The Way To Geometry

by Petrus Ramus

triangles, except a Rightangled-parallelogramme, are in Geometry held to be irrationalls. This is therefore the definition of a proportionall section: The section it selfe followeth, which is by the rate of an oblong with a quadrate.

3. If a quadrate be made of a right line given, the difference of the right line from the middest of the conterminall side of the said quadrate made, above the same halfe, shall be the greater segment of the line given proportionally cut: 11 p ij.

Or thus: If a square be made of a right line given, the difference of a right line drawne from the angle of the square made unto the middest of the next side, above the halfe of the side, shall be the greater segment of the line given, being proportionally cut: H.

Let the right line gived be ae. The quadrate of the same let it be aeio: And from the angle e, unto u, the middest of the conterminal side, let the right line eu, be drawne; Then compare or lay it to the halfe ua; The difference of it above the said halfe shall be ay, This ay, say 1, is the greater segment of ae, the line given, proportionally cut.

For of ya, let the quadrate aysr, be made: And let sr, be continued unto l. Now by the [8 e, xiij]. the oblong of oy, and ay, with the quadrate of ua, is equall to the quadrate of uy, that is by the construction of ue: And therefore, by the [9 e xij]. it is equall to the quadrates ea, and au: Take away from each side the common oblong al, and the quadrate yr, shall be equall to the oblong ri. Therefore the three right lines, ea, ar, and re, by the [8 e xij]. are continuall proportionall. And the right line ae, is cut proportionally.

Therefore

4. If a right line cut proportionally, be continued with the greater segment, the whole shall be cut proportionally, and the greater segment shall be the line given. 5 p xiij.

As in the same example, the right line oy, is continued with the greater segment, and the oblong of the whole and the lesser segment is equall to the quadrate of the greater. And thus one may by infinitely proportionally cutting increase a right line; and againe decrease it. The lesser segment of a right line proportionally cut, is the greater segment, of the greater proportionally cut. And from hence a decreasing may be made infinitely.