Proportional Division

Lines can be divided proportional to each other in the manner described above and 340:3 gives the details where AC is to be divided in the same proportion as AB. The method is also useful in dividing a given line into any number of equal parts (as in setting out dovetails) where AB (340:4), which is assumed to be 51/2 units (inches or millimetres) long, has to be divided into four equal parts. AB is first drawn the correct length and AC any convenient length easily divisible by four. Parallel lines from AC will then equally divide AB.

Where a line has to be divided into two equal parts, i.e. bisected, then 340:7 shows the method in which arcs with equal radii are struck from A and B to intersect at C and D, and a vertical line drawn through CD will then bisect AB. This method can also be used for striking a perpendicular to a given line, and 340:8 shows the method in which a semicircle is described from point A on BC, and using the same radius throughout describing arcs D and E from B and C and intersecting arcs from D and E to F; a line drawn from F to A will then be truly perpendicular. If compasses are not available then perpendiculars can be drawn using the theorem attributed to Pythagoras (scale of equal parts) in which the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. Thus in 340:9 if AB is drawn 3 units long, then measurements of 4 units from B and 5 units from A (or any multiples of all three measurements) will only coincide at point C on a line which is at true right angles to A.

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