arithmetic and geometric means, find a' and A'. In like manner, from a', A' we can find a", A" related to a', A'; as a', A' are to a, A. Therefore, proceeding in this manner until we arrive at values a("), A(n) that will agree in as many decimal places as there are in the degree of accuracy we wish to attain; and since the area of a circle is intermediate between the reciprocals of a(n) and A(n), the area of the circle can be found to any required degree of approximation.

If for simplicity we take the radius of the circle to be unity, and commence with the inscribed and circumscribed squares, we have

=

a = *5,

a' = '3535533,
a"=.3264853,

A = 25,
A'3017766.

A" 3141315.

These numbers are found thus: a' is the geometric mean between a and A; that is, between 5 and 25, and A' is the arithmetic mean between a' and A, or between 3535533 and 25. Again, a" is the geometric mean between a' and A', and A′′ the arithmetic mean between a" and A'. Proceeding in this manner, we find a (13) ·3183099, A(13): 3183099. Hence the area of a circle radius unity, correct to seven decimal places, is equal to the reciprocal of 3183099; that is, equal to 3·1415926; or the value of correct to seven places of decimals is 3.1415926. The number is of such fundamental importance in Geometry that mathematicians have devoted great attention to its calculation. MR. SHANKS, an English computer, carried the calculation to 707 places of decimals. The following are the first 36 figures of his result::

3.141,592,653,589,793,238,462,643,383,279,502,884.

The result is here carried far beyond all the requirements of Mathematics. Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.

CONCLUSION.

In the foregoing Treatise we have given the elementary Geometry of the Point, the Line, and the Circle, and Figures formed by combinations of these. But it is important to the student to remark, that points and lines, instead of being distinct from, are limiting cases of, circles, and points and planes limiting cases of spheres. Thus, a circle whose radius diminishes to zero becomes a point. If, on the contrary, the circle be continually enlarged, it may have its curvature so much diminished that any portion of its circumference may be made to differ in as small a degree as we please from a right line, and became one when the radius becomes infinite. This happens when the centre, but not the circumference, goes to infinity.

THE END.

THIRD EDITION, Revised and Enlarged-3/6, cloth.

A SEQUEL

TO THE

FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID.

BY

JOHN CASEY, LL.D., F.R.S.,

Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy; &c. &c.

Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co.

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EXTRACTS FROM CRITICAL NOTICES.

"NATURE," July 6, 1882.

'This handy book has deservedly soon reached a second edition. In this way it will be seen that it has met a want. The author appears to have thoroughly revised the text, and he has added many notes of interest, a few figures, and several problems. An index has been added at the end. We recommend Dr. Casey's book, with increased confidence, for use in the higher forms of our schools."

"NATURE," April 17, 1884.

"We have noticed ('Nature,' vol. xxiv., p. 52; vol. xxvi., p. 219) two previous editions of this book, and are glad to find that our favourable opinion of it has been so convincingly indorsed by teachers and students in general. The novelty of this edition is a Supplement of Additional Propositions and Exercises. This contains an elegant mode of obtaining the circle tangential