Extract of a letter from J. Gregory to Collins. Et s = r + a3 2a5 a2 5a1 61a6 &c. 17a7 3233a9 + + &c. + &c. 277a8 Sit nunc tangens artificialis = t et secans artificialis = s et integer quadrans = q Erits= a2 at + a6 17a8 3233 a 10 + + + &c. 2r 12r3 4575 2520r7 1814400r9 Sit 2a-q=e Erit te + e3 5e5 61e7 277e9 + 62244 5040767257678 + + Sit nunc secans artificialis 45o = s. Sitque &+L secans artificialis ad libitum. L2 L3 7L4 14L5 452L6 + &c. Erit ejus arcus= + 2 3r2 + + &c. 4575 You shall here take notice that the radius artificialis = 0 and that when you find q> 2a, or the artificial secant of 45° to be greater than the given secant, to alter the signs, and go on in the work according to the ordinary precepts of Algebra. a In Collins's handwriting. CCII. COLLINS TO J. GREGORY. Mr. Gregory, Sir, March 25, 1671. I have yours of the 15th of Feb., for the which I render you hearty thanks; what you write in relation to Dr. Barrow I sent him, but have not heard from him since. I must confess my unwilling neglect in not answering you sooner, but the case is this: His Majesty is pleased to increase the number of the Council of Plantations by adding divers persons of great dignity thereto; to wit, His Highness the Duke of York, Prince Rupert, the Duke of Buckingham, the Duke of Ormond, the Lord Lauderdale, the Lord Culpepper, Mr. Evelyn of the Royal Society; and it fell to my lot to transcribe copies of the Commissions and Instructions for their use. I think the Lord Lauderdale's sister's son is to be his heir, he is now at Oxford in St. John's College, and one Mr. Bernard reads the mathematics to him, likewise the astronomy lecture for Dr. Wren, who is now surveyor of His Majesty's buildings. The said Mr. Bernard is a good mathematician, and understands the Arabic tongue well; he hath found in the libraries there two entire copies of the first seven books of Apollonius his Conics, (and some other tracts of that author,) the one of Ben Musa, the other of Abdelmelech, and one of them hath Eutocius his notes. The three latter books, when translated and put into Dr. Barrow's method, may probably be printed with Dr. Barrow's comment on the first four, and be sold together. As for foreign mathematical intelligence, I have to add, since the last I sent you, that there is lately come out in Italy, Borellius de Liquidis, a physico-math. treatise; Honorati Fabri commentaria in Archimedem; the Comment of Borellius on Archimedes is to be printed at Lyons; Gottigni's Dioptrics; he is accounted a good geometer, and I think so truly by a sight of his small Euclid. In France, A Capuchin hath published Dioptrics, speculative and practical, in folio, of which Berthet the Jesuit writes he hears a good character. Mons. Picart de mensura perimetri terræ. None of these books are yet come over, nor Fermat's Diophantus, which are not to be bought at Paris, (there being none save presents sent from Toulouse,) one was sent from thence, by the Lord Aylesbury's man, intended for Mr. Oldenburg, but the fellow sold the book by the way, and spent the money. I have remitted money to our friend Mr. Vernon, at Paris, from whom I may expect some of those Commentaries on Diophantus, as soon as they are to be had, and, God willing, I shall not fail to send you one. Young Fermat writes there are notable algebraical inventions in it, which I should greedily covet to see; and as for my narrative about finding the roots of equations by tables, I now render you a better account of it. One Mr. Warner deceased, whose Optics you find mentioned in Mersennus, did, about thirty-two years since, spend above an hundred pounds for aid, and took great pains himself, with some assistance from Dr. Pell, to calculate a table, to twelve places of figures, of 100,000 continual proportionals, to wit, to find 99999 mean proportionals between an unit and 100,000. Such a large table, elegantly writ, remains in the hands of Dr. Thorndyke, a prebendary of Westminster; the construction and uses of it, with the tactions of circles rendered analytical, were lent to one Gibson, deceased in anno 1650, author of a book entitled Syntaxis Math., after whose death all his papers were consumed to light tobacco. But this was the Canon Mathematicus intended purposely for the solving of equations; and indeed Vieta shewing the constitution of many equations from continual proportionals, some of the terms whereof shall be the roots of equations, and the sums or differences of other terms the coefficients, renders it probable to me that such a table might be very expedite for tentative work. But Dr. Pell asserts that in any equation, after he hath the limits, (viz. where the serpentine curves for equations cross the base line, when it so happens,) and the greatest ordinate or homogeneum in the said portion of the serpentine curve given, that then he finds the logarithms of the roots so precisely by logarithmical operations, that the logarithm found shall not err an unit in the last figure, and this without tentative work, but admitting trials or the rule of false position in logarithms. I see it may be done, and that the homogenea to a series of roots of an equation may be made up by multiplication, without adding or subtracting of biquadrates, cubes, squares, &c. according to Dr. Barrow's method. Dr. Pell in discourse affirms that in a complete equation of the eighth power, between certain limits, the six intermediate terms may be all taken away; between other limits there can be but four of them taken away; and again between other limits but two of them taken away, Mediis sese mutuo perimentibus, but always any one of them taken away, (as Dulaurens hath shewed,) and the like in all equations, and this he explained long since by instances; yea, our Harriot, printed forty years ago, shews how in a complete biquadratic equation to take away any two of the inferior powers. Dr. Pell communicates nothing: he once refused me a proposition, and I am resolved never to move him more. He doth assert that Frenicle made a large table of figurate numbers, and by help of them solveth equations. I see thereby that to any series of numbers, whose last differences are equal, there may be a common equation found, and hereof I give you an instance, but I cannot perform the converse. As for the musical progression, I now send you my last thoughts about it, which indeed are better prioribus curis. By the same method you will easily obtain the reciprocals of the squares or cubes of an arithmetical progression. From Slusius we have lately received his geometrical method for determining of all equations, of the which, together with other papers of good worth, do not doubt to receive a copy as soon as my leisure can permit. I proposed unto these booksellers, viz. Pitts, Scot, Martin, and Kettleby, the quondam servant of Mr. Thompson deceased, the printing of your intended treatise, but they all refused it, and the like is the fate of Mr. Kersey's Algebra; but at length conferring with one Hickman, (the late servant of Mr. Allestry deceased,) newly set up for himself, he is willing to undertake the same, and promiseth it shall be well done: and I promise you my care and assistance in drawing the schemes and correcting [the press], and would not have you withhold any thing of your former thoughts of enlarging it. Book |