JmfroVtmentJin/Eng!and ithe Rijolutton iEyationt i»Numbers, by AsJ. Collins.n. \6. f. 929-

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Cujus quidem processus totius Ratio his Principiis nititur; nempe, quodfigura ex Triangulis est dimidia figuroe ex Parallelogramms , stipereisdem B;: sibus, wque-altis : (Illam ego appello Figuram Convolutam ■, hancEvolutam-) Et f gura ex Pyramidibus, est, triens figura; ex Parallelepipe-da, super eisdem Basibus sequc-altis: (Illam ego appello Figuram Complica-tam ; hanc Explicatam .) Quse poliunt mille modis accommodari Figuris Cur-viiineis (tum Superficialibus tum Solidis) mirum in modum perplexis.

XVI 11 . i. It ha(h been oblervedfiy diversos this Nation, that in any Equa-^tion, howsoever.affected, if you give a Root, and find the absolute N umheror Resolvend, (which Victq cctls Homogcneum Ccmparationir, ) and again giver ’ Roots and . find more Resolvends; that if these Roots, or rather rank of Roorsbe assumed in Arithmetical Progression, the Resolvends, as to their first, secondor third disserenc.es, &c, imitate the Laws of the pure Powers of an Arithmeti-cal Progreifion of the läme degree, that the highest Power, or first Term ofthe Equation,.is of. .e- g. In this Equation a a a — 3 a a -F 4 <1 = N.

Isi a be =r.

Then N or theAbsolutes or Re-solvends will befoünd to be

740

114

131

t diss.

i dist.

118

170

48

118

41

91

36

3 diss

6

6

}

io.wit, the zd disterences of those Absolutes ave equal, as in the Cubes ofan Arithmerical Progression.

1. To find what habitude thoso disterences have to the Coefficicnt, ofthe Equation, ’tis best to begin ffom an Unite.

3. In any Arithmetical Progression, if you multiply Numbers by Pairs,yoii shall create a rank of Numbers whose second disterences are equal ;and it by Temaries, then the zd disterences of those Products Ihall be equalAnd how to find the greatest Product of. an Arithmetical Progression of anyNumber of Terfns having any common difference- aiiign’d, contain’d in anyN umher propos’d, is shewed by Pascal in his Tract du Triangle Aritbmetique,where he applies it to the Extraction of the Roots of Simple Powers.

4. It appears, how this Rank may be carried easily by Addition, till youfiave a Resolvend either equal or greater or leis than that propoied.

y. When you have a Majus and Minus, you may interpole as manymore terms in the Arithmetical Progression as you will, , that is to luy, Subdi-vide the Common. difference in the Arithmetical Progression, and render itleis; and then renew, and. find the Resolvends, which are easily obtained outof the Powers and their Coefficients, which are suppoled known, and may bereadily raised from a Table of Squares and Cubes, &c. with which kind theReader may be furnistl’d in Guldini Ccntrobaryca, and Babingtcns Fircvoorkj.By this means you tnay obtain divers Figures of the Root j and then the Ge-neral Method of Vieta and Harriot runs away more easily, and is so far im-proved, that aster any Figure is placed in the Root, most certain Characters

are