A New Discovery of the Constructious of all Aequations howso­ever affected, not exceeding the 4th degree; viz. Of Linears, Quadratics, Cubics, and Biquadratics, and the finding of all their Roots as well true as negative, without the use of Mesolabe, and Trisection of Angles; without Reduction, Depression, or any o­ther praevious preparation of Aequations, by a Circle and any (and that but one only) Parabole: and this by one only general Rule. than which a more simple, more perfect, more gene­ral, more easy to be understood, or more fit for practise, can­not be devised or wished for.

Fortified with Demonstrations, Illustrated with Figures to each Aequation, which are Exemplyfied with numeral Aequations, (according to all the varieties of Cases) adapted to each Fi­gure, for the use of young Mathematicians; a work hitherto defired.

THe Treatise consists of about a Quire of Pa­per, the Discourse whereof (but not the Algebraick Calculus) is both in Latin and English, the better to promote its forreign vend; and this doth not render it above three Sheets the larger than it would have been in one of these Languages. Besides which, there is belonging to it diverse Draughts of Schemes to be engraven, and one Folio Draught, whereto the literal Calculus for setling the Center, and finding the [Page 2] Raedius of the Circle that is to intersect the Parabole is expressed in readiness for all Cases.

How Des Cartes and all other famous Analysts came to miss this general Rule, and himself to fall upon it, he acquaints the Reader in the middle of his Discourse; namely, that they considered the Axe of a Parabole and not its Diameter: and affirms, that if it had been his or their hap to have described a Circle from any Point in Position given, passing through the Vertex of any Diameter in the Parabole, and had taken into consideration a certain propriety than which none could so have suited the design) belonging to the Diameter of any Parabole, they could not but with greatest ease, have made a full discovery of the Universal Rule.

The excellency of which Invention appears, in that it discovers not only the Geometrical Construction of all Aequations as above­said, by one only standing measure and Scheme, and that by one only general rule, with the exact number of Roots as well true as negative, but also by giving a fair prospect towards their Arith­metical Calculus, or numerous Resolution, by making a Discovery of their two first figures or numbers; namely, by applying the Compasses to the several Roots Geometrically found in the Scheme, and comparing them with that very Scale from which the said Scheme (suited to the proposed Aequation) was drawn, the resi­due of which roots, (though not precisely, yet sufficient near­ly approximating to the true) may diverse ways in Decimals be found out, which the Author (as he intimated in a Letter of A­pril 1682, to Mr. Collins) is willing to impart; but as to the In­vention of these residuals (to be entail'd to the two first figures or Numbers of this Author thus findable. The Learned Mr. Isaac Newton Professor of Mathematicks in Cambridge (in a Letter long since communicated to the aforesaid Mr. Collins) hath as to this pur­pose performed the same (as is conceived) by a different method, namely, that when a root of any Aequation is by any Method (which by the Authors aforesaid it may be) so near found that it doth not differ above a tenth part of its self from the true root sought, the residue of the root inquired will be easily calcu­lated by aid of some terms or Fractional parts of an infinite Series or rank of continual Proportionals, derived from the difference between the Resolvend of the known part of the Root, and that whose Root is sought. By which means by raising Resolvends out of any assumed Roots with an easy approach, without raising [Page 3]the respective powers of the said Roots, we are delivered from the most toilsom Drudgery of Mathematical Calculations, by finding the Roots of Aequations in numbers, by Vietas general method; a thing utterly unknown to the Ancients. However this is not said to disparage that Method which Victa so greatly esteemed, that when he had obtained it, he gave Algebra this high Encomium, that it did Nullum non Problema solvere, in his numerical Method Mr. Oughtred and Harriot have taken commendable pains. But now last of all, to perform in Species as Mr. Isaac Newton hath done, seems a new Invention never to be sufficiently praised; for out of a literal Aequation of five Dimensions, supposing all the terms ex­tant and affirmed, he hath given a Series for the Root in Species, and such a one as shall serve for finding the Roots of all Equations of 3, 4 or 5 Dimensions, by only altering the signs according as as the Aequation is affected, and expunging such parts as relate to Deficient terms in an incompleat Aequation proposed.

Now that this admirable Doctrine may come to light, and the Learned Author (who hath many other Treatises worthy publick view) may be incited to impart the same, encouragements for the promoting thereof (seeing Undertakers are not to be had with­out) must be propounded.

It is therefore humbly offered, that the Royal Society by their Treasurer &c. enter into Bond to such Bookseller as shall be the Undertaker, to take of 60 of these Books in Quires at 1½ d. each Sheet, and as much each Plate, as soon as printed.

The Treatise it self and the Proposal, is approved and agreed to by the Council of the Royal Society.

And in regard such a Subscription is not sufficient to incite an Undertaker, that the respective Members endeavour by virtue of this Narrative, to obtain as many more Subscribers as they can pro­cure amongst others that are not of the Society, each of them to advance half a Crown in hand, in part of the former price: up­on which encouragements, Robert Clavel Bookseller at the [...] in St. Pauls Church Yard, is ready to give reciprocal [...] for the performance according to this Proposal, hoping the like en­couragement will be given towards printing the rest of the Trea­tise of this most Learned Author, whereof take the ensuing Ca­talogue.

1. The Hyperbolic Key, or the Geometrical Construction of Cubic and Biquadratic Aequations; by a Circle and an equilateral Hyperbole; whereof the one moyety (exactly) viz. of eight Cu­bicks and four and twenty Biquadratics; (as is expressed in the for­mer treatise) namely,

1. Of all those Cubicks; wherein

• 1. The quantity (q) is wanting; and p and r affected with divers Signes.
• 2. The quantity (p) is wanting, and in the Aequation be had − q.
• 3. None of the terms are wanting, and in the Aequation be had + q,

2. Of all those Biquadratic Aequations in which are had + s, by the demission of Perpendiculars, from the points of Intersecti­ons of the Circle and Hyperbole to the Assymptote; part of the other moyety, by the demission of perpendiculars from the afore­said interfections to the axe; &c. with Schemes adapted to each Aequation, &c. with a Synopsis of the whole, wherein the Literal Rule for fixing the Center, and finding the Radius of the Circle, that is to intersect the Aequilateral Hyperbole (the easiest way of the Construction of which is likewise therein discovered) is expressed in readiness for all Cases.

This method of Construction (were it not, that for every Case a new Hyperbole must be described) would not be inferior to that by a Parabole, but rather exceed it; in that the Circle doth not arcuate the same way which the figure doth, but crosses it the other way; by which means a clearer discovery (as to the one moyety) of the points of intersection of the Circle with the Hy­perbole is obtained, than what can possibly be had in any other Coni-section.

3. The Geometrical Construction of some Aequations which ascend to the 5th and 6th power, with the finding of their Roots, by a Curve of the third degree; namely by the first kind (for there are two kinds) of a Paraboleid and a Circle, illustrated with Schemes to each Aequation, and numeral Aequations adapted to them; with a Synopsis to the same for placing the Center and finding the Radius, and a general little Table, for the describing of both kinds of Paraboleids.

4. The Construction of all Cubick Aequations howsoever af­fected by a Circle only, Geometricaliy upon Supposition, that one Postulatum be granted to be Geometrical (which indeed is but a Supplement to Geometrical defects;) viz. That from any point assigned in the circumference of a Circle (that is normaly quadrisected) may be drawn a right Line, so that the parts intercepted both ways by the Circumference and Diameter, may be equal to the Radios of the Circle: this way (though not so purely Geometrical as the rest) is not to be despised, sith that these Lines may sufficient precisely be so drawn.

5. The Geometrical Construction of all Cubic Aequations ac­cording to the Rule found out by Franciscus a Shooten, mentioned in his Commentaries on Des Cartes, Lib. 3. Pag. 328, 329, 330, illustrated with Figures and Numeral Aequations adapted to each Figure, &c.

6. The Resolution of all Cubick Aequations in numbers not only by a general Rule (by the assistance of any Figure resolving them Geometrically &c. but by a more particular method far ex­ceeding any extant in numbers or by help of tables; illustrated with Figures and Examples in numbers, suited to each figure and Aequation.

7. Mixt or Compound Trigonometry; in many instances far exceeding the simple, as finding two Quaesitas (as it were) by one operation, or by two at most; with a Synopsis of the admi­rable harmony between Plain and Sphaerical Triangles: for in­stance,

In plain Rectangular Triangles, the ▭ e under half the sum of the Hypotenuse and one side: and half their difference, is equal to the Square of ½ the other side, so in Sphaerical Rectangular Triangles. The ▭ under the Tangents of half the sum and half the difference of the Hypotenuse and one side: is equal to the square of the Tangent of half the other side.

Again in Obliquangular Plain Triangles.

[...]

Thus likewise in Sphaerical Obliquangular Triangles.

[...]

Again in plain Triangles. ½ Basis. ½ Z crurum, ½ X crurum, ½ X of the Segments of the Base.

In Sphaerical also.

t, ½ Basis. t, ½ Z Crurum, t, ½ X Crurum, t, ½ X Segmentorum basis, with infinite other alike harmonious.

To which is added the Geometrical Construction of all Sphaeri­cal Triangles, by a most plain and easy uniform way, which is indeed of singular use.

Also a discovery of the Method by which Vieta (Lib. 8 p. 431 &c.) found out his Canonical Analogy of Sphaerical Triangles, which he hath left undemonstrated, but in this Treatise is disco­vered.

8. Cardanus Promotus, or Cardans Rules, or Vieta's duplicata Hypostasis, in infinitum, carried on with a Table for the composition in infinitum of such Aequations. By which means such Canons are generally composed for Aequations of two Nomes (and in many Cases for more) equal to a Resolvend given.

9. A Continuation of Vietas Apollonius Gallus, Appendicula; 1. And his Problemes otherwise demonstrated, wherein the Base and Angle opposite to the Base are always two of the Data's, and the other, either the perpendicular or the difference of the Seg­ments of the Base, or the difference of the squares of the sides, or the sum of the Squares of the sides, or the Sum of the sides, or the difference of the sides or their ▭, whose Geometrical effection was altogether unknown to the antient Analysts, Vieta ibid.

10. Vera & Genuina Symmetrica Climactismus, by which means all Asymmetries in Algebraics may be wiped off, and an Aequation found in any one of the unknown magnitudes proposed, which shall never ascend higher, than the double of the highest power first proposed, [...] which also that most peplexing entangling in­extricable way of Vieta may be laid aside as useless, and ineffica­cious, though hitherto it hath been the only remedy. Adver­sus vitium Asymmetriae, this treatise was many years since composed and laid aside; but the Author lately meeting with the opera Posthuma of Monsciur de Fermat, treating on the same Subject in his (Varia opera Mathematica pag. 58, 59, &c.) and finding that though he rightly hits the mark; yet that he goes not in a streight Line to it, hath revised his old Copy, and compared it with Fer­mats; and which of the two, hath gone the Simpler way, the Author leaves to the judgment of others, being loth in the least to take up the Gantlets against such a famous man whom all the world admires.

11. Apollonius magnus Gregorianus, or a Treaiise of four Geo­metrical proportionals, wherein divers ways are found to solve that Grand problem, which hath so amused the world, (viz.)

Having the sum of all the Squares, and the sum of all the Cubes, of four Geometrical Proportionals to find the Proportio­nals themselves; with questions of the like nature, by low Ae­quations, without aid of Analytick Store.

11. Of Triangular Sections by a different method than what Anderson has performed it by, in Vieta, with a discovery of the falshood (as to angular Sections) of Mr. Oughtreds 1st. Rule, in his Clavis Mathem. c. 16. p. 14.

12. The finding out of Aequations which may infinitely ascend, whose Roots are either in Arithmetical or Geometrical propor­tion which may be found out in numbers by extracting the Square and Cube Root, with surd Canons adapted to that purpose, and to many other Aequations.

13. A Miscellancy of the solution of many knotty Problemes, namely, such as have been found difficult to be brought to any Equation, or else would mount very high in Ordine Scalae, with a new method of Depressing them, by aid of one or two Aequations, raised by altering the Data, and putting two unknown quantities, by which means the adjutant Aequations as having the same com­mon root, depress the Aequation that otherwise should be resolved.