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Join the right lines B G, GH,
HI, IC; BF, F E, ED, DA. I

B
say the polygon ADEPBGHICA

G
will have nine equal fides.

For because the four angles
B A G, GAH, HA I, I A C [by E

H
fuppofition) are each equal to the
angle abc at thecircumference,
and also the four angles F CB, D

I E C F, E C D, DCA at the peri

A phery each equal to the same angle A BC. Therefore [by 26. and 29. 3.] the nine

A right lines BG, GH, HI, IC, AC, AD, DE, EF, FB will be equal to one another. And since these are equal to one another, the angles DAC, ACI, D, E, F, FBG, G, H, I will be also equal to one another. Wherefore the polygon ADE FB GHIC A will be a regular nonagon. The demonstration is the same whatever be the number of times that one of the angles at the base of the isosceles triangle A B C contains the angle A B C at the vertex.

Therefore, &c. Which was to be demonstrated.

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SCHOLIUM.
From this proposition it is evident, that a regular polygon
of any number of fides, might be inscribed in a given circle,
if an isosceles triangle could be made such that one angle at
the base should contain the angle at the vertex any given num-
ber of times. But this is not to be done generally by a right
line and a circle ; it must be effected in some cases by a conic
section, and in others by geometrical curves of an higher or-
der. The order of the geometrical curve necessary to the bu-
finess, being higher according as the number of times the an-
gle at the base is greater than that at the vertex. But as
the construction by these curves is useless, by reason of the
difficulty of the solution, the cycloid or spiral of Archimedes, ty
which the problem can be easily resolved, are much preferable.

PROP. IX. THEOR.
Any regular polygon inscribed in a circle, or circum-,

scribed about a circle, will approach nearer to the
circle, according as the number of the sides of the
polygon is greater.

Let

Let ABCDEFGH be a regular octagon inscribed in a circle, whose centre is L.

Bisest [by 10. 1.) the side A B in the point K, and draw the semidiameter L i, and join A I, I B.

Then because K A is equal to K B, the angles AK I, BKI [by 3. 3.) are right angles, and the fide i k is common; the right lines A 1, IB [by 26. 1.] will be equal to one another; and fo [by 28. 3.) the arches A I, IB will be equal to one another, Now the circle will exceed the

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B

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A

D

А

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G R octagon A B CDEF G H by eight equal segments A I B, B C, CD, DE, E F, GF, HG, A H, and the trilineal space A IK contained under A K, IK, and the arch A I will be one half the segment A i B of the circle, which is evidently greater than either of the equal segments A I, IB by the right angled triangles A IK, or I B K. Wherefore sixteen of the segments A 1, whereby the circle differs from an inscribed regular polygon of fixteen equal sides will be less than eight of the segments A B, whereby the circle differs from the regular polygon A B C D E F G H of eight fides. Consequently a regular polygon of fixteen sides is nearer to the circle in which it is inscribed, than that of eight sides. In like manner a regular polygon of thirty two sides infcribed in the circle, will differ less from the circle than one of fixteen fides; and one of fixty four fides will still be nearer to the circle in which it is inscribed, than one of thirty two sides ; and so on ad infinitum. The demonftration is the same, whatever be the number of the sides of the assumed polygon A B C D E F G H inscribed in the circle. So likewise when a regular polygon circumscribes the circle, the demonstration of the theorem is much the same as when the polygon is inscribed in the circle.

Corollary. Hence when the number of the sides of a regular polygon inscribed in a circle, or circumscribed

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about a circle are exceeding great, the difference between the circle, and either of those polygons, will be exceeding small, that is all three of them; the circumscribed polygon, the circle and the inscribed polygon will be so nearly equal to one another, as to differ only by a space less than any assignable one; the same may be said of the circumferences or ambits of either of these polygons, and that of the circle; and hence it is, that some have called a circle a polygon of an infinite number of fides.

SCHOLIU M.
Altho' in the propostion we have only taken a rogular poly-
gon, yet the thing is true of any polygons foever circum-
fcribing or inscribing a circle. But the demonftration is
not
so easy.

PRO P. X. THEO R.
First, Any regular polygon inscribed in a circle will

be equal to a triangle whose base is the circumfe-
rence of the whole polygon, and perpendicular
the right line drawn from the centre of the circle
to the middle point of one of the sides of the poly-
gon. And secondly, Any regular polygon circum-
scribing a circle will be equal to a triangle whose
base is the circumference of the wbole polygon,
and perpendicular the semidiameter of the circle.
Thirdly, A circle is equal to a triangle whose base
is the circumference of the circle, and perpendicular
the femidiameter of the circle.

1. For [see the figure of the last proposition] let ABCDEFGH be a regular octagon, inscribed in or circumscribed about the circle. I say the polygon A B C D E F G H when inscribed in the circle, will be equal to a triangle whose base is the sum of the sides A B, BC, CD, DE, EF, F G, GH, HA, and the perpendicular the right line L K drawn from the centre i to the middle point K of one fide A B of the polygon. And when it is circumscribed, its space will be equal to a triangle whose base will be the fum of the sides of the polygon, and perpendicular equal to the semidiameter 1 l of the circle.

For

For drawing the several semidiameters A L, BL, CL, &c these will divide the polygon into as many equal isosceles triangles as the figure has fides whose common perpendicular altitude will be the right line L K. Wherefore [by 1. 2.) a triangle whose base is the sum of the fides A B, BC, CD, &c. and perpendicular altitude K L will be equal to the inscribed or circum{cribed polygon.

Again, Because [by cor, of the last prop.] a regular polygon of an exceeding great number of fides circumscribing or inscribing a circle differs from the circle itself, by an exceeding small magnitude, viz. less than any asfignable one, as well as the circumference of such a polygon from that of the circle: Therefore [by what has been already demonstratedl a circle will be equal to a triangle whose base is the circumference of the circle, and perpendicular altitude the semidiameter thereof.

Corollary. Hence a sector of a circle is equal to a triangle, whose base is the arch of the sector, and perpendicular the semidiameter of the circle,

This is the famous proposition of Archimedes, in his book of the Circle, which he has demonstrated by the method cf exhaustions, as it is called.

EUCLI D's

1

EU CL I D's
EL
E L E M E N T S.

Μ Ν
BOOK V.

;

DEFINITIONS. 1. A Part is a magnitude of a magnitude, a less of a

greater, when the less measures the greater. 2. A multiple is a greater (magnitude] of a less, when the greater is measured by the less a 3.

Ratio is a certain mutual relation of two magnitudes to one another of the same kind, according to quantity b.

4. Magnia It might perhaps be better to call that magnitude any number of times greater than another a multiple, and that magnitude any number of times less than another a submultiple, and not a part, as Euclid has called it; because part generally signifies any magnitude less than a whole ; and the word to measure, seems as much to want de ining as part. Some call this an aliquot part š.

5 Ratio, according to the etymology of the word, fignifies a judgment, account, or estimation of things known, from a comparison of them. And I think the common english word rate gives a good notion of its meaning.

In every ratio is usually considered how much the antecedent contains of the consequent, or the confequent of the antece. dent. For let the antecedent be either equal to, greater, or less than the consequent, it is the quantum of the consequent contained in the antecedent, which is generally taken for the ratio of the antecedent to the consequent; and as the antecedent contains more or less of its consequent, so it is more or less valued in respect to that consequent.

There is no obtaining an adequate notion of a ratio from Euclid's definition of it; because there are other comparisons of two magnitudes of the same kind, according to quantity, that are not properly ratio, as the excess whereby one magnicude exceeds another, or the defect whereby one magnitude is less than another, is a certain fort of mutual comparison of them, viz. as to excess or defect, but this comparison is not eir ratio.

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