Mr Newton


The Letter herewith sent proclaimes the Authors great worth knowledge and Candour, and the method therein seemes unto me most admirable for Geometrick Curves wherein the Ordinates are expressed by an Æquation, but howit will performe in Mechanick Curves, wherein there is no such habitude expressed I humbly crave your sentiment: at first reading it was not immediately obvious how he came to find y=2zr2rr+zz his designe is first out of these Data AQ=r the Radius and Q1N=z to find the Square of the Chord AD=4r4rr+zz for the finding whereof suppose the Chord of the Complement to the Semicircle to be likewise drawne, and then there is given the ratio of these Chords such as r to z and the Sum of their Squares =4rr: out of such data by an Analyticall processe AD is found =4r4rr+zz

And then it holds
ANNQADD1B1which he callsy2 that isrr+zzzz4r4rr+zz4r4zz2rr+zzthe roote
whereof is 2zr2rr+zz=y as he makes it, and changing zz in the second tearme of the Proportion for rr, by such meanes A1B or x is found =2r3rr+zz as he likewise makes it, excuse me for troubling you with this impertinence I remaine

Your most humble thankfull Servitor

John Collins

Whereas he sayes habita ergo recta B11D=2zr2rr+zz
et recta B21B_______________=4r3zβ2rr+zz
habebitur valor rectanguli, D11B2B multiplicatis eorum Valoribus in
se invicem habebitur inquam8r53r2+z2this should behe sayes the OrdinateNPis8r5z23r2+z2qif this should be       the same}8r5zzβ3r2+zz


Out of r=QA and z=Q1N he finds x=A1B (which is supposed not to fall out of the line AQ though the figure represents otherwise) thus
This squared viz zzxxrr=2rxx2 whence by reduction x=2r3zz+rr his Calculus of the areall ordinate N11P is faulty but I hope ere long to send you the Calculation true.

2rx1x0 2rxxx

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