44 Of Increafing & Decreasing the Valne of Equations.

14

If I would have an Equation which fhall be of this Cubic xxx+bxx+cx=d, multiply by a Series of Proportionals, beginning with 1, whofe fecond Term is, and put y for x, then by the preceding Method we fhall have this new Equation, xxx + byy+cy=d. Or more generally thus, put a for any Number, Integer, or Fraction, and reallume the preceding Equation, & xxx + bxx + cx=d, then is where the Value of y to x, is as a to 1. yyy+abyy+aacy aaad,

y

27

{

The Ufes of this Artifice are very great; fome of which are thefe. Ufe i. To free an Equation from Fractions. Suppofe xxbxd, putting 3, we have yy by 9d. Again, fuppofe this Equation was propos'd, xxxd; that is, xxx +÷xx+ox=d, putting y=ax, we have 33y+byyaaad.

b

Ufe 2. To take away the fecond Term in any Equation. Suppofe xx + bx=d, puty-bx.

=

Then is yy-by+bb = xx? = d.
and-by-bbbx}
yy 4bb=d.

By the fame Method the fecond Term in a Cubic may be deftroy'd; only remember that y the Coefficient in the Square, y the Coefficient in the Cube yt, the Coefficient in the Biquadrate, &c. must be taken with the contrary Sign to the fecond Term; for if the fame Sign be taken, the fecond Term will doubled not taken

away.

Ufe 3. To take away all the Coefficients in a Cubic Equation. Which I think has not yet been done by any one; and perhaps others may make better use of it than I can at prefent. Suppose the second Term in a Cubic Equation taken away, or elfe that this is given xxx+bx=d, affume y=x√(√, being a mean Proportional betwixt and Unity); Then multiply by a term of continual Proportionals in that Series as before; and instead of xxxbxd, we have ixxxx+√÷0+% x=that is by equivalent Substitution yyy + = √

I

b

In all the preceding Cafes, as foon as the Value of y is found, the Value of x is alfo discover'd, y being always taken in fome certain Proportion to x.

L

Of the Punctation of Adfected Equations.

ET a Biquadrate be made from the Binomial x+ap, according to the former Genefis of Powers, and it will arife to this,

xxxx+4bxxx+6bbxx + 4bbbx+bbbb=pppp. Again, let a Biquadratic be found from feveral different Values of x, thus;

In this and the preceding Genefis, (from one Value of x, omitting the diftinction of the Uncia) 'tis evident that the Coefficients encrease in their Power, as the highest unknown Term decreases, and that the laft abfolute known Quantity is of the fame Power as the First. Imagine the fame under different Signs; alfo if one or more of the Terms be intercepted or loft, the remaining will yet keep their Places. Hence in Numeral Equations, fuppofe xxxx+3x2 + 16xx + 125x1000, the Coefficients are first Lateral, then Quadratic, then Cubic, &c. and 'tis no matter whether they are figurate Numbers or not, viz. exact Squares, Cubes, &c. they are always fuppofed of that Nature. Hence we have these two Confectaries.

1. If the Root of the Coefficient (in any Place) be continually multiply'd to its degree of Adfection, and then by it felf again, the last Product shall be of the fame Power with the bigbeft unknown Term in the Equation.

For Inftance, x— mx3 + n2x2 — p3x —— 9999, the Coefficient m is a Lateral, (therefore it self a Root); and if Cub'd, (for fo is the highest Power which it is join'd to, or its degree of Adfection) it becomes mmm; which multiplied again by it felf, becomes m of the 4th Power, as is xxxx. Again, in nnx, the Root of nn is its degree of Affection is a Square nn, which multiply'd into it felf, viz. nn, becomes nnnn a Biquadrate. Lastly, p'x the Root of p Cube, is p, its degree of Affection, is only Lateral: Therefore p (the Side) × ppp (the Coefficient it felf) produces p*.

2. If the abfolute known Power be divided by any one of the Coefficients, the Quote will be of the fame degree of Adfection with the place of fuch Coefficient, viz. Lateral, if the Adfection be Lateral; Quadratic, if the Adfection be Quadratic, &c.

For Inftance, x2+3xxx+16xx+125x=1000: if 1000 be divided by 3, the Quote is Cubic; if by 16, the Quote is Quadratic; if by 125, 'tis Lateral. From hence then, and the preceding Genefis, we have a Method of Pointing any adfected Equation in order to its Extraction.

It's evident from the Formation of Powers, that a Square must be pointed every fecond Figure, (beginning from the Right) a Cube every third, a Biquadrate every fourth; and so on. For fuppofe 57209 was given to be fquar'd, 5 has four places after it; therefore its Square 25 will have eight Ciphers; or if Cub'd, 'tis 125 with twelve Ciphers annex'd; and fo on, every Cipher or Place following a Figure in the Root, making 2 in the Square, 3 in the Cube, &c.

Let any Equation be propos'd, (in order to its pointing). Suppofe without any Coefficient xxx-xx-186494880.. From the preceding Genefis 186494880 is Cubic, because so is the highest unknown Term, therefore let it be pointed at every third Figure thus, 186494880. But fince it also contains in it the Negation of xx, therefore imagine it also pointed as a Square, thus, 186494880, but with the fame Number of Points, because xxx is no oftner contain'd in 186494880, than xx is fuppofed to be taken out of it.

There are three Cafes that can only happen in pointing.

1. Where the number of Figures in the Coefficients are regular, and will admit of an equal Number of Points according to their respective Powers. For Instance, xxxx + 13xxx + 237xx+5927x=100000, where each place has two Points, determining the Value of x to confift of two Integers; fo that we have for the first pointing this Equation + 1x+2xx+5x=10 near; whereby the Value of x in its firft Figure is readily difcover'd. 'Tis the fame if interrupted, viz. xxxx+9532x=123579, then x+9532x=12; or if the natural Order of Places be inverted, viz. x*+3537xx 35xxx=2598372, where x+35xx-3x2 =259.

"

2. The fecond Cafe is call'd a Dovolution, where the Number of Figures is too fhort in 'fome of the Coefficients: And here it's easier than before to find the Value of x as to the first Figure, for we have no more to do than prefix fo many Ciphers in each, as are the number of Figures that are wanting.

For Inftance, xxx +24x587914372; that is, xxx+00024x=587914372. Whence xxx + ox=587, or xxx = 587.

3. The third, which is called a Cafe of Anticipation, where fome of the Coefficients run beyond their juft Number of Punctations, and this is much more troublefome than either of the former.

For Inftance, xxx+4xx +376958x=753922; which pointed as before, would be xxx+04xx +376958x=753922; or x3+3769x=753, where *=.1, whereas 'tis equal to 2 precifely. The best Expedient I can meet with in this, (if the Figures have but one Punctation) is that of Dr. Wallis, who fuppofes the first Figure arifing to be 1, 10, 100, 1000, &c. Or downwards, . 1, .01 001, c. And then having found by trial whether it lies betwixt 1 and 10, or 10 and 100, r. He takes the Medium 5, 50, &c. till he meets with it. But this Expedient will also prove fruitless for the fecond Figure, if the Anticipation be large we have therefore but one way left us that I know of, and that is the incomparable Method of a Converging Series, lately difcover'd by the ingenious Mr. Raphan; which though it might be help'd by Punctation, as in the two preceding Cafes, yet it has no abfolute Neceffity of it, for by affuming any Number at pleasure, no matter how far off the Truth, (as to the poffibility of its accomplishing the End) it will converge to it infinitely quicker than by any Essays or Trials whatever in the most perplex'd Anticipations, therefore extreamly ready and eafy in the other Cafes. The further Ufes of it are beft explain'd by Examples; however I thought it necessary to say something of Punctations here, having occasion to use them hereafter, because Mr. Raphfon has not done it in bis excellent Series, but only refer'd to Vieta, Oughtred, &c. which perhaps every one has not at hand, or can't read them in Latin. But Mr. Raphfon's Bufinefs was not Repetition, or a troubling himself with what was requilite to be known before he could be understood, prefuming reasonably enough, that his Reader was not ignorant of what Advancements had been made before him in the Exegifis Numerofa of the aforementioned Authors.

Of Infinite Approximations, or a Numeral Converging Series for all Adfected Equations what

ever.

Example 1.

ET any Equation be propos'd, fuppofe xx=a.

1. Put a known Quantity) + (an unknown Quantity) equal to x. Then is p+yli, or pp+2py+yy=(xx=) 4.

2. By Transposition 2py=appyy, or y; or which is the fame thing, y=-2. But x=p+y by Pofition. Ergo, x=

in which Equation p+is wholly a known Quantity, and an unknown.

3. Put q=p+, and

99 — 2qz+zz= (xx=) a; and

2

-

༢༢

21

21

12+ But x=9- by Pofition; Ergo, x=q + —9

24

22 24

29

where the known q+, is evidently lefs than the former known Quantity 9, because qqa

25

will be fo very fmall, that we need

make no fubduction for it, if the Value of a be not above 32 Figures, which I think is fufficient for Practice.

Hence then by collecting the whole together we have this Series; p+2 (=9) 9+ m2 (=r) r+77 (=s) s+=

21

a-rr

- pp

2P

=x proximè; or according to Mr. Raphfon's Method, p+ only, repeated at pleasure is equal tox, where the Value of p always changes at every Operation; and the Value of is firft 1 Punctation, then 2, then 4, then 8, then 16, and fo on as far as any one pleases to profecute it, from the fame Confiderations which I mention'd in the preceding Series for Extraction of Square Roots.

Examp.. 2. Let this adfected Quadratic be propos'd, xx + bx=c. 1. Put x=p+y; then is pp +2py + xyS=xx (

x=py; therefore x=p+

bp + by・・・ 2=bx}"

c. That is,

2p+6°

2p+62

Now putting

By the fame Procedure repeated at

but

q=p+c=pp bp

pleasure, we have this Series,

Or according to Mr. Raphson's Method, p+PP-bp only repeated as far as there

is occafion, is equal to x, p acquiring a new Value every Operation; alfob and c changing their Value, reaching firft to one Punctation, then to two, then to four, then to eight, and fo on, or at least as far as there are Punctations; and when there's no more their Value is fix'd,thop will always converge and vary ad infinitum. Examp. 3. Let this adfected Cubic Equation be propos'd, x2+bxx+cx=d.

d_p3_bp2_cp __ 3pyy—byyyyy. Now putting q for all that's known, as be3p2+2bp+c 3PP+2bp+c

fore, and for all that's unknown, and fo repeating the fame Process arbitrarily, we shall have this Series;

d-p3-bp1-cp

3p2+2bp+c

(=q) q+d-q'-bq2-cq
39'+26g+c

(=r)r+

d-r3-br-cr
3r2 +2br+6"

&c.=x.

And fo on in all higher Equations, reprefenting always all those Quantities which are affected with the Powers of the unknown affum'd Quantity by another unknown, thereby driving forward and decreafing that unknown Value, till it becomes fo

fmall that it may fafely be rejected, and what is known taken for Truth it self, as being indeterminately near it: remembring alfo, that all the Values of the Coefficients, as alfo the last abfolute known Number change, fo long as the Punctations last ; but no longer, though the Value of the allum'd Quantity will ever converge and

alter.

Before I proceed to Examples, I fhall give the Series for the four Cafes in Quadratics, as alfo the fourteen useful Cafes in Cubics, leaving others to calculate higher as they have occafion.

No one that understands how to convert a Literal Theorem into a Numeral Ope ration, can be ignorant how to apply the preceding Theoretic Series. Yet notwithstanding the Method is fo very natural and eafy, I fhall give fome Examples.

I have formerly taken notice, that in adfected Quadratics, if the Homogeneum Comparationis arifes high, this Series is preferable to the common Way; but in all higher Equations whatever, it infinitely exceeds any Exegifis Numerofa that has yet been found out.

Let this Adfected Quadratic be propos'd, xx+587x=987459, and let the Value of x be found to five or fix Places in Decimals; which is the fame thing as to find the Value of x (if Rational) where the abfolute known Quantity arises to twelve Places.

Let