Sit n + 1 =p, n − 1 =m, b + c =ƒ.

Si rite examinetur problema, resultat sequens æquatio b"c=b"fa-a". Hæc æquatio duas semper admittit radices veras, modo quæstio sit possibilis, quarum major ad problema non pertinet, cum sit semper=b, et in puncto maximi, puta cum usura nulla sit, utraque æqualis est ipsi b. At in omni casu radix minima nostro proposito inserviens per hanc seriem (modo ita

bc

nominari possit) invenitur; a> =d, item a>

f

bc hp

m

=h, item a> +

que ita de cæteris in infinitum.

The longer the continuance is, and the greater the interest is, this approximation is the nearer, and cometh the sooner to a period; but if the approximation continue any considerable time, ye may observe that the first differences are quam proxime in progressione geometrica majoris inæqualitatis, and therefore the whole sum of them in infinitum easily to be gathered, which, subtracted from the respective approximation, giveth the true root. This method is, of all which I know, the best, especially because of the easy continuation of it in infinitum. I could give some more compendious, but the method of the invention of its terms were infinitely more tedious. If in any thing else I can serve you, ye may command, Sir,

your humble servant,

J. GREGORY.

If ye think not this clear enough, send me any question ye please, and I shall resolve it according to this method. I suppose I need not advertise you that if the continuance be considerable, ye must use logarithms.

Sir,

CCXV.

J. GREGORY TO COLLINS.

Edinburgh, 8 October, 1674.

I received yours just now at my return from the north of Scotland, where some business called me. I thank you for the pains ye have been at in buying the tube and prisms for me, and for these books ye are pleased to bestow on me.

I am exceedingly surprised with these objections, which are moved against the Tentamina de motu project., &c. They that affirm the curve VTP, figure vif, to be no parabola must be grossly ignorant of the doctrine De locis solidis, and therefore no competent judges. Others, who affirm that I suppose an uniform motion, are impudent to admiration, seeing in the viii and ix de motu pend. &c., there is expressly supposed a motus æqualiter retardatus, yea all along that uniform motion supposed by Galilæus and all others hitherto (who have treated of that subject) is declared contrary to the mind of that author, as in 7ma, and manifestly declared to be the principal, if not the only reason of these Tentamina. As for Mr. Anderson, I would have him to prove his first assertion, viz.

x2 r2 = s2 r2 — s2 b2 + b2 r2 — s2 62, and then I shall answer the rest of his learned discourse. If he understood either multiplication of surds or the fourth of the second of Euclid, he had not been so ridiculous. I believe, (until I be otherwise taught by his

He hath

√4r4 s 2 b2 — 4 r2 s4 b2 — 4r2 s2 b1 + 48a ba. manifestly here left out the one half of the equation. If ye would have me examine any one part of his Gunnery, I shall obey you in examining it accurately, but I can hardly spare time for so much pitiful stuff. I satisfy myself with this, that it in general can be to no purpose, seeing he supposeth the motion, abstracting from gravity, to be uniform, which is palpably absurd, seeing the air resists. This (albeit I be thought by some to maintain the same) satisfieth me as much as a million of experiments on Blackheath manifesting its falsehood. That question of Mr. Kersey, to which Dr. Pell finds so many answers, is nothing strange, seeing it ascends to such a high equation; for equations may suffer as many roots as dimensions. No further, but rest,

your most humble servant,

J. GREGORY.

I mind not to renew any correspondence with Mr. Kerre, for I never got any thing by the carrier, which was not in great hazard of miscarriage, by the change of the carriers and opening the packs on the border, for by this small things are neglected. Being here in Edinburgh, I can have things securely that are directed to me by sea, if I be advertised by the post.

I shall go about Sir A. Dick's business as effectually as I can, which I am obliged to do. I wish I could do you more considerable service.

I am daily expecting from my scholar an account of these [sp]eculah ye desire, which I shall immediately transmit to you.

CCXVI.

J. GREGORY TO COLLINS.

Edinburgh, May 26, 1675.

Sir,

I received lately two of yours, one dated April 20th, and the other May 1st. I admire that ye fancy any difficulty in the cubic equation of three roots, seeing Des Cartes long since hath reduced it to the trisection of an angle, and the trisection of an angle can be turned infinite several ways into an infinite series; some of which methods I sent you long ago, not only of trisecting an angle, but also dividing it in ratione data.

--

That which ye intimate of the sum of the squares, cubes, biquadrates of the roots in a biquadratic equation, is pretty obvious in any equation, as for example, let x7 — px6 + q2 x3 — r3 x1 + s1 x3 — ƒ3 x2 + ko x − l2 = 0, the sum of all the roots =p, all their squares = p2 -2q2, their cubes-p3-3pq2 +3r3, their biquadrates=p1— 4p2q2+4pr3+2q4-4s4, their surdesolids =p5-— 5p3 q2 + 5pq1 + 5 p2 r3 − 5q2 r3 — 5ps1+5t5, their

h This word is very doubtful.

This word ought probably to be omitted.

sixth powersk=p — 6p1q2 + 9p2q1 + 6p3μ3 — 12pq2r3 — 6p2s4+6p t5-4q6 + 6 q2 s1 + 3r6-66, the seventh powers=p7-7p5 q2+14p3 q1 +7p1 r3 — 21 p2 q2 r3 — 7p3 s4 +7p2 t5 — 7pq6 + 14pq2 s1 + 7pr6 −7r3 s4 — 7q2 t5 +7q4 r3 —7pk6+717. It is no hard matter to give the rule, whereby to continue this in infinitum; for it is so in all equations, as (I believe) I did intimate to you some time ago, albeit under another dress, viz. to give the rational quantities contained in the potestates of the surd roots of all equations.

But this signifieth nothing to Dr. Davenant's problem, neither (if I be not much mistaken) doth that lemma ye mention, for it presently discovers itself in the natural progress of the solution. I have had some thoughts upon that problem; if any man resolve it by an equation under thirty dimensions, erit mihi magnus Apollonius. It is true the equations I brought it to were so tedious, that my patience would not permit me to apply rules of reduction, but I tried so many several equations, that if they had been capable of reduction, sure some of the reductions had been obvious. It is easy to constitute equations so that either two, three, &c., or all the intermediate terms, may easily go off but to take off even two intermediate terms in an arbitrary equation, without elevating it, is absolutely impossible. By elevating it I can take away all the intermediate terms myself, which (so far as I know) the world is yet ignorant of.

That equation about the rate of interest is composed of a simple lateral equation with a rational root, (which hath nothing to do with the question,) and a long adfected equation with one only root. Always, I

k This is expressed in a symbol, which could not be given in print; the same, in some mea

sure, applies to the following expression for the seventh power.