 n} \times b}{4 n a} \times \frac{m b - 4 n b}{5 n a}$ {illeg}|&|c.

Prop 8^th. $\frac{c}{{\bar{a + b}}}^{\frac{m}{n}}} = \frac{c}{a^{\frac{m}{n}}}$ The truth of this Prop: appeareth by compareing it w^th y^e two former as also by calculation if $\frac{m}{n}$ is a whole & affirmative number, or b over lesse yⁿ an|a| unit

Prop 8th. ${\bar{a + b}}}^{\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} - \frac{m}{n} \times \frac{b}{a} \times \frac{1}{a^{\frac{m}{n}}} . - \frac{m}{n} \times \frac{- m - n}{2 n} \times \frac{b b}{a a}$ $\times \frac{1}{a^{\frac{m}{n}}} - \frac{m}{n} \times \frac{\bar{- m - n}}{2 n} \times \frac{- m - 2 n}{3 n} \times \frac{b^{3}}{a^{3}} \times \frac{1}{a^{\frac{m}{n}}}$ . &c. As may bee dedu{illeg}|c|ed from $\frac{1}{a^{\frac{m}{n}}} \times \frac{- m b}{n a} \times \frac{- m b - n b}{2 n a} \times \frac{- m b - 2 n b}{3 n a} \times \frac{- m b - 3 n b}{4 n a}$ &c.

The truth of this appeares also by y^e 5^t & 6^t proposition, or by calculation If a>b.

The truth of these two prop: is also thus demonstrated If ${\bar{a + b}}}^{\frac{1}{1}} = \frac{1}{a + b}$ I divide, 1 by a+b as in decimall {illeg}|f|ractions & find y^e quote $\frac{1}{a} - \frac{b}{a a} + \frac{b b}{a^{3}} - \frac{b^{3}}{a^{4}} + \frac{b^{4}}{a^{5}}$ &c as appeareth also by multiplying both ꝑts by a+b. So {illeg}if|I| extract y^e {no{illeg}|t|e} of a²+b as if they were decimall numbers & find $\sqrt{a^{2} + b} = a$ $+ \frac{b}{2 a} + \frac{- b b}{8 a^{3}} + \frac{b^{3}}{16 a^{5}}$ &c, as also may appeare by squareing both ꝑts

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\1./{illeg}|I|f two bodys {illeg}|c|, {illeg}|d| describe y^e streight li{illeg}|n|es {illeg}|a|c, bd, in y^e same time, (calling ac=x, bd=y, p=motion of c, q=motion of d) & if I have an equation expressing y^e relation of ac=x & bd=y whose termes are all put equall to nothing. I multiply y^e equation each terme of y^t equation by so many times py \{illeg}|o|r $\frac{p}{x}$ / as x hath dimensions in {illeg}|it|, & againe also by soe many times {illeg}|q|x \o{illeg}|r| $\frac{q}{y}$ / as {illeg}|y| hath dimensions in it. the sum̄e of these products is an equation expresing y^e motions relations of y^e motions of {illeg}|c| & {illeg} |d|. |Example if $a x^{3} + a^{2} y x - y^{3} x + y^{4} = 0$ yⁿ $a p q x^{3} + a p y y x -$ $3 a p x x + a^{2} p y - p y^{3} + a a q x - 3 q y y x + 4 q y^{3} = 0$ .|

2. If an equation expressing y^e relation of their motions bee given, tis more difficult & someti{illeg}|m|es Geometrically impossible, thereby to find y^e relation of y^e spaces described by those motions.

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If $a p x^{\frac{m}{n}} = q$ . _______then $\frac{n a}{m + n} x^{\frac{m + n}{n}} = y$ .

— — — — — — — — — — — — — — — — —

As if m=3. n=2. yⁿ $a p x^{\frac{3}{2}} = q$ , & $\frac{2 a}{3} x^{\frac{5}{2}} = y$ . Soe if $a p x^{\frac{−3}{2}} = q$ $= \frac{a p}{x^{\frac{3}{2}}}$ , yⁿ m=-3. n=2. & $\frac{2 a}{-1} x^{\frac{-1}{2}} = \frac{-2 a}{x^{\frac{1}{2}}} = y$ . If y^e valor of q{illeg} consisteth of severall such termes, consider each terme severall y. as if $a x + b x x = y$ . y^e first terme gives $\frac{a x^{2}}{2}$ y^e 2^d $\frac{b x^{3}}{3}$ . therefore $\frac{a x x}{2} + \frac{b x x x}{3} = y$ .

In generall multiply y^e valor of q|y| by x & divide {it} by y^e logra each terme of it by y^e logarithm{a}|e| of x, in y^t {illeg}|t|er{illeg}|m|e: if y^{e}|t| equation \valor of q/ consist of simple termes.

{illeg}

$\frac{\bar{m a x^{m}} + \bar{n b x^{n}} \times \bar{d x^{r} + e x^{s}} - \bar{r d x^{r} - s e x^{s}} \times a n^{m} + b x^{n}}{x \times d x^{r} + e x^{s} \times d x^{r} + e x^{s}} = \frac{q}{p}$ . $\frac{a x^{m} + b x^{n}}{d x^{r} + e x^{s}} = y$ . $\frac{- r d x^{r - 1}}{d d x^{2 r} + 2 d e x^{r} + e e} = \frac{q}{p}$ . $\frac{2}{d x^{r} + e} = y$ . $\frac{\bar{m - r} \times a d x^{m + r} + \bar{m - s} \times a e x^{m + s} + \bar{n - r} \times b d x^{n + r} + \bar{n - s} \times b e x^{n + s}}{x in d d x^{2 r} + 2 d e x^{r + s} + e e x^{2 s}} = \frac{q}{p} . \frac{a x^{m} + b x^{n}}{d x^{r} + e x^{s}} = y$ . $m b + \bar{3 n - 2 m} \times a x$ $\frac{m a + \bar{3 n - 2 m} \times b x^{n - m}}{x} \times \sqrt{a x^{m} + b x^{n}} = \frac{q}{p} . \bar{a + b x^{n - m}} \times \sqrt{a x^{m} + b x^{n}} = y$ . ____________ or thus $\frac{\bar{m a + \bar{3 n + m} \times b x^{n}}}{2 x} \times \sqrt{a x^{m} + b x^{n + m}} = \frac{q}{p} . \bar{a + b x^{n}} \sqrt{a x^{m} + b x^{n + m}} = y$ . $\frac{\bar{m m + 8 m n + 15 n n} \times d d x^{2 n} - \bar{2 m n - m m} \times e e}{x} \sqrt{e x^{m} + d x^{m + n}} = \frac{q}{p}$ . And ij $\bar{\bar{2 m + 6 n} \times d d x^{2 n} + 2 n d e x^{n} - \bar{2 m - 4 n} \times e e} \times \sqrt{e x^{m} + d x^{m + n}} = y$ . Or thus $\frac{\bar{3 m - 2 n} \times m a a x^{m - n} + \bar{3 n - 2 m} \times n b b x^{n - m}}{2 x} \sqrt{a x^{m} + b b x^{n}} = \frac{q}{p}$ . $\bar{m a a x^{m - n} + \bar{m - n} \times a b - n b^{n} x^{n - m}} \sqrt{a x^{m} + b x^{n}} = y$ .

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+ $\bar{\frac{m b - n b}{m} + a x^{n - m} - \frac{r b b}{m a} x^{m - n}} \times \sqrt{c x^{m} + d x^{n}} = y$ . And $\bar{3 r - 2 m} \times a x^{r - m} - \bar{3 m + 2 r} \times \frac{r b b}{m a} x^{m - r} \times \frac{\sqrt{c x^{m} + d x^{n}}}{2 x} = \frac{q}{p}$ .
$m a b - n a b + m a a x^{n - m} + n b b x^{m - n} \times \sqrt{c x^{m} + d x^{n}} = y$ . And $\bar{\bar{3 n - 2 m} \times n a a x^{n - m} + \bar{2 n - 3 m} \times m b b} \times \sqrt{a x^{n} - b x^{m}} = \frac{q.}{p}$ . $n a a x^{n - m} + \bar{m - n} \times a b - m b b x^{m - n} \sqrt{a x^{n} - b x^{m}} = y$ .

$\bar{m a a x^{m - n} + \bar{m - n} \times a b - n b b x^{n - m}} \sqrt{a x^{m} + b x^{n}} = y$ . And $\frac{3 m - 2 n \times m a a x^{m - n} + 3 n - 2 m \times n d b x^{n - m}}{2 x} \sqrt{a x^{m} + b x^{n}} = \frac{q}{p}$ .

$m a c + \bar{3 r - 2 m} \times a d x^{r - m} + \bar{3 m + 2 n} \times b c x^{m + n} + 3 r + 2 n \times b d x^{n + r} \frac{in \sqrt{c x^{m} + d x^{r}}}{2 x} = \frac{q}{p}$ . $\bar{a c + a d x^{r - m} + b c x^{m + n} + b d x^{r + n}} \times \sqrt{c x^{m} + d x^{r}} = y$ $\bar{3 m - 2 n} \times$ $\frac{\bar{3 m - 2 n} \times \bar{2 n - m} \times m d^{3} x^{m - n} + \bar{3 n - 2 m} \times \bar{5 n - 4 m} \times n e^{3} x^{2 n - 2 m}}{2 x} in \sqrt{d x^{m} + e x^{n}} = \frac{q}{p}$ {illeg} $\bar{3 m - 2 n} \times 2 n - m$ \ $\bar{2 n - m} \times m d^{3} x^{m - n} + \bar{2 n - m} \times \bar{m - n} \times e d d + \bar{n - m} \times n e e d x^{n - m} + \bar{3 n - 2 m} \times n e^{3} x^{2 n - 2 m}$ ×/ × $\sqrt{d x^{m} + e x^{n}} = y$ .

Or more ge{rda}lly generally, $\frac{\bar{3 m - 2 n} \times m c d d x^{m - n} + \bar{2 m - 3 n} \times n c e e x^{n - m} + \bar{3 m + 2 p} \times m b d d x^{m + p} + \bar{3 n + 2 p} \times b e d m x^{n + p}}{2 x}$ in|in| $\sqrt{d x^{m} + e x^{n}} = \frac{q}{p}$ . And $\bar{m d d c x^{m - n} + \bar{m - n} \times c d e -}$ $\bar{- n c e e x^{n - m} + m b d^{2} x^{p + m} + m b d e x^{n + p}} \sqrt{d x^{m} + e x^{n}} = y$ . $\frac{\bar{5 m - 2 n} \times \bar{2 m + n}}{m} \times e^{3} x^{2 m - n} \frac{\bar{+6 n - 3 m}}{4 n - m} \times 9 n d^{3} x^{2 n - m} in \sqrt{\frac{2 m + n}{m} \times e e x^{m} +}$ $\bar{+ \frac{9 n d d}{4 n - m} x^{2 n - m} - 3 d e x^{n}} = \frac{2 q x}{p} . \bar{And \frac{\bar{2 m + n}}{m} \times e^{3} x^{2 m - n} \frac{\bar{+n - m}}{m} \times e e d x^{m} +}$ $\bar{\frac{+m - n}{4 n - m} \times 3 d d e x^{n} \frac{+9 n d^{3}}{4 n - m} \times x^{2 n - m}} in \sqrt{\frac{2 m + n}{m} \times e e x^{m} - 3 d e x^{n} + \frac{9 n d d}{4 n - m} \times x^{2 n - m}} =$ $= y$ .

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$\frac{\bar{5 n - 2 m} \times \bar{2 m - 5 n} \times 3 m n n b^{5}}{\bar{2 n + m} \times \bar{16 n^{4} - 8 n n m m + m^{4}}} \times x^{3 m - 2 n} \frac{\bar{+2 m - 5 n} \times 3 m n b^{4} c}{\bar{2 n + m} \times \bar{4 n n - m m}} x^{2 m - n}$ $\frac{\bar{+2 n - 2 m} \times \bar{5 n - 2 m} \times n b^{3} c c}{\bar{2 n + m} \times \bar{4 n n - m m}} x^{m} \frac{\bar{+2 n - 2 m}}{2 n + m} \times b b c^{3} x^{n} \frac{\bar{+ m - n}}{5 n - 2 m} \times b c^{4} x^{2 n - m}$ $\frac{+3 m - 6 n}{5 n - 2 m} \times c^{5} x^{3 n - 2 m} in \sqrt{\frac{5 n - 2 m \times n b b}{4 n n - m m} x^{m} + b c x^{n} + c c x^{2 n - m}} =$ $= y$ . And, $\frac{\bar{7 m - 4 n} \times \bar{5 n - 2 m} \times \bar{2 m - 5 n} \times 3 m n n b^{5}}{\bar{2 n + m} \times \bar{16 n^{4} - 8 m m n n + m^{4}}} \times x^{3 m - 2 n}$ $\frac{\bar{+4 m - n} \times \bar{2 m - 5 n \times 3 m n b^{4} c}}{\bar{2 n + m} \times \bar{4 n n - m m}} x^{2 m - n} \frac{\bar{+8 n - 5 m} \times \bar{3 m - 6 n}}{5 n - 2 m} \times c^{5} x^{3 n - 2 m}$ $in \sqrt{\frac{\bar{5 n - 2 m} \times n b b}{4 n n - m m} x^{m} + b c x^{n} + c c x^{2 n - m}} = \frac{2 q x}{p}$ .

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sit ab=x. bc=y. d{illeg}|f|=z, de=v. {illeg} ______ $\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} The area abc of y^{e} \\ line whose nature is \end{matrix} & \begin{matrix} is equall to y^{e} area fde \\ of y^{e} line whose nature is \end{matrix} & \begin{matrix} supposeing y^{e} re= \\ lation twixt ab & \\ fd to bee \end{matrix} \end{matrix} \\ \begin{matrix} 2 x x \sqrt{c + d x x} = y . & \sqrt{c z + d z z} = v . & x x = z . \\ \sqrt{c x + d x x} = y . & a \sqrt{c a z + d a a z z} = v . & a x = z . \\ \frac{-1}{x^{3}} \sqrt{c x + d} = y . & \sqrt{c z + d z z} = v . & 1 = z x . \\ \frac{-2}{x^{5}} \sqrt{c x x + d} = y . & \sqrt{c z + d z z} = v . & 1 = z x^{2} . \\ \frac{-3}{x^{7}} \sqrt{c x^{3} + d} = y . & \sqrt{c z + d z z} = v . & 1 = z x^{3} . \\ 3 x^{3} \sqrt{c x + d x^{4}} = y . & \sqrt{c z + d z z} = v . & x^{3} = z . \\ 4 x^{5} \sqrt{c + d x^{4}} = y . & \sqrt{c z + d z z} = v . & x^{4} = z . \\ \frac{1}{2 x} \sqrt{c x^{\frac{3}{2}} + d x x} = y . & \sqrt{c z + d z z} = v . & x = z z . \\ In generall \\ n x^{n - 1} \times \sqrt{c x^{n} + d x^{n + n}} = y . & \sqrt{c z + d z z} = v . & x^{n} = z . \end{matrix} \end{matrix} \\ \begin{matrix} 2 x \sqrt{c + d x^{4}} = y . & \sqrt{c + d z z} = v . & x x = z . \\ 3 x x \sqrt{c + d x^{6}} = y . & \sqrt{c + d z z} = v . & x^{3} = z . \\ \frac{-1}{x^{3}} \sqrt{c x x + d} = y . & \sqrt{c + d z z} = v . & 1 = z x . \\ \frac{-2}{x^{5}} \sqrt{x^{4} + d} = y . & \sqrt{c + d z z} = v . & 1 = z x x . \\ \frac{1}{2 x} \sqrt{c x + d x x} = y . & \sqrt{c + d z z} = v . & x = z z . \\ \frac{3}{2} \sqrt{c x + d x^{4}} = y . & \sqrt{c + d z z} = v . & x^{3} = z z . \\ \frac{- 1}{2 x x} \sqrt{c x + d} = y . & \sqrt{c + d z z} = v . & 1 = x z z . \\ \frac{- 3}{2 x^{4}} \sqrt{c x^{3} + d} = y . & \sqrt{c + d z z} = v . & 1 = x^{3} z z . \\ In generall \\ n x^{n - 1} \sqrt{c + d x^{n + n}} = y . & \sqrt{c + d z z} = v . & x^{n} = z . \end{matrix} \end{matrix}$

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$\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} The area abc of y^{e} \\ line ∼. \end{matrix} & \begin{matrix} is equall to y^{e} area \\ of y^{e} line ∼. \end{matrix} & \begin{matrix} Supposeing y^{t} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} 2 x \sqrt{c + d x x + e x^{4}} = y . & \sqrt{c + d z + e z z} = v . & x x = z . \\ 3 x x \sqrt{c + d x^{3} + e x^{6}} = y . & \sqrt{c + d z + e z^{2}} = v . & x^{3} = z . \\ 4 x^{3} \sqrt{c + d x^{4} + e x^{8}} = y . & \sqrt{c + d z + e z z} = v . & x^{4} = z . \\ \frac{-1}{x^{3}} \sqrt{c x x + d x + e} = y . & \sqrt{c + d z + e z z} = v . & 1 = z x . \\ \frac{-2}{x^{5}} \sqrt{c x^{4} + d x x + e} = y . & \sqrt{c + d z + e z z} = v . & 1 = z x x . \\ \frac{-3}{x^{7}} \sqrt{c x^{6} + d x^{3} + e} = y . & \sqrt{c + d z + e z z} = v . & 1 = z x^{3} . \\ \frac{1}{2 x} \sqrt{c x + d x^{\frac{3}{2}} + e x x} = y . & \sqrt{c + d z + e z z} = v . & x = z z . \\ \frac{-1}{2 x x} \sqrt{c x + d x^{\frac{1}{2}} + e} = y . & \sqrt{c + d z + e z z} = v . & 1 = z z x . \\ In generall. \\ n x^{n - 1} \sqrt{c + d x^{n} + e x^{n + n}} = y . & \sqrt{c + d z + e z z} = v . & x^{n} = z . \end{matrix} \\ \begin{matrix} \frac{b}{a + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x = z . \\ \frac{2 b x}{a + b x x} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x^{2} = z . \\ \frac{3 b x x}{a + b x^{3}} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x^{3} = z . \\ \frac{4 b x^{3}}{a + b x^{4}} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x^{4} = z . \\ \frac{- b}{a x x + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a x + b = z x . \\ \frac{- 2 b}{a x^{3} + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{2} + b = z x^{2} . \\ \frac{- 3 b}{a x^{4} + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{3} + b = z x^{3} . \\ \frac{1}{2 a \sqrt{x} + 2 b x} = y . & ————— & \frac{1}{z} = v . & ————— & a + b \sqrt{x} = z . \\ \frac{3 x b}{2 a \sqrt{x} + 2 b x x} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x^{\frac{3}{2}} = z . \\ \frac{- b}{a x \sqrt{x} + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a + \frac{b}{\sqrt{x}} = z . \\ \frac{- 3 b}{a x x \sqrt{x} + b x} = y . & ————— & \frac{1}{z} = v . & ————— & a + \frac{b}{x \sqrt{x}} = z . \end{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \frac{a + 2 b x}{a x + b x x} = y . & \frac{1}{z} = v . & a x + b x^{2} = z . \\ \frac{a + 3 b x x}{a x + b x^{3}} = y . & \frac{1}{z} = v . & a x + b x^{3} = z . \end{matrix} \end{matrix}$

<157r>

$\begin{matrix} \begin{matrix} \frac{a x x z - b}{a x^{3} + b x x} = y . & ————— & \frac{1}{z} = v . & ————— & a a x + b = z x . \\ \frac{2 a + 3 b x}{a x + b x x} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{2} + b x^{3} = z . \\ \frac{2 a + 4 b x x}{a x + b x^{3}} = y . & ————— & \frac{1}{z} = v . & ————— & a x x + b x^{4} = z . \\ \frac{2 a + 5 b x 3}{a x + b x^{4}} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{2} + b x^{5} = z . \\ \frac{3 a + 4 b x}{a x^{} + b x^{2}} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{3} + b x^{4} = z . \\ \frac{3 a + 5 b x x}{a x^{} + b x^{3}} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{3} + b x^{5} = z . \\ \frac{4 a + 5 b x}{a x^{} + b x x} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{4} + b x^{5} = z . \\ \frac{4 a + 6 b x x}{a x + b x^{3}} = y . & ————— & \frac{1}{z} = v . & ————— & a x^{4} + b x^{5} = z . \\ \frac{- a + b x x}{a x + b x^{3}} = y . & ————— & \frac{1}{z} = v . & ————— & a + b x^{2} = x z . \\ \frac{- a + 2 b x^{4}}{a x x + b x^{5}} = y . & \frac{1}{z} = v . & ————— & a + b x^{3} = x x z . \\ \frac{- 2 a - b x}{a x + b x x} = y . & \frac{1}{z} = v . & ————— & a + b x^{} = x x z . \\ In generall. \\ \frac{m a d x^{m - 1} + n b d x^{n - 1}}{a x^{m} + b x^{n}} = y . & \frac{d}{z} = v . & a x^{m} + b x^{n} = z . \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix}$

note {illeg}|y|^t these are compounde{illeg}|d| onely of y^e first simplest Areas:

<157v>

$\begin{matrix} \begin{matrix} \begin{matrix} The area abc of y^{e} \\ line \end{matrix} & \begin{matrix} is equall to y^{e} \\ area fde of y^{e} \\ line. \end{matrix} & \begin{matrix} Supposeing that \end{matrix} \end{matrix} \\ \begin{matrix} \frac{x x}{\sqrt{d x x - d c}} = y . & \sqrt{c + d z z} = v . & \sqrt{\frac{x x}{d} - \frac{c}{d}} = z . \\ \frac{x}{2 \sqrt{d x x - d c x}} = y . & \sqrt{c + d z z} = v . & \sqrt{\frac{x - c}{d}} = z . \\ \frac{-1}{2 x x \sqrt{d - c d x}} = y . & \sqrt{c + d z z} = v . & \sqrt{\frac{1 - c x}{d x}} = z . \\ \frac{-1}{x^{3} \sqrt{d - c d x x}} = y . & \sqrt{c + d z z} = v . & \sqrt{\frac{1 - c x x}{d x x}} = z . \\ Or generally. \\ \frac{s x^{3 s - 1}}{\sqrt{d x^{2 s} - c d}} = y . & \sqrt{c + d z z} = v . & \sqrt{\frac{x^{2 s} - c}{d}} = z . \end{matrix} \\ \begin{matrix} \frac{b c c + 2 b b c d x + b^{3} d d x x}{\sqrt{2 b c x + b b d x x}} = y . & \sqrt{c c + d z z} = v . & \sqrt{2 c b x + d b b x x} = z . \\ \frac{2 b c c + 4 b b c d x x + 2 b^{3} d d x^{4}}{\sqrt{2 b c + b b d x x}} = y . & \sqrt{c c + d z z} = v . & x \sqrt{2 b c + b b d x x} = z . \\ \frac{- b c c x x - 2 b b c d x - b^{3} d d}{x^{3} \sqrt{2 b c x + b b d}} = y . & \sqrt{c c + d z z} = v . & \sqrt{2 c b x + b b d} = z x . \\ \frac{- 2 b c c x^{4} - 4 b b c d x x - 2 b^{3} d d}{x^{5} \sqrt{2 b c x x + b b d}} = y . & \sqrt{c c + d z z} = v . & \sqrt{2 b c x x + b b d} = z x x . \\ In generall \\ \frac{m b c c x^{m} + 2 m b b c d x^{2 m} + m b^{3} d d x^{3 m}}{x \sqrt{2 b c x^{m} + d b b x^{2 m}}} = y . & \sqrt{c c + d z z} = v . & \sqrt{2 b c x^{m} + b b d x^{2 m}} = z . \end{matrix} \\ \begin{matrix} \frac{b \sqrt{c + a d + b d x}}{2 \sqrt{a + b x}} = y . & \sqrt{c + d z z} = v . & \sqrt{a + b x} = z . \\ \frac{b x \sqrt{c + a d + b d x x}}{\sqrt{a + b x x}} = y . & \sqrt{c + d z z} = v . & \sqrt{a + b x x} = z . \\ \frac{- b \sqrt{c x^{2} + a d x x + b d x}}{2 x x \sqrt{a x x + b x}} = y . & \sqrt{c + d z z} = v . & \sqrt{a + \frac{b}{x}} = z . \\ \frac{- b \sqrt{c x x + a d x x + b d}}{x^{3} \sqrt{a x x + b}} = y . & \sqrt{c + d z z} = v . & \sqrt{a + \frac{b}{x x}} = z . \\ In generall. \\ \frac{m a x^{m} + n b x^{n}}{2 x \sqrt{a x^{m} + b x^{n}}} \times \sqrt{c + a d x^{m} + b d x^{n}} = y . & \sqrt{c + d z z} = v . & \sqrt{a x^{m} + b x^{n}} = z . \end{matrix} \end{matrix}$

<158r>

$\begin{matrix} \frac{- c e}{d \sqrt{e - c x}} \\ \frac{- c e x}{d \sqrt{e x x - c}} = e x = y . & \sqrt{c + d d z z} = v . & \frac{r - c x}{r x x - c d d} = z . \end{matrix}$ $\begin{matrix} \begin{matrix} \frac{e b \sqrt{a c + d e e + c b x}}{\bar{2 a + 2 b x} \times a + b x} = y . & \sqrt{c + d z z} = v . & \frac{e}{\sqrt{a + b x}} = z . \\ \frac{e b x \sqrt{a c + d e e + c b x x}}{\bar{a + b x x} a + b x x} = y . & \sqrt{c + d z z} = v . & \frac{e}{\sqrt{a + b x x}} = z . \\ \frac{- e b \sqrt{a c x x + d e e x x + c b x}}{\bar{2 a x^{2} + 2 b x} a x x + b x} = y . & \sqrt{c + d z z} = v . & \frac{e x}{\sqrt{a x x + b x}} = z . \\ \frac{- e b \sqrt{a c x x + d e e x x + c b}}{\bar{a x^{3} + b x} a x x + b} = y . & \sqrt{c + d z z} = v . & \frac{e x}{\sqrt{a x x + b}} = z . \\ \frac{4 a^{4} c d x x + 4 a a b c d + b b c d}{\bar{2 a a x x + 2 b x} 4 a^{4} d d x x + 4 a a b d d x} = y . & \sqrt{c c + d d z z} = v . & \frac{c b}{2 a d \sqrt{a a x^{2} + b x}} = z . \\ \frac{a^{4} c d x^{4} + 4 a a b c d x x + b b c d}{\bar{a a x^{3} + b x} \times 2 a d a a x^{4} + b x x} = y . & \sqrt{c c + d d z z} = v . & \frac{c b}{2 a d x \sqrt{a^{2} x^{2} + b}} = z . \\ \frac{-4 a^{4} c d - 4 a a b c d x - b b c d x x}{\bar{2 a a x + 2 b x x} \times 2 a d a a + b x} = y . & \sqrt{c c + d d z z} = v . & \frac{c b x}{2 a d \sqrt{a a + b x}} = z . \\ \frac{-4 a^{4} c d x^{2} - 4 a a b c d x^{4} - b b c d x^{6}}{\bar{a a + b x^{2}} \times 2 a d a a + b x x} = y . & \sqrt{c c + d d z z} = v . & \frac{c b x x}{2 a d \sqrt{a a + b x x}} = z . \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} In g enerall. \\ \frac{\bar{e m a x^{m}} \bar{+ e n b x^{n}} \sqrt{c a x^{m} + c b x^{n} + d e e}}{2 a x^{m + 1} + 2 b x^{n + 1} \times a x^{m} + b x^{n}} = y . & \sqrt{c + d z z} = v . & \frac{e}{\sqrt{a x^{m} + b x^{n}}} = z . \end{matrix} \\ \begin{matrix} \frac{e m a x^{m - 1} + e n b x^{n - 1} \sqrt{c a x^{m} + c b x^{n} + d e e}}{2 a a x^{2 m} + 4 a b x^{m + n} + 2 b b x^{2 n}} = y . & \sqrt{c + d z z} = v . & \frac{e}{\sqrt{a x^{m} + b x^{n}}} = z . \end{matrix} \end{matrix} \\ \begin{matrix} \frac{c e b^{3} x x}{a a + 2 a b x x + b b x^{4}} = y . & \sqrt{c c - \frac{a c c}{e e} z z} = v . & \frac{e}{\sqrt{a + b b x x}} = z . \\ In g enerall. \\ \frac{n b^{3} c e x^{\frac{3 n - 2}{2}}}{2 a a + 4 a b x^{n} + 2 b b x^{2 n}} = y . & \sqrt{c c \frac{- a c c}{e e} z z} = v . & \frac{e}{\sqrt{a + b b x^{n}}} = z . \end{matrix} \end{matrix} \end{matrix}$

<159r>

$\begin{matrix} \begin{matrix} \begin{matrix} c b x \frac{c b}{a + b x} = y . & ————— & c + z v = 0 . & ————— & \frac{1}{a + b x} = z . \\ \frac{2 b c x}{a + b x x} = y . & c + z v = 0 . & \frac{1}{a + b x x} = z . \\ \frac{3 b c x x}{a + b x^{3}} = y . & c + z v = 0 . & \frac{1}{a + b x^{3}} = z . \\ \frac{c b}{a x x + b x} = y . & ————— & c = z v . & ————— & \frac{x}{a x + b} = z . \\ \frac{c b}{2 a x^{3} + 2 b x} = y . & ————— & c = z v . & ————— & \frac{x x}{a x^{2} + b} = z . \\ As before, & In generall, \\ \frac{c m a x^{m - 1} + n c b x^{n - 1}}{a x^{m} + b x^{n}} = y . & ————— & c + z v = 0 . & ————— & \frac{1}{a x^{m} + b x^{n}} = z . \\ Also \\ \frac{\bar{r d x^{r - 1} + s e x^{s - 1}} \times \bar{a x^{m} + b x^{n}}}{a x^{m} + b x^{n} \times d x^{r} + e x^{s}} & \frac{- m a x^{m - 1} - n b x^{n - 1}}{a x^{m} + b x^{n}} = y . & c = z v . & \frac{d x^{r} + e x^{s}}{a x^{m} + b x^{n}} = z . \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \frac{c r d x^{r - 1} + c s e x^{s - 1}}{d x^{r} + e x^{s}} & \frac{- c m a x^{m - 1} - c n b x^{n - 1}}{a x^{m} + b x^{n}} = y . & c = z v . & \frac{d x^{r} + e x^{s}}{a x^{m} + b x^{n}} = z . \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \frac{-9 a c}{2 b x^{4} + 2 c x} = y . & a = z v . & \bar{b x^{3} + c} \times \sqrt{b x^{4} + c x} = x^{5} z . \\ \frac{-3 a c}{b x^{3} + c x} = y . & a = z v . & \bar{b x x + c} \sqrt{b x x + c} = x^{3} z . \\ \frac{-3 a c}{2 b x x + 2 c x} = y . & a = z v . & \bar{b x + c} \sqrt{b x x + c x} = x x z . \\ \frac{3 a c}{2 b + 2 c x} = y . & a = z v . & b + c x \sqrt{b + c x} = z . \\ \frac{3 a c x}{b + c x x} = y . & a = z v . & b + c x x \sqrt{b + c x x} = z \\ As be f ore was found, I n generall \\ \frac{\bar{3 m + 2 r} \times a b x^{m + r} + \bar{3 n + 2 r} \times a c x^{n + r}}{2 b x^{m + r + 1} + 2 c x^{n + r + 1}} = y . & a = z v . \\ And & \bar{b x^{m + r} + c x^{n + r}} \times \sqrt{b x^{m} + c x^{n}} = z . \end{matrix} \end{matrix}$

<160r>

$\begin{matrix} \begin{matrix} x x = a y . & ————— & 1 = v . & ————— & x^{3} = 3 a z . \\ x^{3} = a y . & ————— & 1 = v . & ————— & x^{4} = 4 a z . \\ x^{4} = a y . & ————— & 1 = v . & ————— & x^{5} = 5 a z . \\ a = x x y . & ————— & 1 = v . & ————— & - a = x z . \\ a = x^{3} y . & ————— & 1 = v . & ————— & - a = 2 x x z . \\ x^{3} = a y y . & ————— & 1 = v . & ————— & 4 x^{5} = 25 a z z . \\ a = x^{3} y y . & ————— & 1 = v . & ————— & 4 a = z z x . \\ In generall \\ a x^{m} = y . & ————— & 1 = v & ————— & \frac{a}{m + 1} x^{m + 1} = z . \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix}$

Tha{illeg}|t| is.
multiply y^e valor of {illeg}|y|. by x, & each divide y^t each terme in y^t valor by soe many units as x hath dimensions in y^t terme, y^e product is y^e area.

$2 d - e \sqrt{}$ $\begin{matrix} \begin{matrix} \frac{c}{b b + 2 b c x + c c x x} = y . & 1 = v . & \frac{1}{b + c x} = z . \\ \frac{2 c x}{b b + 2 b c x x + c c x^{4}} = y . & 1 = v . & \frac{1}{b + c x x} = z . \\ \frac{3 c x x}{b b + 2 b c x^{3} + c c x^{6}} = y . & 1 = v . & \frac{1}{b + c x^{3}} = z . \\ \frac{- c}{b b x x + 2 b c x + c c} = y . & 1 = v . & \frac{x}{b x + c} = z . \\ \frac{- 2 c x}{b b x^{4} + 2 b c x x + c c} = y . & 1 = v . & \frac{x x}{b x x x + c} = z . \\ In generall \\ \frac{n c x^{n - 1}}{b b + 2 b c x^{n} + c c x^{2 n}} = y . & 1 = v . & \frac{1}{b + c x^{n}} = z . \end{matrix} \\ \begin{matrix} \frac{b + 2 c x}{b b x x + 2 b c x^{3} + c c x^{4}} = y . & 1 = v . & \frac{1}{b x + c x x} = z . \\ \frac{b + 3 c x x}{b b x x + 2 b c x^{4} + c c x^{6}} = y . & 1 = v . & \frac{1}{b x + c x^{3}} = z . \\ \frac{2 b + 3 c x}{b b x^{3} + 2 b c x^{4} + c c x^{5}} = y . & 1 = v . & \frac{1}{b x x + c x^{}} = z . \\ In generall \\ \frac{m b x^{m - 1} + n c x^{n - 1}}{b b x^{2 m} + 2 b c x^{m + n} + c c x^{2 n}} = y . & 1 = v . & \frac{1}{b x^{m} + c x^{n}} = z . \end{matrix} \end{matrix}$

cui eodem modo

<160v>

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{- b + c x x}{b b + 2 b c x x + c c x^{4}} = y . & 1 = v . & \frac{x}{b + c x x} = z . \\ \frac{- b x x - 2 c x}{b b x x + 2 b c x + c c} = y . & 1 = v . & \frac{x x}{b x + c} = z . \\ \frac{- b x^{4} - 3 c x x}{b b x^{4} + 2 b c x x + c c} = y . & 1 = v . & \frac{x^{3}}{b x x + c} = z . \\ \frac{- 2 b x^{3} - 3 c x x}{b b x x + 2 b c x + c c} = y . & 1 = v . & \frac{x^{3}}{b x + c} = z . \\ \frac{- 2 b x^{5} - 4 c x^{3}}{b b x^{4} + 2 b c x^{2} + c c} = y . & 1 = v . & \frac{x^{4}}{b x x + c} = z . \\ \frac{c d - e b}{d d + 2 e d x + e e x x} = y . & 1 = v . & \frac{b + c x}{d + e x} = z . \\ \frac{c d - 2 e b x}{d d + 2 e d x x + e e x^{4}} = y . & 1 = v . & \frac{b + c x}{d + e x x} = z . \\ \frac{2 c d - 2 e b x x}{d d + 2 e d x x + e e x^{4}} = y . & 1 = v . & \frac{b + c x x}{d + e x x} = z . \end{matrix} \\ \begin{matrix} In generall \\ \frac{\bar{m - r} \times b d x^{m + r} + \bar{m - s} \times b e x^{m + s} + \bar{n - r} \times c d x^{n + r} + \bar{n - s} \times c e x^{n + s}}{d d x^{2 r + 1} + 2 e d x^{r + s + 1} + e e x^{2 s + 1}} = y . & 1 = v . & \frac{b x^{m} + c x^{n}}{d x^{r} + e x^{s}} = z . \end{matrix} \end{matrix} \\ \begin{matrix} \frac{- 3 c}{2 x x \sqrt{b x^{4} + c x}} = y . & 1 = v . & b x^{3} + c = z^{2} x^{3} . \\ \frac{- c}{x x \sqrt{b x x + c}} = y . & 1 = v . & b x x + c = z^{2} x x . \\ \frac{- c}{2 x \sqrt{b x x + c x}} = y . & 1 = v . & b x + c = z^{2} x . \\ \frac{c}{2 \sqrt{b + c x}} = y . & 1 = v . & b + c x = z z . \\ \frac{c x}{\sqrt{b + c x x}} = y . & 1 = v . & b + c x x = z z . \\ \frac{3 c x x}{\sqrt{b + c x^{3}}} = y . & 1 = v . & b + c x^{3} = z z . \\ \frac{b x^{4} - 3 c}{2 x x \sqrt{b x^{5} + c x}} = y . & 1 = v . & b x^{4} + c = z^{2} x^{3} . \\ \frac{b x^{3} - 2 c}{2 x x \sqrt{b x^{3} + c}} = y . & 1 = v . & b x^{3} + c = z^{2} x x . \end{matrix} \end{matrix}$

<161r>

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{b x x - c}{2 x \sqrt{b x^{3} + c x}} = y . & 1 = v . & b x x + c = z^{2} x . \\ \frac{b + 2 c x}{2 \sqrt{b x + c x x}} = y . & 1 = v . & b x + c x x = z z . \\ \frac{b + 3 c x x}{2 \sqrt{b x + c x^{3}}} = y . & 1 = v . & b x + c x^{3} = z z . \\ \frac{b + 4 c x^{3}}{2 \sqrt{b x + c x^{4}}} = y . & 1 = v . & b x + c x^{4} = z z . \\ \frac{2 b x^{3} - c}{2 x \sqrt{b x^{4} + c x}} = y . & 1 = v . & b x^{3} + c = z z x . \\ \frac{2 b + 3 c x}{2 \sqrt{b + c x}} = y . & 1 = v . & b x x + c x^{3} = z z . \\ \frac{b + 4 c x x}{2 \sqrt{b + c x x}} = y . & 1 = v . & b x x + c x^{4} = z^{2} . \\ In generall \\ \frac{m b a x^{m} + n a c x^{n}}{2 x \sqrt{b x^{m} + c x^{n}}} = y . & a = v . & \sqrt{b x^{m} + c x^{n}} = z . \\ Also m o re generally. \\ \frac{m a b x^{m} + n a c x^{n} + r a d x^{r}}{2 x \sqrt{b x^{m} + c x^{n} + d x^{r}}} = v . & a = v . & \sqrt{b x^{m} + c x^{n} + d x^{r}} = z . \end{matrix} \\ \begin{matrix} \frac{- a c}{\bar{b x x + c} \sqrt{b x x + c}} = y . & a = v . & \frac{x}{\sqrt{b x x + c}} = z . \\ \frac{- a c}{\bar{2 b x + 2 c} \sqrt{b x x + c x}} = y . & a = v . & \frac{\sqrt{x}}{\sqrt{b x + c}} = z . \\ \frac{a c}{\bar{2 b + 2 c x} \sqrt{b + c x}} = y . & a = v . & \frac{1}{\sqrt{b + c x}} = z . \\ \frac{a c x}{\bar{b + c x x} \sqrt{b + c x x}} = y . & a = v . & \frac{1}{\sqrt{b + c x x}} = z . \\ \frac{a b + 2 a c x}{2 b x + 2 c x x \sqrt{b x + c x x}} = y . & a = v . & \frac{1}{\sqrt{b x + c x x}} = z . \\ In generall \\ \frac{m a b x^{m - 1} + n a c x^{n - 1}}{2 b x^{m} + 2 c x^{n} \times \sqrt{b x^{m} + c x^{n}}} = y . & a = v . & \frac{1}{\sqrt{b x^{m} + c x^{n}}} = z . \end{matrix} \end{matrix} \end{matrix}$

<161v>

$\begin{matrix} \begin{matrix} \frac{3}{2} \sqrt{b + c x} = y . & 1 = v . & \bar{b + c x} \sqrt{b + c x} = z . \\ 3 a c x \sqrt{b + c x x} = y . & a = v . & \bar{b + c x x} \sqrt{b + c x x} = z . \\ \frac{9 a c x x}{2} \sqrt{b + c x^{3}} = y . & a = v . & \bar{b + c x^{3}} \times \sqrt{b + c x^{3}} = z . \\ \frac{-3 a c}{2 x^{3}} \sqrt{b x x + c x} = y . & a = v . & \bar{b x + c} \sqrt{b x x + c x} = z x . \\ \frac{- 3 a c \sqrt{b x x + c}}{x^{4}} = y . & a = v . & \bar{b x x + c} \sqrt{b x x + c} = z x x x . \end{matrix} \\ \begin{matrix} In generall \\ \frac{3 n a c x^{n - 1}}{2} \times \sqrt{b + c x^{n}} = y . & a = v . & \bar{b + c x^{n}} \times \sqrt{b + c x^{n}} = z . \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{-3 a c \sqrt{b x^{3} + c x}}{x^{3}} = y . & a = v . & \frac{\bar{b x^{2} + c}}{x^{5}} \sqrt{b x^{3} + c x} = z . \\ \frac{3 a c x}{2} \sqrt{b x + c x x} = y . & a = v . & \bar{b + c x} \times \sqrt{b x + c x x} = z x x . \\ 3 a c x x \sqrt{b x + c x^{3}} = y . & a = v . & \bar{b + c x x} \times \sqrt{b x + c x^{3}} = z x x . \\ \frac{3 a c x^{3}}{2} \sqrt{b + c x} = y . & a = v . & \bar{b + c x} \times \sqrt{b + c x} = z x^{3} . \\ 3 a c x^{4} \sqrt{b + c x x} = y . & a = v . & \bar{b + c x x} \times \sqrt{b + c x x} = z x^{3} . \\ \frac{3 a c x^{4}}{2} \sqrt{b x + c x x} = y . & a = v . & \bar{b + c x} \times \sqrt{b x + c x x} = z x^{5} . \\ 3 a c x^{5} \sqrt{b x + c x^{3}} = y . & a = v . & \bar{b + c x x} \sqrt{b x + c x^{3}} = z x^{5} . \\ \frac{3 a c \sqrt{b x + c x^{3}}}{x} = y . & a = v . & \bar{b x + c x^{3}} \times \sqrt{b x + c x^{3}} = z . \\ \frac{-3 a c}{2 x^{4}} \times \sqrt{b x + c} = y . & a = v . & \bar{b x + c} \times \sqrt{b x + c} = z . \\ \frac{-3 a c}{x^{6}} \sqrt{b x^{3} + c x} = y . & a = v . & b x^{2} + c \times \sqrt{b x^{3} + c x} = z x x . \\ \frac{-3 a c}{2 x^{6}} \sqrt{b x x + c x} = y . & a = v . & \bar{b x x + c x} \times \sqrt{b x x + c x} = z . \end{matrix} \\ \begin{matrix} In generall \\ \frac{\bar{3 n - 3 m} \times c x^{n - 1}}{2} \times \sqrt{b x^{m} + c x^{n}} = y . & a = v . & \bar{b x^{\overset{—}{2 m}} + c x^{n - 3 m}} \times \sqrt{b x^{m} + c x^{n}} = z . \end{matrix} \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{\bar{2 b a + 5 a c x}}{2} \sqrt{b + c x} = y . & a = v . & \bar{b x + c x x} \sqrt{b + c x} = z . \\ \bar{a b + 4 a c x x} \sqrt{b + c x x} = y . & a = v . & \bar{b x + c x^{3}} \times \sqrt{b + c x x} = z . \\ \frac{\bar{2 a b x - a c}}{2 x x} \times \sqrt{b x x + c x} = y . & a = v . & \bar{b x + c} \times \sqrt{b x^{2} + c x} = z x \end{matrix} \\ \begin{matrix} In generall \\ \frac{\bar{3 m + 2 r} \times b x^{m + r} + \bar{3 n + 2 r} \times c x^{n + r}}{2 x} \times a \sqrt{b x^{m} + c x^{n}} = y . & a = v . \\ and \\ \bar{b x^{m + r} + c x^{n + r}} \times \sqrt{b x^{m} + c x^{n}} = z \end{matrix} \end{matrix} \end{matrix}$

<162r>

$\begin{matrix} \begin{matrix} \begin{matrix} And more generally \\ \frac{\bar{2 m + r} \times b d x^{m + r} + \bar{2 m + s} \times b e x^{m + s} + \bar{2 n + r} \times c d x^{n + r} + \bar{2 n + s} \times c e x^{n + s}}{2 x \sqrt{d x^{r} + e x^{s}}} \times a = y . \end{matrix} \\ \begin{matrix} a = v . & \bar{b x^{m} + c x^{n}} \times \sqrt{d x^{r} + e x^{s}} = z . \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \frac{3 c d x}{2 \sqrt{d x + e}} = y . & 1 = v . & \bar{\frac{-2 c e}{d} + c x} \times \sqrt{d x + e} = z . \\ \frac{3 c d x^{3}}{\sqrt{d x x + e}} = y . & 1 = v . & \bar{\frac{-2 c e}{d} + c x x} \sqrt{d x x + e} = z . \\ \frac{-3 c d}{2 x x \sqrt{d x + e x x}} = y . & 1 = v . & \bar{\frac{-2 c e x}{d} + c} \times \sqrt{d x + e x x} = z x x . \\ \frac{-3 c d}{x^{4} \sqrt{d + e x x}} = y . & 1 = v . & \bar{\frac{-2 c e x x}{d} + c} \times \sqrt{d + e x x} = z x . \end{matrix} \\ \begin{matrix} In generall \\ \frac{\bar{3 m + 3 n} \times c d x^{3 m + 2 n}}{2 x \sqrt{d x^{3 m + n} + e x^{2 m}}} = y . & 1 = v . & \bar{c x^{n} - \frac{2 c e}{d} x^{- m}} \times \sqrt{d x^{3 m + n} + e x^{2 m}} = z . \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \frac{\bar{5 a b + 5 b b x} \times \sqrt{a + b x}}{2} = y . & 1 = v . & \bar{a a + 2 a b x + b b x x} \sqrt{a + b x} = z . \\ 5 a b x + 5 b b x^{3} \sqrt{a + b x x} = y . & 1 = v . & \bar{a a + 2 a b x^{2} + b b x^{4}} \sqrt{a + b x x} = z . \\ \frac{\bar{5 a a x + 10 a b x x + 5 b b x^{3}} \sqrt{a x + b x x}}{2} = y . & 1 = v . & \bar{a^{2} x^{2} + 2 a b x^{3} + b b x^{4}} \sqrt{a x + b x x} = z . \\ \frac{\bar{-5 a b x - 5 b b} \sqrt{a x x + b x}}{2 x^{3}} = y . & 1 = v . & \bar{a a x x + a b x + b b} \sqrt{a x x + b x} = z x^{2} . \\ In generall \end{matrix} \\ \begin{matrix} \frac{5 m a a x^{2 m} + \bar{5 m + 5 n} \times a b x^{m + n} + 5 n b b x^{2 n}}{2 x} \sqrt{a x^{m} + b x^{n}} = y . & 1 = v . \\ And & \bar{a a x^{2 m} + 2 a b x^{m + n} + b b x^{2 n}} \times \sqrt{a x^{m} + b x^{n}} = z . \end{matrix} \end{matrix} \\ \begin{matrix} a x \sqrt{b + c x} = y . & \frac{a}{5 c} = v . & \bar{b c c x x + 2 b c x - 4 b b} \times \sqrt{b + c x} = z . \end{matrix} \end{matrix}$

<162v>

$\begin{matrix} \begin{matrix} \begin{matrix} \frac{+ 15 a e e}{x^{6} \sqrt{d x x + e}} = y . & a = v . & -3 e e + 4 d e x^{2} - 8 d^{2} x^{4} \sqrt{d x x + e} = x^{5} z e . \\ \frac{+ 15 a e e}{2 x^{3} \sqrt{d x x + e x}} = y . & a = v . & -3 e e + e x - 8 d^{2} x x \sqrt{d x x + e x} = x^{3} z e . \\ \frac{+ 15 a e e x x}{2 \sqrt{d + e x}} = y . & a + v = 0 . & -3 e e x^{2} + 4 d e x - 8 d d \sqrt{d + e x} = z e . \\ \frac{+ 15 a e e x^{5}}{\sqrt{d + e x x}} = y . & a + v = 0 . & -3 e e x^{4} + 4 d e x^{2} - 8 d^{2} \sqrt{d + e x x} = z e . \end{matrix} \\ \begin{matrix} In generall \\ \frac{15 n a e e x^{3 n - 1}}{2 \sqrt{d + e x^{n}}} = y & a = v . & And \\ -3 e e x^{2 n} - 4 e d x^{n} + 8 d d \sqrt{d + e x^{n}} = z e . \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} 15 d d x^{} \sqrt{d x + e} = y . & 1 = v . & \bar{6 d d x x + 2 d e x - 4 e e} \sqrt{d x + e} = z . \\ 60 d d x^{3} \sqrt{d x x + e} = y . & 1 = v . & \bar{12 d d x^{4} + 4 d e x x - 8 e e} \sqrt{d x x + e} = z . \\ \frac{-15 d d}{x^{4}} \sqrt{d x + e x x} = y . & 1 = v . & \frac{\bar{-6 d d - 2 d e x + 4 e e x x} \sqrt{d x + e x x}}{x^{3}} = z . \\ \frac{-15 d d \sqrt{d + e x x}}{x^{6}} = y . & 1 = v . & \frac{\bar{-3 d d - d e x x + 2 e e x^{4}}}{x^{5}} \sqrt{d + e x x} = z . \end{matrix} \\ \begin{matrix} In generall \\ \frac{15 n d d x^{2 n}}{x} \sqrt{d x^{n} + e} = y . & 1 = v . & \bar{6 d d x^{2 n} + 2 d e x^{n} - 4 e e} \sqrt{d x + e} = z . \end{matrix} \end{matrix} \\ \begin{matrix} \frac{24 d d x x - 3 e e}{x} \sqrt{d x x + e x} = y . & 1 = v . & \begin{matrix} 8 d d x x \\ +2 d e x \\ -6 e e \end{matrix}\} \times \sqrt{d x x + e x} = z . \\ 77 d d x^{4} - 5 e e \sqrt{d x + \frac{e}{x}} = y . & 1 = v . & \begin{matrix} 14 d d x^{4} \\ +4 d e x x \\ -10 e e \end{matrix}\} \times \sqrt{d x^{3} + e x} = z . \\ \frac{8 d d + e e x x}{x^{3}} \sqrt{d + e x} = y . & 1 = v . & \bar{-4 d d - 2 d e x + 2 e e x^{2}} \sqrt{d + e x} = x x z . \\ \frac{45 d d + 3 e e x^{4} \sqrt{d x + e x^{3}}}{x^{6}} = y . & 1 = v . & \begin{matrix} -10 d d \\ -8 d e x x \\ +6 e e x^{4} \end{matrix}\} \sqrt{d x + e x^{3}} = x^{5} z . \end{matrix} \end{matrix}$

<163r>

$\begin{matrix} \frac{32 d d + 4 e e x^{4}}{x^{5}} \sqrt{d + e x x} = y . & 1 = v . & \begin{matrix} -8 d d \\ -4 d e x x \\ +4 e e x^{4} \end{matrix}\} \sqrt{d + e x x} = x^{4} z . \\ \bar{35 d d x x - 8 e e} \sqrt{d x + e} = y . & 1 = v . & \begin{matrix} 10 d d x x \\ +2 d e x \\ -6 e e \end{matrix}\} \sqrt{d x + e} = z . \\ \bar{96 d d x^{4} - 12 e e} \sqrt{d x x + e} = y . & 1 = v . & \begin{matrix} \bar{16 d d x^{4} +4 d e x^{2}} \\ \underline{-12 e e} \end{matrix}\} \sqrt{d x x + e} = z . \\ \frac{21 d d + 3 d e e x^{4}}{x^{5}} \sqrt{d x + e x^{3}} = y . & 1 = v . & \begin{matrix} -12 d d \\ -4 d e x x \\ \underline{+2 e e x^{4}} \end{matrix}\} \sqrt{d x + e x^{3}} = z x^{4} . \\ \bar{48 d d x x - 15 e e} \sqrt{d x x + e x} = y . & 1 = v . & \begin{matrix} \bar{8 d d x x} \\ +2 d e x \\ \underline{-10 e e} \end{matrix}\} \sqrt{d x^{4} + e x^{3}} = z . \\ \bar{117 d d x^{4} - 21 e e} \sqrt{d x^{3} + e x} = y . & 1 = v . & \begin{matrix} 18 d d x^{4} + 4 d e x x \\ \underline{-14 e e} \end{matrix}\} \sqrt{d x^{5} + e x^{3}} = z . \\ \frac{12 d}{x^{4}} \sqrt{d + e x x} = y . & v = 1 . & \bar{-4 d - 4 e x x} \sqrt{d + e x x} = z x^{3} . \\ \frac{- d d - 8 e e x x}{x x} \sqrt{d x + e x x} = y . & \bar{2 d d - 2 d e x - 4 e e x^{2}} \sqrt{d x + e x^{2}} = z . \\ \bar{63 d d x^{3} - 24 e e x} \sqrt{d x + e} = y . & v = 1 . & \begin{matrix} 14 d d x x \\ +2 d e x \\ -12 e e \end{matrix}\} \sqrt{d x^{5} + e x^{4}} = z . \\ \bar{80 d d x^{3} - 35 e e x} \sqrt{d x x + e x} = y . & v = 1 . & \begin{matrix} 16 d d x \\ +2 d e x \\ -14 e e \end{matrix}\} \sqrt{d x^{6} + e x^{5}} = z . \\ \frac{3 d d - 24 e e x x}{x^{}} \sqrt{d x + e x x} = y . & v = 1 . & \bar{6 d d - 2 d e x - 8 e e x^{2}} \sqrt{d x^{} + e x^{} x} = z . \\ \bar{99 d d x^{4} - 42 e e x x} \sqrt{d x + e} = y . & v = 1 . & \begin{matrix} 18 d d x^{5} + 2 d e x^{4} \\ - 16 e e x x x \end{matrix}\} \sqrt{d x + e} = z . \\ \bar{8 d d - 35 e e x x} \sqrt{d + e x} = y . & v = 1 . & \bar{8 d d x - 2 d e x x - 10 e e x} \sqrt{d + e x} = z . \\ \bar{120 d d x^{4} - 63 e e x x} \sqrt{d x x + e x} = y . & v = 1 . & \bar{20 d d x^{5} + 6 d e x^{4} - 18 e e x^{3}} \sqrt{x x + e x} = z . \\ \frac{-4 d d - 32 e e x^{4}}{x x} \sqrt{d + e x x} = y . & v = 1 . & \bar{4 d d - 4 d e x x - 16 e e x^{4}} \sqrt{d + e x x} = z x . \\ \frac{\bar{24 d d - 3 e e x x} \sqrt{d + e x}}{x^{4}} = y . & v = 1 . & -8 d d - 2 d e x - 6 e e x x \sqrt{d + e x} = z x^{3} . \end{matrix}$

<163v>

In Generall.

$\bar{m m + 8 m n + 15 n n} \times d d x^{2 n - 1} - \bar{m m - 2 m n}$ d $e e x^{- 1} in \sqrt{d x^{m + n} + e x^{m}} = y$ . $1 = v$ . &, $\bar{2 m + 6 n} \times d d x^{2 n} + 2 n d e x^{n} - \bar{2 m - 4 n} \times e e in \sqrt{d x^{m + n} + e x^{m}} =$ {illeg} $z$ .

^[1] prop 12. 13 & I think 11 are trew onely mechanically.