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may be described about the circle, which shall be less than F; join the centre G, and any of the points of contact O; and since it may be assumed that the perimeter of KLMNX is greater than the circumference of the circle, the rectangle contained by the perimeter of KLMNX and GO, which rectangle is the double of KLMNX, is greater than the rectangle contained by the circumference of the circle and GO; .. the circumscribed polygon KLMNX > F; and it is also less; which is absurd. Therefore, the circle ABCD can neither be greater, nor less, than F; i. e. it is equal to F.

103. Cor. The circumferences of circles are to one another as their semi-diameters.

PROP. XC.

104. Theorem. A circle is a mean proportional

between any regular polygon, described about it, and a similar polygon, the perimeter of which is equal to the circumference of the circle.

For if there be taken a straight line (P) equal to the perimeter of the regular polygon described about the circle, and another straight line (p) equal to the perimeter of the similar polygon, or (hyp.) equal to the circumference of the circle, then (E. 20. 6. and E. 22. 6.) the polygon, de

scribed about the circle, is to the similar polygon, as P* is to p* : But (S. 2. 4. cor. 2.) the polygon, described about the circle, is the half of the rectangle contained by P and the circle's semi-diameter ; and (S. 89. 6.) the circle is the half of the rectangle contained by p; and by the circle's semi-diameter; ..(E. 1.6.) that polygon is to the circle, as P is to p; and it has been shewn to be to the similar polygon, as P* is to p ; . it has to the similar polygon a ratio, the duplicate of that which it has to the circle ; •. the circle is a mean proportional between the two similar polygons.

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THE END.

T. Davison, Printer, Whitefriars.

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OF THE SAME PUBLISHERS MAY BE HAD,

The ELEMENTS of LINEAR PERSPECTIVE, designed for the use of Students in the University, by D. CRESSWELL, A. M. Fellow of Trinity College, Cambridge; with Plates, engraved by Lowry.-8vo. 6s.

An ELEMENTARY TREATISE on the Geometrical and Algebraical Investigation of MAXIMA and MINIMA; to which is added, a Selection of Propositions deducible from Euclid's Elements; by D. CRESSWELL, A. M. &c.-Second Edition, corrected and enlarged.-8vo. 128.

An ELEMENTARY TREATISE on the DIFFERENTIAL and INTEGRAL CALCULUS, by S. F. LACROIX, translated from the French, with an Appendix and Notes, by CHARLES BABBAGE, M.A. F.R.S. St. Peter's College; GEORGE PEACOCK, M.A. F.R.S. Fellow of Trinity College; J. W.F. HERSCHEL, M.A. F.R.S. Fellow of St. John's College.-8vo.

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The PRINCIPLES of FLUXIONS, designed for the use of Students in the Universities, by W. DEALTRY, B.D. F.R.S. late Fellow of Trinity College, Cambridge.--Second Edition, with corrections and considerable additions.-8vo. 14s.

A TREATISE on PLANE and SPHERICAL TRIGONOMETRY, by R. WOODHOUSE, A.M. F.R.S. Fellow of Gonville and Caius College.-Third Edition, corrected, altered, and enlarged.--8vo. 98.6d.

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A TREATISE on GEOMETRY, comprising Euclid's Elements of Plane Geometry, methodically arranged, and concisely demonstrated, together with the Elements of Solid Geometry. By D. CRESSWELL, A.M. &c.

A COLLECTION of EXAMPLES of the different Applications of the DIFFERENTIAL and INTEGRAL CALCULUS, by CHARLES BABBAGE, M.A. F.R.S. St. Peter's College; GEORGE PEACOCK, M.A.F.R.S. Fellow of Trinity College, and J. W.F. HERSCHEL, M.A. F.R.Ș. Fellow of St. John's College.

An ELEMENTARY TREATISE on MECHANICS for the use of Students in the University.

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