continuation, with the quadrate of the bisegment, shall be equall to the quadrate of the line compounded of the bisegment and continuation. These were the rates of an oblong with a rectangle.
From hence ariseth the Mesographus or Mesolabus of Heron the mechanicke; so named of the invention of two lines continually proportionall betweene two lines given. Whereupon arose the Deliacke probleme, which troubled Apollo himselfe. Now the Mesographus of Hero is an infinite right line, which is stayed with a scrue-pinne, which is to be moved up and downe in riglet. And it is as Pappus saith, in the beginning of his III booke, for architects most fit, and more ready than the Plato's mesographus. The mechanicall handling of this mesographus, is described by Eutocius at the 1 Theoreme of the II booke of the spheare; But it is somewhat more plainely and easily thus layd downe by us.
9. If the Mesographus, touching the angle opposite to the angle made of the two lines given, doe cut the said two lines given, comprehending a right angled parallelogramme, and infinitely continued, equally distant from the center, the intersegments shall be the meanes continually proportionally, betweene and two lines given.
Or thus: If a Mesographus, touching the angle opposite to the angle made of the lines given, doe cut the equall distance from the center, the two right lines given, conteining a right angled parallelogramme, and continued out infinitely, the segments shall be meane in continuall proportion with the line given: H.
As let the two right-lines given be ae, and ai: And let them comprehend the rectangled parallelogramme ao: And let the said right lines given be continued infinitely, ae toward u; and ai toward y. Now let the Mesographus uy, touch o, the angle opposite to a: And let it cut the sayd continued lines equally distant from the Center.
(The center is found by the [8 e iiij], to wit, by the meeting of the diagonies: For the equidistance from the center the Mesographus is to be moved up or downe, untill by the Compasses, it be found.)
Now suppose the points of equidistancy thus found to be u, and y. I say, That the portions of the continued lines thus are the meane proportionalls sought: And as ae is to iy: so is iy to eu, so is eu, to ai.
First let from s, the center, sr be perpendicular to the side ae: It shall therefore cut the said ae, into two parts, by the [5 e xj]: And therefore againe, by the [7 e], the oblong made of au, and ue, with the quadrate of re, is equall to the quadrate of ru: And taking to them in common rs, the oblong with two quadrates er, and rs, that is, by the [9 e xij], with the quadrate se is equall to the quadrates ru and rs, that is by the [9 e xij], to the quadrate su. The like is to be said of the oblong of ay, and yi, by drawing the perpendicular sl, as afore. For this oblong with the quadrates li, and sl, that is, by the [9 e xij], with the quadrate is, is equall to the quadrates yl, and ls, that is, by the [9 e 12], to ys. Therefore the oblongs equall to equalls, are equall betweene themselves: And taking from each side of equall rayes, by the [11 e x], equall quadrates se and si, there shall remaine equalls. Wherefore by the [27 e x], the sides of equall rectangles are reciprocall: And as au is to ay: so by the [13 e vij], oi, that is, by the [8 e x], ea, to iy: And so therefore by the concluded, yi is to ue; And so by the [13 e vij], is ue to eo, that is, by the [8 e x], unto ai. Therefore as ea is to yi: so is yi to ue; and so is ue, to ai. Wherefore eu, iy, the intersegments or portions cut, are the two meane proportionals betweene the two lines given.