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Tower ye 15 of Ot: 1674

Sr

Although I sent you three papers yesterday, I Cannot refrain from sending you this: I have had fresh Thoughts this morning about those two sorts of Equations which wee have latly Bandied about, and I have attained a|n| {illeg}|U|nivesall {sic} series for any Equation of two Cossique notes. Truly it pleaseth me well. But yet I doe hereby submitt it to your Censure.

To Extract the root of an Equation Consisting
of Two severall powers (or potestates) and
an absolute number; ꝑ an approximation
ea{illeg}|s|ily performed by logarithmes.
As for example $+{z}^{p}=a{z}^{q},n$

In which example and in all such like Equations, you must observe that z is the unknown \symbol/ or root sought, $p=$ the index (or power note) of the highest power, $a=$ the known numb: or Coefficient of the midle term, $q=$ the index of the inferior power; $n=$ the absolute number or resolvend.

The rule is thus: First guess at the root as nearly as you can, the nearer the better (not for nescessity {sic} but for accomodatn) and suppose that guess to be z.

Then observing the following series, you shall approach (from this supposed z) toward the true z which is sought: Because every term in this series brings you nearer and nearer, for if your supposition be too grea{illeg}|t|, every term in this series makes it less and less; or if your supposition be too litle, every term in this series makes it greater and greater. So when you are pleased to make a Ceasation, the last term is that which you seek. The series followes:
$\sqrt[p]{:+a{z}^{q}+n:}=b$ $\sqrt[p]{:+a{b}^{q}+n:}=c$ $\sqrt[p]{:+a{c}^{q}+n:}=d$.
$\sqrt[p]{:+a{d}^{q}+n:}=e$ $\sqrt[p]{:+a{e}^{q}+n:}=f$ $\sqrt[p]{:+a{f}^{q}+n:}=g$, &c:

In which series you must note that + only intimates the retaining of the proper signes whether they be + or −.

Sr pray doe not count mee troublesome for I could not forbear but send this, by Stiles ye Carrier, who is paid for the Carriage. pray Remember mee about the series of Logarithmes.

Yor most humble and oblidged servan{illeg}|t|

Mich: Dary

A true Coppie of that which I sent to Mr Newton this morning.

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