Mathematical Notebook

<2r>

Of the extraction of Pure Square Cubick. Square-square & square-cubick rootes &c.

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

Then out of the figures of the first point next the left hand extract the greatest roote proper to the power of the number & set that downe in the Quotient which is the first side & is called A. (as the roote quintuplicate of 5708 is $\left(5\right)$, & $\left(5\right)$ quintuplicate is 3125 ) then takeing that roote duely multiplied out of the number (as 3125 out of 5708 ) with the rest of the numbers to the next point. seeke the seacond side which is found by divideing that number by another number made out of the first side (which is called the Divisor) & this second side I name E. (thus by divideing 258363524 by $5\mathrm{A}$qq$+10\mathrm{A}c+10\mathrm{A}q+5\mathrm{A}$ after such a maner that $5\mathrm{A}qq\mathrm{E}+10\mathrm{A}c\mathrm{E}q+10\mathrm{A}q\mathrm{E}c+5\mathrm{A}\mathrm{E}qq+\mathrm{E}qc$ may be conteined in the number the product of that division shall be E =

<2v>

The extraction of the square roote

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

The extraction of the cube roote

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

The extraction of the square square roote

$\begin{array}{cc}\hfill \text{The square-square}3\underset{\text{.}}{3}& 177\underset{\text{.}}{6}\text{(24}\hfill \\ \hfill \text{The square-squ: to be subduc:}16& =\mathrm{A}qq\\ \hfill \text{Remainder.}17& 1776\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Divisors for finding}{\text{y}}^{\text{e}}\\ \text{seacond side E.}\end{array}& \{\begin{array}{cc}\phantom{0}& 3\\ \phantom{0}& \phantom{0}\\ \phantom{0}& \phantom{0}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}2& \phantom{0}\\ 24& \phantom{0}\\ \phantom{0}\phantom{0}8& \phantom{0}\end{array}& \begin{array}{c}4\mathrm{A}c\hfill \\ 6\mathrm{A}q\hfill \\ 4\mathrm{A}\hfill \end{array}\end{array}\hfill \\ \hfill \text{Theire su}\overline{\text{m}}\text{e}\phantom{0}3& 448\phantom{0}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Squ-Squares to be sub=}\\ \text{=ducted}\end{array}& \{\begin{array}{cc}& 12\\ & 3\\ & \phantom{0}\\ & \phantom{0}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}8& \\ 84& \\ 512\phantom{0}& \\ \phantom{0}256& \end{array}& \begin{array}{c}4\mathrm{A}c\mathrm{E}\hfill \\ 6\mathrm{A}q\mathrm{E}q\hfill \\ 4\mathrm{A}\mathrm{E}c\hfill \\ \mathrm{E}qq\hfill \end{array}\end{array}\hfill \\ \hfill \text{Theire Su}\overline{\text{m}}\text{e}17& 1776\hfill \end{array}$

<3r>

The extraction of the Square-Cube roote

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

Note that the 3^d 4^th 5^th & other figures are found by the same manner that the seacond figure is found onely makeing all the figures found to stand for A the first side & the figure sought for e or the 2^d side

And if roote is found inexpressible in whole numbers then adding ciphers & pointing them from the unite towards the right {kind} as was before explained & soe hold on the worke in decimalls.

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

If the Number bee of 6.7.8.9.10 &c dimensions The roote may be extracted after the same manner

<4r>

Of the Extraction of Rootes in Affected powers.

The manner of the extraction of rootes in pure & affected powers is very much alike, especially when the affected powers are decently prepared, that is, when theire affections are not over large & those altogether either affirmative or negative, & the power affirmative, affirmations & negations so mixt that there be noe ambiguity & all fractions & Asymmetry taken away

All the figures in the coefficients & affected power are to be pointed (after the manner before explained in the Analisis of pure powers) according to the degree of theire dimensions & the worke onely differs from that in pure powers in that the coefficients enter into the divisors

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

The analysis of Cubick Equations.

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

<4v>

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

But the Coëfficient maybe greater than the Power soe that it cannot be substracted from it which argues that the Cube more propperly affects than is affected. In this case the coëfficient must descend towards the unite soe many points untill it may be substracted, & soe many points as the coëfficient is devolved soe many pricks must be blotted out towards the left hand in the power affected. As the Example shews
$\mathrm{L}c+95400\mathrm{L}$ $=1819459$.

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

<5r>

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

If there be many coefficients in the equation each must be placed according to this rule.

Sometimes the coefficient is under a negative sine as $\mathrm{L}c-10\mathrm{L}=13584$ & the Analysis is as follows

$\begin{array}{cc}\hfill \text{Coëfficiens planum}\phantom{0}-& 1\underset{\text{.}}{0}\phantom{0}\text{sublaterale}\hfill \\ \hfill \text{Cubus resolvendus}+1\underset{.}{3}& 58\underset{.}{4}\text{(24}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Solida ablativa}\end{array}& \{\begin{array}{cc}& +\phantom{0}8\\ \phantom{0}& -\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}\phantom{0}& \\ 20& \end{array}& \begin{array}{c}\mathrm{A}c\hfill \\ \mathrm{A}\mathrm{C}q\hfill \end{array}\end{array}\hfill \\ \hfill \text{Su}\overline{\text{m}}\text{a}+\phantom{0}7& 80\phantom{0}\hfill \\ \hfill \text{Restat}+\phantom{\underset{.}{0}}5& 78\underset{.}{4}\text{resolvendum}\hfill \\ \hfill \text{Divisor}\overline{\text{u}}{\underset{\_}{\text{p}}}^{\text{s}}\text{superior}\phantom{+\phantom{0}0}& -10\text{coëfficiens planum}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Divisor}\overline{\text{u}}{\underset{\_}{\text{p}}}^{\text{s}}\text{inferior}\end{array}& \{\begin{array}{cc}& +\phantom{0}1\\ & +\phantom{01}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}2\phantom{0}& \\ \phantom{0}6& \end{array}& \begin{array}{c}+3\mathrm{A}\mathrm{A}\hfill \\ +3\mathrm{A}\hfill \end{array}\end{array}\hfill \\ \hfill \text{Suma divisor}\overline{\text{u}}+\phantom{0}1& 25\phantom{0}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Solida ablativa}\end{array}& \{\begin{array}{cc}& +\phantom{0}4\\ & +\phantom{0}\\ & +\phantom{0}\\ \phantom{0}& -\phantom{0}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}8\phantom{0}\phantom{0}& \\ 96\phantom{0}& \\ \phantom{0}64& \\ \phantom{0}40& \end{array}& \begin{array}{c}3\mathrm{A}\mathrm{A}\mathrm{E}\hfill \\ 3\mathrm{A}\mathrm{E}\mathrm{E}\hfill \\ \mathrm{E}\mathrm{E}\mathrm{E}\hfill \\ \mathrm{E}\mathrm{C}\mathrm{C}\hfill \end{array}\end{array}\hfill \\ \hfill \text{Eor}\overline{\text{u}}\text{ su}\overline{\text{m}}\text{a}+\phantom{0}5& 784\hfill \\ \end{array}$

But sometimes the square coëfficient hath more paires of figures than the cube to be analysed, hath & then there is præfixing so many ciphers to the cube as figures are wanting, the first side will not much differ from the square roote of the coefficient. as $\mathrm{L}c-116620\mathrm{L}=352947$

$\begin{array}{ccc}\hfill \begin{array}{cc}\begin{array}{c}\end{array}& \begin{array}{ccc}-& 1& \underset{\phantom{.}}{1}\end{array}\end{array}& \begin{array}{ccc}6& 6& \underset{\phantom{.}}{2}\end{array}& \begin{array}{cc}\begin{array}{ccc}\underset{.}{0}& \underset{.}{\phantom{0}}& \underset{.}{\phantom{0}}\end{array}& \begin{array}{l}\text{Coefficiens plan}\overline{\text{u}}\end{array}\end{array}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Cubus resolvendus}\end{array}& \begin{array}{ccc}\phantom{+}& 0& \underset{\phantom{.}}{0}\end{array}\end{array}& \begin{array}{ccc}3& 5& \underset{\phantom{.}}{2}\end{array}& \begin{array}{cc}\begin{array}{ccc}9& 4& \underset{.}{7}\end{array}& \begin{array}{l}\text{(343}\end{array}\end{array}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Sollida Ablativa}\end{array}& \{\begin{array}{ccc}+& 2& 7\\ -& 3& 4\end{array}\end{array}& \begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\\ 9& 8& 6\end{array}& \begin{array}{cc}\begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{l}\mathrm{A}c\\ \mathrm{A}\mathrm{C}q\end{array}\end{array}\hfill \\ \hfill \begin{array}{cc}\begin{array}{c}\text{Restat auferend}\overline{\text{u}}\end{array}& \begin{array}{ccc}\phantom{0}& -& 7\end{array}\end{array}& \begin{array}{ccc}9& 8& 6\end{array}& \begin{array}{cc}\begin{array}{ccc}\phantom{0}& \phantom{0}& \phantom{0}\end{array}& \begin{array}{l}\end{array}\end{array}\hfill \\ \hfill \begin{array}{rr}\begin{array}{c}\text{Reliquum resolvendi}\end{array}& \begin{array}{ccc}\phantom{0}& +& 8\end{array}\end{array}& \begin{array}{ccc}3& 3& 8\end{array}& \begin{array}{cc}\begin{array}{ccc}9& 4& 7\end{array}& \begin{array}{l}\text{Cubi}\end{array}\end{array}\hfill \\ \phantom{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}& \phantom{\_\_\_\_\_\_\_\_\_\_}& \phantom{\_}\end{array}$

<5v>

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

Sometimes though there be as many 2 figures in the coefficient as 3 figures in the cube affected yet the coëfficient may be so greate as to deceive an unwary Analist As in this $\mathrm{L}c-6400\mathrm{L}=153000$. where the roote of 64 is 8 which cubed is 512 which added to 153 makes 665 then whose roote the number immediately greater is 9 which is the first side $=\mathrm{A}$.

But if the coefficient had beene affirmative, then not the aggregate of the facts but the difference must be taken as in this. $\mathrm{L}c+64\mathrm{L}=1024$.

Since the roote of 64 is 8 . which cubed is 512 . & $1024-512=$ $512$. the roote of which is $8=\mathrm{A}$. The like is observable in equations of higher powers

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

☞

Suppose in the former cubick equation the lesse roote be 12. then $\frac{155520}{12}=12960$. or else $13104-12\times 12=12960$. & $\mathrm{L}q+12\mathrm{L}=12960$. where $\mathrm{L}=108$ is the greater roote.

And in the latter equation if the greater roote be 27. & $\frac{217944}{27}=8072$, c. or $-27\times 27\times 27$ $+$ $27755=8072$. $27\times 27=729$. If there be 4 cubes continually proportionall whose greate extreame is $27c=19683$. & the aggregate of the 3 rest is 8072 & $\mathrm{L}c$ the lesse extreame, therefore $\mathrm{L}c+27\mathrm{L}q+729$ $\mathrm{L}$ $=8072$. the roote of which .is 8 the other roote of the equation

☞ Or haveing one roote of an equation the Equation may be lessened by division thus $13104\mathrm{l}-\mathrm{l}c$ $=$ $155520$ or ${\mathrm{l}}^3$ $-13$ $1$ $04\mathrm{l}+155520=0$. & one roote is 12 . therefore divide this equation by $\mathrm{l}-12$ & the Quote is an equation conteining the other roote viz: $\mathrm{l}q+12\mathrm{l}=12960$.

<7v>

<8r>

Propositiones Geometricæ. Franc: Vietæ.

prop 1

$\mathrm{ab}:\mathrm{ac}\colon\colon \mathrm{ce}:\mathrm{bd}$

prop 2

& if $\mathrm{ab}:\mathrm{ac}\colon\colon \mathrm{ac}:\mathrm{bd}$: then $\mathrm{ac}:\mathrm{ab}\colon\colon \mathrm{ab}:\mathrm{ce}$

prop 3. If $\mathrm{ab}\times \mathrm{ac}=\mathrm{bd}\times \mathrm{ce}$. then $\mathrm{bd}:\mathrm{ac}\colon\colon \mathrm{ac}:\mathrm{ab}\colon\colon \mathrm{ab}:\mathrm{ce}$ ∺

prop 3. To find two meane proportionalls {twixt} Bc & IK . On the center a with the radius ai describe the circle ibck . inscribe b c $=\mathrm{cd}$. draw da through the center & bg parallel to it. draw hk through A soe that $\mathrm{gh}=\mathrm{ab}=\text{(}=\mathrm{ai}\text{)}$. & $\mathrm{ik}:\mathrm{hb}\colon\colon \mathrm{hb}:\mathrm{hi}\colon\colon \mathrm{hi}:\mathrm{bc}$. ∺

Prop: 4

If $\mathrm{ad}=\mathrm{db}=\mathrm{cb}$. then the Angle c be is tripple to the Angle abd .

Prop 5

If $\mathrm{ab}=\mathrm{bd}=\mathrm{Rad}$. $3Ang:\mathrm{bad}=\mathrm{cde}$

Prop 6

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

Prop 7

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

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

prop 9 If $\mathrm{ah}=\mathrm{hb}=\mathrm{bf}=\mathrm{fd}$. & $\mathrm{ch}=2\mathrm{eh}$ or $\mathrm{ceh}=3\mathrm{hce}$. then $\begin{array}{}{\mathrm{ac}}^3=3\mathrm{ac}\times {\mathrm{ah}}^2-\mathrm{db}\times {\mathrm{ah}}^2\\ \text{\&}{\mathrm{cb}}^3=3\mathrm{cb}\times {\mathrm{ah}}^2-\mathrm{db}\times {\mathrm{ah}}^2\end{array}\}\boxed{{\mathrm{Z}}^3=\mathrm{a}\mathrm{a}\mathrm{z}-{\mathrm{b}}^3\text{.}}$

<8v>

Prop 10

If $\mathrm{de}=\mathrm{ea}$ & $\mathrm{db}:\mathrm{da}\colon\colon \mathrm{ab}\times \mathrm{ab}:\mathrm{dc}\times \mathrm{dc}$. then be is a side of a 7 equall sided & angled figure. or $7\mathrm{eab}=4\mathrm{right\; angles}$.

prop 10

If $\mathrm{ac}=\mathrm{ef}$. & aef a right angle & ab passes through the center then $\mathrm{cd}:\mathrm{de}\colon\colon \mathrm{de}:\mathrm{df}\colon\colon \mathrm{df}:\mathrm{db}$. And if $\mathrm{cd}:\mathrm{de}\colon\colon \mathrm{de}:\mathrm{df}\colon\colon \mathrm{df}:\mathrm{db}$ then ae is perpendicular to ef . $2\mathrm{hd}$ is the difference of the extreames & $2\mathrm{do}$ is the difference of the meanes. which given the proportionall lines may be found &c.

prop 11

Pseudomesolabium wherby To find 2 meane proportionalls. If, $\mathrm{ae}:\mathrm{ec}\colon\colon \mathrm{ec}:\mathrm{ed}\colon\colon \mathrm{ed}:\mathrm{eb}$. they be inscribed in the circle acbd the diam : being $\mathrm{ae}+\mathrm{eb}$. If twixt g i & i h two meane proportionalls are sought on the same center f with the Rad : $\frac{\mathrm{gi}+\mathrm{ih}}{2}$ describe gkhl & inscribe a line kl parallel to cd cutting ab in the point i & $\mathrm{gi}:\mathrm{ki}\colon\colon \mathrm{ki}:\mathrm{il}\colon\colon \mathrm{il}:\mathrm{ih}$.      Examine it.

^[1]

prop 12

If $\mathrm{do}=\mathrm{dh}$ & ac bisected in b & bd bee drawne rd is the side of a pentagon which may be inscribed in defcro

prop 13.

If rd be the side of a $\left\{\begin{array}{c}\text{octogon}\\ \text{decagon}\end{array}\right\}$ & pd the side of an $\left\{\begin{array}{c}\text{hexagon}\\ \text{octogon}\end{array}\right\}$ the arch rp divided in o , od will be the side of an $\left\{\begin{array}{c}\text{heptagon}\\ \text{enneagon}\end{array}\right\}$ to be inscribed in the circle ord & the arch RP is rightly divided by Bisecting the Line ac . Examine it

<9r>

Of Angular sections.

prop 14

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

prop 15

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

prop 16.

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

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

Prop 17. If {∠} $\mathrm{ab}=\mathrm{bc}=\mathrm{ce}=\mathrm{eg}$ &c: & $\mathrm{ac}=\mathrm{cd}$. & $\mathrm{ae}=\mathrm{ef}$ &c then $\mathrm{ab}:\mathrm{ac}\colon\colon \mathrm{ac}:\mathrm{ad}\colon\colon \mathrm{ae}:\mathrm{af}$ &c & $\mathrm{ed}=\mathrm{ab}$ & $\mathrm{ac}=\mathrm{gf}$ &c. {nam} triangle cde & cba , efg & eac &c: = & sim.

Prop.18.

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

<9v>

Prop 19

If $\mathrm{fa}=\mathrm{ab}=\mathrm{be}=\mathrm{eh}$ &c. &. $\mathrm{af}+\mathrm{ab}+\mathrm{be}+\mathrm{eh}$ are greater than the semiperiphery: & dh is the greatest, db the least line drawn from d to these points $\mathrm{a}$, $\mathrm{b}$, $\mathrm{e}$, $\mathrm{h}$. then $\mathrm{rad}:$ $\mathrm{dh}$ $\colon\colon \mathrm{db}:\mathrm{da}-\mathrm{de}$.

Prop 20

Out of the 18^th & 19^th Propositions To divide An angle into any number of points in the figure of the 18^th prop: $\mathrm{al}=\mathrm{diam}=$ $2$ z . ah is the greatest of the inscribed lines $=\mathrm{B}$: now z ∶ $\mathrm{B}\colon\colon \mathrm{ah}+$ $2\mathrm{z}$. therfore $\mathrm{b}\mathrm{b}=\mathrm{ah}in\mathrm{z}+2{\mathrm{z}}^2$. & $\frac{\mathrm{b}\mathrm{b}-2\mathrm{z}\mathrm{z}}{\mathrm{z}}=\mathrm{ah}$. And $\mathrm{z}:\mathrm{B}\colon\colon \frac{{\mathrm{b}}^2-2{\mathrm{z}}^2}{\mathrm{z}}:\mathrm{b}+\mathrm{ag}$ . therfore $\frac{{\mathrm{b}}^3-2\mathrm{z}\mathrm{z}\mathrm{b}}{\mathrm{z}\mathrm{z}}-\mathrm{b}=\mathrm{ag}$ Likewise $\frac{{\mathrm{b}}^4-4\mathrm{z}\mathrm{z}\mathrm{b}{\mathrm{b}}^2+2{\mathrm{z}}^4}{{\mathrm{z}}^3}=\mathrm{ad}$. & $\frac{{\mathrm{B}}^5-5\mathrm{z}\mathrm{z}{\mathrm{B}}^3+5{\mathrm{z}}^4\mathrm{B}}{{\mathrm{z}}^4}=\mathrm{ab}$ $\frac{{\mathrm{B}}^6-6\mathrm{z}\mathrm{z}{\mathrm{B}}^4+9{\mathrm{z}}^4\mathrm{B}\mathrm{B}-2{\mathrm{z}}^6}{\mathrm{z}\mathrm{z}\mathrm{z}\mathrm{z}\mathrm{z}}=\mathrm{a\delta }$
$\frac{{\mathrm{B}}^7-7\mathrm{z}\mathrm{z}{\mathrm{B}}^5+14{\mathrm{z}}^4{\mathrm{B}}^3-7{\mathrm{z}}^6\mathrm{B}}{{\mathrm{z}}^6}=\text{to a seaventh line}$
$\frac{{\mathrm{B}}^8-8\mathrm{z}\mathrm{z}{\mathrm{b}}^6+20{\mathrm{z}}^4{\mathrm{b}}^4-16{\mathrm{z}}^6\mathrm{b}\mathrm{b}+2{\mathrm{z}}^8}{{\mathrm{z}}^7}=\text{to an eight}$ $\text{line}\genfrac{}{}{0ex}{}{\phantom{0^0}}{\phantom{0^0}}$ $\frac{{\mathrm{B}}^9-9\mathrm{z}\mathrm{z}{\mathrm{b}}^7+27{\mathrm{z}}^4{\mathrm{b}}^5-30{\mathrm{z}}^6{\mathrm{b}}^6+9{\mathrm{z}}^8\mathrm{b}}{{\mathrm{z}}^8}=\text{a}$ $\text{nineth line}\genfrac{}{}{0ex}{}{\phantom{0^0}}{\phantom{0^0}}$ $\frac{{\mathrm{B}}^{10}-10\mathrm{z}\mathrm{z}{\mathrm{b}}^8+35{\mathrm{z}}^4{\mathrm{b}}^6-50{\mathrm{z}}^6{\mathrm{b}}^4+25{\mathrm{z}}^8\mathrm{b}\mathrm{b}-2{\mathrm{z}}^{10}}{{\mathrm{z}}^9}=$ tenth &

Prop 21

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

<10r>

Prop 22.

If $\mathrm{aq}=\mathrm{ab}=\mathrm{bd}=\mathrm{dc}=\mathrm{ch}=\mathrm{hk}=\mathrm{kl}=\mathrm{lf}$. Then $\mathrm{GK\; Rad}:\mathrm{kl}\colon\colon \mathrm{kl}:\mathrm{el}(=\mathrm{al}-\mathrm{ah})\colon\colon$ $\mathrm{hl}:\mathrm{hm}(=\mathrm{qh}-\mathrm{qd}=$ $\mathrm{ak}-\mathrm{ac})\colon\colon \mathrm{dl}:\mathrm{do}(=\mathrm{qd}-\mathrm{qa}=\mathrm{ac}-\mathrm{ab})\colon\colon \mathrm{lc}:\mathrm{cn}(=\mathrm{qc}-\mathrm{qb}=$ $\mathrm{ah}-\mathrm{ad}$ &c ) Soe that the Periph: divided into any number of points. $\mathrm{gl}:\mathrm{lk}\colon\colon \mathrm{lk}:\mathrm{al}-\mathrm{ak}\colon\colon \mathrm{lh}:\mathrm{ak}-\mathrm{ac}\colon\colon \mathrm{lc}:\mathrm{ah}-\mathrm{ad}\colon\colon \mathrm{dl}:\mathrm{ac}-\mathrm{ab}$ &c. & $\mathrm{gl}:\mathrm{lk}\colon\colon \mathrm{ah}:\mathrm{lc}-\mathrm{lk}\colon\colon \mathrm{ac}:\mathrm{ld}-\mathrm{lh}\colon\colon \mathrm{ad}:\mathrm{lb}-\mathrm{lc}\colon\colon \mathrm{ab}:\mathrm{al}-\mathrm{ah}$ &c.

hence Prop 23.

In the former scheame If al $=2\mathrm{x}=\text{hypotenusa}$. $\mathrm{kl}=\mathrm{b}$. $\mathrm{x}:\mathrm{b}\colon\colon \mathrm{b}:2\mathrm{x}-\mathrm{ah}$ & $\frac{-\mathrm{b}\mathrm{b}+2\mathrm{x}\mathrm{x}}{\mathrm{x}}=\mathrm{ah}$ / $\mathrm{x}:\mathrm{b}\colon\colon \frac{-\mathrm{b}\mathrm{b}+2\mathrm{x}\mathrm{x}}{\mathrm{x}}:\frac{2\mathrm{b}\mathrm{x}\mathrm{x}-{\mathrm{b}}^3}{\mathrm{x}\mathrm{x}}(=\mathrm{lc}-$ b ) therefore $\frac{3\mathrm{b}\mathrm{x}\mathrm{x}-{\mathrm{b}}^3}{\mathrm{x}\mathrm{x}}=\mathrm{lc}$. & $\frac{2{\mathrm{x}}^4-4\mathrm{b}\mathrm{b}\mathrm{x}\mathrm{x}+{\mathrm{b}}^4}{{\mathrm{x}}^3}=\mathrm{ad}$ the base of the 4^th triang: & $\frac{5\mathrm{b}{\mathrm{x}}^4-5{\mathrm{b}}^3\mathrm{x}\mathrm{x}+{\mathrm{b}}^5}{{\mathrm{x}}^4}=$ the perpendicular ( bl ) of the 5^t triangle & $\frac{2{\mathrm{x}}^6-9{\mathrm{x}}^4\mathrm{b}\mathrm{b}+6\mathrm{x}\mathrm{x}{\mathrm{b}}^4-{\mathrm{b}}^6}{{\mathrm{x}}^5}=$ base of the 6^t triang. $\frac{7{\mathrm{x}}^6\mathrm{b}-14{\mathrm{x}}^4{\mathrm{b}}^3+7{\mathrm{x}}^2{\mathrm{b}}^5-{\mathrm{b}}^7}{{\mathrm{x}}^6}=$ perpendicular of the 7^th triangle $\frac{2{\mathrm{x}}^8-16{\mathrm{x}}^6\mathrm{b}\mathrm{b}+20{\mathrm{x}}^4{\mathrm{b}}^4-8\mathrm{x}\mathrm{x}{\mathrm{b}}^6+{\mathrm{b}}^8}{{\mathrm{x}}^7}=$ base of the 8^th tri. $\frac{9{\mathrm{x}}^8\mathrm{b}-30{\mathrm{x}}^6{\mathrm{b}}^3+27{\mathrm{x}}^4{\mathrm{b}}^5-9\mathrm{x}\mathrm{x}{\mathrm{b}}^7+{\mathrm{b}}^9}{{\mathrm{x}}^8}=$ perp: of the 9^th tri:

<10v>

Prop 24:

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

Prop 24

If in the circle cfgh be inscribed the helix bedc & ac touch it in the point c then $\mathrm{ab}=$ to the circumference.

Prop 25

If apcr be les than halfe the circle. & $\mathrm{vt}=\mathrm{tp}$. & $\mathrm{vo}=$ to vrap : then $\frac{\mathrm{rq}\times \mathrm{po}}{2}=$ 4 times the section rapc

Prop 26

If $\mathrm{ab}=\mathrm{bd}=\mathrm{ad}$ & bh perpendicular to ad from the angle b . $\mathrm{ce}=\mathrm{ed}$. then $\mathrm{aed}=\mathrm{adi}=3\mathrm{dae}$. & ed is the side of a heptagon

Prop 27.

If a line be cut by extreame & meane proportion the lesse segment almost is to the whole line as the diameter is to 5 times the periphery divided by 6 .

Prop 28

Si secetur linea per extremam & mediam proportionem erit proximè, ut tota linea plus minori segmento ad bis totam lineam, ita quæ potest quadrato sesquialterum semidiametri, ad latus quadrati circulo equalis.
linea secta sit 100,000 . minus segmentum 38,197 . Semidiametrum 100,000 , quæ potest quadrato sesquialterum semidiametri paulo maior est quam 122,474 . Radix Peripheriæ, 177,245 .

<11r>

Prop 28.

If $\mathrm{er}=\mathrm{rh}=\mathrm{or}$. & $\mathrm{ao}=\mathrm{fc}=$ to the side of a decagon; & fn parallell to cd then en shall be almost equall to the fourth parte of a circle for ef is divided in extreame & meane propor in the point c . & $\mathrm{ec}:\mathrm{ef}\colon\colon$ $\mathrm{ef}:\frac{5}{12}$Perimeterhbkfa $\colon\colon \mathrm{hr}:\frac{5}{24}\mathrm{Perim}\colon\colon \frac{6}{10}\mathrm{ef}\left(=\mathrm{de}\right)\colon\colon \frac{1}{4}\mathrm{Perim}$; by the 27^th prop: & $\mathrm{ec}:\mathrm{ef}\colon\colon \mathrm{ed}:\mathrm{en}\left(=\frac{1}{4}\mathrm{Perim}\right)$.

Prop 29.

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

Prop 30

If the line dc touch the helix in the line ag . & the line hf toucheth the beginning of it in the center a & $4\mathrm{ac}=\mathrm{af}$ then $2\mathrm{ad}$ shall bee equall to perim: asr . & ac being the Diameter: the area of the triang acd = to the area of the circle asr

Prop 31

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

<11v>

Prop 32

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