thus: If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. 9 and 10 p vj.
Punctum contingens, A contingent point, that is falling or lighting in some place at al adventurs, not given or assigned.
This is a marvelous generall consectary, serving indifferently for any manner of section of a right line, whether it be to be cut into two parts, or three parts, or into as many parts, as you shall thinke good, or generally after what manner of way soever thou shalt command or desire a line to be cut or divided.
Let the assigned Right line to be cut into two equall parts be ae. And the right line making an angle with it, let it be the infinite right line ai. Let ao, one portion thereof be cut off. And then by the [7 e], let oi, another part thereof be taken equall to it. And lastly, by the [24 e], draw parallels from the points i, and o, unto e, the end of the line given, and to u; a contingent point therein. Now the third parallell is understood by the point a, neither is it necessary that it should be expressed. Therefore the line ae, by the [28], is cut into two equall portions: And as ao, is to oi: So is au, to ue. But ao, and oi, are halfe parts. Therefore au, and ue, are also halfe parts.
And here also is the [12 e] comprehended, although not in the same kinde of argument, yet in effect the same. But that argument was indeed shorter, although this be more generall.
Now let ae be cut into three parts, of which the first let it bee
the halfe of the second: And the second, the halfe of the third: And the conterminall or right line making an angle with the sayd assigned line, let it be cut one part ao: Then double this in ou: Lastly let ui be taken double to ou, and let the whole diagramme be made up with three parallels ie, uy, and os, The fourth parallell in the toppe, as afore-sayd, shall be understood. Therefore that section which was made in the conterminall line, by the [28 e], shall be in the assigned line: Because the segments or portions intercepted are betweene the parallels.
And