COR. 3. If a perpendicular be drawn from any angle of a triangle to the opposite side, that side will be to the sum of the other two sides, as half their difference is to the distance of the perpendicular from the middle of the base.

For 2BC.DE (AB+ AC). (AB-AC), therefore 2BC: AB+ AC :: AB AC: DE (Propor.33), therefore BC: AB+ AC:: (AB-AC): DE (Propor. 30). ED.

Scholium. If the perpendicular be always drawn from the greatest angle of any triangle to the opposite side, the proposition will be reduced to one case only, and its application to practice will be more convenient. The proposition will then be expressed as follows.

If a perpendicular be drawn from the greatest angle of a triangle to the opposite side, the greatest side will be to the sum of the other two sides, as their difference is to the difference of the segments of the greatest side made by the perpendicular.

By means of this proposition, or Cor. 2 or 3, we can divide a triangle, of which all the sides are given, into two right angled triangles, by determining the segments of the base, and thence the position and length of the perpendicular. Thus, by the proposition, half the sum of the base and the fourth proportional is equal to the greater segment (A. 2), and half the difference between the base and the fourth proportional is equal to the less segment.

ED.

THE PRINCIPAL THEOREMS IN BOOK II.

Of any two unequal magnitudes, or quantities, if the sum and difference be added together, half the aggregate will be the greater quantity; and if the difference be subtracted from the sum, half the remainder will be the less quantity.

Note.-This prop. may be briefly expressed by signs as follows.

Of two unequal quantities half the sum + half the difference is equal to the greater quantity, and half the sum-half the difference is equal to the less.

If there be two straight lines, and if one of them be divided into any number of parts, the sum of the rectangles contained by the whole line and the several parts of the divided line is equal to the rectangle contained by the two whole lines.

If a straight line be divided into any two parts, the two rectangles contained by the whole line and each of the parts are together equal to the square of the whole line.

...If a straight line be divided into any two parts, the rectangle contained by the whole line and one of the parts is equal to the rectangle contained by the two parts together with the square of the foresaid part.

If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts together with twice the rectangle contained by the two parts. Or, in other words, the square of the sum of two lines is greater than the sum of their squares by twice the rectangle contained by those lines.

If a straight line be divided into two equal parts, the square of the whole line is equal to four times the square of half the line.

The rectangle contained by the sum and difference of two lines is equal to the difference of their squares. The square of the difference of any two lines is less than the sum of their squares by twice the rectangle contained by those lines.

In any right angled triangle the square described on the hypothenuse is equal to both the squares described on the other two sides.

If two right angled triangles have two sides of one tria gle equal to two corresponding sides of the other, they are equal in all respects.

In any obtuse-angled triangle, if a perpendicular be drawn from either of the two acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the sum of the squares of the sides containing it by twice the rectangle under the base and the distance between the perpendicular and the obtuse angle.

In any triangle the square of the side subtending an acute angle is less than the sum of the squares of the two sides containing that angle by twice the rectangle under either of these sides and the distance between the acute angle and the perpendicular drawn from the opposite angle to that side.

If a straight line be drawn from any angle of a triangle to the middle of the opposite side, the sum of the squares of the other two sides will be double of the sum of the squares of that line and of half the base.

The sum of the squares of the two diagonals of any parallelogram is equal to the sum of the squares of the four sides.

M

If a perpendicular be drawn from the greatest angle of a triangle to the opposite side, the rectangle contained by the sum and difference of the other two sides is equal to the rectangle contained by the sum and difference of the segments of the base made by the perpendicular; or, the greatest side is to the sum of the other two sides, as their difference is to the difference of the segments of the greatest side made by the perpendicular.

If a perpendicular be drawn from any angle of a triangle to the opposite side, the difference of the squares of the other two sides is equal to the difference of the squares of the segments of the base made by the perpendicular.

END OF BOOK II.

2. Circles are said to touch one another when they meet, but

do not cut one another.

3. Straight lines are said to be equally
distant from the centre of a circle,
when the perpendiculars drawn to
them from the centre are equal.
4. And the straight line on which the
greater perpendicular falls is said to
be farther from the centre,

B. An arch of a circle is any part of the
circumference.

And the chord of any arch is the straight
line joining the two extremities of the arch.
5. A segment of a circle is the figure
contained by a straight line and the
arch which it cuts off, or, it is the
space included between an arch and
its chord.

ED.

* Numbers 1, 2, 3, 4, 7, are explanations, not definitions.

ED.