Book V · Proposition 2

V.2

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.Heath, 1908

Every step, checked

• the magnitudes are positivegiven
• the sum of the first and fifth is a multiple of the secondby Def.2
• and the sum of the third and sixth the same multiple of the fourthby V.1

What it needs, and what needs it

Needs: V.1

Rests on: C.N.2, Def.2

Depth: 1 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "V.2",
    THEOREM,
    sample=_two_magnitudes_and_two_multiples,
)
def prop_V_2(b, d, first_times: int, second_times: int) -> Out:
    """A is `first_times` B as C is of D; E is `second_times` B as F is of D."""
    hypothesis("the magnitudes are positive", sign(b) > 0 and sign(d) > 0, guard=True)
    a, c = first_times * b, first_times * d
    e, f = second_times * b, second_times * d
    because(prop_V_1, b, d, first_times)
    because(prop_V_1, b, d, second_times)

    claim("the sum of the first and fifth is a multiple of the second", "Def.2",
          a + e == (first_times + second_times) * b)
    claim("and the sum of the third and sixth the same multiple of the fourth", "V.1",
          c + f == (first_times + second_times) * d)
    return Out(sums=(a + e, c + f))