Letter from Michael Dary to Newton, dated 15 October 1674

<41r>

Tower y^e 15 of O^t: 1674

S^r

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 $+{\mathrm{z}}^{\mathrm{p}}=\mathrm{a}{\mathrm{z}}^{\mathrm{q}},\mathrm{n}$

In which example and in all such like Equations, you must observe that z is the unknown \symbol/ or root sought, $\mathrm{p}=$ the index (or power note) of the highest power, $\mathrm{a}=$ the known numb: or Coefficient of the midle term, $\mathrm{q}=$ the index of the inferior power; $\mathrm{n}=$ 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:
$\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{z}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{b}$ $\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{b}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{c}$ $\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{c}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{d}$.
$\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{d}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{e}$ $\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{e}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{f}$ $\sqrt[\mathrm{p}]{:+\mathrm{a}{\mathrm{f}}^{\mathrm{q}}+\mathrm{n}:}=\mathrm{g}$, &c:

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

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

Yo^r most humble and oblidged servan{illeg}|t|

Mich: Dary

A true Coppie of that which I sent to M^r Newton this morning.

<41v>

About finding the roote of an Æquation of two Nomes.

A true Coppie of M:D. his letter to M^r Isaac Newton of Cambridge.