Definitions

Definitions

1. “Commensurable” are magnitudes measured by the same measure.

“Incommensurable” are magnitudes which have no common measure.

2. Two straight-lines are commensurable when their bounding area are measured by the same area.

They are incommensurable when their area has no common measure.*

3. There is an infinite number of straight-lines which are either commensurable or incommensurable.

• Some are commensurable or incommensurable in length only.
• Others are commensurable or incommensurable in area also, with an assigned straight line.

‘Rational’ lines are:

• the assigned straight line
• those straight lines which are commensurable with the assigned straight line, whether in length and in square, or in square only

‘Irrational’ lines are those which are incommensurable with the assigned straight line.*

4. The [binding] square on the assigned straight line is called ‘rational’

Those areas which are commensurable with it are rational.

But those areas which are incommensurable with it are irrational

• The straight lines which produce those areas are irrational

Proposition 1

Assume there are 2 unequal lines.

• The longer line has more than half of it subtracted
• More than half of the remainder is subtracted

If this happens continually, then some magnitude will eventually be left. It will be less than the lesser laid out magnitude.

Let:

• AB and C are unequal magnitudes
• AB is the longer or greater

We subtract more than half from AB. From the remainder, we also subtract more than half. If this happens continually, there will be some magnitude left. But it will be less than the magnitude C.

If we keep on multiplying C, it will eventually be greater than AB.

Let it be multiplied, and let DE be a multiple of C, and greater than AB.

Divide DE into the parts DF, FG, GE equal to `C.

Subtract:

• from AB, BH greater than its half
• from AH, HK greater than its half

Repeat this process continually until the divisions in AB are equal in multitude with the divisions in DE.

Then, let AK, KH, HB be divisions which are equal in multitude with DF, FG, GE.

The remainder GD is greater than the remainder HA because:

• DE is greater than AB
• from DE there has been subtracted EG less than its half
• from AB, BH greater than its half

GD is greater than HA

The following were subtracted:

• from GD, the half GF
• from HA, HK greater than its half

This is why the remainder DF is greater than the remainder AK.

But DF is equal to C.

• Therefore C is also greater than AK.
• Therefore AK is less than C.

Therefore, there is left of the magnitude AB the magnitude AK which is less than the lesser magnitude set out, namely C.

The theorem can be similarly proved even if the parts subtracted be halves.