Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

INTRODUCTION TO SPECIOUS ANALYSIS.

THE Analytick Art, or Specious Analysis, is solely subvervient to finding of Theorems, and resolving Problems, by leading us from certain Data or given Quantities, into the knowledge of unknown and sought ones, by a Chain of certain and infallible Consequen∣ces: This admirable Artifice may be reduced to four Primary Heads, viz. Denomination, Reduction, Equation, and Effe∣ction (if the Problem be a Geometrical one) or Construction.

I. DENOMINATION.

BY Denomination is understood a preparatory imposition of Names peculiar to each Quantity, whereby every one of the Quantities given or sought, are denoted by one or more peculiar Letters of the Alphabet at pleasure, but with this (arbitrarious) difference, that known or given quantities are mark'd by the former Letters of the Alphabet, a, b, c, &c. and the unknown or sought ones by the latter, z, y, x, &c. But although this imposition of Names, is, as we have said, altogether arbitrarious, yet there often happens not a little faci∣lity to the Solution it self, by its being chosen as accommodate as possible to the conditions of the quantities given and sought; which any one w••ll learn better by Use than Precepts: As we find that both Theorems may be demonstrated, and Problems resolved e. g. by an extraordinary Compendium, if we denote any reason of two given Homogeneous quantities by a and e a, b and i b, d and o d, &c. (v z. by expressing the Names of the Reasons by e, and i, and o, &c.) and continued proporti∣onality

by a, ea, e{powerof2}a, e{powerof3}a, &c. and discontinued or discrete by b i b, c i c, d i d, or after the like manner, as we have done in our Math. Enucl. Lib. 1. Cap. 2, 3, 4, 7. and Lib. 2. Cap. 1. &c.

II. EQƲATION.

HAving thus given each quantity its Name, and making no further distinction between the quantities given and those sought, but treating them all promiscuously, and as already known, you must carefully search into and discuss all the Cir∣cumstances of the Question, and making various Comparisons of the quantities, by adding, substracting, multiplying, and dividing them, &c. 'till at length, which is the chief aim and design of it, you can express one and the same quantity two ways, which is that we call an Equation: And you must find as many of these Equations, or Equalities of literal quantities, (as expressing the same thing) as there are several unknown quantities in the Question, independent on each other, and consequently denominated by so many different Letters, z, y, x, &c. But if so many Equations cannot be found, after hav∣ing exhausted all the Circumstances of the Questions by one or two Equations; that is a sign the other unknown quanti∣ties may be assumed at pleasure: Which the Examples we shall hereafter bring will more fully shew.

But as here also (as likewise in all this Art) Ingenuity and Use do more than Rules and Precepts; yet we will here shew the principal Fountains, for the sake of young Beginners, whence Equations, according to circumstances obvious in the Questi∣on, are usually had. These are partly Axioms self evident, E. g.

That the whole is equal to all its parts taken together.

That those quantities which are equal to one third, are equal among themselves.

That the Products or Rectangles under the Parts or Segments, are equal to the Product of the whole.

Partly some universal Theorems that are certain and already demonstrated, as,

Three(α) 1.1 continual Proportionals being proposed, the Rect∣angle of the Extremes is equal to the Square of the mean.

(β) 1.2Four being proposed, whether in continued or disconti∣nued Proportion, the Product or Rectangle of the Extremes is equal to that of the Means.

And several others such like, which we have demonstrated in Cap. 2, 3, and 4. Lib. 1. of our Mathesis Enucleat. partly in the last place, some particular Geometrical Theorems al∣ready demonstrated, as e. g. that common Pythagorick one.

That in rightangled Triangles(γ) 1.3 the Square of the Hypo∣thenusa is equal to the two Squares of the sides.

That the Square of the Tangent of a(δ) 1.4 Circle is equal to the Rectangle of the Secant and that Segment of it that falls without the Circle; the first whereof, we have demon∣strated, Lib▪ 1. Math. Enuc. Def. 13. Schol. and also Prop. 34. Consect. 8. also Prop 44. after various ways; to which may be numbred Prop. 34. with Schol. 11. n. 3. Prop. 37. and following, Prop. 45. and 46. also the 48. and several others in Lib. 1. Math. Enucl. and likewise Lib. 2 Prop. 1, 2, 3, and se∣veral following. And as for Examples both of Denominati∣on, and Equations found after various ways, you may see them hereafter follow, and some we will here give you by way of Anticipation.

III. REDƲCTION.

An Equation thus found must be reduc'd, i. e. those two e∣qual quantities, which for the most part are very much com∣pounded of the quantities given and sought together, must be reduc'd to such a form, by adding or substracting something to or from each part, or multiplying and dividing by the same, &c. that the unknown or sought quantity alone, or its Square or Cube or Biquadrate, &c. may be found on the one side, and on the other the quantity express'd by meer given or known Letters, or affected with the unknown and sought ones; such are these Forms which follow, distinguish'd by their Names prefix'd to them,

A simple Equation, z=b, or y=〈 math 〉〈 math 〉.

A pure Quadratick, 〈 math 〉〈 math 〉.

A pure Cubick, 〈 math 〉〈 math 〉.

An affected Quadratick, 〈 math 〉〈 math 〉.

An affected Cubick, 〈 math 〉〈 math 〉 &c.

A Biquadratick, 〈 math 〉〈 math 〉 &c.

To one of which, or some other like them, when yo•• reduc'd your Equation first found, there are Rules in rea•• whereby the Value of the unknown or sought quantity z or x, may be either expressed in Numbers, if the Quest•• an Arithmetical one, or geometrically determin'd if it Geometrical one: Which is that we call Effection or Co••¦ction.

Thus therefore the whole, or at least the chief busine•• Analyticks, is conversant in finding a convenient or fit Eq••¦on: For Reduction is very easie, and consisting only in Operations and mere Axioms, as e. g.

If to equal quantities you add or substract equal ones Aggregates or Remainders will be equal;

If equal quantities are multiplyed or divided by the s•• the Products or Quotients thence arising will be equal, 〈◊〉〈◊〉

[illustration]

Pag. 5

Fig. I.

Fig. II.

Fig. III.

Fig. IV.

Fig. V.

VI

VII

VIII

IX

X

XI

XII

XIII

XIV

XV

XVI

XVII

IV. EFFECTION, or CONSTRƲCTION.

1. In simple Equations.

••Uppose z=b, the quantity b is sought.

If z be =〈 math 〉〈 math 〉.... or or x=〈 math 〉〈 math 〉.... or 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉 make

• as c to b so a to z,
• as b to a so a to x,
• as h−1 to b+g so f to y,
• as h+1 to b−g so f to z,

&c. every where according to n. 2.

Schol. 2. Prop. 34. Lib. 1. Math. Enuc.

If z be = 〈 math 〉〈 math 〉, the Resolution of it into Proportionals 〈◊〉〈◊〉 be more difficult, because neither of the Letters are found ••ce in the Numerator. That therefore you may have e. g. twice, you must make as k to n so m to a fourth Proportio∣••l which call p; then will, by vertue of Prop. 18. Lib. 1. ••=mn, and the proposed Equation be changed into this 〈 math 〉〈 math 〉, to be now constructed from the 2d. Case.

Or if 〈 math 〉〈 math 〉, find a mean Proportional between k •• l, which call p; and between m and n, which call q, ••ording to n. 3. of the afore-cited Schol.; and the propo∣sed Equation, by virtue of Prop. 17. will be in this form: 〈 math 〉〈 math 〉. Make therefore in the right-angled ▵ (Fig. 1.) B=p and BC=q; and the □ AC by vertue of the Py∣••gorick Theorem, =pp+qq: Which since it must be di∣••ed by r−s, make further, by Prop. 18 as r−s to the 〈 math 〉〈 math 〉, so is 〈 math 〉〈 math 〉 to y, according to the afore∣••ed.

4 In like manner if x be = 〈 math 〉〈 math 〉, make 1st. as b to m so n to a fourth which call k; and so putting bk for mn, the Equation will be reduced to the second Case under this form: 〈 math 〉〈 math 〉

Or thus: Find a mean Proportional between b and g, which call p, and between m and n, which call q; and the proposed Equation will be in this Form: 〈 math 〉〈 math 〉 Make therefore (in Fig. 2.) AB=p, and having on this described a Semi-circle, apply BC=q; then will, by vertue of Schol. 5. Prop. 34. □ AC=pp−qq: Which since it must be divided by c+d, make farther, as c + d to 〈 math 〉〈 math 〉 so 〈 math 〉〈 math 〉 to x; all from the same Foundations, whence you have the Construction of the third Case.

5. If z be =〈 math 〉〈 math 〉; make first as f to a, so a to a third Proportional m, and you'l have (putting fm for aa) 〈 math 〉〈 math 〉, i. e. 〈 math 〉〈 math 〉. Make secondly as f to m, so b to a fourth n, and by putting fn for mb you'l have z=〈 math 〉〈 math 〉, i. e. 〈 math 〉〈 math 〉, wherefore thirdly you'l have as g to n so c to z.

6. If y be =〈 math 〉〈 math 〉 make first as m to n, so l to a fourth, which call n, and by putting now mn for hl, you'l have 〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉=y. Therefore you'l now have secondly, as m to n, so l to y by Case 2: So that the Construction of the fifth and sixth Cases is nothing but reiterations of the Rule of three, ac∣cording to what we have often inculcated, N. 2. and 3. Schol. 2. Prop. 34.

2. In simple Quadratick Equations.

1.

• IF xx=ab
• or y{powerof2}=1c
• or z{powerof2}=¾dd

you'l have

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉

that is to a mean propor∣tional between

• a and b
• 1 and c
• ¾d and d

and so the Construction will be had from n. 3. Schol. 2. Prop. 34. (see Fig. 3.)

• 2. If y{powerof2}=fg+kl
• or x{powerof2}=fg−kl

you'l have

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉

make there∣fore on the one side the Right-angled Triangle ABC (Fig. 4.) whose side

AB is = to a mean Proportional between f and g,

BC is = to a mean Proportional between k and l;

On the other a Right-angled ▵ (Fig. 5.) whose side AB is = to a mean Proportional between f and g, and the side BC = to a mean Proportional between k and l; and on the one hand the Hypothenusa, on the other the side

• AC will be the va∣lue of y
• AC will be the va∣lue of x

sought.

And all by vertue of the Pythag. Theor. and according to Schol. 5. Prop. 34. or the Consectarys of Prop. 44. See Fig 4. and 5.

3. If z{powerof2}=〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉 extracting the Roots on both sides, z will =〈 math 〉〈 math 〉, and so be the second case of simple Equations.

4. If y{powerof2} be =〈 math 〉〈 math 〉, make first as l to f, so g to a fourth which call n, and by putting ln for fg, you'l have y{powerof2}=〈 math 〉〈 math 〉 i. e. 〈 math 〉〈 math 〉. Make Secondly, as m to n, so h to a fourth, which call p; and by putting mp for nh, you'l have y{powerof2}=

〈 math 〉〈 math 〉 i. e. pk, and so the first case of the present Equations.

5. If 〈 math 〉〈 math 〉 in the first place the Rectangles fg and lm being turned into Squares, and collected into one Sum, make them =nn. Then (since cc and cd are multiplyed by qb+bd) in like manner qb and bd added make pp; and you'l have 〈 math 〉〈 math 〉. Thirdly (since pp is already multiplyed by cc+cd) having added cc+cd into one Sum that they may e. g. make rr; x{powerof2} will =〈 math 〉〈 math 〉, and so be the third case of the present Equations.

3. In affected quadratick Equations.

1. IF zz be =az+bb, then will 〈 math 〉〈 math 〉; which may be thus in short demonstrated a priori; Since z{powerof2}−az=bb per Hypoth. and that first quantity if it be added to ¼aa, it becomes an exact Square, the root where∣of is z−½a; therefore 〈 math 〉〈 math 〉, and consequently 〈 math 〉〈 math 〉; and lastly 〈 math 〉〈 math 〉; which last Root is a false one and less than nothing, but yet gives you the propo∣sed Equation back again as well as the former; as will be evi∣dent to any one who trys, viz. having transferr'd ½a on the other side, and so the two equal quantities z−½a and 〈 math 〉〈 math 〉 being squared. For here will come out ¼aa+bb as well as if the radical Sign were affected with the Sign + be∣cause − by − gives +. Therefore 〈 math 〉〈 math 〉, and taking away on both sides ¼aa, zz−az=bb i. e. zz=az+bb.

The value therefore of this Root will be had geometrically, by making (in Fig. 6.) CD=½a and DE=b, that the Hy∣pothenusa CE may be 〈 math 〉〈 math 〉; and moreover, drawing out on both sides CD, and at the interval CE describing a Se∣mi-circle

AEB: This being done, AD will be the value sought of the true Root z, and DB of the false one.

2. If y{powerof2} be =−ay+bb, then will 〈 math 〉〈 math 〉; which again may thus appear: Since y{powerof2}+ay is =bb per Hy∣poth. adding to both sides ¼aa, the first quantity will be an exact Square, and 〈 math 〉〈 math 〉. Therefore the Roots will be also equal, viz. 〈 math 〉〈 math 〉, and consequently 〈 math 〉〈 math 〉; which is a false Root.

The value of these Roots may be had geometrically, viz. of the true Root DB in Fig. 6. or BE in Fig. 7. and of the false one in the first AD, in the second AE.

3. If xx is =ax−bb, you'l have 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉.

Which may be demonstrated after the same way a priori, as the former Cases, viz. Since x{powerof2}−ax is =−bb, adding on both sides ¼aa, the former quantity will be an exact Square, viz. 〈 math 〉〈 math 〉. Therefore the Root of the one x−½a = to the Root of the other, viz. 〈 math 〉〈 math 〉, and adding on both sides ½a, 〈 math 〉〈 math 〉 which is one of the true Roots. Or 〈 math 〉〈 math 〉 which in this case is al∣so a true one. But the value of each may be obtain'd by making (Fig. 8.) CB=½a and by erecting BD perpendicu∣larly =b, and making the Semi-circle BEA, and drawing DE parallel to CB, and letting fall the Perpendicular EF: For thus CF will be 〈 math 〉〈 math 〉 and consequently AF, 〈 math 〉〈 math 〉, and FB 〈 math 〉〈 math 〉.

Or, with Cartes, making (Fig. 9) CB=½a and BD=b, drawing DF parallel to CB, that FD may be one root and ED the other; as is manifest from the precedent Construction, and its Rule. See also another Deduction from Cartes's Con∣structions, Schol. 1 Prop. 47. Lib. 1. Math. Enucl.

NB. 1. The ingenious Schooten has before shew'd this Me∣thod of demonstrating, and also of finding out these Rules in his Comment on the Geometry of Des Cartes, p. m. 163. and moreover deduces another ingenious Method for all the three Cases of these Equations, by taking away the second term in

the Equation, p. 290, and the following, where we may make only this Remark concerning the third Case; that perhaps the Rule might be better deduc'd, if we make x=½a−z rather than x=z−½a.

NB. 2. If any one has a mind to see the new Constructions of affected quadratick Equations of the Abbot Catelan, he may find them in Acta Erud. Lips. Ann. 1682. p. 86. and in the 27th Journal des Scavans. 1 Dec. 1681.

IV. For Cubick and Biquadratick Equations both simple and affected, and also for all before mentioned, and consequently universally for all not exceeding the fourth Dimension.

THE value of the unknown Quantity or Root may for any Case be determined by one general Rule, found out by Mr. Thomas Baker an English-man, occasioned by what Des Cartes had taught concerning this matter, Lib. 3. Geom. p. 85, and the following; but now very much perfected by this Rule, and made more simple. Now that this Rule may be the better comprehended by Learners, we will premise these following things.

1. That all Equations occurring under those Forms which we have before shewn in the Article of Reduction, or the like, must always for this purpose be so changed as to have all the terms or parts of the Equation both known and unknown, af∣fected and not affected, brought over to one side promiscu∣ously, and so on the other there will stand o or nought, as e. g. let 〈 math 〉〈 math 〉, or 〈 math 〉〈 math 〉; or 〈 math 〉〈 math 〉; or 〈 math 〉〈 math 〉; or 〈 math 〉〈 math 〉 &c. which also was usual to Des Cartes in Lib. 3.

2. In all Equations the known quantity or Co-efficient of the second Term we will generally denote by the Letter p, that of the third Term by the Letter q, of the fourth by r, and the fifth (or absolute Number) by S; according to Cartes, but with some little alteration: So that hence the Equations we have before been treating of, and all others like them (every where denoting the unknown quantity by x) may all be reduced to these forms: 〈 math 〉〈 math 〉, &c. &c.

3. These and the like Equations may either occur whole, or with all their Terms, as here, or depriv'd of one or more of them, as the following Examples will shew, where we will always put an Asterisk in the place of the deficient Term.

〈 math 〉〈 math 〉

4. The unknown quantity in any Equation has† 1.5 as many diverse Roots or Values, as the Equation has Dimensions; which Des Cartes shews, Lib. 3. Geom. p. 69. at the same time evidently demonstrating this, viz. that some of those Roots may be false ones, i. e. less than nothing: Which from him we here suppose.

When therefore Des Cartes in his Construction of Cubick and biquadratick Equations, p. 85, and the following, requires as a necessary Condition, the ejection of the second Term in the given Equation, unless it were already wanting, and so was obliged to shew a way to eject it, with several other Prepa∣rations; and afterwards, when by help of his Rule delivered p. 91. he had found a way of finding two mean Proportionals, and

dividing any given Angle into three equal parts, then he use•• it for solving other solid Problems, or finding two mean Pro∣portionals, or trisecting an Angle. But the general Rule of Baker has no need of these methods or helps, neither of the Ejection of the second Term, nor any other Preparation, but immediately shews us a way, by the help of a Circle and Para∣bola to find all the Roots of any given Equation, both true and false, whether the Equation want any term or not, and howsoever affected, after the way we will now, and perhaps a little more distinctly, shew.

1. It supposes with Cartes a Parabola NAM to be already described, (See Fig. 10. and 11.) whose Latus Rectum shall be L or 1, and its Axe ay; which Des Cartes only making use of, and never thinking of the other Diameters, was forced to take away the second Term of the Equation, &c. Baker therefore (strangely perfecting the Cartesian Geometry by this one thought) if the quantity p or second Term be in the E∣quation applys to the Ax ay (or draws an ordinate to it) BA=〈 math 〉〈 math 〉 i. e. he erects at top of the Ax a on the right hand the perpendicular aE=〈 math 〉〈 math 〉 and from E draws EAy parallel to the Ax ay; whereby he obtains the Diameter Ay sought.

2. Having made this Preparation, the whole business de∣pends on this, to find the Center of the Circle to be described through the Parabola, which (by vertue of some arbitrary suppositions in the beginning) he always seeks on the left side of the Ax or Diameter, by help of two Lines 〈 math 〉〈 math 〉 or b, and DH or d; viz. by placing the former upon the Ax from a to D, if p be wanting in the Equation, or upon the Diameter Ay from A to D if p be there; and letting fall from the point D the latter to aD or AD perpendicularly towards the left hand.

3. He shews how to find the quantity of either of these Line (which is here very requisite) in any given Equation, by •• certain general Rule (which he calls the Central Rule, because it alone helps to find the Center H) comprehended in these terms:

Directions for the Bookbinder.

The Eight half-sheet Plates that are to be folded in, and the single Leaf mark'd Page 89, are to be placed in those Pages of the Introduction to Specious Analysis which the figures at the top of them direct to.

A SYNOPSIS of Mr. Baker's CLAVIS, to be annexed to Page 13, of the Introduction to the Specious Analysis.

Of Aequations.	Of Aequations.	Of Central Rules.
Class. I.	1. 〈 math 〉〈 math 〉	〈 math 〉〈 math 〉.
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉
〈 math 〉〈 math 〉	〈 math 〉〈 math 〉	〈 math 〉〈 math 〉

N B. The Sign ∽ denotes a dubious Case, viz. That either the Antecedent must be Substracted from the Consequent, or the Consequent from the Antecedent, according as the matter will bear.

〈 math 〉〈 math 〉

This Rule as it stands here whole, only answers to those ••••ations wherein are all the Terms p, q, and r; and in the ••••••n time may also be easily accommodated to all other Cases, 〈◊〉〈◊〉 observing these things. 1. Whatever Term, or Quantities ••••r, be wanting in the proposed Equation, that must also be ••••ctively omitted, or put out of the general Central Rule, that 〈◊〉〈◊〉 ••emaining quantities may determine the special or particular ……ral Rule. 2. As for what belongs to the Signs, viz. whether 〈◊〉〈◊〉 ∽ (which latter Sign denotes a dubious Case, either that the 〈◊〉〈◊〉 must be substracted from the latter, or contrary-wise, as 〈◊〉〈◊〉 matter will bear) must be put in the Central Rule, he 〈◊〉〈◊〉 (α) that in the Rule you'l always have 〈 math 〉〈 math 〉, unless 〈◊〉〈◊〉 in the proposed Equation p and r are affected with diverse ……s: (β) By what Sign soever in the proposed Equation it ……ens that the quantity q is marked with, it must be noted 〈◊〉〈◊〉 the contrary one (altho' involv'd with other quantities) ……e Rule; as may be seen in the application of the Rule to ……ecial Cases done by the Author himself for the sake of Be∣……ers, and is exhibited in the Synopsis hereunto adjoyning, ……h yet we have thought fit to give at the end of this Trea∣…… much more contract as to the Central Rules, in a short ……pendium by way of Appendix.

By these Rules therefore, the quantities of the Lines aD ……D and DH will be so determined, that the parts in the 〈◊〉〈◊〉 marked with the Sign + (taken either aggregately or 〈◊〉〈◊〉) will be put downwards from a to A towards y, and on ……ft hand of D; but the negative Parts, or those affected …… the Sign—, will be cut off, on the one part above, on ……her on the right hand: Which being done the Center H ……e found.

From the Center H thro' the Vertex of the Ax a (if the ……D is found in the Ax) or in the other Case thro' the 〈◊〉〈◊〉 of the Diameter A, you must draw a Circle which by ……g or touching the Parabola will deermine the Roots

sought, if the Equation be not a Biquadratick i. e. has not the quantity S; otherwise another Point L or Z must be found, (vid. Fig. 12. and 13.) and a Circle described on the Radius HL or HZ, according to Des Cartes p. 86, and following, of his Geometry.

7. Viz. If you have−S, you must take on the Line Ha or HA produced, on the one side AI=L or 1, and on the other AK = 〈 math 〉〈 math 〉, and describing a Semi-circle on IK, draw AL perpendicular to AH, to obtain the point L. (see Fig. 12.) But if you have +S, then in another Semicircle described on AH, apply the Line AZ = to AL found, thereby to obtain Point Z, (see Fig. 13.)

8. A Circle therefore described from H through a or A, if S be wanting, but thro' L if there be−S, and thro' Z if +S, may touch or cut the Parabola either in 1, 2, 3 or 4 Points; from which if you let fall Perpendiculars to the Ax or Diame∣ter, you will obtain all the Roots of your Equation both true and false.

9. And, 1. If in the Equation p be wanting and−r be there, the true Roots will be on the left side of the Ax, as NO, and the false ones as MO on the right side. 2. But if there be in the Equation p and−p, the true Roots will fall on the left side of the Diameter, and the false ones on the right; but if +p, on the contrary the true will be on the right hand and the false on the left.

10. But if the Circle neither touches nor cuts the Parabola in any point, it is a sign that the Equation is impossible, and has no Root either true or false, but only imaginary ones. All which, how they may be found out, and that they are undoub∣tedly true, are demonstrated a posteriori, in an easie and plain way by the Author, wherefore we shall not give the Demon∣strations of them here; but remit the Reader, after he has made a little progress in this Art, to the Author himself.

11. Wherefore now, (omitting also in this place the Do∣ctrine of the Composition of the plain and solid Geometrical Loci, or Places, which would serve for a Complement of the Analytick Art) we will shew the Practice of these Rules al∣ready delivered, premonishing only this from Mr. Baker, if the Latus Rectum be made Unity, that L in the Central Rules

and all its Powers may be omitted, and so the Rules exhibited more compendiously, as we have already done in our Synop∣sis, and may be seen from the form of a general Central Rule hereunto annexed.

〈 math 〉〈 math 〉

To which Premonition of Baker we may also add this, if any given Line in the Problem it self be taken for Unity, which may be often very commodiously done [as a in the former Problem p. 91. Geom. Cartes, and the Line NO in the lat∣ter, and a again in the Equation p. 83. the last line] and then the same Line also may be taken for the Latus Rectum of the Parabola to be described, if we have a mind to make use of this Compendium for abbreviating the Central Rules. For otherwise if we would construct all Problems, as Baker rightly asserts we may, by only one Parabola, we shall fall often into very tedious Prolixities.

Notes

• (α) 1.1

  Eucl. 6.17.
• (β) 1.2

  id. 6.16.
• (γ) 1.3

  id. 1.47.
• (δ) 1.4

  id. 3.36.
• † 1.5

  N. This is an error, as has been prov'd by Dr. Wallis.