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and they are upon equal straight lines BC, CK: But similar Book VI. segments of circles upon equal straight lines, are equals to one another: Therefore the segment BXC is equal to the segment & 21. 3. COK; and the triangle BGC is equal to the triangle CGK; therefore the whole, the sector BGC, is equal to the whole, the sector CGK: For the same reason, the sector KGL is equal to each of the sectors BGC, CGK: In the same manner, the sectors EHF, FHM, MHN may be proved equal to one another : Therefore, what multiple soever the circumference BL is of the circumference BC, the same multiple is the sector BGL of the sector BGC: for the same reason, whatever multiple the circumference EN is of EF, the same multiple is the ctor EHN of the sector EHF: And if the circumference
BL be equal to EN, the sector BGL is equal to the sector EHN; and if the circumference BL be greater than EN, the sector BGL is greater than the sector EHN; and if less, less : Since then, there are four magnitudes, the two circumferences BC, EF, and the two sectors BGC, EHF, and of the circumference BC, and sector BGC, the circumference BL and sector BGL are any equimultiples whatever; and of the circumference EF, and sector EHF, the circumference EN, and sector EHN, are any equimultiples whatever; and it has been proved, that if the circumference BL be greater than EN, the sector BGL is greater than the sector EHN: if equal, equal; and if-less, less. Therefore , as the circumference BC is to b 5. def.5. the circumference EF, so is the sector BGC to the sector EHF. Wherefore, “ in equal circles,” &c. Q. E. D.
PROP. B. THEOR.
If an angle of a triangle be bisected by a straight line which likewise cuts the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.
Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD; the rectangle BA, AC is equal to
the rectangle BD, DC, together with the square of AD. a 5. 4. Describe the circle a ACB about the triangle, and produce
AD to the circumference in E,
the angle CAE, and the angle b 21. 3. ABD to the angle AEC, for
B they are in the same segment,
D the triangles ABD, AEC are
equiangular to one another : c4. 6.
Therefore as BA to AD, so is c
EA to AC, and consequently d 16. 6. the rectangle BA, AC is equala
E to the rectangle EA, AD, that e 3. 2.
is, to the rectangle ED, DA, together with the square of f 35. 3. AD: But the rectangle ED, DA is equal to the rectangle
BD, DC. Therefore, the rectangle BA, AC is equal to the rectangle BD, DC, together with the square of AD. Wherefore, “ if an angle," &c. Q. E. D.
PROP. C. THEOR.
If from any angle of a triangle a straight line be drawn perpendicular to the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular, and the diameter of the circle described about the triangle.
Let ABC be a triangle, and AD the perpendicular from the angle A to the base BC; the rectangle BA, AC is equal to the rectangle contained by AD, and the diameter of the circle described about the triangle.
a 5. 4.
Describe a the circle ACB
Book VI. about the triangle, and draw its
А. diameter AE, and join EC: Because the right angle BDA is equal to the angle ECA in a B
b 31. 3. semicircle, and the angle ABD
D to the angle AEC in the same segment", the triangles ABD,
c 21. 3. AEC are equiangular: Therefore as d BA to AD, so is EA to AC;
d 4. 6. and consequently the rectangle
E BA, AC is equal to the rectan.
e 16. 6. gle EA, AD. “If, therefore, from an angle,” &c. Q. E. D.
PROP. D. THEOR.
b 4. 6.
The rectangle contained by the diagonals of a See N. quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.
Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD; the rectangle contained by AC, BD is equal to the two rectangles contained by AB, CD, and by AD, BC*
Make the angle ABE equal to the angle DBC: add to each of these the common angle EBD, then the angle ABD is equal to the angle EBC: and the angle BDA is equal a to the an- a 21. 3. gle BCE, because they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE: Wherefore 6, as BC is to CE, so is BD
B to DA; and consequently the
C rectangle BC, AD is equal to the rectangle BD, CE: Again, because the angle ABE is equal to the angle DBC, and the anglea BAE to the angle BDC,
E the triangle ABE is equiangular
D to the triangle BCD: As there. A fore BA to A E, so is BD to DC; wherefore, the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been shown equal to the rectangle BD, CE; therefore, the whole rectangle AC, BD d is equal to the rectangle AB, DC, together with the d l. 2. rectangle AD, BC. Therefore, “the rectangle,” &c. Q. E. D.
• This is a Lemma of Ci, Ptolomæus, in p. 9. of his psyon. n ourTušus.
c 16. 6. THE
ELEMENTS OF EUCLID.
when it makes right angles with every straight line meeting
drawn in one of the planes perpendicularly to the common
The inclination of a straight line to a plane is the acute angle
contained by that straight line, and another drawn from the
tained by two 'straight lines drawn from any the same point
Book XI. Two planes are said to have the same, or a like inclination to
one another, which two other planes have, when the said angles of inclination are equal to one another.
VIII. Parallel planes are such as do not meet one another though produced
IX. A solid angle is that which is made by the meeting of more See N.
than two plane angles, which are not in the same plane, in one point.
X. • The tenth definition is omitted for reasons given in the notes. See N.
XI. Similar solid figures are such as have all their solid angles See N.
equal, each to each, and which are contained by the same
stituted betwixt one plane and one point above it in which
two that are opposite are equal, similar, and parallel to one
XIV. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.
through the centre, and is terminated both ways by the
angled triangle about one of the sides containing the right
angle, which side remains fixed. If the fixed side be equal to the other side containing the
right angle, the cone is called a right angled cone; if it be