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- An AccountConcerning the Resolution of Equations in Numbers; imparted by Mr. Iohn Collins.
This Account should have been annex'd to what was discoursed of Monsieur Slusius his Mesolabe in the precedent Tract, if then we had found room for it. For, the Reader having there understood, how farr the Geometrick part of Algebra is advanc'd by that excellent person, 'twas likely, he would be inquisitive to hear somewhat concerning the Exegeis Numerosa, or the Resolution of AEquations in Numbers. For whose satisfaction herein, we shall here insert the Account then omitted, being part of a narrative, formerly made by M. Iohn Collins touching some late Improvements of Algebra in England, upon the occasion of its being alledged, that none at all were made since Des Cartes.
1. It hath been observ'd by divers of this Nation, than in any Equation, howsoever affected, if you give a Root, and find the Absolute number or Resolvend (which Vieta calls Homogeneum Comparationis) and again give more Roots and find more Resolvends, that if these Roots or rather rank of Roots be assum'd in Arithmetical progression, the Resolvends, as to their first, second or third differences, &c., imitate the Laws of the pure Powers of an Arithmetical progression of the same degree, that the higest Power or first term of the Equation is of. e. g. In this Equation aaa—3 aa + 4a = N,
| 1. dif | 2. dif | 3. dif. | ||||||
| If a be = | 10 | Then N. or the | 740 | 218 | ||||
| 9 | Absolutes or Re- | 522 | 48 | |||||
| 8 | 352 | 170 | 42 | 6 | ||||
| solvends will be | 128 | 6 | ||||||
| 7 | found to be | 224 | 92 | 36 | ||||
| 6. | 132 |
To wit the 3d. differences of those Absolutes are equal, as, in the Cubes of an Arithmetical Progression.
2. To find, what habitude those differences have to the Coefficient, of the Equation, 'st best to begin from an Unit.
3. In any Arithmetical Progression, if you multiply Num-
