The former consectary was of a single mesolabium; this is of a double, whose use in making of solids, to this or that bignesse desired is notable.
Let the two lines assigned be ae, and ei; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be as ae, is unto
one of the lines found; so the same may be unto the second line found. And as that is unto this, so this may be unto ei. Let therefore ae, and ei, be joyned rightanglewise by their ends at e; and let them be infinite continued, but vertically, that is, from that their meeting from the lines ward, from ei, towards u, but ae, towards o. Now for the rest, the construction; it was Plato's Mesographus; to wit, a squire with the opposits parallell. One of his sides au, moueable, or to be done up and downe, by an hollow riglet in the side adjoyning. Therefore thou shalt make thee a Mesographus, if unto the squire thou doe adde one moveable side, but so that how so ever it be moved, it be still parallell unto the opposite side [which is nothing else, but as it were a double squire, if this squire be applied unto it; and indeed what is done by this instrument, may also be done by two squires, as hereafter shall be shewed.] And so long and oft must the moveable side be moved up and downe, untill with the opposite side it containe or touch the ends of the assigned, but the angles must fall precisely upon the continued lines: The right lines from the point of the continuation, unto the corners of the squire, are the two meane proportionals sought.
As if of the Mesographus auoi, the moveable side be au;
thus thou shalt move up and downe, untill the angles u, and o, doe hit just upon the infinite lines; and joyntly at the same instant ua, and oi, may touch the ends of the assigned a, and i. By the former consectary it shall be as ei, is to eo, so eo, shall be unto eu: and as eo, is to eu, so shall eu, be unto ea.
And thus wee have the composition and use, both of the single and double Mesolabium.
35. If of foure right lines, two doe make an angle, the other reflected or turned backe upon themselves, from the ends of these, doe cut the former; the reason of the one unto his owne segment, or of the segments betweene themselves, is made of the reason of the so joyntly bounded, that the first of the makers be joyntly bounded with the beginning of the antecedent made; the second of this consequent joyntly bounded with the end; doe end in the end of the consequent made.
Ptolomey hath two speciall examples of this Theorem: to those Theon addeth other foure.
