and they are upon equal straight lines BC, BK: But similar seg- Book VI.' ments of circles upon equal straight lines, are equal 8 to one ano. m ther : Therefore the segment BXC is equal to the segment COK: 8 24. 3. 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 foever the circumference BL is of the circumference BC, the fame multiple is the sector BGL of the sector BGC: For the same reason, whatever multiple the circumfe. rence EN is of EF, the same multiple is the sector EHN of the sector EHF: And if the circumference BL be equal to EN, the

feltor 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, lefs : Since then, there are four magnitudes, the two circumferences BC, EF, and the two fectors BGC, EHF, and of the circumference BC, and sector BGC, the circumference BL and sector BGL are any equi. multiples whatever; and of the circumference EF and sector EHF, the circumference EN and sector EHN are any equimultiples whatever ; and that it has been proved, if the circumference BL be greater than EN, the sector BGL is greater than the sector EHN; and if equal, equal ; and if less, less. Therefore, as the

circumference BC is to the circumference EF, fo b 5. def. s. is the sector BGC to the sector EHF. Wherefore, in equal circles, &c. Q. E. D.

Book VI.

See N.

PRO P. B. THE OR.
IF an angle of a triangle be bisected by a straight line,

which likewife cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the fegments of the base, together with the square of the straight line bisecting the angle.

a

Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD, the rectangle BĂ, AC is equal to the rectangle BD, DC, together with the square of AD.

Describe the circle a ACB about the triangle, and produce
AD to the circumference in E,

A
and join EC: Then, because the
angle BAD is equal to the angle
CAE, and the angle ABD to the
angle AEC, for they are in the

с same segment; the triangles ABD,

B

D
AEC are equiangular to one an-
Other : Therefore as BA to AD,
so is EA to AC, and conse-
quently the rectangle BA, AC is
equal to the rectangle EA, AD,

E
that is, to the rectangle EDDA
together with the square of 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.

See N.

7F 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 AEC be a triangle, and AD the perpendicular from the angle Authe kae: PC; the rectangle BA, AC is equal to the redan, a coi ained or AD and the diameter of the circle de scrised at the triangic.

Describe

Describe the circle ACB about the triangle, and draw its diameter AE, and join EC: Because the . right angle BDA is equal to the angle ECA in a semicircle, and the angle ABD to the angle’AEC B in the same fegment; the triangles ABD, AEC are equiangular : Therefore as d BA to AD, fo is EA to AC; and consequently the rectangle BA, AC is equal E to the rectangle EA, AD. If therefore, from an angle, &c. Q. E. D.

D

C 21. 3.

d 4. 6.

€ 16. 6.

PROP. D. THEOR.
HE rectangle contained by the diagonals of a qua-

drilateral 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 to the a 21. 3. angle BCE, because they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE: Wherefore b as

b 4. 0. BC is to CE, fo is BD to DA; and consequently the rectangle

BC, AD is equal to the rectangle BD, CE: Again, because the angle ABE is equal to the angle DBC, and the angle · BAE to the angle BDC, the triangle ABE is equiangular to the triangle BCD: As therefore BA to AE, To is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been shewn equal to the rectangle BD, CE; therefore the whole rectangle AC, BD d is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E D. N2

THE

C 16. S.

† This is a Lemma of al Poplomagus, io page 9. of his

μεγαλη συνταξις.

E Ε L Ε Μ Ε Ν Τ S

M T S

A .

I.
Solid is that which hath length, breadth, and thicknefs.

II.
That which bounds a solid is a superficies.

NII,
A straight line is perpendicular, or at right angles to a plane,

when it makes right angles with every straight line meeting
it in that plane.

IV.
A plane is perpendicular to a plane, when the straight lines

drawn in one of the planes perpendicularly to the common
section of the two planes, are perpendicular to the other
plane.

V.
The inclination of a straight line to a plane is the acute angle

contained by that straight line, and another drawn from the
point in which the first line meets the plane, to the point in
which a perpendicular, to the plane drawn from any point of
the first line above the plane, meets the same plane.

VI.
The inclination of a plane to a plane is the acute angle contain-

ed by two straight lines drawn from any the same point of
their common section at right angles to it, one upon one
plane, and the other upon the other plane.

VII. Two

VII.

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 which 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.

See N. Similar folid figures are such as have all their solid angles equal,

each to each, and which are contained by the same number
of fimilar planes.

XII.
A pyramid is a solid figure contained by planes that are con-

stituted betwixt one plane and one point above it in which
they meet.

XUI.
A prism is a solid figure contained by plane figures of which

two that are opposite are equal, similar, and parallel to one
another; and the others parallelograms.

XIV.
A sphere is a solid figure described by the revolution of a semi-
circle about its diameter, which remains unmoved.

XV.
The axis of a sphere is the fixed straight line about which the
femicircle revolves.

XVI.
The center of a sphere is the same with that of the semicircle.

XVII.
The diameter of a sphere is any straight line which palles chro'

the center, and is terminated both ways by the superficies of
the sphere.

XVIII.
A cone is a solid figure described by the revolution of a right

angled triangle about one of the sides containing the right

angle, which fide remains fixed.
If the fixed fide be equal to the other side containing the right

angle, the cone is called a right angled cone; it it be less
than the other fide, an obtuse angled, and if greater, an acute
angled cone.

N 3

XIX. The