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Proposition 28

To apply a parallelogram equal to a given rectilinear figure to a given straight line but falling short by a parallelogram similar to a given one; thus the given rectilinear figure must not be greater than the parallelogram described on the half of the straight line and similar to the given parallelogram. | VI.27
Let C be the given rectilinear figure, AB the given straight line, and D the given parallelogram, and let C not be greater than the parallelogram described on the half of AB similar to the given parallelogram D. It is required to apply a parallelogram equal to the given rectilinear figure C to the given straight line AB but falling short by a parallelogram similar to D.
Bisect AB at the point E. Describe EBFG similar and similarly situated to D on EB, and complete the parallelogram AG. | I.9 VI.18
If then AG equals C, that which was proposed is done, for the parallelogram AG equal to the given rectilinear figure C has been applied to the given straight line AB but falling short by a parallelogram GB similar to D.
But, if not, let HE be greater than C. Now HE equals GB, therefore GB is also greater than C.
Construct KLMN equal to GB minus C and similar and similarly situated to D. | VI.25
But D is similar to GB, therefore KM is also similar to GB. | VI.21
Let, then, KL correspond to GE, and LM to GF. Now, since GB equals C and KM, therefore GB is greater than KM, therefore also GE is greater than KL, and GF than LM.
Make GO equal to KL, and GP equal to LM, and let the parallelogram OGPQ be completed, therefore it is equal and similar to KM.
Therefore GQ is also similar to GB, therefore GQ is about the same diameter with GB. | VI.21 VI.26
Let GQB be their diameter, and describe the figure. Then, since BG equals C and KM, and in them GQ equals KM, therefore the remainder, the gnomon UWV, equals the remainder C. And, since PR equals OS, add QB to each, therefore the whole PB equals the whole OB.
But OB equals TE, since the side AE also equals the side EB, therefore TE also equals PB. | I.36
Add OS to each. Therefore the whole TS equals the whole, the gnomon VWU. But the gnomon VWU was proved equal to C, therefore TS also equals C. Therefore there the parallelogram ST equal to the given rectilinear figure C has been applied to the given straight line AB but falling short by a parallelogram QB similar to D.
Q.E.F.

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When this proposition is used, the given parallelgram D is a square, and that case is much less complicated.

| In that case this proposition applies a rectangle ASQT to a given straight line AB equal to a given rectilinear figure C but falling short by a square (BSQR in the figure). So the rectangle being laid alongside the line doesn't reach the end of the line AB, but misses by a square. For the construction, bisect AB at E, construct a square GFBE, then make the square GPQO equal to the square GFBE minus the figure C, and complete the figure. The verification in this case is much easier in this case.

We can understand the meaning of this construction more easily if we interpret it algebraically. Let a stand for the known quantity AB and x stand for the unknown quantity SB. Then this construction finds x so that (a - x) x = C. In other words it solves the quadratic equation ax - x2 = C.

The next proposition solves a similar quadratic equation: ax + x2 = C.

This constuction in this proposition is used in propositions X.33 and X.34.

Book VI Introduction - Proposition VI.27 - Proposition VI.29.

© 1996
D.E.Joyce
Clark University