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To describe a square that shall be equal to a given rectilineal figure. LET a be the given rectilineal figure; it is required to describe a square that shall be equal to a
Describe (i. 45) the rectangular parallelogram bcde equal to the rectilineal figure a If then the sides of it, be, ed, are equal to one another, it is a square, and what was required is now done. But if they are not equal, produce one of
them be to f, and make ef equal to ed, and bisect bf in g; and from the centre at g the distance gb or gf, describe the semicircle bhf, and produce de to h, and join gh: therefore because the straight line bf is divided into two equal parts in the point g,
and into two unequal
at e, the rectangle be, ef, together with the square of eg, is equal (ii. 5) to the square of gf: but gf is equal to gh: therefore the rectangle be, ef, together with the square of eg, is equal to the square of gh: but the squares of he, eg are equal (i. 47) to the square of gh: therefore the rectangle be, ef, together with the square of eg, is equal to the squares of he, eg; take away the square of eg, which is common to both; and the remaining rectangle be, ef is equal to the square of eh: but the rectangle contained by be, ef is the parallelogram bd, because ef is equal to ed; therefore bd is equal to the square of eh; but bd is equal to the rectilineal figure a; therefore the rectilineal figure a is equal to the square of eh. Wherefore a square has been made equal to the given rectilineal figure a, viz. the square described upon eh. Which was to be done.
EXERCISES ON BOOK II.
1. Given the difference between two squares, to find a line whose square shall be equal to that difference.
2. Given two lines, to produce one of them, so that the rectangle contained by the two given lines shall be equal to the square of the part produced.
3. Given a straight line, to describe on it, as the hypotenuse, a rightangled triangle such that the sum of the hypotenuse and the lesser of the other two sides shall be double of the remaining side of the triangle.
4. Given the sum of the squares of any number of lines, to find a line whose square shall be equal to the given sum.
SECT. II. THEOREMS.
5. If a straight line be drawn from the vertex of an isosceles triangle to any point in the base, the square described on this line, together with the rectangle contained by the segments of the base, is equal to the square described upon either of the equal sides.
6. The difference between the squares of two unequal straight lines is equal to the rectangle contained by their sum and difference.
7. If a straight line be divided into five equal parts, the square of the whole line is equal to the square of the straight line which is made up of four of those parts, together with the square of the straight line which is made up of three of those parts.
8. If a perpendicular be drawn from the right angle of a right-angled triangle to the hypotenuse, the square of the perpendicular is equal to the rectangle contained by the segments of the hypotenuse.
9. If the base of a triangle be bisected, the sum of the squares of the other two sides is equal to twice the square of half the base and twice the square of the line drawn from the point of bisection to the vertical angle.
10. If the middle points of the opposite sides of a quadrilateral figure be joined, the sum of the squares of the joining lines is equal to one-half the sum of the squares of the diagonals of the quadrilateral.
11. If the sides of a triangle be bisected and lines drawn from the points of bisection to the opposite angles, four times the sum of the squares of these lines shall be equal to three times the sum of the squares of the three sides of the triangle.
I. EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal.
This is not a definition, but a theorem, the truth of which is evident; for, if the circles be applied to one another, so that their centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal.
II. A straight line is said to touch a circle when it meets the circle, and being produced does not cut it.
III. Circles are said to touch one another, which meet but do not cut one another.
Definitions II. and III.
IV. 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. V. And the straight line on which the greater perpendicular falls is said to be farther from the centre.
Definitions IV. and V.
VI. A segment of a circle is the figure contained by a straight line and the circumference it cuts off.
VII. The angle of a segment is that which is contained by the straight line and the circumference.
VIII. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment to the extremities of the straight line which is the base of the segment.
IX. And an angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle.
Definitions VIII. and IX.
X. The sector of a circle is the figure contained by two straight lines drawn from the centre, and the circumference between them.
XI. Similar segments of a circle are those in which the angles are equal, or which contain equal angles.
To find the centre of a given circle.
LET abc be the given circle; it is required to find its centre.
Draw within it any straight line a b, and bisect (i. 10) it in d; from the point d draw (i. 11) dc at right angles to ab, and produce it to e, and bisect ce in f: the point f is the centre of the circle a bc.
For, if it be not, let, if possible, g be the centre, and join ga, gd,
Then, because da is equal to db, and dg common to the two triangles a dg, bdg, the two sides a d, dg, are equal to the two b d d g, each to each; and the base ga is equal to the base g b, because they are drawn from the centre g therefore the angle ad g is equal (i. 8) to the angle gdb. But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of the angles is a right angle (i. def. 10): therefore the angle gdb is a right angle: but a fdb is likewise a right angle: wherefore the angle fdb is equal to the angle g db, the greater to the less, which is impossible: therefore g is not the centre of the circle a bc. In the same manner it can be shewn, that no other point but fis the centre; that is, f is the centre of the circle a b c. Which was to be found. COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.
PROPOSITION II. THEOREM.
If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.
LET abc be a circle, and a, b any two points in the circumference; the straight line drawn from a to b shall fall within the circle.
For, if it do not, let it fall, if possible, without, as aeb; find (i. 3) d the centre of the circle abc; and join a d, db, and produce df, any straight line meeting the circumference a b, to e: then because da is equal to db, the angle dab is equal (i. 5) to the angle dba; and because a e, a side of the triangle da e, is produced to b, the angle de b is greater (i. 16) than the angle dae; but dae is equal to the angle dbe; therefore the angle de b is greater than the angle dbe but to the greater angle the greater side is opposite (i. 19); db is therefore greater than de but db is equal to df; wherefore df is greater than de, the less than the greater, which is impossible: therefore the straight line drawn from a to b does not fall without the circle. In the same manner, it may be demonstrated that it does not fall upon the circumference; it falls therefore within it. Wherefore, if any two points, Q. E. D.
* Whenever the expression "straight lines from the centre," or "drawn from the centre," occurs, it is to be understood that they are drawn to the circumference.