CHAP. I. Of Equation, and the true stating and purging of a Mathematical question, whereby (if need be) to bring it to one Equation.

EQuation is when one Quantity is equal to one Quantity, or when one Quantity is equal to many Quantities, or when many Quan­tities are equal to many Quantities.

2. Any Equation may be re­duced to a new Homogeneal Equation, if each side be augmented or diminished equally by Addition, Substraction, Multiplication, Division, Involution or Devolution.

3. Any Mathematical Question may be propo­sed in an Equation, or in several Equations, cohe­rent in respect of the unknown quantities.

4. Therefore when a Question of this nature is proposed, and Resolution desired, let there be put (Symbolically) for every quantity therein men­tioned some Letter of the Alphabet, (for the known quantities those Letters towards the beginning of the Alphabet; and for the unknown, those Letters towards the later end of the Alphabet:) then shall [Page 2] you state the whole business of the Question by Ra­tiocination thereto requisite, (for which purpose you ought to be sufficiently furnished with Analytical store) into an Equation, or into several coherent Equations: which coherent Equations ought to be one to another irreducible or heterogeneal.

5. If (in a question so stated) the unknown Symbols be more then the Equations, that questi­on is impertinent, and capable of an infinite num­ber of Answers: But if the unknown Symbols be not more then the Equations, that question is pertinent, and not capable of an infinite number of Answers.

6. If the question proposed be pertinent, and stated into several Equations, you shall (by the 2. hereof) purge those Equations of all the un­known Symbols, excepting that which is desired to be known; and so the question will stand in one Equation.

CHAP. II. Of clearing and trimming an Equation, whereby to know (what Roots and how many it hath, viz.) the Constitution thereof.

1. IF there be any Fractions in the Equation, let them be cleared of their Nominators by Ch. 1. Sect. 2. Also by the same let the Surd Roots be taken away if there be any.

2. If the unknown Symbol be found in all the Terms, you shall clear as many of the Terms of it as may be cleared by Chap. 1. Sect. 2. But if all the Terms may be cleared, that Equation is identical.

3. If there be a Cofactor in the first Term, let it be cleared away, (if it may be done without Fractionizing any other Term) by Chap. 1. Sect. 2.

4. Let all those Terms on the right side of the Equation be transposed to the left by Chap. 1. Sect. 2.

5. Let all Homogeneal Terms be gathered into one Term by the prescription of their Signs and Cofactors: placing the highest Potestate of the un­known Symbol first, (most sinisterly) and the rest of the inferiour Potestates succeeding in their order, the absolute being last; and on the right side of the Equation put a Cypher.

6. If the Absolute be affirmed, you shall make it denied by changing all the Signs throughout the [Page 4] Equation; so is the Equation cleared.

7. This done, you shall expugne all the known Symbols, and put their respective Numbers in their rooms; and now you may call it a trimm'd Equation.

8. Every trimm'd Equation hath so many Roots as is intimated by the Index of the unknown Sym­bol in the first Term, whereof there are so many true Roots, as the Signs (in their order) have Changes, and the rest are false Roots.

9. A [ true / false ] Root of a trimm'd Equation is a number or quantity [ greater / lesser ] then nothing, which wrought according to the Conscription of the un­known Symbol, the result is [ alwayes / sometimes ] equal to no­thing.

10. In a trimm'd Equation having all the Pote­states of the unknown Symbol extant, if you change the Signs in even places, the true Roots are turned to false Roots, and the false Roots to true.

11. In a trimm'd Equation, having all the Po­testates of the unknown Symbol extant, if you change the Signs in even places,

• The Cofactor in the second Term is equal to the summe of all the single Roots. Multiplied by the Co­factor in the first term.
• The Cofactor in the third Term is equal to the summe of the Facts of every two Roots. Multiplied by the Co­factor in the first term.
• The Cofactor in the fourth Term is equal to the summe of the Facts of every three Roots, &c. Multiplied by the Co­factor in the first term.

12. In every trimm'd Equation the unknown Symbol is arbitrarily equal to any one of the Roots (whether true or false) as constituting the Equa­tion proposed.

13. If therefore to the unknown Symbol a true Root be connexed with the Sign − or a false Root with the Sign +; every such Connexion may not unfitly be called an equated Binomial.

14. Every trimm'd Equation is constituted by the Fact of all its equated Binomials; multiplied by the Cofactor in the first Term.

15. All trimm'd Equations, wherein the Signs are not changed, are always impossible: yea and sometimes (also) although the Signs are changed, as in some pretended trimm'd Equations.

CHAP. III. Of the construction and use of the Canon, whereby the possibility or impossibility of any trimm'd Equa­tion is discovered, as also one of the true Roots [...] educed in number, and consequently all the rest of the true Roots.

1. LEt the Absolute Number be transposed to the right side of the Equation, and then you may call it the Resolvend.

2. Now if you substitute in the room of the unknown Symbol, any Binomial; as (if you please) l+n: then are all the Potestates of l+n equal to all the like Potestates of the unknown Symbol, which Potestates of l+n being orderly placed do make up the Canon for the Equation proposed.

3. Let the Resolvend be placed on the right side of the Canon, with the Sign of Equati­on between them, and draw a curved line on the right side of the Resolvend for a Quotient or Re­ceptacle for the quesitious root.

4. Beginning at the place of Units, let the Re­solvend be pointed into Periods, as it intimated by the several Potestates of the unknown Symbol, viz. for a Root every single Figure, for a Square every second Figure, for a Cube every third Fi­gure, and so on; extending toward the left hand for a whole Number, but towards the right hand for a Decimal Fraction; completing the Resol­vend [Page 7] with Cyphers if need be, for there must be as many Periods for one Potestate as another, and no more; of which Periods so seated, one of every sort of the same gradual distance from the place of Units, are called a congruent set.

5. Having thus limited the Resolvend by Pun­ctation, you shall put for the first Figure of the Root, (viz. for l) the greatest Figure that may be; always provided that those parcels in the Ca­non which have not the Secundary Root n in them, being collected according to the first, (if need be, second or third, &c.) congruent set of Periods into one Result, may be subducted from the correspon­dent part of the Resolvend limited by Punctation; and the Remainer retaining the same Sign with the Resolvend, must be always less then the succeeding Divisor.

6. (Subduction being made, let all those par­cels in the Canon which have not the Secundary Root n in them be expugned, for they are no more to be used.)

7. Now and henceforward in the enquiry for every succeeding Figure of the Root, l shall repre­sent (or be equal to) that part of the Root already found; and n shall represent that next succeeding Figure of the Root which is enquired for.

8. To the Remainer you shall regularly draw down so much of the Resolvend as is limited by the last Period of the next congruent set of Periods, makes your Minorand, having the same Sign with the Resolvend.

9. That there may be some light exhibited fo [...] the chusing of the next succeeding Figure of th [...] Root, the known part of the Canon (for therein i [...] nothing unknown but [...]) being collected accor­ding to the precedent Congruent set of Periods in­to one Result makes your Divisor, having the same Sign with the Resolvend; the last Remainer decuplicated being your Dividend.

10. (But if there cannot be such a Divisor made as is required by the 5. hereof, that Equation is im­possible; if there may, it is possible.)

11. By enquiry and some trial you shall put for the next succeeding Figure of the Root, (viz. for n) the greatest Figure that may be; always provided that l being decuplicated, the whole Canon (for now therein is nothing unknown) being collected according to the present congru­ent set of Periods into one Result, may make a Subducend, having the same Sign with the Resol­vend; which being subducted from the foresaid Minorand, the Remainer retaining the same Sign, must be always less then the succeeding Di­visor.

12. And so you may proceed to find every Fi­gure of the Root, repeating the 8, 9, and 11. here­of, leaving out the 10, for if the first Figure of the Root be possible, all the rest are possible.

13. Now if it be required to find another of the unknown true Roots, (if there be any) let the Equa­tion as it stood trimm'd by Chap. 2. Sect. 7. be di­vided [Page 9] by the equated Binomial, viz. the unknown Symbol less the new-known Root: and the Quo­tient is a trimm'd Equation, (one degree low­er then the Dividend) constituted (Ch. 2. Sect. 14.) by the Fact of all the remaining equated Binomials, multiplied by the Cofactor in the first Term.

Therefore you may proceed to find that other true Root required, (and consequently all the rest of the true Roots one after another, if need be) by repeating this whole Chapter, excepting the 10 hereof; for if one true root be possible, all the rest are possible.

Laus Deo.