Page:Philosophical Transactions - Volume 004.djvu/43

 kind the Reader may find Examples in Briggii Arithmetica Logarithmica et Trigonametria Britannica, relating to Logarithmes, Sines, and the Powers of an Arithmetical Progression: But the method there deliver'd may be rendred more easy and general, viz. by aid of a Table of figurat Numbers, by deriving Generating differences sought, from those given; a doctrine, that easily flows from Mercators Logarithmotechnia, and of use in the Case in hand, should we suppose these Powers and their Coefficients vnknown, or a Table of Squares and Cubes wanting, and give nothing more, than a few Resolvends belonging to equal Moments or Spaces. And this may likewise be of good use in Guaging, when having the Contents of a Solid, for every 3. Inches, more or lesse, given, without knowing the dimensions of the Figure, and even in most Cafes, when the differences are Progressive of one kind, without knowing the Figure it self, having nothing given but its Contents at several equal Parallel distances, each such distance may be subdivided, and made as many as you please, and the respective Contents found by this general Method of Interpolation.*

After one Root is obtained, the Methods of Huddenius and others will depresse the Equations so as to obtain more, and consequently all of them.

6. It is easy, by a Table of figurate Numbers to give the sum of any such Rank or any term in it relating to a known part of the Series of Equals or Roots; but è coverso, giving the Resolvend to find the Root, coms to an Eqnation as difficult as that proposed; as in D. Wallis his Chapter of Figurate Numbers.

7. Some affirm, they can give good Approaches for the obtaining a Root of any pure power, affected Equation, or for the finding of any of the mean Proportionals in any Rank between two extreams given.

8. Others pretend to have found out a method (incited thereto an example in Albert Gerards Invention Nouuelle en Algebre à Amsterdam 1629.) so much, by comparing of Equations, to Rh