(10)1
Cambridge Aug 20 1672.
SrSir
Since yoryour last I have tryed the calculation for finding by an infinite
series the content of yethe second segments of an Ellipsoid. The first series ytthat
I met with was this Which upon comparison proved yethe same wthwith Mr Gregories & therefore I have exprest it in yethe same letters. I tryed two or thre others, but could finefinde none more simple. Wherefore since I understand your designe is to get a rule for guageing vessells, Ithis Problem having so bad success for ytthat end I shall in its stead present you wthwith this following expedient.
Let represent the vessell viewed endwise its perimeter at the middle being , & at yethe end , & yethe top of yethe liquor . Also let be a circle whose semidiameter is equall yto yethe difference of yethe semidiameters of yethe other two circles & , & the whole length of yethe vessel. Find by a table or instrument composed for that purpose, the segments , & , & the whole circle ; & shall be the whole quantity of the liquor in yethe vessel. This rule is not exact but approaches the content of yethe Parabolick spindle exactly enough for practice when the top of yethe liquor buts upon yethe end of yethe vessell. If yethe vessel be just half full tis exact; if more then half full, tis something to little; if lesse ynthan half full, too much.
The approximating yethe roots of affected Æquations by Gunters line is thus. Let yethe æquation for instance be : to resolve wchwhich place thre of Gunters Rulers , , & parallel & equally distant from one another, & to any line wchwhich crosseth then all apply yethe number of the tfhirstd Ruler , yethe number of yethe second, , & yethe number of yethe first , accordingly as yethe coefficients of yethe æquation are. Then in yethe cross line taking yethe point as far from as is from apply any ruler to that point , about wchwhich whilst you turne it slowly observe the numbers where it cuts the other rulers untill you see yethe summ of yethe other numbers on the first & third ruler equall to yethe sum of yethe resolvend & the number on yethe other ruler, wchwhich when it happens the number on yethe 3d ruler shall be yethe cube of yethe desired root. Thus in this case you see , & therefore the number on yethe 3d ruler is yethe cube of yethe desired root wchwhich consequently is .
The application of this to any æquations of higher dimensions is obvious. As also so to proportion yethe rulers , , , &c ytthat yethe line may be carried over them with parrallel motion.
The description of a Conick section wchwhich shall pass through five given points is this. Let the five points be , , , , & any three of wchwhich joyn to as , , & joyn to make yethe a rectilinear triangle , to any two angles of wchwhich as & apply two sectors, their poles to yethe angular points, & their leggs to the sides of yethe triangle. And so dispose them that they may turne freely abouth their poles & without varying the angles they are thus set at. Which done, apply to yethe other two points & successively their two leggs & wchwhich were before applyed to (wchwhich leggs for distinction sake may be called their describing leggs & the other two & wchwhich were applyed to , their describing directing leggs,) & marke the intersections of their directing leggs, wchwhich intersections suppose to be when yethe application was made to , & when made to E. Draw the right line & produce it infinitely both ways. And then if you move the rulers in such manner that their directing leggs doe continually intersect one another at the line , the intersection of their other leggs shall describe the conic section wchwhich will pass through all the said five given points.
If three of the given points lye in the same streight line tis impossible for any conick section to pass through them all, And in that case you shall have instead thereof two streight lines.
Much after the same manner a Con. Sect. may be described wchwhich shall pass through 4 given points & touch a given line, or pass through 3 given points & tougch two given lines, whether those lines be right or curved. &c
I presume it will not be an unpleasing speculation to yoryour Mathematicians to find out yethe Demonstration of this Theorem. As also to determin the centers, diameters, axes, vertices, & Asymptotes of yethe Con. sect. thus described or to describe a Parabola wchwhich shall pass through 4 given points. And there I omit them.
I herewith send you a set of Problems for construing æquations wchwhich that I might not forget them I heretofore set down rudely as you'le find. And therefore I think 'em not fit to be seen by any but yoryour selfe, wchwhich therefore you may return when you have perused them. How yethe afforesaid descriptions are to be applyed to thof yethe Conick sections are to be applyed to these constructions I need not tell you.
Borellius, of whom you desire my opinion, I esteem among the middle sort of Authors. I find not that he hath added any thing considerable to yethe sciences of motion but onely proved things already evidently known. Nor hath he done ytthat wthwithout some Paralogisms, as in yethe proofs of yethe 2079th, 212th, & 260th Propositions. And some of them are not onely proved parallogistically but are also fals as 2 the 233d & those ytthat depend on it, but yet he may be of good iuse to young students in Mechanicks.
SrSir I shall trouble you no further at prsentpresent, but subscribe my self
yoryour very humble servant
I. Newton