An Account Concerning the Resolution of Equations in Numbers; Imparted Parted by Mr. John Collins

An Account Concerning the Resolution of Equations in Numbers; parted by Mr. John 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 far the Geometrick part of Algebra is advanc'd by that excellent person, 'twas likely, he would be inquisitive to hear somewhat concerning the Exegesis Numerosa, or the Resolution of Equations in Numbers. For whose satisfaction herein, we shall here insert the Account then omitted, being part of a narrative, formerly made by M. John 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, that 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 highest Power or first term of the Equation is of, e.g. In this Equation aaa—3 aa+4 a=N, | If a be = | Then N, or the Absolutes or Resolvends will be found to be | |-----------|-------------------------------------------------------------| | 10 | 740 | | 9 | 522 | | 8 | 352 | | 7 | 224 | | 6 | 132 | 1.dif | 2.dif | 3.dif | |------|-------|-------| | 218 | 48 | 6 | | 170 | 42 | 6 | | 128 | 36 | | 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, 'tis best to begin from an Unit. 3. In any Arithmetical Progression, if you multiply Numbers... bers by pairs, you shall create a rank of Numbers whose 2d differences are equal; and if by ternaries, then the 3d differences of those Products shall be equal. And how to find the greatest Product of an Arithmetical Progression of any number of terms having any common difference assign'd, contain'd in any Number proposed, is shew'd by Pascal in his Tract du Triangle Arithmetique, where he apply's it to the Extraction of the Roots of simple powers. 4. It appears, How this rank may be caried easily by Addition, till you have a Resolvend either equall or greater or lesse, than that proposed. 5. When you have a Majus and Minus, you may interpole as many more termes in the Arithmetical Progression as you will, that is to say, Subdivide the Common difference in the Arithmetical Progression, and render it lesse; and then renew, and find the Resolvends, which are easily obtain'd out of the Powers and their Coefficients, which are suppos'd knowne, and may be readily rais'd from a Table of Squares and Cubes, &c., with which kind the Reader may be furnish'd in Guldini Centrobaryea and Babingtons Fireworks: By this means you may obtain divers Figures of the Root; and then the General Method of Vieta and Harriot runs away more easily, and is so far improv'd, that after any figure is plac'd in the Root, most certain Characters are given to know by aide of the subsequent Dividend and Divisor, Whether the figure before assum'd be too great or too small: or lastly it may well be concluded, that, as in Logarithmes, when you propose such an one as is not absolutely given in the Canon, you doe by Proportional Work, using the aid of their first differences (when their Absolute Numbers differ by Unit) find the absolute Number true to 5. or 6. places further than the Canon gives it (the reason whereof is, that the first Differences doe likewise agree to about the same Number of places;) that I say, the like may be done in Equations, after divers of the first figures of the root are found; provided there be the like agreement in the first differences of the interpoled Resolvends. Moreover we ought here to take notice of a more subtile kind of Interpolation, common to all gradual Ranks or Progressions of Numbers, wherein Differences happen to be equal: Of which kind kind the Reader may find Examples in Briggit Arithmetica Logarithmica et Trigonometria 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 Logarithmotecnia, and of use in the Case in hand, should we suppose these Powers and their Coefficients unknown, 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 leffe, given, without knowing the dimensions of the Figure, and even in most Cases, when the differences are Progressive of one kind, without knowing the Figure itself, 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.* *Nota. The Author (M. Collins) having explain'd Mercators Logarithmotecnia in English, and illustrated the elegant Doctrine thereof with Examples, hath likewise handled this work of Interpolation, and makes the Logarithmes true to 25. or 30. places of figures by meer Division (or Proporation;) having herein advanc'd that Author's doctrine thereof by Division, which (as 'tis there illustrated) did not seem to extend farr enough. This hath already been communicated, some Months since, to some of the Members of the R. Society, and may be expected hereafter. After one Root is obtained, the Methods of Huddenius and others will depress 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 converso, giving the Resolvend to find the Root, comes to an Equation 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 extremes given. 8. Others pretend to have found out a method (incited thereto by an example in Albert Gerards Invention Nouelle en Algebre à Amsterdam 1629.) so much, by comparing of Equations, to increase crease or diminish the unknown Root of Equation, as to render it a whole number (or less differing therefrom, than any Error assign'd,) and by Albert Gerards Method of Aliquot parts to find the same, and thereby the Root sought, although it be a Mixt Number, Fraction, or Surd. Probably this may sympathise with what is promised by the Learned Huddenius in Annexis Geometrica Cartesiana, where he faith, he intended not then to publish certain Rules, he had ready; whereof one was, To find out all the irrational Roots both of Literal and Numerical Equations. This must be understood when such Roots are possible; for 'tis certain, there are infinite Equations, whose Roots are no ways explicable, either in whole or mixt numbers, Fractions or Surds, and can be no otherwise explain'd, but by a quam proxime. 9. The Author of this Narrative considering, that the Conick Sections may be projected from lesser Circles placed on the Sphere, and thence easily (otherwise than hitherto hath been handled) described by Points, and that by their Intersections some Spherick Problem is determined, accordingly he found, that this following Problem according to the various Scituation of the Eye, and of the Projecting Plain, would take in all Cases. The Distances of an unknown Star are given from two Stars of known Declination and Right Ascension; the Declination and Right Ascension of the unknown Star is required. And faith, he hath observed, that, admitting the Mechanisme of dividing the Periphery of a Circle into any number of equal parts, or (which is equivalent) the Use of a Line of Chords, that this Problem, wherever the Eye be plac'd, may be resolved by Plain Geometry, and yet the Ey shall be so plac'd, as to determin it by the Intersections of the Conick Sections; consequently those Points of Intersection (the Species and Position of the figures being given) may be found without describing any more Points than those sought; and the Lengths of Ordinates falling from thence on the Axes of either figure calculated by mixt Trigonometry, and hence likewise the Roots of all Conick and Bi-quadratick Equations found by Trigonometry. For giving from the Mesolabe mention'd the Scheme that finds these Roots, it will then be required to fit those Sections into Cones, which have their Vertex either in the Center, or an assigned point in the Surface of the Sphere, to which they relate. relate as projected, and proceed to the resolution of the Problem proposed: And how to fit in those Sections, see the 7th books of Apollonius, Mydergius, the 3rd Volume of Des Cartes's Letters, Leontaudi Geometrica practica, Andersonii Exercitata Geometrica. As to the Problem itself, it is determined on the Sphere by the Intersections of the two lesser Circles of Distance, whose Poles are the known Stars. And this Problem hath divers Geometric ways of resolution. 1. By Plain Geometry (in the sense before mentioned;) Supposing a Plain to touch the Sphere at the North-pole: if the Eye be at the South-pole, projecting those Circles into the said Plain, they are still Circles (by reason of the sub-contrary Sections of the Visual Cones) whose Centers fall in the sides of the Right-lin'd Angle made by the Projected Meridians, that pass through the known Stars; and thus the Problem is easily solv'd in this manner. 2. If it be required to be performed by Conick Geometry; in one case it may be done, by placing the Eye at the Center of the Sphere, and projecting as before; to wit, when the longer Axes of the figures being produced concur above the Vertex; Here the Problem is determined by the Intersections of two Conick Sections (whereof a Circle cannot be one, unless its Center be in the Axis of the other figure.) And in this second Case these points of Intersection fall in the same right line or projected Meridian, they did before, but at a more remote distance from the Pole-point, to wit, in the former Supposition, the Solar distance was measured by a Right line, that was the double Tangent of half the Arch; here it is the Tangent of the whole Arch. Hence it is evident, how one Projection may beget another, yea infinite others, altering the Scale; and how the lesser Circles in the Stereographick Projection help to describe the Conick Sections in the Gnomonick Projection: But (to reduce the matter to one common radius) if we suppose two Spheres equal, and so placed about the same Axis, that the Pole-point of the one shall pass through the Center of the other, and the Touch-plain to pass through the said Center or Pole-point, and that a lesser Circle hath the same position in the one as in the other; Then, if the Eye be at the South-Pole of the one, it is at the Center of the other; and any projected Meridian drawn from the projected Pole-point to pass through both the projections of these le- for Circles, the distances of the Points of intersection are the Tangents of the half and the whole Arch of the Meridian so intersected. But as to the Points of Intersection, which determine the Problem proposed, they may be found without the aid of the former way, from a Gnomonick and Stereographick method of measuring and setting off the sides and angles of Spherical Triangles in those Projections, which is necessary in what follows. 3. If the Problem is to be perform'd by Mixt Geometry, as by a Circle and either a Parabola, Hyperbola, or Ellipsis, the Circle may be conceived to be the South-contrary Section of a Cone projected by the Eye at the South-pole, and any of the rest of the Sections by the Eye at the Center of the Sphere. 4. If by any of the Conick Sections however posited; the projecting Plain may remain the same, but the Eye must be in some other part of the Surface of the Sphere, and not in the Axis. These things were mentioned to invite the Learned to their Consideration: I shall only further add, that we cannot say what may be expected from the labours and endeavors of divers Learned men of this Nation, particularly from Dr. Wallis, who hath so excellently resolved and constructed all Cubick Equations at the end of the first Treatise of his Opera Mathematica by aide of a Cubick Parabolaster, mentioning, that by such Curves the Roots of all Equations may be found: And who hath promised a Treatise of Algebra and Angular Sections, wherein the Reader need not doubt to meet with satisfaction in these Mysteries. Nor ought we to omit the mentioning of the Modest and Learned Mr. Barrow, who (among many other excellent Subjects, and particularly his Opticks now, at the Press) hath perform'd, what the famous Italian Geometer Mich. A. Ricci hath promis'd in Exercitatus Geometrica (printed at Rome 1666, and lately reprinted here) about Curves of several degrees, that serve to determine and resolve all Equations: which hath likewise been done by other Learn'd men of this Nation. An Account of Books. I. PRÆLUDIA BOTANICA Roberti Morison Scotti Aberdonensis. Londini, impensis Fac. Allestry, 1669. in8o. This Prelude of this Excellent Botanist hath two parts; The first gives us an Alphabetical Catalogue of all the Plants in the Royal