If a pointe be assigned without a circle, and from that pointe .ij. right lynes be drawen to the circle, so that the one doe crosse the circle, and the other dooe ende at the circumference, and that the longsquare of the line which crosseth the circle made with the portiõ of the same line beyng without the circle betweene the vtter circumference and the pointe assigned, doe equally agree with the iuste square of that line that endeth at the circumference, then is that lyne so endyng on the circumference a touche line vnto that circle. _Example._ In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right line A.B, be equall to the longsquare made of the whole line A.C, as one of his lines, and of his portion A.D, as his other line, then must it nedes be, that the lyne A.B, is a right touche lyne vnto the circle D.B.C. And thus for this tyme I make an ende of the Theoremes. +FINIS,+ _IMPRINTED at London in Poules churcheyarde, at the signe of the Bra- senserpent, by Reynold Wolfe._ Cum priuilegio ad imprimen- dum solum. +ANNO DOMINI+ .M.D.L.I. [Illustration: Publisher’s Device (brazen serpent, NVM. XXI.)] * * * * * * * * * _Errors and Inconsistencies:_ Unless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were regularized only where there was a very large disparity between the expected form and the apparent errors (for example, a thousand “A.B” against a dozen “A,B”), or a flagrant misprint such as “cnt” for “cut”. The letters u and v follow the conventional “initial v, non-initial u” pattern except in numbers (xv, iv). The lower-case j form occurs only as the last digit of a number (ij, xxj); upper-case I and J share a form, always read as I. Italic double s was printed as an ſ+s ligature, similar to the German ß; it is shown as simple “ss”. Capital and lower-case “w” were often used interchangeably. Words split across line breaks may or may not have a hyphen. _Language:_