PROPOSITION V.

If four straight lines are proportional the rectangle contained by the extremes is equal to that contained by the means.

Let the four straight lines a, b, c, d be proportional.
Then rect. (a, d) is = rect. (b, c).

On a, b construct rectangles, the altitude of each being equal to c, and on c, d construct rectangles, the altitude of each being equal to a.

... rect. (a, c): rect. (b, c) as rect. (a, c) : rect. (a, d); (v. 3)

.•. rect. (b, c) = rect. (a, d).

(v. 4)

Conversely,

If a, b, c, d are four straight lines, such that

rect. (a, d) = rect. (b, c),

then ab as c: d.

For the same construction being made,

..: (a,d)
rect. (a, d) = rect. (b, c);

.. rect. (a,c): rect. (b, c) as rect. (a, c): rect. (a, d); (v. 2)

In like manner it may be demonstrated that, if the rectangle contained by two straight lines is equal to the rectangle contained by two other straight lines, then any proportion having the sides of one rectangle for the extremes and the sides of the other for the means is true.

then

and

Thus if

rect. (a, d)=rect. (b, c),

iba as dc,

a: c as b: d.

COR. If cb, then rect. (b, c) = square on b.

the straight lines a, b, d are then said to be proportional.

Hence, if three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean: and conversely.

Hence may be easily established the following Pro

positions :

PROPOSITION VI.

If the four straight lines a, b, c, d are proportional,

then a cas b d.

For a b as c: d;

... rect. (a, d) = rect. (b, c);

.. ac as b: d.

PROPOSITION VII.

If a b as c: d,

then ba as d c.

For rect. (a,d) = rect. (b, c),
and .. ba as d: c.

PROPOSITION VIII.

If a b as c: d,

then a is > = or < b, according as c is > = or < d.

For rect. (a, d) = rect. (b, c);

.. if c be> or < d, a must be accordingly > =

PROPOSITION IX.

If a b as c: d,

= or < b.

then a is >= or < c, according as b is > = or < d.

This follows in a similar manner to the preceding demonstration.

PROPOSITION X.

If a b as c: d,

then ab as a + c : b + d.

For rect. (a, d) = rect. (b, c);

=

.. rects. (a, b) (a, d) are together rects. (b, a) (b, c);
.. rect. (a, b + d) = rect. (b, a + c);
.. ab as a + c : b + d.

ference of b and d.

(II. 1)

b as the difference of a and c .the dif

PROPOSITION XI.

If a basc : d,

and as e :f,
and also as g: h,

then ab as c+e+g : d+f+h.

a b as cd;
.:rect. (a, d) = rect. (b, c);
and ·.·a : b as e : f;
... rect. (a, f) = rect. (b, e),
so rect. (a, h) = rect. (b, g).

Hence rects. (a, d) (a, f) (a, h) are together

= rects. (b, c) (b, e) (b, g); .. rect. (a, d+f + h) = rect. (b, c + e +g); .. ab as c+e+g: d+f+h.

PROPOSITION XII.

If a b as c: d, and b h as d: k,

(II. 1)

Since any lines whatever are equal to certain straight lines,.. the above propositions (6-12) will hold for any lines whatever (v. 2).

The preceding Propositions (6-12) may be easily extended to areas,

thus:

Let the four rectilineal figures A, B, C, D be proportional, and let a, b, c, d be the bases of rectangles having the same altitude and areas respectively A, B, C, D.

=

... rect. (p, a);: rect. (p, b) as rect. (p, c) : rect. (p, d); (v. 2)

Again, A> or < B, according as a > = or < b;

=

and ... as c >= or < d;

and .. as C> or < D.

(v. 8)

=

Similarly, A > = or < C, as B > = or < D.

(v. 9)

Again, since

a b as cd;

..a: b as a + c : b + d.

(v. 10)

Hence A : B as A + C : C+D.